UNIVERSIDAD DE CANTABRIA
Escuela T?ecnica Superior de
Ingenieros de Caminos Canales y Puertos
A STATISTICAL FATIGUE MODEL COVERING
THE TENSION AND COMPRESSION WO?HLER
FIELDS AND ALLOWING DAMAGE ACCUMULATION
Author
Mar??a Luisa Ruiz Ripoll
Directors
Prof. Dr. Enrique Castillo Ron (Universidad de Cantabria)
Prof. Dr. Alfonso Fern?andez Canteli (Universidad de Oviedo)
Santander, 2008
ii
iii
A mis padres,
a mi hermana,
a ?Alvaro...
\No hay enigmas, si un problema puede plantearse, es que puede resolverse"
Ludwig Wittgenstein, fll?osofo (1889{1951)
\En el fondo, los cient??flcos somos gente con suerte: podemos jugar a lo que queramos
durante toda la vida"
Lee Smolin, f??sico (1955{today)
\A theory can be proved by experiment; but no path leads from experiment to the birth of
a theory"
Albert Einstein, cient??flco (1879-1955)
iv
v
Acknowledgements
It is well known that these pages of a PhD thesis are the most widely read pages of the entire
publication. I will try to write these lines as emotional as possible, because the conclusion of all
this work, all this time has to be sincere.
Thanks to my supervisors Enrique and Alfonso. They have given me the opportunity of be
the scientist that I am today, and have the pleasure of knowing many new.
Thanks to Hern?an and Paola, good mates during this long travel.
Thanks for the flnancial, academic and technical support of Empa (Swiss Federal Labora-
tories for Testing and Research at Du?bendorf (Switzerland)). Thanks to Roland for his daily
supervision of my work and all the help that he gave me during this year. Thanks to Evelyne,
for the good moments expended in our o?ce. Thanks to the durability group, because they
were with me always. Thanks to G. Terrasi for his supervision. Thanks to Sandro and Walti,
my best German professors, my Empa uncles, for the personal support. Thanks to Christian,
Alex, Berni, Ali, Peter, each one has a important contribution during the time I have been in
Swizterland.
Thanks to my friends, Luba, Vanessa, Claire and Martin, for the personal support. During
my time in Zu?rich I have shared my best time with all of you giving me advices, surprises and
making me smile when I needed it, always perfect!.
Thanks to the existence of the cities Santander and Zu?rich. In them I discovered myself as
the happiest person in the world. They are the most beautiful cities that I know, and I wish to
come back to each one during all my life to experience between their fantastic beaches, forests,
mountains and lakes fantastic moments.
Thanks to my family. To my parents, Javier and Luisa, because they have been always there,
even thousand of kilometers apart. I am the person that I am because of them. Thanks to my
sister Lidia, during all my life she has been the perfect sister and friend. Thanks \peque".
Thanks to ?Alvaro, my motivation. He has sufiered with me all the good and bad moments
during this stage, and always he has given me the best. Today I am here because of you. You
are my energy.
And I would like to thank to the rest of the people that should be here, but I have forgotten
and they supported me in one or another way.
M. Luisa Ruiz Ripoll
October 30, 2008
Ciudad Real, Spain
vi
Contents
I Sinopsis de la Tesis Doctoral en Castellano 1
1 Sinopsis de la Tesis Doctoral en Castellano 3
1.1 Objeto y objetivos de la investigaci?on . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Introducci?on, hip?otesis y objetivos de la investigaci?on . . . . . . . . . . . 4
1.2 Planteamiento y metodolog??a utilizada . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Aportaciones originales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Los modelos Weibull y Gumbel basados en el campo S{N . . . . . . . . . 6
1.3.2 Validaci?on experimental del modelo . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 An?alisis del da~no acumulado . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Conclusiones y futuras l??neas de investigaci?on . . . . . . . . . . . . . . . . . . . . 19
1.4.1 Conclusiones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.2 Futuras l??neas de investigaci?on . . . . . . . . . . . . . . . . . . . . . . . . 20
II Presentation and State of the Art 21
2 Presentation 23
2.1 Justiflcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 State of the Art 27
3.1 Introduction to the fatigue problem . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 The fatigue problem in engineering . . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Fatigue concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Fatigue under the fracture mechanics point of view . . . . . . . . . . . . . . . . . 29
3.2.1 Deflnitions for fatigue crack growth . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 Describing material fatigue crack growth behavior . . . . . . . . . . . . . 30
3.2.3 In uence of difierent parameters on fatigue crack growth . . . . . . . . . . 31
3.2.4 Fatigue laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Stress based approach to fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.1 Estimated S{N curve of a component based on ultimate tensile strength . 33
3.3.2 Fatigue strength testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.3 Mean stress efiect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.4 Trends in S{N curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.5 Variable amplitude loading . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Strain based approach to fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
vii
viii CONTENTS
3.4.1 Analysis of monotonic and cyclic stress-strain behavior of materials . . . . 42
3.4.2 Mean stress correction methods . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Models Used in Fatigue 47
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Models used in fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.1 The Basquin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.2 The Palmgren{Miner rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.3 The up{and{down method . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.4 The Bastenaire model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.5 The Spindel and Haibach model . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.6 The Pascual and Meeker model . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.7 The Kohout and Vechet models . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 New statistical models: Castillo?s models . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.1 The general model for lifetime evaluation: The Weibull model . . . . . . . 65
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Use of Functional Equations 69
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1.1 One example of functional equation: area of a rectangle (Legendre [89]) . 70
5.2 History of functional equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Basic concepts and deflnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Some methods for solving functional equations . . . . . . . . . . . . . . . . . . . 76
5.4.1 Replacement of variables by given values . . . . . . . . . . . . . . . . . . . 76
5.4.2 Transforming one or several variables . . . . . . . . . . . . . . . . . . . . 77
5.4.3 Transforming one or several functions . . . . . . . . . . . . . . . . . . . . 77
5.4.4 Using a more general equation . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4.5 Treating some variables as constants . . . . . . . . . . . . . . . . . . . . . 77
5.4.6 Inductive methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4.7 Iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4.8 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.4.9 Reduction by means of analytical techniques . . . . . . . . . . . . . . . . 78
5.4.10 Mixed methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5 Functional equations with two variables . . . . . . . . . . . . . . . . . . . . . . . 79
5.5.1 Cauchy?s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5.2 Jensen?s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5.3 Cauchy?s exponential equation . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5.4 Pexider?s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.5.5 Vincze?s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.5.6 Euler?s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.5.7 D?Alambert?s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5.8 Equations involving functions of two variables . . . . . . . . . . . . . . . . 82
5.6 Functional equations with one variable . . . . . . . . . . . . . . . . . . . . . . . . 83
5.6.1 Basic families of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.6.2 Conjugate equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.6.3 Functional equations and nested radicals . . . . . . . . . . . . . . . . . . . 84
5.7 Functional equations in probability theory . . . . . . . . . . . . . . . . . . . . . . 85
5.7.1 Integrated Cauchy functional equations on R+ . . . . . . . . . . . . . . . 85
CONTENTS ix
5.7.2 Integrated Cauchy functional equation with error terms on R+ . . . . . . 85
5.8 Applications to science and engineering . . . . . . . . . . . . . . . . . . . . . . . 86
5.8.1 A statistical model for lifetime analysis . . . . . . . . . . . . . . . . . . . 86
5.8.2 Statistical models for fatigue life of longitudinal elements . . . . . . . . . 89
III Theoretical Contributions. Proposed Model 97
6 The Weibull and Gumbel S{N Field Stress Based Fatigue Models 99
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 Derivation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3 Some properties of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3.1 The regression equation for max-logN for difierent stress ratios R . . . . 105
6.4 Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4.1 Physical restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4.2 Statistical restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4.3 Range of the problem. Simpliflcations of constraints . . . . . . . . . . . . 108
6.5 Resulting models and submodels . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.5.1 General model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.5.2 Submodel Nr.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.5.3 Submodel Nr. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.5.4 Submodel Nr. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.5.5 Submodel Nr. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.5.6 Submodel Nr. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.6 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.6.1 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . . . . . 112
6.6.2 Parameter estimation by regression . . . . . . . . . . . . . . . . . . . . . . 113
6.7 Use of the model in practise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.7.1 Some difierent representations of the Gumbel model . . . . . . . . . . . . 114
6.8 Example of application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.8.1 Validation using data in the existing literature . . . . . . . . . . . . . . . 114
6.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
IV Experimental Validation of the Models 121
7 Experimental Validation of the Model 123
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.3 Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.4 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.5 Type of load and testing strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.6 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.6.1 General information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.6.2 Parameter estimation for the 42CrMo4 steel . . . . . . . . . . . . . . . . . 130
7.6.3 Parameter estimation for the AlMgSi1 alloy . . . . . . . . . . . . . . . . . 131
7.7 Parameter validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.7.1 Validation of the theoretical example . . . . . . . . . . . . . . . . . . . . . 133
7.7.2 Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
x CONTENTS
7.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8 Damage Measures and Damage Accumulation 143
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.2 Damage measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.2.1 Some requirements for a damage measures . . . . . . . . . . . . . . . . . . 144
8.2.2 Some damage measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.3 Cumulate damage associated with a general load history . . . . . . . . . . . . . . 148
8.3.1 Procedure to perform a damage analysis . . . . . . . . . . . . . . . . . . . 149
8.4 Example of applications. Validation of damage accumulation . . . . . . . . . . . 152
8.4.1 Constant loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.4.2 Variable loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
V Conclusions 161
9 Conclusions 163
9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
9.2 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
9.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A Derivation of the Model 167
B Specimen Characterization 171
B.1 Material characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
B.1.1 Metallographic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
B.1.2 Static tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
B.2 Geometric specimens deflnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.2.1 42CrMo4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.2.2 AlMgSi1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
C Experimental Protocols 179
C.1 Constant load tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
C.1.1 42CrMo4 steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
C.1.2 AlMgSi1 alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
D SISIFO program 181
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
D.2 Background of SISIFO program: the new fatigue Castillo?s model . . . . . . . . . 181
D.2.1 Simpliflcation of Castillo?s model . . . . . . . . . . . . . . . . . . . . . . . 182
D.2.2 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
D.2.3 Validation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
D.2.4 Damage accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
D.3 Progam SISIFO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
D.3.1 Introduction and general organization . . . . . . . . . . . . . . . . . . . . 182
D.3.2 Program installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
D.3.3 Menu 1: parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . 186
D.3.4 Menu 2: damage analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
CONTENTS xi
D.4 Example of application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
D.4.1 Estimation of the Castillo model parameters . . . . . . . . . . . . . . . . 209
D.4.2 Analysis of the damage accumulation . . . . . . . . . . . . . . . . . . . . 216
D.4.3 Rain ow analysis in a quasi-random load history . . . . . . . . . . . . . . 218
E Nomenclature 223
Bibliography 227
xii CONTENTS
List of Figures
1.1 Metodolog??a de trabajo seguida en la elaboraci?on de la tesis doctoral. . . . . . . 6
1.2 Representaci?on esquem?atica del signiflcado f??sico de las variablesB y C del modelo
de Castillo et al. [39]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Esquema de las curvas de Wo?hler para los percentiles f0:01; 0:05; 0:5; 0:95; 0:99g
con un ?max = 1 y ?max = 1:5, y ?min = 0:4 y ?min = 0:8 [42]. . . . . . . . . . . 9
1.4 Representaci?on esquem?atica de la distribuci?on tensional de los ensayos. A la
izquierda el acero 42CrMo4, a la derecha la aleaci?on AlMgSi1. (a) l??mite de
plasticidad, (b) l??mite de endurancia del material. . . . . . . . . . . . . . . . . . . 10
1.5 Campo de Wo?hler resultante para el acero 42CrMo4 con los par?ametros obtenidos
por el m?etodo de m?axima verosimilitud (flgura de la derecha) y regresi?on por
m??nimos cuadrados (flgura de la izquierda). . . . . . . . . . . . . . . . . . . . . . 11
1.6 Campo deWo?hler resultante para la aleaci?on AlMgSi1 con los par?ametros obtenidos
por el m?etodo de m?axima verosimilitud (flgura de la derecha) y regresi?on por
m??nimos cuadrados (flgura de la izquierda). . . . . . . . . . . . . . . . . . . . . . 11
1.7 Represesntaci?on del campo de isoprobabilidad P{S{N para la aleaci?on AlMgSi1.
Los percentiles representados corresponden con los valores 0:01,0:05, 0:50, 0:95 y
0:99. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Diferentes historias de carga utilizadas para el an?alisis del da~no acumulado, cuando
? = 1050 MPa ! ? ? = 1:099 ( 0 = 955:5 MPa). (a) ?m = ?0:500,
?M = 0:599; (b) ?m = ?0:399, ?M = 0:700; (c) ?m = ?0:500, ?M = 0:599
para los ciclos impares y, ?m = ?0:399, ?M = 0:700 en los ciclos pares. . . . . . . 14
1.9 Funciones de distribuci?on obtenidas para las distintas historias de cargas uti-
lizadas en el da~no acumulado cuando ? = 1050 MPa! ? ? = 1:099 ( 0 = 955:5
MPa). (a) ?m = ?0:500, ?M = 0:599; (b) ?m = ?0:399, ?M = 0:700; (c)
?m = ?0:500, ?M = 0:599 para los ciclos impares y, ?m = ?0:399, ?M = 0:700
en los ciclos pares. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.10 Historias de cargas utilizadas para el an?alisis de la in uencia de existencia de
discontinuidades dentro de un espectro de carga constante. De arriba hacia abajo
y de izquierda a derecha: secuencia origial (sin discontinuidades), secuencia con
discontinuidad situada en N? = 10, secuencia con discontinuidad situada en N? =
100 y secuencia en la que las discontinuidades se sit?uan en N? = 10 y N? = 30
(para un valor de N0 = 532000 ciclos). . . . . . . . . . . . . . . . . . . . . . . . . 15
xiii
xiv LIST OF FIGURES
1.11 Funciones de probabilidad obtenidas del an?alisis de la in uencia de existencia de
discontinuidades dentro de un espectro de carga constante. De arriba hacia abajo
y de izquierda a derecha: secuencia original (sin discontinuidades), secuencia
con discontinuidad situada en N? = 10, secuencia con discontinuidad situada en
N? = 100 y secuencia en la que las discontinuidades se sit?uan en N? = 10 y
N? = 30 (para un valor de N0 = 532000 ciclos). . . . . . . . . . . . . . . . . . . . 16
1.12 Historias de carga variable analizadas en el estudio del da~no acumulado: (a1); (a2)
constante ?m y variable ?M , (b1; b2) constante ?M y variable ?m, (c1; c2) variable
?M y ?m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.13 Esquema representativo del problema sim?etrico. . . . . . . . . . . . . . . . . . . . 17
1.14 Funciones de distribuci?on obtenidas tras el an?alisis de diversas historias de carga
variable: (a1); (a2) constante ?m y variable ?M , (b1; b2) constante ?M y variable
?m, (c1; c2) variable ?M y ?m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.15 Funciones de distribuci?on obtenidas tras el an?alisis del da~no acumulado en difer-
entes historias de carga: (a1); (a2) constante ?m y variable ?M , (b1; b2) constante
?M y variable ?m, (c1; c2) ?M y ?m variables. . . . . . . . . . . . . . . . . . . . . 18
2.1 Schematic representation of a typical fatigue plot of the fatigue life data, and
some difierent flts: (a) linear flt, (b) piece wise linear flt, (c) nonlinear flt. . . . . 24
3.1 Illustration of the constant stress amplitude test and three difierent cases: (a)
mean = 0, (b) mean 6= 0 and (c) min = 0 [59]. . . . . . . . . . . . . . . . . . . . 28
3.2 Example of a S{N curve [59]. The left side corresponds to a representation on
an arithmetic scale of N . The right side shows a representation on a logarithmic
scale of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Schematic representation of crack propagation. Typical Paris curve [106]. . . . . 31
3.4 Schematic representation of the Yokobori problem, [134]. . . . . . . . . . . . . . . 33
3.5 Schematic of a S{N curve for steels [59]. . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Difierent diagrams of mean stress corrections. Gerber?s and Goodman?s diagrams
(left) and Haigh?s plot for Gerber?s and Goodman?s diagrams (right) [135]. . . . . 35
3.7 Difierent diagrams of mean stress corrections. Comparison between Goodman?s
and Morrow?s mean stress models (left side). Models for combined fatigue limit
and yield in ductile materials (right side)[135]. . . . . . . . . . . . . . . . . . . . 36
3.8 Mean stress sensitivity factors [135]. . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Mean stress sensitivity factors [135]. . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.10 Temperature and frequency efiects on the S{N curve for a nickel-base alloy Inconel
[32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.11 Rules of the three{point rain{ ow cycle counting [135]. . . . . . . . . . . . . . . . 40
3.12 Rules of the four{point rain{ ow cycle counting [135]. . . . . . . . . . . . . . . . 40
3.13 Example of four{point rain ow cycle counting [135]. (a) original loading, (b)
extraction of cycle,(c) reconstruction of the load. . . . . . . . . . . . . . . . . . . 41
3.14 Concept of the local strain-life approach. . . . . . . . . . . . . . . . . . . . . . . . 42
3.15 Engineering stress{strain curve (left) and true stress-strain curve (right) [135]. . . 42
3.16 Hysteresis loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.17 Schematic total strain-life curve [135]. . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 Use of the Palmgren{Miner rule for life prediction for variable amplitude loading
which is completely reversed [59]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
LIST OF FIGURES xv
4.2 Illustration of the up-and-down method using the Dixon and Mood data and
showing the flve stress levels [48]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Schematic diagram of equiprobability of fracture curves [22]. . . . . . . . . . . . . 52
4.4 S{N and S{? diagrams. Letters a, b, c correspond to S{N fleld. Letters a?, b?, c?
correspond to the ?{N fleld [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Bastenaire schematic curve [126]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.6 Experimental S{? diagram for flve difierent steels. flgure from [22]. . . . . . . . . 56
4.7 Cumulative frequency curves of ?(NCF ) for steel 33CD4 (treated to 80kg=mm2).
This curve correspond with the third curve in flgure 4.6. flgure from [22]. . . . . 57
4.8 Shapes of S{N curve considered: (a) simple model, (b) extended model [126]. . . 59
4.9 S{N curve established by 20 tests per stress level [126]. . . . . . . . . . . . . . . . 59
4.10 Best supported S{N curves: (a) changing slope abruptly to the horizontal; (b)
changing slope continuously to the horizontal as fltted to the set of data from
flgure 4.9. flgure from [126]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.11 Regions of validity of its simplifled forms [86]. . . . . . . . . . . . . . . . . . . . . 63
4.12 Comparison of regressions by the Basquin function (Equation (4.1)) for N <<
106, the Stromeyer function (Equation (4.36)), and new function (Hx) with its
asymptotes. flgure from [86]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.13 Graphical representation of the Weibull model. Percentiles curves representing
the relationship between lifetime, N?, and stress range, ? ?, in the S{N fleld [44]. 67
5.1 Basic rectangles. flgure from [46]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Schematic deflnition of linear functions [120]. . . . . . . . . . . . . . . . . . . . . 72
5.3 Schematic deflnition of logarithmic functions [120]. . . . . . . . . . . . . . . . . . 72
5.4 Some solutions of D?Alambert?s equation. The functions presented are cosnx and
coshnx for n = 1; 2; 3 [121]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.5 Regression model. flgure from [46] . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.6 Wo?hler fleld of model 1. flgure from [46] . . . . . . . . . . . . . . . . . . . . . . . 88
5.7 Wo?hler fleld of model 2. flgure from [46] . . . . . . . . . . . . . . . . . . . . . . . 88
5.8 Illustration of the hypothesis of independence. flgure from [46] . . . . . . . . . . 89
5.9 Experimental and theoretical survivor functions for lengths 30, 60 and 90 cm.
(from [26]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.10 Illustration of separate and consensus proposals. Image from [46] . . . . . . . . . 96
6.1 Schematic representation of the physical meanings of B and C model parameters. 102
6.2 Wo?hler curves for difierent percentiles: (a) for constant max, (b) for constant
min. Image from [42] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 Schematic representation about the compatibility in Castillo?s models (see [42],
[40]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.4 Schematic Wo?hler curves for percentiles f0:01; 0:05; 0:5; 0:95; 0:99g for ?max = 1
and ?max = 1:5, and ?min = 0:4 and ?min = 0:8, illustrating the compatibility
condition. Dashed lines refer to Wo?hler curves for constant ?min, and continuous
lines refer to Wo?hler curves for constant ?max. Image from [42]. . . . . . . . . . . 104
6.5 Schematic representation of the range of the problem. . . . . . . . . . . . . . . . 108
6.6 Schematic representation of difierent sub-models. . . . . . . . . . . . . . . . . . . 111
xvi LIST OF FIGURES
6.7 S{N curves for notchedKt = 3:3, AISI 4340 Alloy steel bar fltted by three difierent
methods. The upper corresponds to the MIL-HDBK-5G, the intermediate to
the proposed model without constraints and the lower to the proposed model
including all the constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.8 S{N curves for notched Kt = 3:3, AISI 4340 Alloy steel bar fltted using the
proposed regression model with constraints and including the runouts. In the
lower flgure the outlier has been removed. . . . . . . . . . . . . . . . . . . . . . . 117
6.9 S{N curves for constant R = 0:5; 0:1;?0:5 and ?1 (From top to bottom and left
to right). The percentiles 0:01, 0:05, 0:50, 0:95 and 0:99 are represented. . . . . . 118
6.10 S{N curves for constant M = 0:8, m = 0, mean = 0 and R = ?1 (From top
to bottom and left to right). The percentiles 0:01, 0:05, 0:50, 0:95 and 0:99 are
represented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.1 Difierent geometries for fatigue tests: (a) Starke et al. specimen [127], (b) Wolf
et al. specimen [133]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2 Geometry of the testing specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.3 Schenk Machine, used for 42CrMo4 tests. (a) General sight of the machine, (b)
detail of the machine, specimen?s grips. . . . . . . . . . . . . . . . . . . . . . . . 126
7.4 Rumul Machine, used for AlMgSi1 tests. (a) General sight of the machine, (b)
detail of the machine, specimen testing. . . . . . . . . . . . . . . . . . . . . . . . 127
7.5 Distribution of the difierent tests loads. The left flgure corresponds to the 42CrMo4
steel, and the right flgure to the AlMgSi1 alloy. . . . . . . . . . . . . . . . . . . . 128
7.6 Schematic representation of the difierent between both distributions of the difier-
ent tests loads. The left flgure corresponds to the 42CrMo4 steel, and the right
flgure to the AlMgSi1 alloy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.7 S{N curves for constant ?M for Gumbel model with constraints for the 42CrMo4
steel using difierent methods: least squares (left side) and maximum likelihood
(right side). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.8 S{N curves for constant ?M for Gumbel model with constraints for the AlMgSi1
alloy using difierent methods: least squares (left side) and maximum likelihood
(right side). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.9 PP{plot for difierent analysis cases of the 42CrMo4 steel: least squares (left side),
maximum likelihood (right side). . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.10 PP{plot for difierent analysis cases of the AlMgSi1 alloy: least squares (left side),
maximum likelihood (right side). . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.11 Representation of the Gumbel Model obtained in the example analyzed in section
6.8.1 (left side) and its PP{plot (right side). . . . . . . . . . . . . . . . . . . . . . 135
7.12 S{N curves representing constant ?M = 0:98; 0:9; 0:8 and 0:7Ry for the 42CrMo4
steel (from top to bottom and left to right). The percentiles 0:01, 0:05, 0:50, 0:95
and 0:99 are represented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.13 S{N curves representing constant ?M = 0:9; 0:8; 0:7 and 0:6Rp0:2 for the AlMgSi1
alloy (from top to bottom and left to right). The percentiles 0:01, 0:05, 0:50, 0:95
and 0:99 are represented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.14 S{N fleld extrapolated to M = 0:9Ry (blue lines). The other S{N flelds represent
the curves obtained with the 0:8; 0:7 and 0:6Ry series data. . . . . . . . . . . . . 138
7.15 S{N fleld extrapolated to M = 0:8Ry (blue lines). The other S{N flelds represent
the curves obtained with the 0:9; 0:7 and 0:6Ry series data. . . . . . . . . . . . . 138
LIST OF FIGURES xvii
7.16 S{N fleld extrapolated to M = 0:7Ry (blue lines). The other S{N flelds represent
the curves obtained with the 0:9; 0:8 and 0:6Ry series data. . . . . . . . . . . . . 139
7.17 S{N fleld extrapolated to M = 0:6Ry (blue lines). The other S{N flelds represent
the curves obtained with the 0:9; 0:8 and 0:7Ry series data. . . . . . . . . . . . . 139
7.18 Representation of the PP{plot obtained for difierent estimations. From the top
to bottom, and left to right: without 0:9Ry, without 0:8Ry, without 0:7Ry and
0:6Ry original series data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.1 Illustration of the isodamage curves. flgure from [36]. . . . . . . . . . . . . . . . . 150
8.2 Theoretical example of application of the procedure for damage analysis. . . . . . 151
8.3 Load histories for the damage analysis, case of constant ? = 1050 MPa !
? ? = 1:099 ( 0 = 955:5 MPa). (a) ?m = ?0:500, ?M = 0:599; (b) ?m = ?0:399,
?M = 0:700; (c) ?m = ?0:500, ?M = 0:599 for odd cycles and ?m = ?0:399,
?M = 0:700 for even cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.4 Load histories for the damage analysis, case of constant ? = 1050 MPa !
? ? = 1:099 ( 0 = 955:5 MPa). (a) ?m = ?0:500, ?M = 0:599; (b) ?m = ?0:399,
?M = 0:700; (c) ?m = ?0:500, ?M = 0:599 for odd cycles and ?m = ?0:399,
?M = 0:700 for even cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.5 Load sequences used for the analysis of damage accumulation when a discontinuity
appear in the sequence. From top to the bottom, and left to right: original
sequence (without discontinuity), discontinuity situated at N? = 10, discontinuity
situated at N? = 100 and punctual cycles situated at N? = 10 and N? = 30 (for
N0 = 532000 cycles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.6 Load sequences used for the analysis of damage accumulation when a discontinu-
ity appear in the sequence. From top to the bottom, and left to right: original
sequence (without discontinuities), discontinuity situated at N? = 10, disconti-
nuity situated at N? = 100 and discontinuities situated at N? = 10 and N? = 30
(for N0 = 532000 cycles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.7 Variable load histories analyzed in the damage accumulation: (a1); (a2) constant
?m and variable ?M , (b1; b2) constant ?M and variable ?m, (c1; c2) variable ?M
and ?m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.8 Schematic representation of the symmetrical problem. . . . . . . . . . . . . . . . 157
8.9 Variable load histories analyzed in the damage accumulation: (a1); (a2) constant
?m and variable ?M , (b1; b2) constant ?M and variable ?m, (c1; c2) variable ?M
and ?m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.10 Variable load histories analyzed in the damage accumulation: (a1); (a2) constant
?m and variable ?M , (b1; b2) constant ?M and variable ?m, (c1; c2) variable ?M
and ?m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
B.1 Tension testing machine. (a) machine, (b) detail of specimens in the machine, (c)
detail of the broken specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
B.2 Geometric deflnition of the tension test specimen for the 42CrMo4 steel. . . . . . 172
B.3 Geometric deflnition of the tension test specimen for the AlMgSi1 alloy. . . . . . 173
B.4 Static test result for the 42CrMo4 steel. Specimen Nr. 1. . . . . . . . . . . . . . 174
B.5 Static test result for the 42CrMo4 steel. Specimen Nr. 2. . . . . . . . . . . . . . 174
B.6 Static test result for the 42CrMo4 steel. Specimen Nr. 3. . . . . . . . . . . . . . 175
B.7 Static test result for the AlMgSi1 alloy. Specimen Nr. 1. . . . . . . . . . . . . . . 175
B.8 Static test result for the AlMgSi1 alloy. Specimen Nr. 2. . . . . . . . . . . . . . . 176
xviii LIST OF FIGURES
B.9 Static test result for the AlMgSi1 alloy. Specimen Nr. 3. . . . . . . . . . . . . . . 176
B.10 Geometric fatigue test deflnition for the 42CrMo4 steel. . . . . . . . . . . . . . . 177
B.11 Geometric fatigue test deflnition for the AlMgSi1 alloy. . . . . . . . . . . . . . . . 177
D.1 Start screen of the SISIFO program. . . . . . . . . . . . . . . . . . . . . . . . . . 183
D.2 Main windows of the program: the parameter estimation window (top flgure) and
the damage analysis window (bottom flgure). . . . . . . . . . . . . . . . . . . . . 184
D.3 Representation of the difierent steps to use the Castillo model in the SISIFO
program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
D.4 Representation of the difierent steps to be followed for the installation of the
SISIFO program. Steps 1 to 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
D.5 Representation of the difierent steps to be followed for the installation of the
SISIFO program. Steps 5 to 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
D.6 Entering the Parameter estimation menu of SISIFO. . . . . . . . . . . . . . . . . 188
D.7 Panels that form the Parameter estimation window of SISIFO. . . . . . . . . . . 189
D.8 Deflnition of the maximum number of cycles (run outs). Left side, a correct
deflnition; right side, bad input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
D.9 Example of the testing option chosen by the user for the data analysis. . . . . . . 190
D.10 Example for loading data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
D.11 Example of data flle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
D.12 Deflnition of the data sets labels. . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
D.13 The two plots obtained with SISIFO: upper flgure, plot of the initial data, and
lower flgure, plot of the normalized data. . . . . . . . . . . . . . . . . . . . . . . . 193
D.14 Plot options in the Save plot window. . . . . . . . . . . . . . . . . . . . . . . . . 194
D.15 Parameter estimation methods represented in the corresponding pop-up menu in
SISIFO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
D.16 SISIFO Check boxes to choose the flnal data for the parameter estimation, with
or without run-outs and/or outliers. . . . . . . . . . . . . . . . . . . . . . . . . . 195
D.17 Deflnition of the parameter estimation options. . . . . . . . . . . . . . . . . . . . 195
D.18 Dialog box: Error on initial parameters. . . . . . . . . . . . . . . . . . . . . . . . 196
D.19 Parameters calculated by the SISIFO program. . . . . . . . . . . . . . . . . . . . 196
D.20 Difierent plot options of the estimation results. . . . . . . . . . . . . . . . . . . . 197
D.21 Plot ranges of the chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
D.22 Probability deflnition of the P{S{N curves. . . . . . . . . . . . . . . . . . . . . . 198
D.23 Example of graphics obtained after the parameter estimation: (a) S{N curves;
(b) P{S{N curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
D.24 Dialog box conflrming the creation of the flle Param.xls. . . . . . . . . . . . . . . 199
D.25 PP{plot obtained by the SISIFO program. . . . . . . . . . . . . . . . . . . . . . . 199
D.26 Resulting of the model validation performed by the SISIFO program. . . . . . . . 199
D.27 Conflrmation after creating the validation flle. . . . . . . . . . . . . . . . . . . . . 200
D.28 Example of the ValidationTest.xls flle. . . . . . . . . . . . . . . . . . . . . . . . . 200
D.29 Deflnition of the case study in the extrapolation panel. . . . . . . . . . . . . . . . 201
D.30 Example of extrapolation: left side, S{N curves for a M = 900MPa; right side,
P{S{N curves for the same value and case of study. . . . . . . . . . . . . . . . . . 201
D.31 Entering the Damage analysis menu of SISIFO. . . . . . . . . . . . . . . . . . . . 202
D.32 Loading a load sequence in the SISIFO program. . . . . . . . . . . . . . . . . . . 202
D.33 Difierent panels in the Damage analysis menu of the SISIFO program. . . . . . . 203
LIST OF FIGURES xix
D.34 Difierent types of load histories: (a) ? = 900 MPa,(b) ? = 0:1 ? N + 900
MPa,(c) ? = 0:4 ?N + 600 MPa. . . . . . . . . . . . . . . . . . . . . . . . . . . 204
D.35 Representation of a quasi-random load history in SISIFO. . . . . . . . . . . . . . 205
D.36 Error dialog box used to tell the user that the sequence flle has not been created. 206
D.37 Rain ow matrix in color scale. The color scale of the right side represents the
frequence of cycles in the rain ow matrix. . . . . . . . . . . . . . . . . . . . . . . 207
D.38 Parameter deflnition for the Rain ow fllter analysis. . . . . . . . . . . . . . . . . 208
D.39 Dialog box indicating the absence of the parameter flle. . . . . . . . . . . . . . . 208
D.40 Representation of the probability of failure calculated with the model parameters
for a certain point of study. The red point corresponds to the point of study, the
blue line corresponds to the cdf of the model for all number of cycles with this load.209
D.41 Deflnition of the data flle example1.xls. . . . . . . . . . . . . . . . . . . . . . . . . 211
D.42 Graphics representing the original and normalized data. . . . . . . . . . . . . . . 212
D.43 Parameter estimation with all the data, using the maximum likelihood method. . 212
D.44 File saved by SISIFO with the estimated parameters. . . . . . . . . . . . . . . . . 213
D.45 S{N curves and P{S{N curves obtained with the parameter estimation. . . . . . 213
D.46 Obtained results for the model validation. . . . . . . . . . . . . . . . . . . . . . . 214
D.47 P{S{N curve obtained for R = ?1, extrapolated from the original data. . . . . . 214
D.48 Representation of the Menu 1: Parameter estimation after using all the panels. . 215
D.49 (a) Load history deflnition: ? = 1250 MPa (constant load history), with m =
?250 MPa and M = 1000 MPa. (b) Error dialog box created by SISIFO when
the flle sequence.xls can not be created. . . . . . . . . . . . . . . . . . . . . . . . 217
D.50 Error dialog box created by SISIFO when the rain ow analysis cannot be carried
out. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
D.51 Damage results obtained with SISIFO, for the load history ? = 1250 MPa in
the point N = 4000, m = ?250 MPa and M = 1000 MPa . . . . . . . . . . . . 219
D.52 Representation of Menu 2: Damage Analysis after using it. . . . . . . . . . . . . 220
D.53 Random spectrum used for the example of rain ow analysis. . . . . . . . . . . . . 221
D.54 Comparison between the rain ow matrix (a) and the flltered rain ow matrix (b). 221
xx LIST OF FIGURES
List of Tables
1.1 Par?ametros obtenidos para el acero 42CrMo4. . . . . . . . . . . . . . . . . . . . . 10
1.2 Par?ametros obtenidos para la aleaci?on AlMgSi1. . . . . . . . . . . . . . . . . . . 10
3.1 Monotonic mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Cyclic material properties: steady{state cyclic behavior . . . . . . . . . . . . . . 44
3.3 Cyclic material properties: constant-amplitude fatigue behavior . . . . . . . . . . 45
4.1 Historical evolution of principal fatigue models. . . . . . . . . . . . . . . . . . . . 48
4.2 Jointly best supported parameters of the S{N curve as a function of the cutofi
points derived for the data from flgure 4.9. flgure from [126] . . . . . . . . . . . . 60
5.1 Some example of homogeneous functions. . . . . . . . . . . . . . . . . . . . . . . 82
6.1 Parameter estimates cases 1 to 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Estimated percentile values associated with the difierent data points using the
Gumbel fltted model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.1 Static characteristics of the metallic alloys used. . . . . . . . . . . . . . . . . . . . 124
7.2 Chemical compositions of 42CrMo4 steel according to DIN EN 10083-3:2007-01
and AlMgSi1 alloy according with DIN EN 573-3 respectively. . . . . . . . . . . . 125
7.3 Specimen?s dimensions for each material. . . . . . . . . . . . . . . . . . . . . . . . 125
7.4 Resulting lifetimes for 42CrMo4 and AlMgSi1. . . . . . . . . . . . . . . . . . . . 129
7.5 Parameter estimates for difierent estimation methods for the 42CrMo4 steel. . . . 130
7.6 Parameter estimates for difierent estimation methods for the AlMgSi1 alloy. . . . 131
7.7 Goodness of flt tests for the 42CrMo4 steel (in parenthesis the p value). . . . . . 133
7.8 Goodness of flt tests for the AlMgSi1 alloy (in parenthesis the p value). . . . . . 134
7.9 Goodness of flt tests for the theoretical example shown in section 6.8.1. . . . . . 135
7.10 Estimated percentile values associated with the difierent data points using the
Gumbel fltted model for the 42CrMo4 steel. . . . . . . . . . . . . . . . . . . . . . 135
7.11 Estimated percentile values associated with the difierent data points using the
Gumbel fltted model for the AlMgSi1 alloy. . . . . . . . . . . . . . . . . . . . . . 136
7.12 Estimated parameter for difierent cases of study. . . . . . . . . . . . . . . . . . . 140
7.13 Goodness of flt test for the difierent cases of study. . . . . . . . . . . . . . . . . 140
8.1 Properties and characteristics of difierent damage measures for the case of con-
stant stress levels (Legend: ? ? ?? very good, ? ? ? good, ?? medium and ? bad)
[115]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.2 Characteristics of the three difierent load histories analyzed. Case of constant ? . 152
xxi
xxii LIST OF TABLES
8.3 Values of the number of cyclesN? for the probabilities of p = 0:01; 0:1; 0:5; 0:9; 0:99
for constant ? and cases (a), (b) and (c). . . . . . . . . . . . . . . . . . . . . . 153
8.4 Values of number of cycles N? for the probabilities of p = 0:01; 0:1; 0:5; 0:9; 0:99
when there exits punctual high cycles in the load sequence. . . . . . . . . . . . . 153
8.5 Parameter for the deflnition of the load histories expressions. . . . . . . . . . . . 156
8.6 Values of number of cycles N? for the probabilities of p = 0:01; 0:1; 0:5; 0:9; 0:99
for difierent variable load histories. . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.1 Result of metallographic test for the 42CrMo4 material . . . . . . . . . . . . . . 171
B.2 Static tests results for the 42CrMo4 steel. . . . . . . . . . . . . . . . . . . . . . . 173
B.3 Static tests results for the AlMgSi1 alloy. . . . . . . . . . . . . . . . . . . . . . . 173
C.1 Experimental protocol for the 42CrMo4 steel. Constant load tests . . . . . . . . 179
C.2 Experimental protocol for the AlMgSi1 alloy. Constant load tests . . . . . . . . . 180
D.1 Useful symbols in MATLAB code. . . . . . . . . . . . . . . . . . . . . . . . . . . 190
D.2 Resulting lifetimes from the laboratory tests. . . . . . . . . . . . . . . . . . . . . 210
D.3 Resulting lifetimes from the laboratory tests for the Example Nr.2. . . . . . . . . 216
D.4 Parameters estimated with SISIFO for Example Nr.2. . . . . . . . . . . . . . . . 216
Part I
Sinopsis de la Tesis Doctoral en
Castellano
1
Chapter 1
Sinopsis de la Tesis Doctoral en
Castellano
1.1 Objeto y objetivos de la investigaci?on
El presente trabajo pretende aportar un avance en el conocimiento del comportamiento de
materiales met?alicos sometidos a cargas c??clicas mediante el desarrollo de un nuevo modelo
probabil??stico de fatiga de materiales.
De acuerdo al Cap??tulo 8 de la Normativa de los Estudios de Doctorado de la Universidad
de Cantabria, al Real Decreto 778/1998 y a la normativa de la Universidad de Cantabria, los
principales requisitos para la obtenci?on del t??tutlo de Doctor son:
? Estar en posesi?on de t??tulo de Licenciado, Arquitecto, Ingeniero o equivalente u homolo-
gado a ellos.
? Realizar y aprobar los cursos, seminarios y trabajos de investigaci?on tutelados del programa
de doctorado correspondiente.
? Presentar y aprobar una tesis doctoral consistente en un trabajo original de investigaci?on.
En el presente documento la doctoranda Mar??a Luisa Ruiz Ripoll presenta las contribuciones
necesarias que avalan el cumplimiento de estos requisitos optando al t??tulo de Doctor por la
Universidad de Cantabria.
Esta tesis doctoral ha sido realizada dentro del Departamento de Matem?atica Aplicada y
Ciencias de la Computaci?on de la Universidad de Cantabria, dentro del Programa Interdepar-
tamental I04: Desarrollo y Aplicaci?on de Modelos en Ingenier??a Civil.
Esta tesis doctoral a su vez opta al t??tulo de doctorado europeo, habi?endose realizado las es-
tancias m??nimas fuera de Espa~na en un centro de investigaci?on de otro Estado europeo cursando
estudios o realizando trabajos de investigaci?on que le hayan sido reconocidos por el Departa-
mento responsable del programa, de acuerdo con el Cap??tulo 15 de la Normativa de los Estudios
de Doctorado de la Universidad de Cantabria, al Real Decreto 778/1998 y a la normativa de la
Universidad de Cantabria.
Debido a que la tesis doctoral est?a escrita en una lengua distinta a la castellana, debe
contener un apartado suflcientemente amplio, escrito en castellano en el que se incluyan las
contribuciones, resultados y conclusiones obtenidas durante la elaboraci?on de la misma, siendo
este el objetivo del cap??tulo en el que nos encontramos.
3
4 1.1. OBJETO Y OBJETIVOS DE LA INVESTIGACI?ON
La presente tesis doctoral, titulada:\A Statistical Fatigue Model Covering the Tension and
Compression Wo?hler Fields and Allowing Damage Accumulation" (Modelo Estad??stico de Fatiga
para el An?alisis del Campo de Wo?hler Bajo Tensi?on y Compresi?on, Incluyendo el C?alculo de
Acumulacin de Da~no) est?a redactado en lengua inglesa, por ello es necesario que contenga un
apartado en castellano que se compone de las siguientes secciones:
1. Objeto y objetivos de la investigaci?on.
2. Planteamiento y metodolog??a utilizada.
3. Aportaciones originales.
4. Conclusiones y futuras l??neas de investigaci?on.
1.1.1 Introducci?on, hip?otesis y objetivos de la investigaci?on
Los trabajos existentes en la literatura de fatiga no han resuelto todav??a el problema de la
acumulaci?on de da~no debida a una historia de cargas arbitraria, ni desde el punto de vista
te?orico ni desde el punto de vista pr?actico y de laboratorio. Este problema es sin embargo
crucial en la ingenier??a pr?actica para dise~no de estructuras. Se plantea el desarrollo de un modelo
probabil??stico que sirva como base para el an?alisis a fatiga de materiales met?alicos sometidos a
cualquier espectro de carga en cualquier rango de la misma (tensi?on, compresi?on o mixto).
Se pretende por tanto, dar una respuesta te?orica y pr?actica a un campo de investigaci?on
ampliamente explorado antes, pero que no ha llegado a obtener una expresi?on matem?atica que
involucre conceptos estad??sticos, f??sicos y del comportamiento del material para el tipo de carga
descrito anteriormente, de manera que sirva como base para el comienzo de una nueva forma de
abordar la fatiga de los materiales basada en un nuevo modelo.
La principales motivaciones de esta tesis doctoral son dos. La primera de ellas es el ahorro
econ?omico que supone un buen conocimiento del comportamiento de los materiales frente al
fen?omeno de la fatiga. Los da~nos por fatiga siguen siendo de los m?as elevados en el dise~no
ingenieril al igual que las inversiones realizadas para su prevenci?on. El coste anual debido a la
fatiga de los materiales en Estados Unidos es de alrededor de un 3% del producto interior bruto
(GON) [59], [94]. El otro motivo es el aumento de la flabilidad en el c?alculo de la vida ?ultima
(comportamiento a fatiga) de un material sometido a cargas c??clicas.
Esta tesis doctoral es la continuaci?on del trabajo realizado por los directores de tesis de la
autora del trabajo, quienes desde los a~nos ochenta del pasado siglo desarrollan, implementan
y mejoran modelos para la fatiga de los materiales. Son destacables algunos de los ?ultimos
trabajos de estos autores, en los que mediante un modelo tipo Weibull se predice la vida ?ultima
a fatiga y su da~no acumulado, de un material sometido a cualquier historia de carga.
Las hip?otesis utilizadas para el desarrollo de esta investigaci?on son las siguientes:
1. Los modelos existentes para el an?alisis a fatiga hoy en d??a no satisfacen todos las condi-
ciones f??sicas, experimentales y estad??sticas para un buen modelo.
2. Un modelo ?util para deflnir el comportamiento de materiales debe satisfacer estas condi-
ciones.
3. Los materiales utilizados en esta investigaci?on son normalmente met?alicos y longitudinales.
4. Se supone el principio del eslab?on m?as d?ebil : \Si un elemento longitudinal es dividido en
n subelementos, la vida ?ultima del material vendr?a dada por la vida ?ultima del elemento
m?as d?ebil".
CHAPTER 1. SINOPSIS DE LA TESIS DOCTORAL EN CASTELLANO 5
5. Se supone que la acumulaci?on del da~no en un material comienza cuando la probabilidad
de fallo es nula, y el fallo total se produce cuando dicha probabilidad es igual a uno.
Tras haber realizado una peque~na introducci?on al problema y haber dispuesto las hip?otesis
de trabajo, los objetivos que se plantean en esta tesis doctoral son los siguientes:
1. Desarrollar un estado del arte de la fatiga de los materiales donde se muestren los princi-
pales modelos usados hasta la fecha para la estimaci?on y el an?alisis del comportamiento
del material sujeto a cargas c??clicas. Realizar una discusi?on de las ventajas y desventajas
de cada uno de ellos.
2. Detectar los principales problemas existentes en estos modelos y en las estrategia de ensayos
utilizados en el an?alisis a fatiga de los materiales.
3. Desarrollar un nuevo modelo para la fatiga de los materiales que cubra todo el rango de
cargas (tensi?on y compresi?on) para el campo de Wo?hler para cualquier historia de carga.
La derivaci?on del modelo estar?a basada en la teor??a de ecuaciones funcionales.
4. Realizar una discusi?on de las medidas de da~no actuales y proponer nuevas, seleccionando
las m?as adecuadas para el an?alisis a fatiga.
5. Analizar el da~no acumulado con el modelo probabil??stico propuesto.
6. Proponer una buena estrategia de ensayos para la obtenci?on de los mejores resultados
de laboratorio posibles y por lo tanto, las mejores estimaciones para los par?ametros del
modelo.
7. Dise~nar una serie de ensayos experimentales para la evaluaci?on y validaci?on del modelo
te?orico.
8. Analizar los resultados obtenidos para validar el modelo.
9. Bas?andose en los resultados obtenidos y su analisis, presentar las principales conclusiones
del trabajo realizado.
1.2 Planteamiento y metodolog??a utilizada
La deflnici?on de una buena metodolog??a trae consigo la obtenci?on de buenos resultados y una
buena organizaci?on en tiempo y trabajo. En esta tesis doctoral se han seguido los siguientes
pasos (ver flgura 1.1):
1. An?alisis del problema: cu?ales han sido los hechos a lo largo de la historia que han pro-
piciado que el hombre se interese por ese problema y quiera solucionarlo?. Corresponde al
cap??tulo 3 de la tesis doctoral.
2. Comprensi?on del problema: En el que se realiza un an?alisis de las causas que provocan esas
situaciones y entendiendo qu?e factores entran a tomar parte de su evoluci?on y desarrollo.
Deflnici?on del t?ermino fatiga de materiales. Todos estos conceptos se analizan en los
cap??tulos 3 y 4 de esta tesis doctoral.
3. Desarrollo: donde se establecen las bases y se desarrollan las aportaciones cient??flcas nece-
sarias para la mejora y el an?alisis de materiales sujetos a este tipo de solicitaciones. Los
cap??tulos 5 y 6 sientan las bases necesarias para la derivaci?on y aplicaci?on del modelo de
fatiga propuesto.
6 1.3. APORTACIONES ORIGINALES
Figura 1.1: Metodolog??a de trabajo seguida en la elaboraci?on de la tesis doctoral.
4. Ensayos experimentales: cuyo objetivo es la obtenci?on de datos que posteriormente se
analizar?an para validar el modelo te?orico propuesto en el punto anterior. Cap??tulos 7 y 8
de la tesis doctoral.
5. An?alisis de los resultados obtenidos: cuyo objetivo es la validaci?on del modelo te?orico.
Cap??tulos 7 y 8 de la tesis doctoral.
6. Publicaci?on: de las principales contribuciones al campo cient??flco internacional. Este
?ultimo paso de la investigaci?on se ha ido realizando a lo largo de la elaboraci?on de la
tesis doctoral y queda plasmado en art??culos enviados a revistas internacionales con ??ndice
de impacto.
1.3 Aportaciones originales
1.3.1 Los modelos Weibull y Gumbel basados en el campo S{N
(chapter 6: The Weibull and Gumbel S{N fleld stress based fatigue models)
A continuaci?on se presenta, desarrolla y analiza de manera te?orica el modelo propuesto
por los autores. Algunos de los aspectos que se estudiar?an son las condiciones tanto f??sicas,
como estad??sticas y de compatibilidad que deflnen el problema, las restricciones del modelo, las
CHAPTER 1. SINOPSIS DE LA TESIS DOCTORAL EN CASTELLANO 7
propiedades del mismo, los submodelos resultantes y los m?etodos de estimaci?on propuestos para
el conocimiento de los par?ametros del modelo.
Se propone en primer lugar un modelo general de fatiga que incluye la consideraci?on de la
tensi?on media, con un modelo probabil??stico tipo Weibull de 9 par?ametros. Posteriormente,
tras el an?alisis del mismo se llega a la expresi?on flnal de un modelo probabil??stico tipo Gumbel
basado en 8 par?ametros caracter??sticos del material.
Las principales aportaciones de este modelo son:
? De acuerdo con el Teorema de Buckingham [31], solo deben utilizarse variables adimensio-
nales en modelos de regresi?on. Esto implica que el modelo sea sencillo y que los par?ametros
obtenidos a su vez sean variables adimensionales.
? El modelo no se basa en hip?otesis arbitrarias, sino en propiedades f??sicas y estad??sticas que
son necesarias en cualquier modelo de fatiga.
? El modelo revela informaci?on estad??stica que no s?olo incluye valores medios, sino que
analiza la variabilidad del modelo y con ello, el conocimiento de la probabilidad en cada
momento.
? El modelo puede usarse para todo el rango de cargas: tensi?on, compresi?on y/o mixto.
? El modelo puede extrapolarse a cualquier otra condici?on de cargas tras la estimaci?on de
los par?ametros caracter??sticos del material.
Consid?erese un material sujeto a unas cargas constantes m??nimas y m?aximas que denomi-
naremos m y M por simplicidad (secci?on 6.2). Sabiendo que N es el n?umero de ciclos al que
este material rompe, el objetivo es conocer la probabilidad de rotura p del material sujeto a
dichas cargas. Para la derivaci?on del modelo se tiene en cuenta que p est?a relacionada con los
t?erminos N; m y M . Con estos conceptos en mente, el procedimiento es el siguiente: Primero,
se parte del modelo presentado por Castillo et al. [39]:
p = 1? exp
(
?
?(logN? ?B) ( ?M ? ?m ? C)? E
D
?A)
; (1.1)
donde A representa el par?ametro de forma de la funci?on de distribucion tipo Weibull; B, re-
presenta el valor l??mite en la vida ?ultima del material, as??ntota vertical de la curva S{N; C, el
l??mite de endurancia; E, deflne la posici?on en la que se encuentra el percentil cero de la hip?erbola
y D, representa de escala de la funci?on de distribuci?on de Weibull (flgura 1.2).
A continuaci?on se aplica la condici?on de compatibilidad en el sentido de que: Si se ejecuta
un ensayo de fatiga a carga constante oscilante entre m y M , podemos derivar el modelo
correspondiente a dos casos particulares, (a) con m constante o (b) con M constante, pero en
ambos casos el modelo es el mismo [42]. Esta condici?on de compatibilidad se representa en la
flgura 1.3, donde las intersecciones entre curvas con m y M constantes se deflnen como l??neas
horizontales.
Sabiendo que si se toma A ! 1 (see [37] and [40]), el modelo pasa a ser tipo Gumbel, la
expresi?on flnal del modelo propuesto es:
p = 1?exp f? exp [C0 + C1 ?m + C2 ?M + C3 ?m ?M + (C4 + C5 ?m + C6 ?M + C7 ?m ?M ) logN?]g ; (1.2)
donde C0; C1; C2; C3; C4; C5; C6 y C7 son los par?ametros que deflnen el modelo y las variables
?m; ?M y N? se deflnen como ?m = m= 0; ?M = M= 0 y N? = N=N0, donde 0 y N0 son
variables utilizadas para adimensionalizar el problema.
8 1.3. APORTACIONES ORIGINALES
Log N
??
C
B
Figura 1.2: Representaci?on esquem?atica del signiflcado f??sico de las variables B y C del modelo
de Castillo et al. [39].
Para hacer el modelo estad??stica y f??sicamente compatible, deben tenerse en cuenta algunas
restricciones (secci?on 6.4):
? Restricciones f??sicas: Las as??ntotas en ambos ejes deben ser positivas y decrecientes con
respecto a las variables del modelo ( m, M y logN).
? Restricciones estad??sticas: la funci?on de distribuci?on debe ser creciente con respecto a las
variables del modelo y la curvatura del modelo debe ser siempre positiva.
La estimaci?on de los par?ametros del modelo se lleva a cabo mediante dos m?etodos diferentes:
m?axima verosimilitud y regresi?on por m??nimos cuadrados (secci?on 6.6.1 y 6.6.2). Para el caso de
m?axima verosimilitud, el problema se reduce a maximizar la funci?on objetivo (1.3) con respecto
a los par?ametros del modelo Ci, sujeta a las restricciones descritas enteriormente, desarrolladas
en la secci?on 6.4
L =
X
i2I1
?H(N?i ) + log
?C4 + C5 ?mi + C6 ?Mi + C7 ?mi ?Mi
?? log(N?i )
??
X
i2I1[I0
exp(H(N?i )); (1.3)
donde H(Ni) = C0+C1 ?mi +C2 ?Mi +C3 ?mi ?Mi +
?
C4 + C5 ?mi + C6 ?Mi + C7 ?mi ?Mi
?
logN?i .
En el caso de la regresi?on, la funci?on objetivo es ahora
Q =
nX
i=1
?
logN + C0 + C2
?max + C1R ?max + C3R ?max2 +
C4 + C6 ?max + C5R ?max + C7R ?max2
!2
; (1.4)
sujeta tambi?en a las restricciones descritas en la secci?on 6.4, deflni?endose R como R = m= M .
Con el fln de usar el modelo en la pr?actica, se deflnen ahora una estrategia basada en los
siguientes pasos:
Paso 1: Dise~no de la estrategia de ensayos. Se deflnen el n?umero de series as?? como la dis-
tribuci?on tensional de los distintos puntos a ensayar, intentando cubrir el mayor rango
tensional posible.
Paso 2: Elecci?on de las variables normalizadas N0 y 0. Con ellas se realiza la adimensiona-
lizaci?on de las variables.
CHAPTER 1. SINOPSIS DE LA TESIS DOCTORAL EN CASTELLANO 9
Log N
??
?
?max=1.5
?min=0.8
*
*
?max=1
*
?min=0.4
*
Figura 1.3: Esquema de las curvas de Wo?hler para los percentiles f0:01; 0:05; 0:5; 0:95; 0:99g con
un ?max = 1 y ?max = 1:5, y ?min = 0:4 y ?min = 0:8 [42].
Paso 3: Estimaci?on de los par?ametros del modelo. Utilizando para ello cualquiera de los dos
m?etodos deflnidos anteriormente (Ecuaciones (1.3) y (1.4)).
Paso 4: Extrapolaci?on del problema a otras condiciones tensionales cualesquiera. Empleando
el modelo (1.2) y los par?ametros C1; C2; C3; C4; C5; C6; C7, logN0 y 0 para cualquier otra
condici?on de ensayos.
1.3.2 Validaci?on experimental del modelo
(chapter 7: Experimental validation of the model)
La validaci?on experimental del modelo se ha llevado a cabo a partir de los resultados
obtenidos tras varias series de ensayos realizados en el departamento de Materials Science and
Technology del laboratorio federal suizo de ensayos e investigaci?on Empa-Du?bendorf.
Se estudia el comportamiento de dos aceros de caracter??sticas est?aticas y mec?anicas diferentes
(tabla 7.1):
? Acero con baja aleaci?on de cromo, 42CrMo4, (DIN-1.7225) con Rm = 1067MPa y Ry =
975:3MPa.
? Aleaci?on de aluminio, AlMgSi1 (DIN-3.2315) con Rm = 391:7MPa y Rp0:2 = 364:3MPa.
Las muestras son descritas en la secci?on 7.3. Los ensayos se enmarcan dentro de la normativa
ASTM E606 [5], manteni?endose constante M . Para cada material se eligieron cuatro series con
diferentes M = cte y variables m teniendo en cuenta la posici?on del l??mite de endurancia
(flgura 1.4, (a)) y el l??mite de plasticidad de cada material (flgura 1.4, (b)). Para la realizaci?on
de los ensayos se utiliz?o una m?aquina vibrophore de alta frecuencia, con capacidad de 150 Hz
de frecuencia.
10 1.3. APORTACIONES ORIGINALES
42C rMo4
0.7?Ry
0.8?Ry
0.9?Ry
0.98?Ry
650
700
750
800
850
900
950
1000
-700 -500 -300 -100 100 300
?min (MPa)
?
m
ax
(M
Pa
)
AlMgSi1
0.6?Ry
0.7?Ry
0.8?Ry
0.9?Ry
150
200
250
300
350
400
-400 -300 -200 -100 0 100 200 300
?min (MPa)
?
m
a
x
(M
Pa
)
(a) (a)
(b)(b)
Figura 1.4: Representaci?on esquem?atica de la distribuci?on tensional de los ensayos. A la
izquierda el acero 42CrMo4, a la derecha la aleaci?on AlMgSi1. (a) l??mite de plasticidad, (b)
l??mite de endurancia del material.
La adimensionalizaci?on se lleva cabo deflniendo 0 = max( Mi), que corresponde con los
valores 0 = 0:98Ry MPa para el 42CrMo4 y 0 = 0:9Rp0:2 MPa para el AlMgSi1. En el caso de
N0 = max(Ni) los valores elegidos corresponden con N0 = 532000 ciclos para el 42CrMo4 and
N0 = 526500 ciclos para el AlMgSi1.
Los resultados obtenidos se presentan en las tablas 1.1, 1.2 y las flguras 1.5 y 1.6 (secciones
7.6.2 y 7.6.3):
Acero 42CrMo4 La tendencia resultante en las curvas S{N es lineal ya que C4 = C5 = C7 = 0.
Los datos tienen una gran dispersi?on y ello diflculta la estimaci?on de los par?ametros del
modelo, aunque los resultados son aceptables (flgura 1.5, tabla 1.1).
Aleaci?on AlMgSi1 En este caso, la tendencia de las curvas que deflnen el campo de Wo?hler
no es lineal. S?olo los par?ametros C4 y C5 son nulos. La dispersi?on es menor que en el
primer material y la estimaci?on parece haber tenido mejor resultado (flgura 1.6, tabla 1.2).
Table 1.1: Par?ametros obtenidos para el acero 42CrMo4.
Case C0 C1 C2 C3 C4 C5 C6 C7
Max. Ver. -79.066 -63.141 85.309 38.297 0.000 0.000 2.394 0.000
Min. Cuad. -77.338 -53.530 83.913 26.824 0.000 0.000 2.570 0.000
Table 1.2: Par?ametros obtenidos para la aleaci?on AlMgSi1.
Case C0 C1 C2 C3 C4 C5 C6 C7
Max. Ver. -78.507 -31.357 101.460 -34.960 0.000 0.000 13.302 -8.642
Min. Cuad, -24.737 -16.091 31.803 -4.569 1.619 -1.619 2.837 -0.377
Para validar el modelo se utilizan los m?etodos de Kolmogoronov-Smirnov, el Test de uni-
formidad ?2 y el an?alisis de los gr?aflcos PP y QQ. La conclusi?on flnal a la que se llega tras
CHAPTER 1. SINOPSIS DE LA TESIS DOCTORAL EN CASTELLANO 11
S-N field
L.S.
0.80
0.95
1.10
1.25
1.40
0.001 0 .01 0 .1 1 10 100
Log N/N0
(M
Pa
)
Da ta 0.98? Ry
Da ta 0.9?Ry
Da ta 0.8?Ry
Da ta 0.7?Ry
0,98? Ry
0,9?Ry
0,8?Ry
0,7?Ry
S-N field
M.L.
0.80
0.95
1.10
1.25
1.40
0.001 0 .01 0 .1 1 10 100
Log N/N0
(M
Pa
)
Da ta 0.98? Ry
Da ta 0.9?Ry
Da ta 0.8?Ry
Da ta 0.7?Ry
0,98? Ry
0,9?Ry
0,8?Ry
0,7?Ry
??/
? 0
??/
? 0
Figura 1.5: Campo de Wo?hler resultante para el acero 42CrMo4 con los par?ametros obtenidos
por el m?etodo de m?axima verosimilitud (flgura de la derecha) y regresi?on por m??nimos cuadrados
(flgura de la izquierda).
S-N fie ld
L.S .
0.70
0.85
1.00
1.15
1.30
1.45
1.60
0.01 0.1 1 10
Log N /N0
(M
Pa
)
S-N fie ld
M.L.
0.70
0.85
1.00
1.15
1.30
1.45
1.60
0.01 0.1 1 10
Log N /N0
(M
Pa
)
??/
? 0
Da ta 0.9?R?0.2
Da ta 0.8?R?0.2
Da ta 0.7?R?0.2
Da ta 0.6?R?0.2
0,9?R?0.2
0,8?R?0.2
0,7?R?0.2
0,6?R?0.2
Da ta 0.9?R?0.2
Da ta 0.8?R?0.2
Da ta 0.7?R?0.2
Da ta 0.6?R?0.2
0,9?R?0.2
0,8?R?0.2
0,7?R?0.2
0,6?R?0.2??/
? 0
Figura 1.6: Campo de Wo?hler resultante para la aleaci?on AlMgSi1 con los par?ametros obtenidos
por el m?etodo de m?axima verosimilitud (flgura de la derecha) y regresi?on por m??nimos cuadrados
(flgura de la izquierda).
la validaci?on experimental del modelo es que para ambos materiales los resultados obtenidos
muestran una buena estimaci?on de los par?ametros, aunque en el caso de la aleaci?on AlMgSi1 los
resultados son mejores, debido quiz?as a la dispersi?on de los datos.
Por ?ultimo, conociendo los par?ametros del modelo para cualquiera de los materiales, se puede
extrapolar el problema a otras condiciones tensionales. Como ejemplo se puede predecir la vida
?ultima asociada a las cargas conocidas representando el campo de isoprobabilidad P{S{N (ver
flgura 1.7).
1.3.3 An?alisis del da~no acumulado
(chapter 8: Damage measures and damage accumulation)
Existen diversas formulaciones para el an?alisis del da~no acumulado, pero como vemos en la
tabla 8.1 la mayor??a de ellas no son una buena herramienta para medir el da~no en un cierto
12 1.3. APORTACIONES ORIGINALES
6
5
4
3
2
1
0.65
0.85
1.05
1.25
1.45
1.65
0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Dat a
S-N fie ld
=0.8?R?0.2
7
9
8
11
10
0.65
0.85
1.05
1.25
1.45
1.65
0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Dat a
S-N fie ld
=0.7?R?0.2
12 13
14
15
16
0.65
0.85
1.05
1.25
1.45
1.65
0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Dat a
S-N fie ld
=0.6?R?0.2
23
17
18
19
21
20
22
0.65
0.85
1.05
1.25
1.45
1.65
0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Dat a
??
/?
0
?M
?M?M
??
/?
0
??
/?
0
??
/?
0
S-N f ield
0.9?R?0.2?M=
Figura 1.7: Represesntaci?on del campo de isoprobabilidad P{S{N para la aleaci?on AlMgSi1. Los
percentiles representados corresponden con los valores 0:01,0:05, 0:50, 0:95 y 0:99.
momento. Algunas de las consideraciones/propiedades que una medida de da~no debe cumplir
son:
Propiedad 1.- Incrementar con el da~no: El da~no incrementa cuando incrementa la medida
del da~no analizada.
Propiedad 2.- Interpretabilidad: La medida de da~no debe ser clara y comprensible.
Propiedad 3.- Medida adimensional: Para evitar problemas de unidades en la estimacion
de la vida ?ultima a fatiga de un material es mejor trabajar con variables adimensionales.
Propiedad 4.- Conocimiento del rango: El rango de variaci?on de la medida de da~no debe
ser fljo, conocido, independiente de la carga y si puede ser, tambi?en del material.
Propiedad 5.- Funci?on de distribuci?on conocida: Para conocer la probabilidad de fallo de
una pieza, es necesario conocer la funci?on de distribuci?on de la probabilidad de esa medida
de da~no.
Por lo tanto, despu?es de analizar la tabla 8.1 en funci?on de las propiedades descritas anterior-
mente, queda claro que la probabilidad de da~no, derivada de la expresi?on del modelo Gumbel
(6.13) derivado en el cap??tulo 6 es la mejor medida que podemos utilizar para analizar el da~no
acumulado de un material sujeto a cargas c??clicas.
CHAPTER 1. SINOPSIS DE LA TESIS DOCTORAL EN CASTELLANO 13
La flgura 1.3 es un ejemplo de curva de isoprobabilidad del campo de Wo?hler, donde se
representa el da~no (probabilidad de fallo) para un material sometido a distintos rangos de carga.
Existen dos reglas que son ?utiles a la hora de evaluar el da~no en cualquier espectro de carga:
1. Regla de la iso-probabilidad: Dos historias de carga producen el mismo da~no si la
probabilidad de fallo es la misma para ambos casos.
2. Regla de la proporcionalidad: El da~no producido por debajo del cero por ciento es
proporcional al n?umero de ciclos, con un m?aximo valor de uno.
Para evaluar la acumulaci?on del da~no, el procedimiento es el siguiente (ver flgura 8.2):
1. El da~no inicial es nulo (p = 0).
2. El da~no p tras el primer ciclo es calculado con el nivel de cargas considerado para ese ciclo.
3. El n?umero equivalente de ciclos asociados a un da~no p para un nivel de cargas ( ?m = ?m(N)
y ?M = ?M (N)), para N? = 2, utilizando la funci?on inversa de (6.13) es:
logN?eq =
log(? log(1? p))? (C0 + C1 ?m + C2 ?M + C3 ?m ?M )
C4 + C5 ?m + C6 ?M + C7 ?m ?M
(1.5)
4. El da~no acumulado, representado mediante la probabilidad de fallo, es calculado mediante
la f?ormula recursiva:
PN+?N = F (N?eq +?N; ?m(N); ?M (N)); (1.6)
que obtiene el da~no acumulado tras N +?N ciclos cuando la unidad esta sujeta a un nivel
de cargas dado por ?m(N) y ?M (N).
5. Tras este punto, el da~no puede ser calculado en base a los percentiles, solo repitiendo los
pasos 3 y 4 sucesivamente, hasta que se alcance el nivel de da~no requerido.
A continuaci?on, diferentes tipos de cargas son analizados para validar la capacidad del modelo
descrito en el cap??tulo 6. Se han estudiado dos casos:
? Da~no acumulado frente a carga constante.
? Da~no acumulado frente a carga variable, de variaci?on lineal.
Da~no acumulado ante cargas constantes
Se analizan a continuaci?on tres historias de cargas diferentes. Todos los casos tienen un valor
constante ? (N) = 1130MPa, que corresponde con un ? ? = 1:099 ( 0 = 955:5 MPa), pero
cada uno de ellos tiene diferente valores de M , m y mean. Las caracter??sticas de cada una
de las historias de carga se muestran en la tabla 8.2. La flgura 1.8 muestra las distintas cargas
aplicadas.
Los resultados se muestran en la tabla 8.3 y flgura 1.9. En la tabla 8.3 se muestran los
diferentes valores de n?umero de ciclos para diversas probabilidades (p = 0:01; 0:1; 0:5; 0:9 y
p = 0:99). Se puede apreciar el efecto de mean en el da~no acumulado, mayores valores de mean
corresponden con mayores valores de probabilidad de fallo.
mean?(a) < mean?(c) < mean?(b) ! N
?
(a) > N?(c) > N?(b)
14 1.3. APORTACIONES ORIGINALES
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 10 20 30 40 50
N*
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 10 20 30 40 50
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 10 20 30 40 50
?
*
(a) (b) (c)
N*N*
Figura 1.8: Diferentes historias de carga utilizadas para el an?alisis del da~no acumulado, cuando
? = 1050 MPa ! ? ? = 1:099 ( 0 = 955:5 MPa). (a) ?m = ?0:500, ?M = 0:599; (b)
?m = ?0:399, ?M = 0:700; (c) ?m = ?0:500, ?M = 0:599 para los ciclos impares y, ?m = ?0:399,
?M = 0:700 en los ciclos pares.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000 100000
N*
Pr
ob.
(p)
)
(a) ?M*=0.598; ?m*=-0.5
(b) ?M*=0.7; ?m*=-0.399
(c) mix between (a) and (b)
Figura 1.9: Funciones de distribuci?on obtenidas para las distintas historias de cargas utilizadas
en el da~no acumulado cuando ? = 1050 MPa ! ? ? = 1:099 ( 0 = 955:5 MPa). (a)
?m = ?0:500, ?M = 0:599; (b) ?m = ?0:399, ?M = 0:700; (c) ?m = ?0:500, ?M = 0:599 para
los ciclos impares y, ?m = ?0:399, ?M = 0:700 en los ciclos pares.
Se ha realizado tambi?en un estudio de la in uencia de la existencia de discontinuidades con
mayor amplitud dentro de un espectro de carga constante. El estudio se ha llevado a cabo
mediante el an?alisis de tres casos diferentes (ver flgura 1.10):
CHAPTER 1. SINOPSIS DE LA TESIS DOCTORAL EN CASTELLANO 15
? Caso a: La discontinuidad se sit?ua al comienzo de la secuencia de carga (N? = 10).
? Caso b: La discontinuidad se sit?ua en la mitad de la secuencia de carga (N? = 100).
? Caso c: Cuando exsiten dos discontinuidades, situadas en N? = 10 y N? = 30.
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 10 20 30 40 50
N*
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
75 85 95 105 115 125
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
?
*
Original (a) Discontinuity at N*=10
(b) Discontinuity at N*=100 (c) Discontinuities at N*=10 and N*=30
N* N*
N*
Figura 1.10: Historias de cargas utilizadas para el an?alisis de la in uencia de existencia de
discontinuidades dentro de un espectro de carga constante. De arriba hacia abajo y de izquierda
a derecha: secuencia origial (sin discontinuidades), secuencia con discontinuidad situada enN? =
10, secuencia con discontinuidad situada en N? = 100 y secuencia en la que las discontinuidades
se sit?uan en N? = 10 y N? = 30 (para un valor de N0 = 532000 ciclos).
Se observa que la existencia de discontinuidades con mayor amplitud de carga (un aumento
de un 10% sobre el resto de los ciclos) in uye en la probabilidad de fallo. Esta probabilidad se
ve modiflcada en funci?on de donde se encuentre el ciclo en cuesti?on. En todos los casos en los
que existe una discontinuidad de este tipo el especimen rompe antes que en el caso original sin
discontinuidades (ver tabla 8.4 y flgura 1.11). Por otro lado, el peor caso es en el que existen
16 1.3. APORTACIONES ORIGINALES
m?as de una discontinuidad durante la secuencia de carga. El incremento de la probabilidad
es funci?on del n?umero equivalente de ciclos (N?eq) que depende a su vez de la probabilidad de
rotura en el ciclo anterior (Equaciones (1.5) y (1.6)), por ello, la evoluci?on de las probabilidades
es diferente aunque el incremento en todos los casos sea el mismo.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000
N*
Pr
ob.
(p)
)
Original
(a) N*=10
(b) N*=100
(c) N*=10, and N*=30
Figura 1.11: Funciones de probabilidad obtenidas del an?alisis de la in uencia de existencia
de discontinuidades dentro de un espectro de carga constante. De arriba hacia abajo y de
izquierda a derecha: secuencia original (sin discontinuidades), secuencia con discontinuidad
situada en N? = 10, secuencia con discontinuidad situada en N? = 100 y secuencia en la que
las discontinuidades se sit?uan en N? = 10 y N? = 30 (para un valor de N0 = 532000 ciclos).
Da~no acumulado ante cargas variables
Se ha realizado el estudio de tres grupos diferentes de historias de carga variable (ver flgura
1.12). A saber:
1. Historia de carga con ?m constante y variable ?M .
2. Historia de carga con ?M constante y variable ?m.
3. Historia de carga con ?M y ?m variables.
La forma general de estas expresiones es ? ? = m ? N? + n = ?M ? ?m, donde ?M =
m1 ?N?+n1 y ?m = m2 ?N?+n2. Los valores de los par?ametros m1;m2;m; n1; n2 y n se deflnen
en la tabla 8.5.
Como se puede observar en la flgura 1.12 hay dos espectros diferentes en cada grupo de carga.
El objetivo es conocer la variaci?on del da~no cuando se tienen espectros con la misma forma pero
sim?etricos, tal y como se muestra en la flgura 1.13, donde la duda es si pA y pB ser?an iguales.
Los resultados se muestran en la flgura 1.14. Se puede apreciar como en el segundo espectro
de cada grupo (designado con las letras a2; b2 y c2) el da~no acumulado crece m?as r?apidamente.
CHAPTER 1. SINOPSIS DE LA TESIS DOCTORAL EN CASTELLANO 17
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 50 100 150 200
Ncycles
?
*
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
Ncycles
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 50 100 150 200
Ncycles
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 50 100 150 200
Ncycles
?
*
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 50 100 150 200
Ncycles
?
*
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 50 100 150 200
Ncycles
?
*
(a1) (b1) (c1)
(a2) (b2) (c2)
Figura 1.12: Historias de carga variable analizadas en el estudio del da~no acumulado: (a1); (a2)
constante ?m y variable ?M , (b1; b2) constante ?M y variable ?m, (c1; c2) variable ?M y ?m.
N
?
??1 ??2
?N
m1
prob = pA
N
?
??1??2
?N
m1
prob = pB
(a) (b)
Figura 1.13: Esquema representativo del problema sim?etrico.
Esto es debido a que el da~no en estos espectros comienza por un valor no nulo como en el caso
primero (denominados por a1; b1 y c1).
La comparaci?on de los resultados entre todas las funciones de distribuci?on obtenidas muestra
que el da~no acumulado incrementa con el valor de mean (ver flgura 1.15): Por lo tanto, si la
18 1.3. APORTACIONES ORIGINALES
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000 100000
Pr
ob.
(p)
)
?M*=4.5?10 ?N +0.50;
?m*=-0.50;
?M*=-4.5?10 ?N +0.66;
?m*=-0.50;
-4
-4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000 100000
N*
Pr
ob.
(p
))
?M*=0.75;
?m*=-4.5?10 ?N +0.10;
?M*=0.75;
?m*=4.5?10 ?N -0.25;
-4
-4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000 100000
Pr
ob.
(p)
)
?M*=1.8?10 ?N +0.25;
?m*=-9?10 ?N -0.45;
?M*=1.8?10 ?N +0.59;
?m*=9?10 ?N -0.62
-4
-5
-5
-4
(b) (c)
N* N*
(c2)
(c1)
(a2)
(a1) (b1)
(b2)
Figura 1.14: Funciones de distribuci?on obtenidas tras el an?alisis de diversas historias de carga
variable: (a1); (a2) constante ?m y variable ?M , (b1; b2) constante ?M y variable ?m, (c1; c2)
variable ?M y ?m.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000 100000
N*
Pr
ob.
(p)
)
a1
a2
b1
b2
c1
c2
Figura 1.15: Funciones de distribuci?on obtenidas tras el an?alisis del da~no acumulado en diferentes
historias de carga: (a1); (a2) constante ?m y variable ?M , (b1; b2) constante ?M y variable ?m,
(c1; c2) ?M y ?m variables.
relaci?on entre las tensiones medias es
meanc2 > meana2 > meana1 > meanb2 > meanb1 > meanc1
la probabilidad de fallo se relaciona del siguiente modo
pfailurec2 > pfailurea2 > pfailurea1 > pfailureb2 > pfailureb1 > pfailurec1
y por conclusi?on
N?failurec2 < N
?
failurea2 < N
?
failurea1 < N
?
failureb2 < N
?
failureb1 > N
?
failurec1
En la tabla 8.6 se representan los valores de n?umero de ciclos N? para las distintas proba-
bilidades de fallo (p = 0:01; 0:1; 0:5; 0:9 y 0:99).
CHAPTER 1. SINOPSIS DE LA TESIS DOCTORAL EN CASTELLANO 19
1.4 Conclusiones y futuras l??neas de investigaci?on
1.4.1 Conclusiones
(chapter 9: Conclusions)
Las principales conclusiones obtenidas tras la realizaci?on de este trabajo de investigaci?on
son:
Modelo de fatiga propuesto
? Se ha desarrollado un nuevo modelo que analiza la vida a fatiga para cualquier tipo de
carga y para cualquier rango de la misma (tensi?on, compresi?on y/o mixto). Adem?as la
derivaci?on del modelo se lleva a cabo teniendo en cuenta condiciones f??sicas, estad??sticas
y de compatibilidad, resultanto por tanto un modelo potente y con gran capacidad de
adaptaci?on ante todo tipo de datos y materiales met?alicos.
? El modelo depende de 8 par?ametros Ci que pueden ser estimados mediante diferentes
m?etodos. Por otro lado, el modelo incluye una gran informaci?on estad??stica, entre la que
podemos destacar la capacidad para conocer la probabilidad de fallo para cualquier estado
de carga y n?umero de ciclos.
Validaci?on experimental del modelo
? Se ha validado el modelo mediante la realizaci?on y an?alisis de resultados con dos materiales
met?alicos de diferentes comportamientos a fatiga.
? Los resultados obtenidos muestran la capacidad de adaptaci?on del modelo a diferentes
materiales en diferentes estados de carga, mostr?andose diferentes tendencias en las curvas
S{N (tensi?on{n?umero de ciclos).
? La distribuci?on de los puntos no afecta a la estimaci?on de los par?ametros, pero es necesario
tener un conjunto de datos que se distribuyan al m?aximo posible en todo el rango de cargas.
? Diferentes casos (submodelos) han sido obtenidos gracias a la libertad del modelo para la
elecci?on de los mejores par?ametros para cada material analizado.
? Se ha realizado tambi?en la validaci?on del modelo gracias a datos obtenidos de otras refe-
rencias bibliogr?aflcas ajustadas anteriormente a otros modelos de fatiga. Estas validaciones
ten??an en cuenta diferentes condiciones de carga, por ejemplo con niveles constantes de
diferentes ratios de tensiones R.
? Se ha validado la capacidad de extrapolaci?on del modelo mediante la elaboraci?on de los
campos P{S{N o mediante la reestimaci?on de los par?ametros tras la eliminaci?on de algunas
de las series originales.
An?alisis del da~no acumulado
? La probabilidad de fallo es un buena medida del da~no acumulado ya que permite la eva-
luaci?on de la probabilidad de fallo directamente. De hecho, las curvas P{S{N de Wo?hler
interpretan de una manera sencilla el da~no del material en cada momento.
20 1.4. CONCLUSIONES Y FUTURAS L?INEAS DE INVESTIGACI?ON
? El modelo propuesto es ?util para el an?alisis del da~no en cualquier tipo de historia de carga:
constante o linealmente variable, para todo el rango de tensiones.
? El factor que tiene m?as in uencia en la evoluci?on del da~no es la tensi?on media. Conociendo
su distribuci?on a lo largo del tiempo y conociendo los par?ametros del modelo resulta sencillo
conocer el da~no acumulado para un momento preciso.
1.4.2 Futuras l??neas de investigaci?on
(chapter 9: Conclusions)
Tras la elaboraci?on de la tesis doctoral nuevos temas de investigaci?on surgen derivados de los
analizados en el presente proyecto de investigaci?on. Por ello, las futuras l??neas de investigaci?on
que se plantean son:
? Realizar la validaci?on del modelo te?orico para otros materiales met?alicos, ampliando el
rango de materiales que cuentan con el conocimiento de los par?ametros caracter??sticos del
modelo propuesto.
? Comprobar la validez del nuevo modelo para las zonas de bajo y alto n?umero de ciclos,
en las cuales el comportamiento del material puede diferente. Con ello se conocer??a con
mayor grado de seguridad el grado de utilidad del modelo en dichos rangos, b?asicos para
distintas aplicaciones ingenieriles como el sector aeroespacial o automovil??stico.
? Investigar en la estrategia de ensayos para desarrollar una metodolog??a de ensayos ?optima.
? Estudiar el efecto de otros factores en el proceso del da~no acumulado. A lo largo de la
historia la in uencia de la frecuencia se supone nula en la vida ?ultima de material [102],
pero esto no es completamente cierto cuando se analizan otros factores como corrosi?on,
altas temperaturas o cargas aleatorias. Esta in uencia depende del material analizado y
ensayo realizado (ver [14], [16], [102] o [119]). El efecto de las altas temperaturas es tambi?en
un factor del cual se debe conocer su in uencia. Es conocido en el dise~no de materiales
met?alicos utilizados en la industria aeroespacial que los materiales est?an sujetos a cargas
variables en combinaci?on con altas temperaturas.
? Comprobar el modelo para otro tipo de materiales (no met?alicos), materiales heterog?eneos
(hormig?on), o nuevos materiales en los que algunas propiedades se han modiflcado para
obtener mejores respuestas a este tipo de solicitaciones.
Part II
Presentation and State of the Art
21
Chapter 2
Presentation
The present doctoral thesis aims at contributing to the advance in the state of knowledge
of the fatigue behavior of materials. New models are proposed for analyzing fatigue failure
and predicting fatigue damage as a function of lifetime, stress level and stress range.
The document is addressed to engineers and researches who want to improve their knowledge
with a new model that covers the tension and compression Wo?hler flelds, so as to scientists
who want to explore a new research fleld.
2.1 Justiflcation
In material science, fatigue is the progressive and localized structural damage that occurs when
a material is subjected to cyclic loading. The flrst researches in this fleld (19th. century)
discovered, that if they represented load versus time to failure (lifetime), a relation between
these two variables would exist. This relation can be represented by difierent mathematical
expressions such as those in the right part of flgure 2.1 where it is possible to observe some of
these difierent representations. Find the best expression to deflne the lifetime of a piece subject
to fatigue is one of the most important aims of these researches.
The fatigue mechanism is present practically in all structures, machine components and
vehicles, which are subjected to repeated loads and can lead to damage of the material involved,
until it develops into the component failure.
Fatigue failures continue to be a major concern in engineering design. The economic costs of
fracture and its prevention are quite large, and they involve situations in which cyclic loads are
at least a contributing factor. But these costs arise from the occurrence or prevention of fatigue
failure. So one of the flrst motivations for this doctoral thesis is to flnd a new model which saves
cost to engineers working with this problem.
Taking into account all the previous researches in this fleld, the author of this doctoral thesis
tries to flnd, with a new statistical model, the best expression to flt the fatigue life data, subject
to physical, statistical and other type of conditions.
This doctoral thesis must be considered as one step further in the line developed by the
thesis directors. In particular, we refer to previous works such as those dealing with Weibull
23
24 2.2. HYPOTHESES
time
lo
a
d
time
lo
a
d
time
lo
a
d
(a)
(b)
(c)time
lo
ad
Figure 2.1: Schematic representation of a typical fatigue plot of the fatigue life data, and some
difierent flts: (a) linear flt, (b) piece wise linear flt, (c) nonlinear flt.
models used to predict fatigue lifetime and damage accumulation in a material subject to any
stress history.
2.2 Hypotheses
The hypotheses utilized for the development of this research were the followings:
1. The models used for fatigue analysis at present time do not satisfy all required physi-
cal, experimental and statistical conditions for a satisfactory model from a physical and
engineering points of view.
2. A model able to deflne the fatigue behavior of the material and satisfying these conditions
can be obtained.
3. The materials used in this research fleld of fatigue are typically metallic and some impor-
tant structural elements are longitudinal.
4. It is assumed that the weakest link principle is satisfled: If a longitudinal element is divided
into n sub-elements, its fatigue life must be the fatigue life of its weakest component.
2.3 Objectives
In the following lines, the objectives of this research are described. These objectives deflne the
characteristics, methods and developments of this doctoral thesis, and are as follows:
1. Perform a state of the art on fatigue of materials, showing the main models used to estimate
and analyze the fatigue behavior of materials and discuss the advantages and shortcomings
of each one.
CHAPTER 2. PRESENTATION 25
2. Detect the main problems of existing models and testing strategies used in fatigue analysis.
3. Develop a new fatigue model covering the tension and compression Wo?hler fleld for any
stress history, based on the application of functional equation theory for its deflnition.
4. Discuss existing damage measures and propose new ones, selecting the most adequate to
be used in fatigue.
5. Study and analyze damage accumulation with the proposed statistical models and damage
measures.
6. Propose a good testing strategy to obtain the best laboratory results, that is, those leadings
to the best possible estimates of the model parameters.
7. Design and perform a series of experimental tests to evaluate and validate and the theo-
retical model.
8. Run the test and analyze the test results to validate the model.
9. Based on the results and their analysis, present the main conclusions with respect to the
model and testing strategies.
2.4 Introduction
This doctoral thesis has been done at the Department of Applied Mathematics and Computa-
tional Sciences of the University of Cantabria, inside the program: Development and Applications
of Models in Civil Engineering.
The directors are Professors Enrique Castillo Ron and Alfonso Fern?andez Canteli.
The experimental tests have been done inside an experimental fatigue program launched at
Empa (Swiss Federal Laboratories for Testing and Research at Du?bendorf (Switzerland)).
The doctoral thesis is organized in four parts as follows:
1. Presentation and state of the art: In this part, an introduction and the state of the
art about fatigue is discussed.
2. Theoretical contributions: The proposed fatigue models are presented, and some as-
pects about them are analyzed, including their derivations model, restrictions and prop-
erties, resulting sub-models and parameter estimation methods.
3. Experimental validation of the models: Here the model is validated based on labo-
ratory tests at constant stress range and variable load histories. Furthermore, damage
accumulation theories are analyzed and a comparison among these theories using the new
models is made.
4. Conclusions: Finally, the conclusions of the doctoral thesis are given, together with,
some proposals for the future lines of research in this fleld.
26 2.4. INTRODUCTION
Chapter 3
State of the Art
The aim of this Chapter is to introduce the readers in the fatigue world and understand
its difierent approaches. In this Chapter, an introduction to the fatigue problem is presented.
In the flrst section, some history issues are analyzed and some basic concepts of fatigue are
introduced. In sections 3.2, 3.3 and 3.4 the basic approaches to fatigue, the fracture
based, stress based and strain based approaches are presented.
3.1 Introduction to the fatigue problem
3.1.1 The fatigue problem in engineering
The history of the mechanical failures due to fatigue has been studied or analyzed for more than
150 years. August Wo?hler was one of the flrst who tested mine hoist chains under cyclic loading
in Germany around 1828, but started fatigue work in Germany in the 1850s and, motivated by
railway axle failures, he began to test irons, steels and other materials to design strategies for
avoiding fatigue failure. He demonstrated that fatigue was afiected not only by cyclic stresses,
but also by the accompanying steady (mean) stresses. More detailed studies by the Wo?hler
followers, like Gerber and Goodman, together with the early work on fatigue and subsequent
efiorts up to the 1950s are reviewed in a paper by Mann [91].
Actually, fatigue failures continue to be a major concern in engineering design. The economic
costs of fracture and its prevention are quite large, and 80% of these costs involve situations in
which cyclic loads are at least a contributing factor to failure. For example, the anual cost of
fatigue materials in the United States is about 3% of the gross national product (GPN) (see [94]
and [59]). But these costs arise from the occurrence or prevention of fatigue failure for ground
and rail vehicles, aircrafts, bridges, cranes, equipment, ofishore oil well structures and everyday
household items like toys and sport equipments.
At present, there are three difierent approaches to fatigue mechanisms and failures. The
stress-based approach was developed in 1955, and the corresponding analysis is based on the
nominal stresses in the afiected region of the engineering component. The second approach is
the strain-based approach, which involves a more detailed analysis of the localized yielding that
may occur at stress raises during cyclic loading. The last one is the fracture mechanics approach,
which speciflcally treats growing cracks by the methods of fracture mechanics.
27
28 3.1. INTRODUCTION TO THE FATIGUE PROBLEM
3.1.2 Fatigue concepts
Description of cyclic loading
The constant stress amplitude test involves cycling between minimum and maximum stress
constant levels [59] (see flgure 3.1).
?
??
time
a
one cycle
0
a
max?
min?
?
?
??
max?
min?
a
?
a
?
mean?
?
0
?
max?
?
?
= ??
a
a
(a)
(c)
(b)
0
Figure 3.1: Illustration of the constant stress amplitude test and three difierent cases: (a)
mean = 0, (b) mean 6= 0 and (c) min = 0 [59].
The stress range ? = max? min is the difierence between the maximum and the minimum
stress values. Averaging the maximum and the minimum values we obtain the mean stress,
mean:
mean = max + min2 : (3.1)
Another important parameter is the stress amplitude, a, that is, the half range of stress range:
a = ? 2 =
max ? min
2 : (3.2)
Another two parameters useful in the fatigue problems, are the stress ratio, R, and the amplitude
ratio, A [59]:
R = min max ; A =
a
mean : (3.3)
Some interesting relationships derived from all the above equations are:
a = ? 2 =
max
2 (1?R); mean =
max
2 (1 +R)
R = 1?A1 +A ; A =
1?R
1 +R (3.4)
CHAPTER 3. STATE OF THE ART 29
Stress versus lifetime (S{N) curves
A stress-life curve, also called S{N curve is the representation of fatigue tests from a number
of difierent stress levels [59]. If a test specimen of a material or an engineering component is
subjected to a su?ciently severe cyclic stress, a fatigue crack (or other damage) will develop,
leading to complete failure of the member. If the test is repeated at a higher stress level, the
number of cycles to failure will be smaller. Since the number of cycles to failure reaches very
high values data are usually plotted on a logarithmic scale (see flgure 3.2).
0
200
400
600
0 5000000 10000000 15000000 20000000
0
200
400
600
100
N cycles to Failure
? ma
x
N cycles to Failure
? ma
x
100000010000 100000000
Figure 3.2: Example of a S{N curve [59]. The left side corresponds to a representation on an
arithmetic scale of N . The right side shows a representation on a logarithmic scale of N .
The endurance limit or fatigue limit [59] is the lower stress amplitude below which fatigue
failure does not occur under ordinary conditions. This occurs with materials like plain carbon
and low alloy steels.
The term fatigue strength [59] is used to specify a stress amplitude value from an S{N curve
at a particular life of interest. Hence, the fatigue strength at 105 cycles is simply the stress
amplitude corresponding to N = 105.
3.2 Fatigue under the fracture mechanics point of view
Engineering analysis of crack growth is often required, and can be done using the stress intensity
factor, K, of fracture mechanics [59]:
K = F p?a; (3.5)
where a is crack length, is stress, usually deflned based on the gross area of the un-cracked
member, and F is a dimensionless function depending on the geometry. The value of F is
afiected by the relative crack length, fi = a=b, where b is a width dimension of the member such
that fi = 1 for complete cracking. The rate of fatigue crack growth is controlled by K. Hence,
under constant amplitude cycling loading, the dependence of K on a and F causes cracks to
accelerate as they grow.
3.2.1 Deflnitions for fatigue crack growth
Consider a growing crack that increases its length by an amount ?a due to the application of
a number of cycles ?N . The rate of crack growth with cycles can be characterized by the ratio
?a=?N , for small intervals, i.e. by the derivative da=dN [59].
Assume that the applied loading is cyclic with constant values of the loads min and max.
For fatigue crack growth, it is conventional to use the stress range ? and the standard stress
30 3.2. FATIGUE UNDER THE FRACTURE MECHANICS POINT OF VIEW
ratio R. The primary variable afiecting the growth rate of a crack is the range of the stress
intensity factor. This can be calculated from the stress range as:
?K = F? p?a: (3.6)
The value of F depends only on the geometry and the relative crack length just as if the loading
were not cyclic. Since K and are proportional for a given crack length, according to Equation
(3.5), the maximum, minimum, range and ratio R for K during a loading cycle [59], are given
by:
Kmax = F max
p?a; Kmin = F min
p?a
?K = Kmax ?Kmin; R = KmaxKmin :
(3.7)
3.2.2 Describing material fatigue crack growth behavior
For a given material and set conditions, the crack growth behavior can be described by the
relationship between cyclic crack growth rate da=dN and stress intensity range ?K [59]. Some
test data and the corresponding fltted curve for a material are shown on a log-log plot in flgure
3.3. There are three difierent states in a material during crack growth, in which geometry,
environment conditions, stresses and material have a big in uence on the material behavior:
1. Initialization,
2. Propagation,
3. Unstable rapid growth and fail of structure.
At low growth rates, the curve generally becomes step and appears to approach a vertical
asymptote denoted ?Kth, which is called the fatigue crack growth threshold [59]. This quantity
is interpreted as a lower limiting value of ?K below which crack growth does not ordinarily
occur.
At intermediate values of ?K, there is often a straight line type of behavior on the log-log
plot as in this case. A relationship representing this line is:
da
dN = C(?K)
m; (3.8)
where C is a constant and m is the slope on the log-log plot. This equation is attributed to P.C.
Paris, who flrst used it and who was in uential in the flrst application of fracture mechanics to
fatigue in the early 1960s.
At high growth rates, the curve may again become steep. This is due to rapid unstable crack
growth just prior to flnal failure of the test specimen. Such behavior can occur where the plastic
zone is small, in which case the curve approaches an asymptote corresponding to Kmax = Kc,
the fracture toughness of the material and thickness of interest. Rapid inestable growth at high
?K sometimes involves fully plastic yielding. In such cases, the use of ?K for this portion of
the curve is improper because the theoretical limitations of the K concept are exceeded. Figure
3.3 is a typical representation of this kind of curves.
Constant C and m for the intermediate region, where Equation (3.8) applies, have been
suggested by Barson for various classes of steel [19]. The value of m is important as it indicates
the degree of sensitivity of the growth rate to stress.
CHAPTER 3. STATE OF THE ART 31
da/d
N
ln ?K?Kth
Kc
I
Discontinuous
Mechanisms
II
Continuous
Mechanisms
III
Static
Mechanisms
m
1
Figure 3.3: Schematic representation of crack propagation. Typical Paris curve [106].
3.2.3 In uence of difierent parameters on fatigue crack growth
Efiects of R = SminSmax on fatigue crack growth: An increase in the R ratio of the cyclicloading causes growth rates for a given ?K to become larger [59]. The efiect is generally more
pronounced for more brittle materials.
Various empirical relationships are employed for characterizing the efiect of R on da=dN vs.
?K curves. One is the Walker Equation:
?K = Kmax(1?R) ; dadN = C
?K
(1?R)1?
?m
; (3.9)
where is a constant for the material and ?K is an equivalent zero-to-tension (R = 0) stress
intensity that causes the same growth rate as the actual Kmax, R combination.
Efiect on ?Kth: The R ratio generally has a strong efiect on the material behavior at low
growth rates, hence also on the threshold value ?Kth [59]. This occurs even for low strength
metals where there is little efiect at intermediate growth rates. The lower limit of the scatter
shown corresponds to ?Kth as follows:
?Kth = 7:0(1? 0:85R)MPa
pm; R ? 0:1: (3.10)
Based on Barson [19], these equations appear to represent a reasonable worst-case estimate for
a wide range of steels. However, lower values of ?Kth may apply for highly strengthened steels
(similar trends occur for other classes of metals).
Environmental efiects: Similar considerations of inspection for cracks, and a similar need for
life estimates, exist where crack growth is caused by a hostile chemical environment, a situation
32 3.2. FATIGUE UNDER THE FRACTURE MECHANICS POINT OF VIEW
termed environmentally assisted cracking (EAC) [59]. There are several physical mechanisms
that occur.
In situations of environmental crack growth during an unchanging static load, the crack
growth life can be estimated based on fracture mechanics, is a manner analogous to the proce-
dures described above for fatigue crack growth under constant amplitude loading. The parameter
controlling crack growth is simply the static value K of the stress intensity factor, as determined
from the applied static stress and the current crack length. Growth rates for the material are
characterized by the use of da=dt versus K curve, where da=dt is the time-based growth rate, or
crack velocity, also denoted _a.
_a = dadt = AK
n; (3.11)
where A and n are material constants that depend on the particular environment and are afiected
by temperature.
3.2.4 Fatigue laws
There are two difierent types of parameters that can afiect fatigue crack growth [23]:
? Intrinsic parameters of material: Young?s modulus, yield stress, cyclic and metallurgic
properties of material, etc.
? Extrinsic parameters: Test conditions, temperature, geometry, R ratio, etc.
Models based on crack propagation: In 1963 Paris and Erdogan [106] proposed the most
commonly used law for crack propagations (Equation (3.8) and flgure 3.3). In this flgure three
difierent parts can be analyzed:
? Part I: In which a big in uence of microstructure, mean stresses and environment condi-
tions takes place (discontinuos mechanisms).
? Part II: Paris?s Law is described here. There is an in uence of microstructure, stresses,
environment conditions and geometry of specimen (continuos mechanisms).
? Part III: Finally, in this part, previous parameters have a big in uence.
Broek and Schijve [117] proposed an empirical equation to describe the crack growth:
da
dN = C1
?K
1?R
?3
exp(?C2R); (3.12)
where C1 and C2 are characteristic parameter of the material, and R is the stress ratio.
Forman et al. [67] assuming that crack failure occurs when K = Kc, derivated from Equation
(3.12), the following expression:
da
dN =
C?Km
(1?R)Kc ??K : (3.13)
Frost and Dugdale [69] analyzed diverse material taking into account dimension considera-
tions and experimental results:
da
dN = (P +Q )?
3a; (3.14)
where P and Q are material parameters.
CHAPTER 3. STATE OF THE ART 33
?
?
bebs
?
b
slip plane
Figure 3.4: Schematic representation of the Yokobori problem, [134].
Models based on theory of dislocation: This model is based on the crack growth when a
movement on dislocation at the head of the crack is produced.
The most used model was deflned by Bilby et al. [25], which supposed a plain strain.
The stress concentration derivates in the growth of a plastic region. Furthermore, there is a
friction strength opposite to this movement. This situation leads to equilibrium when there is a
movement at the head of the crack.
Yokobori [134] deflnes a cinetic theory to describe the strain between dislocations.
f? = ?p;'be = KIbp8?p sin' cos
'
2 cos?; (3.15)
fi = ? ?b
2
4??
1
1?
?
; (3.16)
where ? is the distance from the head of crack to the dislocation, ' is the angle formed with the
propagation direction, f? is the strength applied and fi is the image strength (see flgure 3.4).
Models based on the material behavior: This is the third mode of studying crack growth.
The models here are more complex than others based on the theories exposed above.
Pook and Frost [109] deflne the stress distribution at the crack head, when there is a crack
of a length 2a and subject to a cyclic load (from 0 to ):
da
dN =
9
?
K2I
E ; (3.17)
da
dN =
9
?
KI(1? 2)2
E ; (3.18)
where Equation (3.17) corresponds to plain stress and Equation (3.18) corresponds to plain
strain.
3.3 Stress based approach to fatigue
3.3.1 Estimated S{N curve of a component based on ultimate tensile strength
In the event that experimental S{N data are not available, methods for estimating the S{N
behavior of a component becomes useful and crucial for the design process. Large amounts of
34 3.3. STRESS BASED APPROACH TO FATIGUE
Fatigue limit HCFLCF
N cycles
?
Low-cycle fatigue
High-cycle fatigue
k
1
103 1061
Figure 3.5: Schematic of a S{N curve for steels [59].
S{N data have been historically generated based on fully reversed rotating bending testing on
standard specimens. The standard test specimen is described in [135].
The S{N curve derived from the standard specimens under loads can be constructed as a
piecewise-continuous curve. As shown schematically in flgure 3.5, there are two inclined linear
segments and one horizontal segment in a typical logS-logN curve. The two inclined linear
segments represent the low-cycle fatigue (LCF ) and high-cycle fatigue (HCF ) regions, and the
horizontal asymptote represents the fatigue limit.
For specimens made of steels, the fatigue strength values at 1, 103, 106 cycles deflne an S{N
curve.
3.3.2 Fatigue strength testing
The objective of the fatigue strength test (also called the fatigue limit test, the strength test,
or the response test [59]) is to estimate a statistical distribution of the fatigue strength at a
speciflc fatigue life. The mean fatigue limit has to be flrst estimated, and a fatigue life test is
then conducted at a stress level a little higher than the estimated mean.
Two typical data reduction techniques, the Dixon-Mood [57] and the Zhang-Kececioglu [136]
methods, are used to determine the statistical parameters of the test results.
3.3.3 Mean stress efiect
Fatigue damage of a component correlates strongly with the applied stress amplitude or applied
stress range and is also in uenced by the mean stress (see [59] and [135]). If mean stress is
higher, the possibilities of stay near to tensile stress are bigger, and in this moment, the failure
can be produced by the static parameter of the material. There is very little or no efiect of mean
stress on fatigue strength in the low-cycle fatigue region in which the large amounts of plastic
deformation hide any beneflcial or detrimental efiect of a mean stress.
Early empirical models by Gerber [73], Goodman [74], Haigh [79], and Soderberg [124] were
proposed to compensate for the tensile normal mean stress efiects on high{cycle fatigue strength.
CHAPTER 3. STATE OF THE ART 35
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
?min/?u
?
m
a
x
/?
u
0
0.1
0.2
0.3
0.4
0 0.2 0.4 0.6 0.8 1
?m /?u
?
a
/?
u
?a/?u
?m/?u
?e/?u
Figure 3.6: Difierent diagrams of mean stress corrections. Gerber?s and Goodman?s diagrams
(left) and Haigh?s plot for Gerber?s and Goodman?s diagrams (right) [135].
In 1874, Gerber proposed a parabolic representation of the Wo?hler fatigue limit data on a
plot of max= u versus min= u as shown in flgure 3.6, where u is the stress failure. In 1899,
Goodman introduced a theoretical line representing the available fatigue data and justifled the
use of the impact criterion on the basis that it was easy, simple to use, and provided a good
flt to the data. In 1917, Haigh flrst plotted fatigue data for brasses on a a versus m plot.
flgure 3.6 illustrates the Haigh plot of the Gerber and the Goodman mean stress corrections.
The ordinate of the Haigh plot is the normalized fatigue limit, and the maximum mean stress
is limited to the ultimate strength Su. The curve connecting these two points on the two axes
represents combinations of stress amplitudes and means stresses given at the fatigue limit life.
Mathematically, the Gerber parabola and the Goodman line in Haighs coordinates can be
expressed as the following expressions:
? Gerber?s mean stress correction:
e = a
1? m u
2 : (3.19)
? Goodman?s mean stress correction:
e = a1? m u
: (3.20)
? Soderberg?s mean stress correction: In this case, the maximum normal stress should be
limited to the yield strength y:
e = a1? m y
: (3.21)
It is conservative to assume that for most ductile materials, the compressive normal mean
stress does not beneflt fatigue strength. This means that the fully reversed stress amplitude is
the same as the stress amplitude if the normal mean stress is negative. A modifled Goodman
diagram for both tensile and compressive normal mean stresses is schematically illustrated in
36 3.3. STRESS BASED APPROACH TO FATIGUE
-?m +?m
-?y 0 ?y ?u
?a
?y
?e
Yield envelope
Fatigue Limit envelope
Safe for yield and fatigue
?u ??f
?a
?e
?m
1
-M
Goodman line
Morrow line
Figure 3.7: Difierent diagrams of mean stress corrections. Comparison between Goodman?s and
Morrow?s mean stress models (left side). Models for combined fatigue limit and yield in ductile
materials (right side)[135].
the Haigh plot at the fatigue limit as shown in flgure 3.7. Wilson and Haigh [131] introduced
the line of constant yield strength as an additional constraint for ductile materials on the safe
design stress region, named the safe design region for fatigue limit and yield strength, shown in
flgure 3.7.
?m
?a
R=-1
R=-inf
R=0
R=1
?e.R=-1
?e.R= 0
?m,R= 0
-M
-M3
-M2
Figure 3.8: Mean stress sensitivity factors [135].
Morrow [97] suggested that the stress amplitude plus the mean stress could never exceed the
fatigue strength coe?cient, the fatigue strength at one reversal:
ar = a1? m 0f
: (3.22)
For relatively small mean stress loading, the Morrow approach is considered better than the
Goodman method. The Goodman mean stress correction formula should only be used if none
of the fatigue properties are available [135]. For relatively large mean stress conditions, an
empirical model based on the concept of the mean stress sensitivity factor was introduced. As
illustrated in flgure 3.8, M factors were found to vary in difierent mean stress levels (Radaj
and Sonsino, [111]). For example, the mean stress sensitivity factor for low mean stress loading
CHAPTER 3. STATE OF THE ART 37
(?1 ? R < 0) denoted by M , is deflned as follows:
M = e;R=?1 ? e;R=0 e;R=0 : (3.23)
The mean stress sensitivity factor for loading with low, compressive mean stress levels (?1 ?
R < 1) denotes M2 and varies from 0 to M . The mean stress sensitivity factor for higher mean
stress levels (0 ? R ? 1 or Sm > Sa) denoting M3 is usually lower than M by a factor of 3
(M3 ? M=3). This is based on the empirical observation that loading with high mean and small
amplitude shows higher damaging efiects than that predicted by M. It is also found by Schu?tz
[118] that the M factor for a material increases with a higher ultimate strength, as illustrated
in flgure 3.9.
If the baseline S{N curve was generated by specimens under R = 0 loading, it is required
to convert any positive mean ofiset loading to an equivalent R = 0 loading. For a given mean
stress sensitivity factor, the following conversion formula is used:
Sar;R=0 = a +M ? SmM + 1 : (3.24)
This equation is popular in spot welded fatigue life prediction because single spot weld laboratory
specimens cannot resist any compression that leads to local buckling of the metal sheet. Thus,
these specimens are often subjected to R = 0 loading for the generation of a baseline S{N curve.
Any shear mean stress can be considered positive because signs of shear are arbitrarily chosen.
Experimental fatigue data indicate that shear mean stress has little efiect on fatigue strength
of un-notched members under torsion [135]. Where signiflcant stress raises are present in a
component subjected to torsional loading, the state of stress at high-stress-concentration areas
deviates from pure shear. Thus, experimental results under these conditions show a shear
mean stress detrimental to the fatigue strength approximately as signiflcant as that observed
for bending stresses in other load cases. It is recommended to use the Goodman equation in
?a ? ?m for notched torsion members for which the ultimate shear strength Sus is given.
Al-alloys Cast steel
Steel & Ti-
alloys
0
0.2
0.4
0.6
0.8
0 1000 2000
?u [MPa]
M
fa
ct
o
r
Figure 3.9: Mean stress sensitivity factors [135].
38 3.3. STRESS BASED APPROACH TO FATIGUE
3.3.4 Trends in S{N curves
S{N curves are characteristic for each material, and they are afiected by a variety of factors. In
this subsection environment, frequency of cycling, microstructure, residual stresses and surface
efiects will be studied [59].
Now, some of these aspects are analyzed to know the in uence in the trends of S{N curves,
when one or more of these aspects are presented in material tests.
Efiects of environment and frequency of cycling: Hostile chemical environments can
accelerate the initiation and growth of fatigue cracks. One mechanism is the development of
corrosion pits by chemical reactions and dissolution of material at the crack tip. Gasses in air
can act as a hostile environment, especially at high temperature.
Fatigue life varying with frequency of cycling in such situations, the life in cycles being
shorter for slower frequencies (see flgure 3.10).
16
18
20
22
24
26
28
30
1.00E+03 1.00E+04 1.00E+05 1.00E+06
Ncycles
St
re
ss
[K
si]
1300?F
60 cpm
1500?F
60 cpm
1500?F
6 cpm
Figure 3.10: Temperature and frequency efiects on the S{N curve for a nickel-base alloy Inconel
[32].
Efiects of microstructure: A change in the microstructure or surface condition has the
potencial of altering the S{N curve, especially at long fatigue lives. In metals resistance to
fatigue is generally enhanced by reducing the size of inclusions and voids.
Microstructure of materials often vary with direction, such as the elongation of grains and
inclusions in the rolling direction of metal plates. Fatigue resistance may be lower in directions
where the stress is normal to the long direction of such and elongated or layered grain structure.
CHAPTER 3. STATE OF THE ART 39
Efiects of residual stresses and surface efiects: The internal stresses in the material are
called residual stresses. This can be beneflcial or not depending on the type of stresses. In the
case of compressive residual stresses they are beneflcial because the material attempts to recover
its original size by elastic deformation.
Smoother surfaces that result from more careful machining in general improve resistance
to fatigue, although some machining procedures are harmful, as they introduce tensile residual
stresses.
3.3.5 Variable amplitude loading
Practical applications usually involve stress amplitudes that change in an irregular manner. This
doctoral thesis has the objective (see Section 2.3) to validate the proposed model (see Chapter
6) for all kinds of spectra (see Chapter 8).
Cycles can be counted using time histories of the loading parameter of interest, such as
force, torque, stress, strain, acceleration, or de ection. In the next lines several cycle counting
techniques will be analyzed to reduce a complicated variable amplitude loading history into a
number of discrete simple constant amplitude loading events, which are associated with fatigue
damage.
There are difierent methods for obtaining life estimates for such loadings. One of them
is the Palmgren-Miner Rule (see Section 4.2.2), developed in the 1920s for predicting the life
of ball bearings and then, until 1945 with the appearance of Miner?s paper. For cases that
exhibit constant amplitude loading with or without mean ofiset loading, the determination of the
amplitude of a cycle and the number of cycles experienced by a component is a straightforward
exercise.
There are two difierent groups of cycle counting [135]: One-parameter cycle counting methods
and two parameter cycle counting methods. The flrst one, called peak-valley, have been commonly
used for extracting the number of cycles in a complex loading history. These methods are
unsatisfactory for the purpose of describing a loading cycle and fail to link the loading cycles to
the local stress - strain hysteresis behavior that is known to have a strong in uence on fatigue
failure. Thus, these methods are considered inadequate for fatigue damage analysis and we only
study a method of the second group.
Two{parameter cycle counting methods, such as the rain{ ow cycle counting method, can
faithfully represent variable-amplitude cyclic loading. Dowling [58] states that the rain- ow
counting method is generally regarded as the method leading to better predictions of fatigue
life. It can identify events in a complex loading sequence that are compatible with constant-
amplitude fatigue data. Matsuishi and Endo [93] were the flrst in developing the rain{ ow cycle
counting method based on the analogy of raindrops falling on a pagoda roof and running down
the edges of the roof. A number of variations of this original scheme have been published in
various applications.
Three{point cycle counting method SAE [114] and the ASTM [1] standards, use this
kind of counting method. The three{point cycle counting rule uses three consecutive points in
a load{time history to determine whether a cycle is formed (see flgure 3.11):
? 1 = j 1 ? 2j ; ? 2 = j 2 ? 3j (3.25)
? 1 ? ? 2 ! One cycle is counted
? 1 > ? 2 ! No cycle is counted
40 3.3. STRESS BASED APPROACH TO FATIGUE
1
2
3
1
2
3
1
2
3
?
??1>??2
No cycle
??1 0:2) will harden whereas those with a low monotonic strain
hardening exponent (n < 0:1) will cyclically soften. A rule of thumb (Bannantine [18]) is that
the material will harden if u= y > 1:4 and the material will soften if u= y < 1:2.
The properties that are determined from stabilized hysteresis loops and strain-life data are
deflned in table 3.3: Based on the proposal by Morrow [98], the relation of the total strain
CHAPTER 3. STATE OF THE ART 45
Table 3.3: Cyclic material properties: constant-amplitude fatigue behavior
Parameter Deflnition
0f Fatigue strength coe?cient
b Fatigue strength exponent, usually varying between -0.04 and -0.05 for metals
?0f Fatigue ductility coe?cient
c Fatigue ductility exponent, usually varying between -0.3 and -1.0 for metals
2NT Transition fatigue life in reversals
Strain amplitude
Fatigue life 2 NT
b
1
c
1 Total
Plastic
Elastic
Figure 3.17: Schematic total strain-life curve [135].
amplitude (?a) and the fatigue life in reversals to failure (2Nf ) can be expressed in the following
form:
?a =
0f
E (2Nf )
b + ?0f (2Nf )c: (3.32)
Equation (3.32), called the strain-life equation, is the foundation for the strain - based approach
for fatigue. This equation is the summation of two separate curves for elastic strain amplitude
life (?ea ? 2Nf ) and for plastic strain amplitude - life (?pa ? 2Nf ). Dividing the Basquin [20]
equation by the modulus of elasticity gives the equation for the elastic strain amplitude{life
curve (flrst part in Equation (3.32)). Both Manson [92] and Co?n [50] simultaneously proposed
the equation for the plastic strain amplitude{life curve (second part of Equation (3.32)). When
plotted on log-log scales, both curves become straight lines as shown in flgure 3.17.
3.4.2 Mean stress correction methods
Here mean stress correction methods are studied, as in Section 3.3.3. In designing for durability,
the presence of a nonzero mean normal stress can in uence fatigue behavior of materials because
a tensile or a compressive normal mean stress has been shown to be responsible for accelerating
or decelerating crack initiation and growth (like in stress approach to fatigue (Section 3.3.3).
Many models have been proposed to quantify the efiect of mean stresses on fatigue behavior.
46 3.4. STRAIN BASED APPROACH TO FATIGUE
The commonly used models in the ground vehicle industry are those by Morrow [97] and by
Smith, Watson, and Topper [123].
Morrow?s mean stress correction method: Morrow has proposed the following relation-
ship when a mean stress is present:
?a =
0f ? m
E (2Nf )
b + ?0f (2Nf )c: (3.33)
This equation implies that the mean normal stress can be taken into account by modifying the
elastic part of the strain-life curve by the mean stress ( m).
This correction is used for steels and used with considerable success in the long-life regime
when plastic strain amplitude is of little signiflcance.
Smith{Watson{Topper (SWT) model: Smith, Watson, and Topper [123] proposed a
method that assumes that the amount of fatigue damage in a cycle is determined by max?a,
where max is the maximum tensile stress and ?a the strain amplitude. Also, the SWT param-
eter is simply a statement that max?a for a fully reversed test is equal to max?a for a mean
stress test. Thus, this concept can be generalized and expressed in the following mathematical
form (Langlais and Vogel, [88]):
max?a = a;rev?a;rev ! max > 0; (3.34)
where a;rev and ?a;rev are the fully reversed stress and strain amplitudes, respectively, that
produce an equivalent fatigue damage due to the SWT parameter.
The SWT parameter predicts no fatigue damage if the maximum tensile stress becomes
zero and negative. The SWT formula has been successfully applied to grey cast iron, hardened
carbon steels and micro-alloyed steels.
Chapter 4
Models Used in Fatigue
The aim of this Chapter is to explain the difierent models used in fatigue from the stress
approach point of view. The advantages and disadvantages of each model are analyzed,
discovering which of them predict better fatigue lifetime. The Chapter is organized as
follow: flrst, a short overview is given in section 4.1. In sections 4.2 and 4.3 difierent
models are analyzed. In section 4.4 an analysis and discussion about the models
presented above is made.
4.1 Overview
From the physical point of view, fatigue failure (fracture) is the result of plastic strain accumu-
lation [86]. However, fatigue processes preceding fracture are too complicated and it is therefore
not possible to describe fatigue curves using only simple physical conceptions. On the other
hand, all fatigue properties associated with fatigue curves (fatigue limit, the slope of the fa-
tigue curve in the range of flnite life, etc.) are not strictly deflned physical quantities but only
engineering properties of materials. The result of these two facts is only a phenomenological
description of fatigue curves and it seems that a purely physical approach will not be successful
for the next few years.
All the important functions for the description of fatigue curves can be divided into several
groups according to the geometrical shapes of their graphs. No special attention will be paid to
the classical trivial models of the fatigue curves covering the regions of flnite life and permanent
fatigue limit. To this purpose a broken straight line with two arms (oblique and horizontal) is
used.
The fatigue model historical evolution is shown in table 4.1. The very flrst ideas were arisen
from August Wo?hler, who started work in Germany in the 1850s. Then others, like Basquin,
Palmgrem, Miner, Dixon and Mood, Castillo et al. developed new models to describe this
material behavior ([20], [22], [37]{[49], [57], [107] and [126]).
In the following sections, the most important models are studied and analyzed to show the
evolution of the state of knowledge in this fleld.
47
48 4.2. MODELS USED IN FATIGUE
Table 4.1: Historical evolution of principal fatigue models.
Year Author Main contribution
1850s August Wo?hler First approximation to fatigue of materials
1910 Basquin Fatigue strain approach model
1924 Palmgren Creation of one of the most used rule in fatigue
1945 Miner Development of Palmgren ideas ! Palmgren{Miner rule
1948 Dixon & Mood Use of Up & Down method in fatigue
1972 Basteraine Statistical evaluation of fatigue models
1981 Spindel & Haibach Statistics applied to shape of S{N curves
1985 Castillo et al. Hyperbolic approximation to the lifetime stress level curves
1999 Pascual & Meeker Use of the random fatigue{limit method
2001 Kohout & Vechet Create a model covering low, medium and high cycles region
2001 Castillo & Fern?andez-Canteli A Weibull model for lifetime evaluation and prediction
2006 Castillo & Fern?andez-Canteli A Weibull model for fatigue damage due to any stress history
4.2 Models used in fatigue
In this section the most important models used in fatigue are described.
4.2.1 The Basquin model
Since A. Wo?hler begun to study what happened with the material subject to cyclic loads, many
functions have been suggested to present the material behavior. The Basquin function [130]
represents a hyperbola with an exponent parameter (b is a negative number). Other models,
such as the Stromeyer, Palmgren or Weibull [130] are modiflcations or generalizations of the
Basquin model
(N) = aN b: (4.1)
The simple Basquin function [20] describes the dependence of fatigue limit of flnite life N on
the number of cycles N . This, can be extended also to the low{cycle region
(N) = a(N +B)b; (4.2)
as well as to the high{cycle region
(N) = aN b + 1; (4.3)
a function which is called the Stromeyer function [128], where 1, is the permanent fatigue
limit. A model valid in both the low- and the high{cycle regions:
(N) = a(N +B)b + 1; (4.4)
which is called the Palmgren function.
4.2.2 The Palmgren{Miner rule
This model is one of the most used in fatigue of materials. The model was created by Palmgren
in 1924, but it begun to be popularized by Miner in 1945.
To understand the problem, consider a situation of variable amplitude loading, as illustrated
in flgure 8.6. A certain stress amplitude a1 is applied for a number of cycles N1, where the
number of cycles to failure from the S{N curve for a1 is Nf1. The fraction of life used is then
CHAPTER 4. MODELS USED IN FATIGUE 49
N1=Nf1 . Now let another stress amplitude a2, corresponding to Nf2 on the S{N curve, be
applied for N2 cycles. An additional fraction of the life N2=Nf2 is then used. The Palmgren-
Miner rule [59], simply states that fatigue failure is expected when such life fractions sum to
unity, when 100% of the life is exhausted:
N1
Nf1 +
N2
Nf2 +
N3
Nf3 + ::: =
X Ni
Nfi = 1: (4.5)
?
? a1
? a2
? a3
N1 N2 N3
N
Nf1Nf2Nf3
?
N
? a3
? a2
? a1
Figure 4.1: Use of the Palmgren{Miner rule for life prediction for variable amplitude loading
which is completely reversed [59].
Some cycles of the variable amplitude loading may involve mean stresses. Equivalent com-
pletely reversed stresses then need to be calculated before applying a completely reversed S{N
curve, or else a life equation applied that already incorporates mean stress efiects, studied in
section 3.4.2. In addition, the stress range caused by changing the mean level also needs to be
considered in summing cycle ratios.
4.2.3 The up{and{down method
In daily practice some continuous variables appear such that they cannot be measured directly.
This is for example the case of the fatigue endurance limit, i.e., the stress level below which the
fatigue failure does not occur. Unfortunately, once an specimen subject to a fatigue experiment
at the stress level ? has failed it cannot be tested again at a lower stress level to see if failure
occurs at that level. Thus, the population variable is characterized by a continuous variable (the
endurance limit) which cannot be measured in practice. All we can do is to select some stress
level and determine whether the endurance limit is below or above such a level.
This type of situation arises in many flelds of actual research. There are several common
procedures to deal with the problem of estimation, as for example the up-and-down method.
Originally, this method was used for obtaining sensitivity data and developed and used in
explosives research. But the method may be employed in any sensitivity experiment. The
technique consists of choosing several stress levels:
: : : ;? 2;? 1;? 0;? ?1;? ?2; : : :
and start the test at level ? 0. If the specimen fails, we move downward to ? ?1, and upwards
to ? 1, otherwise. Then, the process is repeated a number of times t. The results of the
experiment consists of the stress levels ? i and the binary values i where 1 means survival and
0, failure.
50 4.2. MODELS USED IN FATIGUE
0.8
1
1.2
1.4
1.6
1.8
2
0 10 20 30 40 50 60
??
t
Runout
Failures1
2
3
4
5
10
7
8
9
11
12 14 16 18 20 30 32 34
35
38 40 44 46
47
50 52 54
21 23 25 27 29 31 33 37 39 41 43 45 49 51 53
56
55 57
58 60
59
22 24 28 36 42 48
6
13 15 17 19
26
Figure 4.2: Illustration of the up-and-down method using the Dixon and Mood data and showing
the flve stress levels [48].
The result of such a experiment can be represented as in flgure 4.2, where we have used
data reported by Dixon and Mood [57] and the failures (asterisks) are the data points followed
by data points in an upper level, and the survivals (squares) are the data points followed by
another data points in a lower level.
The principal advantages proposed by the authors were two: the primary advantage of this
method was that it automatically concentrates testing near the mean so, that this increases the
accuracy with which the mean can be estimated. A second advantage is that the statistical
analysis was quite simple in certain circumstances whereas the analysis for the ordinary method
is rather tedious. But, the method had one obvious disadvantage in certain kind of experiments
because it requires that each specimen be tested separately.
Some conditions on the experiments These conditions are:
1. The analysis requires that the variate under analysis be normally distributed. It is therefore
necessary that the natural variate be transformed to one which does have the normal
distribution.
2. The sample size must be large if the analysis to be described is to be applicable. As it
turns out, the efiective sample size is only about half the actual sample size. The statistical
analysis is based on large sample theory so that if one uses the analysis on a sample of size
forty, he will in efiect be using large sample theory on a sample of size twenty. Measures
of reliability may well be very misleading if the sample size is less than forty or flfty.
3. The interval between testing levels should be approximately equal to the standard devia-
tion. This condition will be well enough satisfled if the interval actually used is less than
twice the standard deviation.
Statistical analysis The simple method of analysis given in this section is applicable only
when all the conditions described in the preceding lines are fulfllled.
To describe the statistical analysis the authors again revert to the explosives experiment in
describing the method. Suppose it is known for the given type of explosive that the logarithms
CHAPTER 4. MODELS USED IN FATIGUE 51
of the critical heights are normally distributed. Letting h represent the height, y = log h will
then be the normally distributed variate. We shall call y the normalized height, and represent
the mean and variance of its distribution by ? and 2. The experiment is performed by choosing
an initial height for the flrst test, say h0. This should be chosen near the anticipated mean.
The other testing levels are determined so that the values of the normalized height y are equally
spaced. If d is the preliminary estimate of a, and if y0 = log h0, then the actual testing heights
are obtained by putting log h = y0 ? d, y0 ? 2d,..., and solving for h. The heights will then be
so spaced that the transformed variate is equally spaced with spacing equal to its anticipated
standard deviation. All computations are done in terms of y.
The estimates of ? and are based on the flrst two moments of the y values using the
frequencies ni. But since the y values are equally spaced, the moments are more easily computed
in terms of the two sums:
A =
X
ini;
B =
X
i2ni: (4.6)
That is,
? = y0 + d
A
N ?
1
2
?
; (4.7)
where y0 is the normalized height corresponding to the lowest level. The standard deviation is:
= 1:62d
?
NB ?A2
N2 + 0:29
!
: (4.8)
4.2.4 The Bastenaire model
The author [22], presents a probabilistic description of constant stress amplitude fatigue-test
results, using the experimental and theoretical results of previous research work. This description
includes the S{N curves (or equiprobability of fracture curves), P{S curves (or stress{response
curves), P{N curves (cumulative distribution functions of fatigue endurance), and accounts
for the occurrence of run-outs. The application of this method is demonstrated through flve
examples for each of which several hundred test results are available.
The purpose of this model [22], is to demonstrate the application of a new statistical method
of evaluation based on the same underlying mathematical model to the experimental data col-
lected. Furthermore, this model is one of the flrst models in which the P{S{N flelds are presented,
being useful in the deflnition of the model proposed in this doctoral thesis and described totally
in Chapters 5, 6 and 7.
General expression for the probability of fatigue failure: It has long been known that
the scatter of fatigue lives in constant stress amplitude tests could be represented using a set
of equiprobability curves in a P{S{N diagram [68]. The P{S{N diagram is a simple method of
representing the following relationship between the probability of fracture p, the stress amplitude
S, and the number of load cycles N
p = F (S;N): (4.9)
The two quantities S and N are assumed to be independent variables to which the experimenter
can assign any values since he is at liberty to carry out a test that may last up to N cycles
under stress S and examine whether or not the specimen sustains this number of cycles without
52 4.2. MODELS USED IN FATIGUE
fracturing, p is the probability of fracture regarded as one of two alternatives. However, F (S;N)
is also the cumulative distribution function (CDF ) of the number of cycles to fracture (NCF )
regarded as a random variate.
If a group of specimens is allocated to each stress level, it is possible to estimate the prob-
ability of fracture F (Si; N) from the proportion of specimens broken in the ith group before N
cycles are completed.
For any given value of S, the value of N which is such that F (S;Np) = p is the pth quantile
of the distribution of the NCF which can be denoted by Np
F (Sp; N) = p: (4.10)
If it is assumed that p increases when either S or N increases then
@F
@S > 0;
@F
@N > 0 (4.11)
Difierentiating Equation (4.11), we obtain:
p = @F@S dS +
@F
@N dN: (4.12)
In the case of dp = 0,
dSp
dN = ?
@F
@N =
@F
@S : (4.13)
Using the above equations, we flnd that dSp=dN < 0. For constant p, Sp is, therefore, a
decreasing function of N , as shown by the schematic equiprobability curves of flgure 4.3.
F(S,N)=p 1
F(S,N)=p
F(S,N)=p 0
Stress amplitude, S
N cycles
SP
SP, inf
NP
Figure 4.3: Schematic diagram of equiprobability of fracture curves [22].
The following remarks are useful for the determination of the limit to the stress{response
curves when N tends to inflnity:
1. S is a stress amplitude and, therefore, cannot be negative.
2. A zero stress amplitude can be assumed to produce no fatigue efiects in a material. Sp
being a bounded decreasing function of N , a limit Ep ? 0 to Sp always exists (except,
perhaps for p = 1, for which Sp may not be deflned).
CHAPTER 4. MODELS USED IN FATIGUE 53
3. That a limiting stress{response curve certainly exists, though it may possess two difierent
shapes.
Bastenaire shows that it is advantageous to plot the proportions of specimens failing before
N load cycles have been completed on a normal or logistic probability scale. If a straight line
is obtained on probability paper, it can be conc1uded that the threshold stresses are normally
distributed with a standard deviation shown by the slope of the straight line.
However, the stress-response curves do not have to be normal. This concept will be analyzed
in Section 4.3 and Chapter 6. The failure of a material is an extremal process and there are other
distributions (like Gumbel or Weibull distributions) that can represent better this behavior of
the material.
Making the hypotheses that the stress{response curves are normal, one only needs to assume
that two parameters, a location parameter ? and a scatter parameter , are su?cient to represent
these curves which may equally well derive from the normal, logistic, extreme value or some other
distribution.
For any given value of N, ? and depend on N and should really be regarded as two functions
?(N) and (N). Denoting the cumulative distribution function of the reduced variate by F , the
probability of fracture can be expressed by the following equation
p = F
S ? ?(N)
(N)
?
: (4.14)
Representation of the equiprobability curves using a transformed variate: Now, the
aim of this subsection is to obtain the general expression that can describe the problem and
the better solution. For this, it is necessary to take into account the boundary conditions to
transform Equation (5.36) into a general function.
Equation (5.36) shows that P=constant if (S ? ?(N))= (N) is constant. In particular, if
S??(N) = 0 then P = F (0). It has been shown that SP tends to a limit EP ? 0 when N !1
(endurance limit).
A new function
?(N) = ?(N)? ?(1); (4.15)
is introduced, one can express ?(N) as
?(N) = ?(N) + ?(1); (4.16)
where ?(N) is a decreasing function of N tending to zero as N ! 1. Using Equations (5.36)
and (4.16), and knowing that ?(1) = E,
p = F
S ? ?(N)?E
(N)
?
: (4.17)
This equation can be used to account for the main features of the NCF distributions.
A contribution of Bastenaire is the representation of the S{? fleld, in where is represented
the variable ?(NCF ) in a diagram. Replacing the usual S{N system of coordinates by S; ?,
the equiprobability curves can be represented in a new diagram. The equation of the F (0)
equiprobability curve is
SF0 = ?(N) + E: (4.18)
54 4.2. MODELS USED IN FATIGUE
This equation shows that by plotting ? instead of N as the abscissa, the F (0) equiprobability
curve is represented by a straight line.
More generally, assuming that F is the cdf of a continuous random variate, F is monotonic,
and P will be constant in Equation (4.17) if, and only if, the argument of F is constant, say
u =
S ? ?(N)? E
(N)
?
; (4.19)
where u is a constant.
Stress amplitude, S
N, ?
a b c a?
b?
c?
Figure 4.4: S{N and S{? diagrams. Letters a, b, c correspond to S{N fleld. Letters a?, b?, c?
correspond to the ?{N fleld [22].
The general equation of the equiprobability curves can be written as
Sp = ?(N) + E + u (N): (4.20)
The S{N and S{? systems of coordinates are superimposed in flgure 4.4 to show how they are
related. flgure 4.4 also shows how the distributions of S and ? difier. For any given value of
N , there is no practical limit to the stress amplitude which can be applied to specimens and
the stress{response curve can be explored up to stresses at which the probability of fracture is
nearly unity. In contrast with this, an increase in the number of cycles for a given stress value
produces a decrease in ?(N) which is bounded at zero and the proportion of failures tends to a
limit.
As a flnal remark, Equation (4.17) shows that the value of ? for which p = F (0) is ? = S?E.
When a value of S less than E is chosen as a test stress, this value is negative though the observed
values of ? are all positive. This is so because the central value of a distribution can be outside
the range of the observed values.
The distributions of `(N) at difierent stress levels difier mainly in location and in the pro-
portion of observations cut ofi at ` = 0 but can be expected to be very similar in shape. This
is in contrast with the distributions of fatigue lives which (even though plotted on a logarith-
mic scale) change markedly in shape when the stress changes. Bastenaire explains [22], that
the logarithm of the NCF is distributed normally only at intermediate stress levels. When
the stress decreases, the scatter and skewness of these distributions increase to a considerable
degree, and the proportion of broken specimens tends to a limit as the number of applied load
cycles increases.
CHAPTER 4. MODELS USED IN FATIGUE 55
More general transformation of the number of cycles to fracture: Bastenaire studied
difierent real cases [21], but, unfortunately, SF (0) = (A=N) + E is not always a convenient
equation for the median equiprobability curve. This is unfortunate, not only because A=N is
a simple function, but also because, if A=N is distributed normally, so is 1=N . Similarity, if
a linear relationship holds between A=N and S, it also holds between 1=N and S. No prior
knowledge of coe?cient A is, therefore, necessary to check the validity of this relationship, and
A can be estimated later by means of a linear regression.
For a number of materials and testing conditions [21], the median equiprobability curve can
be represented by the following equation:
N = A exp[?c(S ? E)](S ?E) ; (4.21)
in which A, c, and E are coe?cients. Making c = 0 we obtain SF (0) = (A=N) + E.
A more general formula is
N +B = A exp[?c(S ? E)](S ? E) ; (4.22)
with an additional coe?cient B (see flgure 4.5). If it is assumed that Equation (4.22) can be
Stress amplitude, S
N cycles
mean
Figure 4.5: Bastenaire schematic curve [126].
used to represent the F (0) equiprobability curve, and knowing that when N ! 1, SF (0) ! E,
we can obtain
N +B = A exp[?c?]? : (4.23)
Finally, sorting elements of Equation (4.23) we can obtain the general expression of the
Bastenaire Model
N =
A exp
"
?
S ?E
B
?C#
S ?E : (4.24)
56 4.2. MODELS USED IN FATIGUE
Example of application: The schematic diagram of flgure 4.4 can now be used to plot real
values. This is done in flgure 4.6 in which the results for flve materials are plotted together. The
estimated means of ?(NCF ) are plotted in the abscissas axis and the stresses in the ordinates
axis.
Figure 4.6: Experimental S{? diagram for flve difierent steels. flgure from [22].
The main purpose of this diagram is not to plot each S{N curve in the form of a straight line:
indeed, nearly any relationship can be represented by a straight line using suitable coordinates.
flgure 4.6 represent estimated mean values. Plotting hundreds of individual test results for
each material was not feasible on the S{? diagram. A better image of their distributions is given
by the cumulative frequency curves shown in flgure 4.7.
4.2.5 The Spindel and Haibach model
The method proposed by the authors [126] tries to flnd the cutofi point, that is, the endurance
at which conventionally shaped S{N curves change to the horizontal.
The determination of the shape of S{N curves is not a purely academic exercise. It is found
in cumulative damage calculation that the position of the cutofi point has a considerable efiect
on the stress that is calculated as tolerable for a given load spectrum when such calculations are
CHAPTER 4. MODELS USED IN FATIGUE 57
Figure 4.7: Cumulative frequency curves of ?(NCF ) for steel 33CD4 (treated to 80kg=mm2).
This curve correspond with the third curve in flgure 4.6. flgure from [22].
based on Miner?s rule [76]. Considerable difierences are found, not only in the stress at the en-
durance limit, but also in the fatigue strength for loading spectra calculated for high{endurance
values, that is, values above 107 cycles where experimental veriflcation is not practicable because
of the testing time and cost.
The authors developed a new method that would determine the best common slope to flt a
number of sets of individual test data.
Possible deflnitions of the S{N curve: Any statistical analysis must be based on some
assumptions about the shape of the S{N curve, however large or small the number of parameters
used to deflne it. The straight{line approximation may be found to give a poor flt in the case
of good data. The assumption that the logarithm of the endurance are normally distributed
with the same standard deviation at all stress levels is evidently untrue. The S shaped types
of S{N curves, like those suggested by Weibull [130] or Bastenaire [22] (Section 4.2.4), may be
considered to provide a more appropriate representation.
Possible statistical methods: The method of analysis appropriate in any given case should
be selected on the principle that the simplest statistical model that will flt the available data
with acceptable precision is the right one to use. Precision refers to the inferences drawn about
the parent population that might have produced the data rather that to a description of the
individual set of data.
Multiple regression may be appropriate for well documented data. Otherwise the regression
and the confldence limits calculated will be distorted. In any case, if the data contain run-outs,
a method based on maximum likelihood is needed to take account of these.
58 4.2. MODELS USED IN FATIGUE
A graphical analysis developed on the assumption of a uniform shape of the S{N curves for
comparable test series but allowing for variation in the parameter Sa (stress amplitude) that
deflnes the fatigue strength at 2? 106 cycles has been described elsewhere in detail [78].
In the next lines, two difierent forms for determining the parameters of the distributions are
analyzed. The flrst is the combination of set of data, used with normal distribution data. The
second one is the maximum likelihood method.
Combining set of data: This technique can be illustrated by applying it to samples drawn
from a normally distributed population: If the population means is ? and its variance is
2, the probability of drawing a sample value yi is
1
p2? exp
"
?12
yi ? ?
?2#
dy; (4.25)
the logarithm of which is
? 12
yi ? ?
?2
? ln + ln dyp2? : (4.26)
The last term in the expression is constant for all values of ? and and, therefore, of no
interest in likelihood ratio
L = ?12
?
1
2
nX
1
(yi ? ?)2 + n ln 2
!
= ?n2
1
2 ((?y ? ?)
2 + s2) + ln 2
?
: (4.27)
For run-outs in fatigue test, the probability is given by the condition that no failure has
occurred up to certain value of yi, where yi is the logarithm of the endurance. The
probability is
1p2?
Z (yi??)=
?1
exp
?
?12?
2
?
d? (4.28)
and the likelihood is
ln
?
1p2?
Z (yi??)=
?1
exp
?
?12?
2
?
d?
!
: (4.29)
The Likelihood method applied to S{N curves: The only difierence between this calcu-
lation and a form of regression analysis is that both ? and become functions of the
independent variable x.
When the prevoius method is applied to the analysis to S{N data, the transformed variables
logN and logSa are most commonly used. To specify the shape of the S{N curve the
following information is needed:
1. The fatigue strength SA at 2? 106 cycles,
2. The slope, k,
3. The standard deviation, s, of logN and logSa, and
4. The position of change of slope NE at stress SE are to be considered as parameters
of a simple model (see flgure 4.8)
An additional parameter, fi, was introduced to specify the mode of transition and the
typical values are in the interval fi 2 [10; 100]. For fi = 1, the model degenerates to the
CHAPTER 4. MODELS USED IN FATIGUE 59
St
re
ss
a
m
pl
itu
de
, S
N cycles
Slope k
Slope k1
(SE,NE) (SE,NE)
?
s
Slope k
s
s
s
(a) (b)
Figure 4.8: Shapes of S{N curve considered: (a) simple model, (b) extended model [126].
simple one changing slope abruptly. Analytically, the S{N curve for the extended model
is given by the equation
Y = 0:5(k + k1)x+ 0:5(k ? k1)
jxj+ 1fi ln [1 + exp(?2fijxj)]
?
; (4.30)
where Y = logN=NF , and x = logS=SE .
Example of application: The described method of determining the shape of the S{N curves
will be illustrated in the next lines. The data is presented in table 4.2 and in flgure 4.9.
Figure 4.9: S{N curve established by 20 tests per stress level [126].
60 4.2. MODELS USED IN FATIGUE
Table 4.2: Jointly best supported parameters of the S{N curve as a function of the cutofi points
derived for the data from flgure 4.9. flgure from [126]
Run Cutofi Slope SD of logS Endurance Likelihood Additional
No. NE k s Limit, SE method criterium
Slope changing abruptly to the horizontal
1 0:6 ? 106 4.00 0.0326 98.1 213.0
2 1:0 ? 106 5.00 0.0228 94.8 203.9
3 1:5 ? 106 5.50 0.0176 91.2 236.5
4 2:0 ? 106 6.00 0.0162 89.5 236.5
5 3:0 ? 106 6.50 0.0175 86.7 221.6
6 10:0 ? 106 7.75 0.0246 79.5 130.7
Slope changing continuously to the horizontal (fi = 20)
7 2:0 ? 106 4.00 0.0215 76.8 249.5 249.0
8 3:0 ? 106 5.00 0.0194 75.2 251.7 250.9
9 5:0 ? 106 5.50 0.0176 72.6 250.8 248.2
10(fi = 25) 3:0 ? 106 5.25 0.0185 77.8 251.1
The determination of the cutofi point depends of the type of curve we chose. In the case
of an S{N curve with the slope changing abruptly to the horizontal, the best supported cutofi
point is found by interpolation between points 3 and 4 (see table 4.2). And better flt is obtained
with a continuously changing slope (see flgure 4.9).
The value of NE is found to be slightly lower, and the confldence limits become narrower
when the additional criterion is used (see flgure 4.10).
4.2.6 The Pascual and Meeker model
This model [107], describes the relationship between fatigue life and applied stress and provides
and illustrates the corresponding data analysis methods. This work is motivated by the need to
develop and present quantitative fatigue{life information used in the design of jet engines.
Fatigue data are often presented in the form of a median S{N curve, a log{log plot of cyclic
stress or strain s versus the median fatigue life N . An extension of this concept is the p quantile
S{N curves, also called S{N{P curves, a generalization that relates the p quantile of fatigue life
to the applied stress or strain. Thus, each curve represents a constant probability of failure p, as
a function of s. We shall use the 0:05 and 0:95 quantile S{N curves to illustrate the variability
of fatigue life.
Fatigue data on ferrous and titanium alloys indicate that experimental units tested below
a particular stress level are unlikely to fail [107]. The S{N curve for these materials exhibits
a strong curvature and an asymptotic behavior near the fatigue limit. Most nonferrous metals
such as aluminum, copper, and magnesium appear not to have a fatigue limit.
For nonferrous materials, it is common practice to deflne the \fatigue strength" to be the
stress level below which failure will not occur before an arbitrary large number of cycles. Collins
[52] and Dieter [56] deflned fatigue strength as such and used the term fatigue limit to imply
inflnite life. On the other hand, to represent fatigue strength at a prescribed long but flnite life,
Nelson [101], and Colangelo and Heiser [51], used the term endurance limit, whereas others use
the term fatigue limit.
There are two main considerations in modeling the relationship between the applied stress
CHAPTER 4. MODELS USED IN FATIGUE 61
Figure 4.10: Best supported S{N curves: (a) changing slope abruptly to the horizontal; (b)
changing slope continuously to the horizontal as fltted to the set of data from flgure 4.9. flgure
from [126].
and fatigue life. First, often the standard deviation of fatigue life decreases as the applied stress
increases. Second, curvature in fatigue curves suggests the inclusion of a fatigue limit in the
statistical model for fatigue life. The random fatigue-limit model describes both characteristics.
Let Y be the fatigue life and s the stress level. We model Y as
log(Y ) = fl0 + fl1 log(s? ) + ?; (4.31)
where fl0, and fl1 are fatigue curve coe?cients, is the fatigue limit of the specimen, ? is the error
term, and log denotes natural logarithm. Let V = log( ), and suppose that V has probability
density function (pdf)
fV ( ;? ; ) = 1 `V
?
? ?
!
; (4.32)
with location and scale parameters ? and , respectively. `V () is either the standardized
smallest extreme value (sev) or normal pdf.
Let x = log(s) and W = log(Y ). Assume that, conditioned on a flxed value of V < x;W jV
has pdf
fW jV (w; fl0; fl1; ;x; ) =
1
`W jV
w ? (fl0 + fl1 log(expx? exp ))
?
(4.33)
with location parameter fl0 + fl1 log(expx ? exp ) and scale parameter . `W jV either the
standardized sev or normal pdf . The marginal pdf of W is given by
fW (w;x ) =
Z 1
?1
1
`W jV
w ? ?(x; ; )
?
`V
?
? ?
!
d ; (4.34)
62 4.2. MODELS USED IN FATIGUE
where = (fl0; fl1; ;? ; ) and ?(x; ; ) = fl0+ fl1 log(expx? exp ).The marginal cumulative
distribution function (cdf) of W is given by
FW (w;x ) =
Z 1
?1
1
'W jV
w ? ?(x; ; )
?
`V
?
? ?
!
d ; (4.35)
where 'W jV is the cdf of W jV . We will refer to this statistical model as the random fatigue-limit
model. There are no closed forms for the density and distribution functions of W .
To estimate the parameters of the random fatigue-limit model the authors use ML methods.
Statistical theory suggests that ML estimators, in general, have favorable asymptotic (large-
sample) properties. For \large" sample sizes and under certain conditions on the fatigue life
distribution, the distribution of ML estimators is approximately multivariate normal with mean
vector equal to the vector of true values being estimated and standard deviations no larger than
that of any other competing estimators.
When fatigue limits exist, plots of fatigue life versus stress{strain often exhibit curvature
at lower stress{strain levels. Moreover, in most fatigue experiments, the variance of fatigue life
decreases as stress{strain increases and the standard deviation is often modeled as a monotonic
function of stress{strain.
4.2.7 The Kohout and Vechet models
The authors present a complex function that can be used in all range of lifetime (from low{
cycle fatigue region to high{cycle region). But this function can be reduced in other simpler
expressions to make the use of the model easier in one special part.
A new function is proposed for the description of fatigue curves in both low and high{cycle
fatigue regions [86], i.e. for the whole region of cycles from tensile strength to permanent fatigue
limit (usually described by the Palmgren function). In each cycle region it can be simplifled;
in fact it changes into the Basquin function in the region of flnite life and an analogy of the
Stromeyer function for the high{cycle fatigue region can also be obtained (Equations (4.2) and
(4.4)).
The contribution of the present authors (Kohout and Vechet [85] and [84]) consists in extend-
ing the Basquin function to the low and the high{cycle regions symmetrically, besides replacing
N with N +B for the extension to the low-cycle region:
(N) = aN b ! (N) = a(N +B)b; (4.36)
where a is a parameter of the Basquin and some other functions (extrapolated value of function
or of the tangent in the point of in exion for N = l)[MPa], b a parameter of the Basquin and
some other functions (in log-log flt the slope of oblique asymptote or of the tangent in the point
of in exion), and B;C is a parameter of new function (see flgure 4.11).
For its extension to the high{cycle region, in some cases 1=N has been replaced with 1=N +
1=C, obtaining:
(N) = aN b = a
1
N
??b
! (N) = a
1
N +
1
C
??b
= a
NC
N + C
?b
: (4.37)
Finally, both extensions can be made simultaneously, but because B << C (their values difier
by many orders of magnitude) quantity B can be neglected in the sum B + C resulting the
function:
(N) = a
(N +B)C
N + C
?b
(4.38)
CHAPTER 4. MODELS USED IN FATIGUE 63
It represents the new function for the description of fatigue curves in both the low- and the
high{cycle fatigue regions, i.e. over the whole range of lifetime.
Very low Low Finite life High Very high
Basquin
Kohout & Vechet
Log C Log B
Log ?1
Log ? inf
Log N
Log ?
Lx
Hx
ALx
AHx
Figure 4.11: Regions of validity of its simplifled forms [86].
The new function (Equation (4.38)) can be simplifled for various regions of the number of
cycles. The regions of the number of cycles where the above relations are valid are shown in
flgure 4.11. Region named LX represents the very low{cycle region, which corresponds with
Equation (4.2). Similarly, region HX represents the very high{cycle region (Equation (4.37))
and regions ALX and AHX , which corresponds with asymptotes of Equations (4.2) and (4.37),
that is:
(N) = aBb = 1 ! ALX ;
(N) = aCb = 1 ! AHX : (4.39)
Comparing the curves corresponding to the new (Hx) and the Stromeyer functions in flgure
4.12 it can be seen that in the range of experimental values the curves difier in their positions
only unsubstantially and the values of fatigue limit for 10 ? 107 cycles are practically identical.
But the curves difier substantially in their directions (slopes) in the margins of measured range.
For a very high number of cycles the curve corresponding to the new function is practically
constant with stress values only slightly lower than the fatigue limit for 10? 107 cycles.
For a low number of the cycles the curve of new function merges with the oblique asymptote
while the slope of the Stromeyer curve substantially increases with decreasing number of cycles.
In both cases the extrapolation by means of the new function is substantially closer to reality
than the extrapolation by the Stromeyer curve. The asymptotes of the new function can be
used for estimation as well as extrapolation of the fatigue curve and they are nearly identical to
the broken straight line used earlier as the fatigue curve.
64 4.3. NEW STATISTICAL MODELS: CASTILLO?S MODELS
All the above means that the new function (Hx) is very suitable for extrapolating out of the
range of measured data while the use of the Stromeyer function cannot be recommended to this
purpose.
Figure 4.12: Comparison of regressions by the Basquin function (Equation (4.1)) for N << 106,
the Stromeyer function (Equation (4.36)), and new function (Hx) with its asymptotes. flgure
from [86].
4.3 New statistical models: Castillo?s models
In 1985, a hyperbolic approximation of the lifetime stress level curves was developed by Castillo
et al. [37]. starting with this model, several methods have been presented by this researcher
with the aim of flnd the best model to deflne the lifetime (stress approach) of a material ([35],
[39], [42], [36], [37] and [44]). These models for the analysis of lifetime data are derived based
on physical and statistical considerations.
When modeling fatigue and other similar lifetime data, models such as those in table 4.1 are
selected mainly because of their mathematical tractability, simplicity and/or concordance with
the data. But, models should be derived based on physical and statistical considerations. These
considerations require that fatigue models should satisfy the following considerations (see [37]
and [44]):
1. Weakest link principle: If a longitudinal element is divided into n sub-elements, its
fatigue life must be the fatigue life of the weakest element [62]. The weakness of a sub-
piece is determined by the size of its largest crack and the stress it is subjected to.
2. Independence: The fatigue strengths of two non-over lapping subelements are indepen-
dent random variables.
3. Stability: The cumulative distribution function (cdf) model must be valid for all lengths,
but with difierent parameters.
4. Limit value: The cdf should encompass extreme lengths. Thus, the cdf must belong to
a family of asymptotic functions.
CHAPTER 4. MODELS USED IN FATIGUE 65
5. Limited range of the random variables involved: The variables have a flnite lower
end, which must coincide with the theoretical lower end of the selected cdf.
6. Compatibility: The distribution of lifetime given stress level should be compatible with
the distribution of the stress level given lifetime; that is, if FX(x; y) is the cdf of X given
y, and FY (y; x) is the cdf of Y given x then
FX(x; y) = FY (y; x): (4.40)
With all these model conditions (see [35], [39], [42], [36], [37] and [44]) the authors proposed a
new solution to the problem. In particular they dealt and gave adequate answers to the following
questions [36]:
Question 1: Is it possible to extrapolate S{N curves derived from laboratory results for a given
stress level to another stress level conditions?
Question 2: We can one predict the fatigue damage associated with a given arbitrary pair
max; min from laboratory results obtained for a given stress level?
Question 3: If the answers to the above two questions are negative, we can ask the following
question: what is the minimum information required to make the above extrapolation
possible?
Question 4: Are the existing fatigue models able to adequately use this information to achieve
the above aims? If the answer is negative, what changes are required in the models to
solve the problem?
Question 5: What is the associated testing strategy (test design) able to produce the data
required by the valid model?
Question 6: Once the lab tests have been conducted and the model selected, are there models
and explicit formulas to perform the above extrapolation or interpolation?
4.3.1 The general model for lifetime evaluation: The Weibull model
In this subsection we introduce the Weibull model (see [35], [38], [36], [48]) with some of its
properties, applicable to lifetime problems.
Like the Bastenaire model, studied in Section 4.2.4, this model is based on the implemen-
tation of a probability distribution to deflne the relation between lifetime and loads (P{S{N
curves). But in this case, contrary to Bastenaire and Spindel and Haibach hypothesis (see [22],
[126]), the probability function is not a normal distribution. In fatigue the probability of failure
can be assumed ro be an extremal distribution, in this case a Weibull distribution.
The cumulative distribution function (cdf) of the three{parameter Weibull family is given
by:
F (x;?; ?; fl) = 1? exp
"
?
x? ?
?
?fl#
x ? ? ; ?1 < ? < 1; ? > 0; fl > 0; (4.41)
where F (x;?; ?; fl) represents the probability of the event X ? x, and ?; ? and fl are the scale,
the location (minimum possible value of the random variable X), and the shape parameter, re-
spectively. When X has the cumulative distribution function in 4.41 we write X ? W (x;?; ?; fl).
66 4.3. NEW STATISTICAL MODELS: CASTILLO?S MODELS
Its mean and variance are:
? = ?+ ??[1 + 1=fl]
2 = ?2[?[1 + 2=fl]? ?2[1 + 1=fl]]; (4.42)
and the corresponding percentiles are:
xp = ?+ ?[? log(1? p)]1=fl ; 0 ? p ? 1: (4.43)
Before selecting a model to solve an engineering problem, the relevant variables involved must
be identifled. By previous experience, accumulated in the study of the fatigue phenomenon, we
know that the flve variables initially involved in the fatigue problem are: p;N;N0;? and ? 0,
where p is the probability of fatigue failure of a piece when subject to N cycles at a stress range
? , N0 is the threshold value for N , the minimum lifetime for any ? , and ? 0 is the endurance
limit, below which fatigue failure does not occur.
However, this initial number of variables can be reduced. The Q Theorem allows rep-
resenting any existing relation among the initial variables in terms of another smaller set of
non-dimensional variables. In fact, a dimensional analysis of the initial set of 5 variables leads
to a set of 3 non-dimensional variables. It seems convenient to choose N=N0, ? =? 0 and P as
these variables.
Solving the problem with functional equation ([49] and [45]), the resulting model is:
F (N?;? ?) = 1? exp
"
?
?(N? ?B)(? ? ? C)? ?
?
?fl#
; N? ? B + ?? ? ? C ; (4.44)
where N? = N=Nref and ? ? = ? =? ref , Nref and ? ref are the number of cycles and the
stress level of reference, respectively, B;C; ?; ? and fl are the non-dimensional model parameters.
Their physical meanings (flgure 4.13) are the following:
B: threshold value of lifetime.
C: endurance limit.
?: position of the corresponding zero-percentile hyperbola.
?: scale factor.
fl: Weibull shape parameter of the whole cdf in the S{N fleld.
We note that the percentile curves are hyperbolas which share the asymptotes.
The principal conclusions of this model may be summarized as follows:
1. The use of non-dimensional variables simplifles the problem under consideration and clar-
ifles what is the minimal set of variables, or functions of them, which are relevant to the
problem, and allows working with non-dimensional parameters that have many important
advantages, apart from independency of the set of selected units, as a better numerical
behavior.
2. Physical and engineering considerations allow rejecting many models non satisfying the
associated constraints. These considerations can be written, in many cases, in terms
of functional equations, that lead to explicit forms for the mathematical and statistical
models.
CHAPTER 4. MODELS USED IN FATIGUE 67
??
*
N*
B
C
p=0
p=0.5
p=0.05
p=0.95
Figure 4.13: Graphical representation of the Weibull model. Percentiles curves representing the
relationship between lifetime, N?, and stress range, ? ?, in the S{N fleld [44].
3. AWeibull based model for the S{N fleld has been obtained by solving a functional equation.
This model is useful not only to flt fatigue data, but also to explain the fatigue behavior
of longitudinal elements.
4. There are two types of parameters. One is related to the non-dimensional variables,
and used for normalization purposes and includes the threshold parameters N0 and ? 0.
Other type of parameters are statistical parameters, as the location parameter ?, the scale
parameter ?, and the shape parameter fl.
5. The probabilities of failure associated with a given load history can be easily calculated.
The authors have been working with other aspect in the model. A comparison between
the up{and{down method (Section 3.4) and the Weibull model has been made by the authors,
concluding that the up-and-down method neglects important information contained in the data
making it very expensive and ine?cient when compared with other alternative models [48].
Furthermore, the problem of estimating the S?N fleld based on samples with difierent lengths
and testing the hypothesis of length independence of fatigue lifetimes has also been analyzed by
the authors, concluding that the length independence assumption cannot be accepted for the
prestressing wires data, while it is a reasonable assumption for the prestressing strands data [?].
Model (4.41) has been studied and successfully applied to difierent cases of lifetime problems
such as plain concrete, prestressed wires and strands with difierent lengths, etc.
4.4 Discussion
The last two sections study the principal models used in fatigue to predict and estimate the
lifetime of a material. In this section, the advantages and shortcomings of these models are
analyzed and discussed.
This chapter has analyzed nine models used in fatigue. The main conclusions after comparing
them are:
68 4.4. DISCUSSION
? The Basquin model [20] and the Palmgren{Miner rule [59] make an approximation to
fatigue law easier.
? There are models which analyze the fatigue in all the fatigue regions (low{cycle region and
high{cycle region). Kohout & Vechet model [86] are part of this group. But the parameter
estimation is complex.
? Some of these models, such as the up{and{down method or the Spindel & Haibach model
[126] are used to determine the endurante limit of a material.
? The Palmegren{Miner?s rule allows us analyze a variable load history, but the model needs
to know the behavior of the material in each stress level for a constant load.
? The relation between probability of failure, stress level and number of cycles to failure
(P{S{N) is provided in Bastenaire [22], Pascual & Meeker [107] and Castillo?s models ([38]
and [36]).
? Only Castillo?s model and Pascual and Meeker Model, deflne the probability of failure
with an extremal distribution. In the other cases the authors choose normal distributions,
but this distribution is not valid for the fatigue of material (extremal process).
? The only model that can be extrapolated to other range of loads after the model parameters
have been estimated is the Castillo model. Remember that in the Bastenaire model, when
the author represents points outside the estimation range, he obtains negative values of ?.
An evolution of fatigue models has been presented in these sections. All of them allow
analyzing difierent stress histories, some constant and other variable load histories, but none of
them can be used to study the behavior of material in all the load range, i.e. mixing tension
and compression loads.
Actually, the most useful models are based on statistic and probabilistic concepts, like
Castillo?s Model. So, the principal aim of this thesis is to develop a new statistic and prob-
abilistic model covering the tension and compression Wo?hler flelds.
Chapter 5
Use of Functional Equations
The aim of this Chapter is to introduce readers into the fleld of functional equations which
have shown very useful in some applications, such as model design. We are aware that this
is not an easy task, and that any efiort to bring together mathematicians and engineers, as
experience shows, has many related di?culties.
The model derived in chapter 6 has been obtained thanks to this theory.
5.1 Introduction
The modeling or idealization of the problem under consideration (structure, road, harbor, water
supply system, etc.) should be su?ciently simple, logically irrefutable, admitting a mathematical
solution, and, at the same time, represent su?ciently well the actual problem. Experienced
engineers and scientists know how a successful design depends on an adequate selection of
the model and method of analysis. As in any other branch of knowledge, the selection of the
idealized model should be achieved by detecting and representing the essential flrst-order factors,
and discarding or neglecting the inessential second order factors.
Functional equations are a tool that avoids arbitrariness and allows selection of models to
be based on adequate constraints.
Though the theory of functional equations is very old (some examples of functional equations
appear in Oresme (see [103], [104]), Napier (see [100], [101]), Kepler [82], Galileo [71], Abel (see
[8],[9] and [10]), not only technicians but many mathematicians are still unaware of the power of
this important fleld of Mathematics and in many flelds of Applied Science. This chapter is based
on the Castillo et al.?s book (see [46]) and others as Acz?el [12], Rassias [113], Balasubrahmayan
[17], Anatolij [15], Small [120], Balasubrahmayan [17] and Smital [121].
One of the most appealing characteristics of functional equations is their capacity for model
design. In fact, conditions required by many models to be adequate replicas of reality can be
written as functional equations. Thus, the engineer flnds there an appropriate tool for his design
purposes. In this manner, functions are not arbitrarily chosen; on the contrary, they appear as
the only solutions to the adequate set of requirements.
We can also cite other ideas about functional equations: \the theory of functional equations
is fascinating because of its instrinsic mathematical beauty as well as its applications.[...] In this
69
70 5.1. INTRODUCTION
fleld one deals with mathematical identities where the solutions strongly depend upon the domains
and the regularity assumptions required for the unknows. Proofs are usually clear, clean, short:
elegant arguments come up. Sometimes the equations give you just a little information, but, by
using the powerful methods that the theory provides, you can say quite a lot about the general
solutions." said by Cl. Alsina [12].
There exist diverse bibliography where we can flnd new ideas and methods to solve new
problems. We cite here some examples where an extensive study of topics related with functional
equations and inequalities is made (see [12] and [113]).
5.1.1 One example of functional equation: area of a rectangle (Legendre [89])
Assume that the formula of the area of a rectangle is unknown but given by f(a; b), where f is an
unknown function, b is its basis and a is its height. Consider flgure 5.1(left) in which the rectangle
of basis b and height a has been horizontally divided in two difierent sub-rectangles with the
same basis b and heights a1 and a2, respectively. According to our assumptions, the areas of
the sub-rectangles and the initial rectangle cannot be calculated, but they can be expressed in
terms of our unknown f function as f(a1; b), f(a2; b), and f(a1 + a2; b), respectively. Similarly,
we can perform the division vertically, as shown in the right rectangle of the same flgure, and
write the areas of the resulting rectangles as f(a; b1), f(a; b2), and f(a; b1 + b2), respectively.
f(a1 + a2 ,b)
f(a1, b) f(a2, b)
a1
a2
b b1 b2
f(a , b1 + b2)
a
f(a, b1) f(a, b2)
Figure 5.1: Basic rectangles. flgure from [46].
Stating that the areas of the initial rectangles must be equal to the sum of the areas of the
sub-rectangles, we get the functional equations
f(a1 + a2; b) = f(a1; b) + f(a2; b)
f(a; b1 + b2) = f(a; b1) + f(a; b2): (5.1)
Because b is constant in the flrst equation and a is constant in the second, both equations
become Cauchy?s Equation, we have
f(a; b) = c1(b)a = c2(a)b;
where c1(b) and c2(a) are initially arbitrary functions, but due to the second identity, they must
satisfy the condition
c1(b)
b =
c2(a)
a = c;
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 71
which implies
f(a; b) = cab; (5.2)
where c is an arbitrary positive constant.
As a consequence, the area of a rectangle is the product of its basis a, its height b and a
constant c.
This proves that the area of a rectangle is not the well known \basis ? height", but \a
constant ? basis ? height". The constant takes care of the units we use for the basis, the height
and the area. This means that if b is measured in inches, h in feet, and we want f in square
miles, the constant must be difierent from the constant required for the case of b measured in
meters, h in kilometers, and f in square meters.
The interesting result is that functional equations discover the need to consider the units of
measure.
5.2 History of functional equations
The use of the functional equations comes from much longer period that the formal mathematic
discipline has existed.
One of the flrst mathematician who worked with functional equations was Nicole Oresme
(1323-1382), who provided an indirect deflnition of linear functions by means of a functional
equation. In 1352, Oresme wrote the Tratatus de conflgurationibus qualitatum et motuun in
which the deflnition of a functional relationship between two variables is deflned, and furthermore
he deflned the idea of that one can express this relationship geometrically by what we would
now call a graph (well ahead of R?en?e Descartes) [120].
We have three distinct real numbers x, y and z, which describe a linear functional equation.
Associated with x, y and z we have a variable that can be writen as f(x), f(y) and f(z),
respectively. The function f is deflned to be linear (i.e. a quality which is uniform) if:
y ? x
z ? y =
f(y)? f(x)
f(z)? f(y) ; (5.3)
for all distinct values of x, y and z.
What makes Oresme?s deflnition a functional equation is that f is treated abstractly: one
may plug any function into this equation to see whether the equation is satisfled for all possible
values of x, y, and z. We can compare this with the standard deflnition to be found in most
modern textbooks which say that a linear funciton is one of the form:
f(x) = ax+ b; (5.4)
for some a and b.
Over the next few hundred years, functional equations were used but no general theory of such
equation arose. Other important mathematician was Gregory of Saint{Vicent (1584{1667),
whose work on the hyperbola made implicit use of the functional equation f(xy) = f(x)+ f(y),
pioneered the theory of the logarithm.
In the year 1647 he wrote Opus Geometricum quadraturae circuli et sectionum coni, in
which the deal is to present methods for calculating areas and the properties of conic sections.
He made great progress on the problem of logarithm using purely geometric arguments: \If a
planar region is stretched horizontally by a given factor, and simultaneously shrunk vertically by
72 5.2. HISTORY OF FUNCTIONAL EQUATIONS
x z y
f(x)
f(y)
f(z)
Figure 5.2: Schematic deflnition of linear functions [120].
x1 x
y
f(xy)=f(x)+f(y)
area=f(x)
Figure 5.3: Schematic deflnition of logarithmic functions [120].
the same factor, then the resulting region will have an area which is equal to that of the original
region", both regions have the same area. Thus, using the scaling argument we have:
f(x) = f(xy)? f(y); (5.5)
or equivalently,
f(xy) = f(x) + f(y); (5.6)
That is a functional equation for the family of logarithms. However, the theoretical work which
links this functional equation to the family of logarithms had to wait for the work of Augustin-
Louis Cauchy [120].
The subject of functional equations is more properly dated from the work of A. L. Cauchy,
born in 1789. The functional equation that is particularly associated with Cauchy is:
f(x+ y) = f(x) + f(y); (5.7)
for a real x and y, and is now called Cauchy?s equation. It is required to flnd all real-valued
functions f satisfying Equation (5.7), but any function of the form
f(x) = ax; (5.8)
can satisfled the Cauchy?s equation (always with a a real number).
Historically, Jean d?Alambert precedes Cauchy. However in the context of functional
equations, it seems more natural to consider his contribution after Cauchy. In his efiorts to
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 73
understand the principles of combinations of forces, d?Alambert was led to the equation:
g(x+ y) + g(x? y) = 2g(x)g(y); (5.9)
where 0 ? y ? x ? ?=2. This equation is called d?Alambert equation. Find the solution for
this equation is not easy. The Equation (5.9) is reminiscent of a trigonometric identity, thus we
need to look inside trigonometric functions to look for the solution [121]. Finally, the solution
to this equation has the form:
g(x) = b cos ax; (5.10)
for suitable chosen constant a; b. However, letting x = y = 0 in Equation (5.9) reduces it to the
equation g(0) = g(0)2, telling us that g(0) = 0 or g(0) = 1, that correspond with b = 0 or b = 1
respectively. The constant a turns out to be arbitrary.
To flnish with this review of the flrsts mathematicians who worked with functional equations,
we present next Charles Babbage. One property that both Cauchy?s and d?Alambert?s equa-
tions have in common is that, although they involve functions of a single variable, the equations
are formulated using two variables, namely x and y, but Babbage investigated other class of
completely difierent functional equations [120].
Babbage is the founder of modern computing and with a difierent engine. One of his inven-
tions is the cowcatcher, a remarkable device that was attached to the front of trains to remove
obstacles (such as cows) that might cause the train to be derailed. On 1815, one paper changes
the life of this mathematician. This paper deflnes some mathematical calculus about direct and
inverse functions. Babbage deflnes a set of functions that satisfled this calculus, thus, a set of
functional equations and their solutions increasing generality and complexity to the functional
equation theory.
5.3 Basic concepts and deflnitions
It is not easy to give a precise deflnition of functional equation. Castillo et al. in [46] deflne
some concepts to make the understanding of the problem easier.
Deflnition 5.1 Functional equation: In a broader sense, a functional equation can be consid-
ered as an equation which involves independent variables, known functions, unknown functions
and constants; but we exclude difierential equations, integral equations and other kinds of equa-
tions containing inflnitesimal operations. In our equations, the main operation is the substitution
of known or unknown functions into known or unknown functions.
Deflnition 5.2 System of functional equations: A system of functional equations is a set
of n ? 2 functional equations.
Deflnition 5.3 Domain of a functional equation: Given a functional equation, the set of
all values of the variables, on which it is supposed to hold, is called its domain (not to be confused
with the domain of deflnition of each known or unknown function appearing in it).
If the functional equation comes from a physical problem we can talk about its natural
domain, as the set of values of the variables with a physical sense.
Sometimes we flnd functional equations which are stated on a restricted domain, that is,
restricted when compared with their natural or initial domain. In this case two difierent names
have been proposed: \functional equations on restricted domains", [87], and \conditional func-
tional equations", [55].
74 5.3. BASIC CONCEPTS AND DEFINITIONS
It is interesting to point out that the domain of the functional equation can be independent
of the unknown functions or dependent on them.
Deflnition 5.4 Particular solution We say that a function or a set of functions is a particu-
lar solution of a functional equation or system if, and only if, it satisfles the functional equation
or system in its domain of deflnition.
Deflnition 5.5 General solution Given a class of functions F , the general solution of a
functional equation or system is the totality of particular solutions in that class.
To obtain the general solution of a functional equation (or system), the following considera-
tions must be taken into account:
1. The general solution of the functional equations can depend on one or more arbitrary
constants.
2. In addition to arbitrary constants, arbitrary functions can appear in the general solution.
Thus, an inflnite number of point conditions could be necessary to get a unique solution.
3. Unlike any other kind of equations, a single equation can determine several unknown
functions.
4. To have a well deflned equation, its domain of deflnition (integer, real, complex, etc.) and
the domains and ranges of the functions appearing in the functional equation or system
should be clearly established. It is important to mention that the general solution of a
given functional equation is strongly dependent on its domain of deflnition.
5. To have a well deflned equation, the class (continuity, measurability, difierentiability, in-
tegrability, etc.) of admissible functions should be given.
To solve a functional equation we have to solve three important points:
1. The equation E[f ;x] = 0.
2. Its domain D.
3. The class F of admissible functions (including domains and ranges).
Thus, a functional equation can be considered as a triplet (E;D;F). This means that we
are interested in all functions f 2 F such that
E[f ;x] = 0; 8x 2 D;
where f and x are the vectors of unknown functions and variables, respectively.
Sometimes it is interesting to compare the general solutions of the same functional equation
but with difierent domains and/or classes of admissible functions; that is, the solution of the
equation with restricted or enlarged domains or classes of admissible functions. In other words,
we want to compare the general solutions of
E [f ;x] = 0; 8x 2 D1; f 2 F1 (5.11)
and
E [f ;x] = 0; 8x 2 D2; f 2 F2: (5.12)
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 75
If the sets of all solutions of (5.11) and (5.12) are denoted by S1 and S2, respectively, we
have
F2 F1; and D1 D2; ) S2 S1: (5.13)
These implications are obvious because, on the one hand, an enlargement of the admissible
functions allows us an enlargement of the set of all solutions but never a reduction. On the
other hand, an enlargement of the equation domain implies a more restrictive equation because
this implies more restrictions on the unknown functions (conditions at more points); hence, the
set of all solutions may be reduced.
To continue solving the problem we have several methods, based in three steeps:
1. Enlarge the domain og the functional equation.
2. Find its general solution on the enlarged domain.
3. The last steep can be:
a.- Use the obtained solutions as particular solutions of the equation on the initial do-
main.
b.- Use the obtained solutions as particular solutions of the equation in the unrestricted
class.
c.- Find which of the solutions of (5.11) are true solutions of the equation with the initial
or unrestricted domain.
d.- Determine which of the solutions of (5.11) belong to the initial class.
Of course, many other methods can be used, too. A description of the main methods for
solving functional equations will be given in section 5.4.
Note that methods (a) and (b) lead to particular solutions of the given functional equation
for the initial domain and class. On the contrary, methods (c) and (d) allow the general solution
to be obtained in a more restrictive situation. So, the obtained solution may not actually be a
solution for the specifled domain and class and an additional test is required. In such a case, we
shall refer to them as candidate solutions. These concepts of general, particular and candidate
solutions can be used to obtain computer solutions, even under changes in the domain and/or
class of the equation.
Given a functional Equation (E;D;F), it is interesting to flnd triplets (E;D1, F1) and
(E;D2;F2) with D1 ? D ? D2 and F2 ? F ? F1, such that they have the same general
solution as the initial equation. In particular, one of the most interesting results is obtained
when the sets D1; D2;F1 and F2 cannot be improved; that is, when they lead to an optimum (a
minimum of restrictions) characterization of the general solution set.
Deflnition 5.6 Equivalent functional equations. Let
E1 [f ;x] = 0; 8 x 2 D; f 2 F1 (5.14)
E2 [f ;x] = 0; 8 x 2 D; f 2 F2 (5.15)
be two functional equations. We say that equations (5.14) and (5.15) are equivalent if and only
if their general solutions coincide.
For a detailed treatment of restricted domains see [13], [55] and [87].
76 5.4. SOME METHODS FOR SOLVING FUNCTIONAL EQUATIONS
5.4 Some methods for solving functional equations
There exist several methods to solve functional equations. Now, in this section, a sumary of
these methods are given.
Unlike the fleld of difierential equations, where a clear methodology to solve them exists,
in functional equations such a methodology does not exist. In fact, in many cases \ad hoc"
methods are required. This represents a great shortcoming and perhaps one of the reasons
explaining why engineers and applied scientists have not incorporated functional equations into
their daily work. To facilitate the use of functional equations we can:
? Elaborate a list including the main functional equations and their corresponding solutions.
? Identify the sets of equations which can be solved using the same methods.
The main methods for solving functional equations to be analyzed are [46]:
1. Replacement of variables by given values
2. Transforming one or several variables
3. Transforming one or several functions
4. Using a more general equation
5. Treating some variables as constants
6. Inductive methods
7. Iterative methods
8. Separation of variables
9. Reduction by means of analytical techniques (difierentiation, integration, etc.)
10. Mixed methods
5.4.1 Replacement of variables by given values
If we replace one or several variables appearing in the functional equation by carefully selected
values, some mathematical relations that give some of the unknown functions or simpler func-
tional equations can be obtained. This method requires a flnal check of the resulting solutions
because the previous replacement leads to equations associated with a set of necessary, but not
su?cient, conditions for the functions to be solutions of the initial equation.
Section 5.5.6 Euler?s equation represent a similar way to solve the functional equation [120].
Theorem 5.1 Homogeneous equations: The most general solution of the equation
f(yx) = ykf(x); x; y 2 R+; (5.16)
where f is a real function of a real variable and k is a given constant, is
f(x) = cxk; (5.17)
where c is an arbitrary constant.
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 77
With x = 1 in (5.16) we get f(y) = cyk, where c = f(1). This solution, taking into account
the commutativity of the product of real numbers, satisfles (5.16) and then (5.17) is proved.
Note that this proof only requires the existence of a unit element and the commutativity of the
product. Thus, the same solution is valid for many other domains and classes of functions [46].
5.4.2 Transforming one or several variables
By transforming one or several of the variables appearing in the functional equation we can
transform the given equations in others, the solutions of which are known.
Section 5.5.3 shown the Cauchy?s exponential equation, clear example of transformation of
several variables to solve a functional equation.
5.4.3 Transforming one or several functions
Similarly, we can transform one or several functions and get some equations with known solutions
[46].
5.4.4 Using a more general equation
Assume that we know the general solution of a functional equation with say n unknown functions.
Assume also that we are asked about the general solution of an equation which is a particular
case of the initial equation where some of the n functions are known. The general solution of
this new equation can be easily obtained by forcing the known functions to flt into the general
format solution of the starting equation. Some useful equations to be used in this group of
methods are
nX
k=1
fk(x)gk(y) = 0; (5.18)
F [G(x; y);H(u; v)] = K[M(x; u); N(y; v)]; (5.19)
G(x; y) = H[M(x; z); N(y; z)]: (5.20)
For more information about how can solve functional equation using this method see [46].
5.4.5 Treating some variables as constants
If, after considering as constants some of the variables appearing in a functional equation, we are
able to solve the resulting functional equation, then, by making the arbitrary constants and/or
the functions in the resulting general solution depend on those variables, we obtain the general
solution of the initial problem.
5.4.6 Inductive methods
The induction method allows us to solve some functional equations.
5.4.7 Iterative methods
Some techniques related to iterative methods are also useful to solve some functional equations.
78 5.4. SOME METHODS FOR SOLVING FUNCTIONAL EQUATIONS
5.4.8 Separation of variables
If we can force some variables to appear on the right hand side of the equation and some others
on its left hand side, then neither side must depend on the non common variables. This leads
to new and normally simpler functional equations.
These methods (Inductive and iterative method and separation of variable method) are
explained with more details in the bibliography (see [46], [120], [121] and [17]).
5.4.9 Reduction by means of analytical techniques
Some other useful techniques are [46]:
? Transformation of a functional equation into a difierential equation.
? Transformation of a functional equation into an integral equation.
? Finding the solution over dense sets and extrapolating solutions by continuity.
? Use of characteristic mappings and invariants.
An example of this type of functional equation is presented in section 5.5.7, with d?Alambert
equation.
5.4.10 Mixed methods
By mixed methods we understand a combination of the previous methods, as for example:
1. Multiple replacements.
2. Transforming variables and functions.
3. Replacements and changes of variables.
4. Replacements and changes of functions.
Theorem 5.2 Cauchy?s equation III: The most general solutions, which are continuous-at-
a-point, of the functional equation [46]
f(xy) = f(x) + f(y); x; y 2 T (5.21)
are
f(x) =
8
<
:
c log(x) if T = IR++;
c log(jxj) if T = IR ? f0g;
0 if T = IR :
(5.22)
For positive x and y, we can make the following change of variables
?u = log(x) , x = exp(u);
v = log(y) , y = exp(v); (5.23)
and get
f(euev) = f(eu+v) = f(eu) + f(ev); (5.24)
which is equivalent to
g(u+ v) = g(u) + g(v); (5.25)
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 79
where g(x) = f[exp(x)].
Thus, we obtain again a Cauchy?s equation (see section 5.5.1). So, under some mild regularity
conditions, we can write
g(x) = cx ) f(x) = c log(x); x 2 IR++: (5.26)
If Equation (5.21) is satisfled for y = 0, then f(0) = f(x) + f(0), which implies f(x) = 0 for all
x.
Finally, if Equation (5.21) is satisfled for all x 6= 0 and y 6= 0, then we have 2f(t) = f(t2) =
2f(?t) and then f(x) = f(?x) = c log(jxj).
5.5 Functional equations with two variables
In this section diverse functional equation with two variables are deflned [120].
5.5.1 Cauchy?s equation
Let us begin by restating and solving Cauchy?s equation. Let f : R ! R be a continuous
function satisfying
f(x+ y) = f(x) + f(y); (5.27)
for a real x and y. We show that there exists a real number a such that f(x) = ax for all x 2 R
(see [121] and [46]).
A special case of this is found by setting x1 = ::: = xn = x, then becomes
f(nx) = nf(x); (5.28)
for all positive integers n and for all real number x.
But what happen when a = 0?, in this case
f(y) = f(y + 0)
= f(y) + f(0): (5.29)
So f(0) = 0. Similarly, the case of a < 0 gives like a solution
0 = f(0)
= f(a+ (?a))
= f(a) + f(?a)
f(ax) = f(?(?a)x)
= af(x); (5.30)
for all real values of x and rational values of a.
We can summarize all these informations in two propositions:
Proposition 1 Let f : R ! R satisfy Cauchy?s equation
f(x+ y) = f(x) + f(y)
for all real values x and y. Then there exists a real number a such that
f(q) = aq
for all rational numbers q.
80 5.5. FUNCTIONAL EQUATIONS WITH TWO VARIABLES
Proposition 2 Suppose that f : R ! R and g : R ! R are continuos functions such that
f(q) = g(q) for all rational numbers q. Then f(x) = g(x) for all real number x.
5.5.2 Jensen?s equation
Jensen?s equation is of the form:
f
x+ y
2
?
= f(x) + f(y)2 ; (5.31)
that is a version of the Cauchy?s equation using averages [121].
The solution to the equation is easy to obtain. Let g(x) = f(x)? f(0), that is
g
x+ y
2
?
= g(x) + g(y)2 : (5.32)
If y = 0, knowing that Equation (5.32) is continuous, and substituting x by x+ y we obtain
g
x+ y
2
?
= g(x+ y)2 : (5.33)
Substituting this equation in Equation (5.32) and working in the simpliflcation of the equa-
tion obtained, the result is
g(x+ y) = g(x) + y(y); (5.34)
which is Cauchy?s equation (section 5.5.1).
5.5.3 Cauchy?s exponential equation
This equation has the form
f(x+ y) = f(x)f(y); (5.35)
where f : R ! R, satisflying
f(x) = exp(cx) and f(x) = 0; (5.36)
where c is an arbitrary constant.
Replacing x and y by t=2 in (5.36) we obtain
f(t) = f
t
2
?2
) f(t) ? 0; 8t:
Now, if f(t0) = 0 for some t0 then f(t) = f(t ? t0 + t0) = f(t ? t0)f(t0) = 0 for all t. Thus,
either f(t) > 0 for all t, or f(t) ? 0.
If f(t) > 0 for all t, then we can take logarithms on both sides in (5.35) and get
log f(x+ y) = log [f(x)f(y)] = log f(x) + log f(y);
which, using the notation g(x) = log f(x), leads to
g(x+ y) = g(x) + g(y);
which is Cauchy?s equation with solution
g(x) = cx:
Thus, we flnally get
f(x) = exp(cx):
Following the steps realized in [121], the conclusion is that there exist a real number b < 0
such that f(x) = bx for all real x.
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 81
5.5.4 Pexider?s equation
Pexider?s equation has the form
f(x+ y) = g(x) + h(y): (5.37)
We need to discover the solutions for all the functions, f; g; h : R ! R satisfying Equation
(5.37) for all real number x and y [121].
This equation is a generalization of Cauchy?s equation. Putting y = 0 and setting h(0) = b,
similarly putting x = 0 and g(0) = a, we have
f(x+ y) = f(x) + f(y)? a? b: (5.38)
Let f0(z) = f(z)?a?b for all real z, f0 satisfles f0(x+y) = f0(x)+f0(y), which is Cauchy?s
equation. So, the solution for f0 is f0(z) = cz, and the solution to Pexider?s equation is
f(z) = cz + a+ b
g(x) = cx+ a
h(y) = cy + b: (5.39)
5.5.5 Vincze?s equation
This equation is considered a generalization of Pexider?s equation. Consider the function f; g; h
f(x+ y) = g(x)k(y) + y(y) (5.40)
for a real x and y [121].
Taking k(0) = a and h(0) = b. If a = 0, f is a constant function, but in this case consider
a 6= 0. Putting y = 0 in Equation (5.40) we get
g(x) = f(x)? ba : (5.41)
Deflning `(y) = k(y)=a and ?(y) = h(y) ? bk(y)=a, and substitutions of `(0) = 1 and
?(0) = 0, leads to the flnal equation
(`(y)? 1)?(x) = (`(x)? 1)?(y); (5.42)
where ?(y) = f(y)? f(0).
For the complete analysis of Vincze?s equation see [121].
5.5.6 Euler?s equation
The Euler?s equation is characteristic for its property of homogeneity (see [121] and [46]). Let
k be a real number, the equation is
f(tx; ty) = tkf(x; y) (5.43)
A function f(x) satisfying a Euler?s equation is called homogeneous function of degree k. In
table 5.1 some examples of these type of equations are shown.
82 5.5. FUNCTIONAL EQUATIONS WITH TWO VARIABLES
Table 5.1: Some example of homogeneous functions.
Function Degree
f(x; y) = (x+ y)=2 1
f(x; y) = x=y 0
5.5.7 D?Alambert?s equation
D?Alamber?s equation is:
f(x+ y) + f(x? y) = 2f(x)g(y); (5.44)
where
f(x) = 1; f(x) = 0; f(x) = cosh(Bx); f(x) = cos(Bx); (5.45)
and B is an arbitrary constant.
To solve D?Alembert?s equation we initially set y = 0 and then x = 0 to obtain
f(0) = 1 or f(x) = 0 and f(y) = f(?y): (5.46)
Then we difierentiate twice with respect to y and set y = 0 and we get
f 00(x) = kf(x) ) f(x) =
8
<
:
a cosh (pkx) + b sinh(pkx) if k > 0;
a+ bx if k = 0;
a cos (p?kx) + b sin(p?kx) if k < 0;
(5.47)
where we have made k = f 00(0). Using now (5.46) we get a = 1 and b = 0. Thus, the general
difierentiable solution of (5.47) becomes
f(x) =
8
>>><
>>>:
cosh(pkx) if k > 0;
1 if k = 0;
0
cos(p?kx) if k < 0:
(5.48)
The derivation of this equation is complex, we can flnd the solution in books such as [121].
flgure 5.4 shows some of the functions that can be found as a solutions to d?Alambert?s equations
in this section. The example corresponds to cosnx and coshnx for n = 1; 2; 3.
5.5.8 Equations involving functions of two variables
One way to extend Cauchy?s equation is to consider functions of two or more variables [121].
Suppose f(x; y) is a real function dependent of two variables x and y, which satisfles
f(x1 + x2; y) = f(x1; y) + f(x2; y); (5.49)
f(x; y1 + y2) = f(x; y1) + f(x; y2); (5.50)
where x1; x2; y1 and y2 are real values. If f is also a continuous function, when y is flxed, the
function f can be written in the form
f(x; y) = c(y)x; (5.51)
Entering in Equation (5.50), we get
c(y1 + y2)x = c(y1)x+ c(y2)x; (5.52)
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 83
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3
x
y
cos x
cos 2x
cos 3x
cosh x
cosh 2x
cosh 3x
Figure 5.4: Some solutions of D?Alambert?s equation. The functions presented are cosnx and
coshnx for n = 1; 2; 3 [121].
Making x = 1 we obtain the Cauchy?s equation. So, the conclusion is that \there exists some
real number c0 such that c(y) = c0y for all y" [121]
f(x; y) = c0xy: (5.53)
for a real x and y.
5.6 Functional equations with one variable
5.6.1 Basic families of equations
The simplest families of functional equations with one variable that we can write has the form
f(x) = f(fi(x)); (5.54)
for real x values, where fi : R ! R is a specifled function [121].
There exist two difierent ways of analyzing this equation, assuming that f is a continuous
function or not. If we do not assume that f is continuous, the solution for the equation is the
family of funtions
fi1 = fi; fin+1(x) = fi(fin(x)); (5.55)
for n = 1; 2; :::. Assuming that fi0(x) = x, the solution of the problem is the sequence
f(x) = f(fin(x)): (5.56)
84 5.6. FUNCTIONAL EQUATIONS WITH ONE VARIABLE
However, if f is continuous function, the limit of the equation gives us
f(x0) = f( limn!1 fi
n(x))
= limn!1 fi
n(x)
= f(x) (5.57)
for all x. The conclusion is that f must be constant, and to avoid problems, x0 must to be in
the domain of f .
For the complete derivation of this equation see [121].
5.6.2 Conjugate equations
As Equation (5.54) there are one-to-one functions which are solutions to it. In this subsection
some examples are presented.
The equation
f(fi(x)) = sf(x); (5.58)
is called Schro?der0s equation. We assume that fi(x) = x has not solutions on some interval
on which the function f(x) is to be constructed. With this, we avoid the problem about the
domain of deflnition of any solution in Equation (5.58). To solve the problem we can use difierent
methods, but here we present a kind of inverse equation to Scho?der0s equation.
Let f be a solution to Equation (5.58), g = f?1 such that f(x) = y if and only if g(y) = x.
g(sx) = fi(g(x)): (5.59)
Equation (5.59) is known as Poincare?s equation.
The equation
f(fi(x)) = f(x) + a; (5.60)
where a is a real number (nonzero), is called Abel?s equation. As in Scho?der0s equation, we
need to deflne the domain of f very carefully.
The equation
f(fi(x)) = (f(x))p; (5.61)
is called Bo?ttcher0s equation. The value of p 6= 1. This equation is used to analyze nonnegative
functions of f(x).
5.6.3 Functional equations and nested radicals
A nested radical function,
f(s) =
r
1 + x
q
1 + (x+ 1)p:::; (5.62)
can be solved squaring both sides
(f(x))2 = 1 + xf(x+ 1); (5.63)
where f(x) ? 0.
This equation does not flt into any family of equations studied above. To determine the
solution we need to guess it. If we deflne f(x) which a polynomial equation, that is f(x) = ax+b,
we get
(ax+ b)2 = 1 + x(ax+ a+ b); (5.64)
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 85
that is true for all x. Taking f(x) = x+1, a = 1; b = 1, and bounding both sides of the equations
[121], we have
x+ 1
2 ? f(x) ? 2(x+ 1): (5.65)
5.7 Functional equations in probability theory
The analysis of functional equations in probability theory is an important subject in engineer
problems. In this section several results on the Cauchy functional equation and on distribution
functions are analyzed [17].
5.7.1 Integrated Cauchy functional equations on R+
An extension of the classical Cauchy?s equation is the integral equation
Z
S
(f(x+ y)? f(x)f(y))d (y) = 0; 8 x 2 S; (5.66)
where f : s ! R, S is a semigroup of R, and is a positive measure of S. If RS(f(y)d (y)) is
nonzero and flnite, the Equation (5.67) reduces to
f(x) =
Z
S
f(x+ y)d (y) = 0; 8 x 2 S; (5.67)
where is related to . This equation, with f and positive, appear repeatedly in analytical
probability theory. We call these type of equations as integrated Cauchy functional equation
(ICFE), and normally are written as f = f ? on S for simplicity. The aim of this subsection
is to shown some applications of the solution of the ICFE.
The lack of memory property of the exponential and geometric laws
Consider a real variable X with an exponential distribution function, the lack of memory prop-
erty is
P [X > y + xjX > y] = P [X > x]; 8x; y ? 0: (5.68)
In the same way, if this property holds, let T = 1?F , where F is the distribution function of
X. T satisfles the Cauchy?s equation, so F is a exponential distribution function. If we replace
y > 0 in the Equation (5.68) by a real variable Y > 0, assuming that P [X > Y ] > 0, we obtain
P [X > Y + xjX > Y ] = P [X > x]; for x ? 0: (5.69)
If G is the distribution function of Y , then
cT (x) =
Z
[0;1)
T (x+ y)dG(y); 8x ? 0; (5.70)
where c = PX > Y .
5.7.2 Integrated Cauchy functional equation with error terms on R+
Similarly to Equation (5.68), the ?? ICFE( ; S) equations are deflned by
f(x) =
Z 1
0
f(x+ y)d (y) + S(x); 8x ? 0; (5.71)
where the error is S and is such that jS(x)j ? Ce??x for all x ? 0.
86 5.8. APPLICATIONS TO SCIENCE AND ENGINEERING
Characterization of the Weibull distribution
In this subsection we want to present the steps to obtain the Weibull distribution and its
characterization.
Consider g be a bounded, real value, continuous function on [0;1), that satisfy the next
relation for some values of fi > 0
exp e?fixg(x) =
Z
!
exp[e?fix
Z 1
0
g(x+ y)u(dy;w)]dP (w); (5.72)
where (!;B; P ) is a probability space in R, and g is inside of the bounds.
For more information about this type of equations see [17].
5.8 Applications to science and engineering
In this section some examples of application of functional equations to science and engineering
are shown. These applications solve problems of difierent flelds of science, like tra?c, network,
fatigue, etc.
5.8.1 A statistical model for lifetime analysis
In many practical engineering situations, the lifetime variable, T , appears as a random variable
which depends on one regressor variable, X. This is, for example, the case of the fatigue life of
wires, strands or tendons, the time up to breakdown in solid dielectrics or the time up to failure
of marine breakwaters, which depend on the regressor variables stress range, voltage stress or
wave height, respectively [46].
As a consequence, a regression model as shown in flgure 5.5, could be a convenient approach
to the problem. The model is completely established as soon as the cumulative conditional
distribution function of lifetime, F (t; x), is deflned for every value x of X.
X
xi
No fatigue failure zone
T
xj
Threshold curve
Figure 5.5: Regression model. flgure from [46]
Two difierent ways in which the engineer can tackle the problem are:
1. Using standard linear regression models in order to flt the experimental data.
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 87
2. Creating adequate models not only to flt the experimental data but also to satisfy physical
and theoretical considerations.
By the flrst approach we mean the use of ready-made regression models, i.e., models not specially
designed for the problem under consideration, but very well recognized by statisticians and
experienced engineers. This is the most generally accepted approach because of its simplicity,
its widespread use and the possibility of performing many standard and simple analysis, such
as confldence limit analysis for example.
In the cases in which the application of the flrst approach is not satisfactory, the engineer
tries to develop tailor{made regression models. These models can either re ect his experience
and feeling about the problem or be based on physical and theoretical considerations.
In the following paragraphs, we derive a statistical model for lifetime analysis related to the
weakest link principle with a wide applicability to engineering problems.
This division into two approaches can sound artiflcial to the reader and in some way it is
artiflcial, in the sense that methods in the second group can go into the flrst as soon as they
are investigated, experimented and widely recognized. This makes it di?cult to deflne a clear
cut between the two groups. However, the main distinction between the two groups is that in
the flrst case the engineer only wants to adjust the trend of the experimental data but in the
second, he also wants to satisfy some extra physical conditions such as compatibility, feasible
range, stability, etc., which must hold for the problem under consideration.
In the following paragraphs, the second approach will be used to derive a statistical model
for lifetime analysis related to the weakest link principle with a wide applicability to engineering
problems.
Derivation of the fatigue model: The following assumptions are made [37]:
1. Weakest link principle: This principle establishes that the fatigue lifetime of a longitu-
dinal element is the minimum fatigue life of its constituent pieces.
2. Stability: The selected distribution function type must hold (be valid) for difierent spec-
imen lengths.
3. Limit behavior: To include the extreme case of the size of the supposed pieces consti-
tuting the element going to zero, or the number of pieces going to inflnity, it is convenient
for the distribution function family to be an asymptotic family (see [70] and [33]).
4. Limited range: Experience shows that the lifetime T and the stress range X, have a
flnite lower end, which must coincide with the theoretical lower end of the selected cdf.
This implies that the Weibull distribution is the only one satisfying requirements 1 to 4.
5. Compatibility: In the X{T fleld, the cumulative distribution function of the lifetime given
stress range, F (t;x), should be compatible with the cumulative distribution function of
the stress range given lifetime, E(x; t).
These conditions lead to the following functional equation
F (t; x) = 1? exp
(
?
? t? t(x)
fit(x)
?flt(x))
= 1? exp
(
?
?x? x(t)
fix(t)
?flx(t))
= E(x; t);
(5.73)
88 5.8. APPLICATIONS TO SCIENCE AND ENGINEERING
where t(x), fit(x) and flt(x) are the location, scale and shape parameters of the Weibull laws for
given x, and x(t), fix(t) and flx(t) are the location, scale and shape parameters of the Weibull
laws for given t.
Expression (5.73) is equivalent to the functional equation
? t? t(x)
fit(x)
?flt(x)
=
?x? x(t)
fix(t)
?flx(t)
: (5.74)
[43] have shown that the only feasible solutions of Equation (5.73) are the three models:
? Model 1 (see flgure 5.6):
F (t; x) = 1? e?[(t?A)(x?B)=C+D]E : (5.75)
Lifetime T
?
?
p = 0.95
p = 0.90
p = 0.50
p = 0.10
St
ress
L
ev
el
X
p = 0
Figure 5.6: Wo?hler fleld of model 1. flgure from [46]
? Model 2 (see flgure 5.7):
F (t; x) = 1? e?[C(t?A)E(x?B)D]: (5.76)
Lifetime T
?
?
p = 0.95
p = 0.90
p = 0.50
p = 0.10
p = 0St
ress
L
ev
el
X
Figure 5.7: Wo?hler fleld of model 2. flgure from [46]
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 89
? Model 3:
F (t; x) = 1? e
?
?
C(t?A)E(x?B)DeF log(t?A) log(x?B)
?
; (5.77)
where A,B, C;E > 0, D and F are arbitrary constants.
5.8.2 Statistical models for fatigue life of longitudinal elements
One of the most important problems when dealing with the statistical analysis of the fatigue
life of longitudinal elements is the size efiect; that is, the in uence of length on the survivor
function [46].
By longitudinal element we understand an element satisfying the following two conditions:
? only one dimension is important in the behavior of the element and
? if the element is longitudinally divided into imaginary pieces (see flgure 5.8) all pieces are
subject to the same external action (stress, force, etc.)
Several models have been given in the past to solve this problem, but, unfortunately, most of
them are based on the assumption of independence of the fatigue life of non-overlapping pieces.
This assumption states that if an element of length s, such as that shown in flgure 5.8, is
hypothetically divided into several pieces of lengths s1; s2; :::; sn, then the survivor function of
the element S(s; z) must satisfy the equation
S(s; z) =
nY
i=1
S(si; z):
s
s s s1 2 n
Figure 5.8: Illustration of the hypothesis of independence. flgure from [46]
Here we shall abandon the independence assumption and, making use of the functional
equations theory, we shall state the problem in a very difierent way.
We shall assume here that a team of three members is required to design a consensus model
for the analysis of the fatigue life of longitudinal elements. However, they are required to
give separate proposals before joining together and reaching a consensus. The three proposals
associated with the three members will be denoted by models 1, 2 and 3, respectively.
Model 1
For the sake of simplicity we assume n = 2, that is, the element of length x + y is divided
into two non-overlapping pieces of lengths x and y. We also assume that there exists a function
S(x; z) that gives the survivor function of a piece of length x and that the survivor function of
90 5.8. APPLICATIONS TO SCIENCE AND ENGINEERING
the element can be calculated in terms of that of the two pieces. In other words, S(x; z) must
satisfy the following functional equation
S(x+ y; z) = H[S(x; z); S(y; z)]; (5.78)
where the function H indicates how the survivor function of the element can be obtained from
those of the pieces [46].
It is worthwhile mentioning that Equation (5.78) implies the associativity and commutativity
character of the H function and the dependence of the survivor function S on the total length
of the element. In fact we can write
S(x+ y + z; t) = H[S(x+ y; t); S(z; t)]
= H[H[S(x; t); S(y; t)]; S(z; t)]
= H[S(x; t); S(y + z; t)]
= H[S(x; t);H[S(y; t); S(z; t)]]
and
S(x+ y; t) = H[S(x; t); S(y; t)] = S(y + x; t) = H[S(y; t); S(x; t)]:
Thus, the survivor function of an element of length s is independent of the number and size of
the sub-elements into which it is divided in order to calculate it, using (5.78).
In the following paragraphs we solve functional Equation (5.78) in two difierent forms.
The functional Equation (5.78) is a particular case of the functional equation
S[G(x; y); z] = H[M(x; z); N(y; z)]; (5.79)
with M = N = S and G(x; y) = x+ y.
Is easily satisfy all regularity conditions, because S(x; 0) = 1 and we can choose families of
survivor functions such that S1(x; c) 6= 0 and functions H such that H1 6= 0 and H2 6= 0. The
general solution of (5.79) is
S(x; z) = l[f(z)g?1(x) + fi(z) + fl(z)];
G(x; y) = g[h(x) + k(y)];
H(x; y) = l[m(x) + n(y)];
M(x; z) = m?1[f(z)h(x) + fi(z)]
N(x; z) = n?1[f(z)k(x) + fl(z)]
(5.80)
where g; h; k; l;m and n are arbitrary strictly monotonic continuously difierentiable functions
and f; fi and fl are arbitrary continuously difierentiable functions.
Thus, for Equation (5.78) we have
S(x; z) = l[f(z)g?1(x) + fi(z) + fl(z)]
= m?1[f(z)h(x) + fi(z)]
= n?1[f(z)k(x) + fl(z)]
g[h(x) + k(y)] = x+ y;
(5.81)
from which
g?1(x+ y) = h(x) + k(y);
which is Pexider?s equation I with the general continuous-at-a-point solution (see [46] chapter
4)
g?1(x) = Ax+B + C; h(x) = Ax+B; k(x) = Ax+ C:
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 91
With this, expressions (5.81) become
S(x; z) = l[f(z)(Ax+B + C) + fi(z) + fl(z)]
= m?1[f(z)[Ax+B] + fi(z)]
= n?1[f(z)[Ax+ C] + fl(z)];
and making Af(z)x = u we obtain
S(x; z) = l[u+ (B + C)f(z) + fi(z) + fl(z)]
= m?1[u+Bf(z) + fi(z)]
= n?1[u+ Cf(z) + fl(z)];
(5.82)
and, we get
u = cu+ a ) c = 1; a = 0;
l(x) = m?1
x? a? b
c
?
= m?1(x? b)
(B + C)f(z) + fi(z) + fl(z) = Bf(z) + fi(z) + b ) fl(z) = b? Cf(z):
Then, (5.82) becomes
S(x; z) = m?1[u+Bf(z) + fi(z)] = n?1(u+ b);
which implies
u = c1u+ a1 ) c1 = 1; a1 = 0;
m?1(x) = n?1
x? a1 ? b1
c1
?
= n?1(x? b1);
Bf(z) + fi(z) = c1b+ b1 = b+ b1 ) fi(z) = b+ b1 ?Bf(z);
and flnally we get the desired solution
Model 1: The general solution of (5.78) is:
S(x; z) = w[f(z)x]; H(x; y) = w[w?1(x) + w?1(y)];
where we have made w?1(x) = [n(x)? b]A .Due to the weakest link principle and because S(x; z) is a survivor function, it must be
non-increasing in z and x. Then, in addition, we must have
S(x; 0) = 1 ) w[f(0)x] = 1 )
8
<
:
[f(0) = 0 ; w(0) = 1]; or
[f(0) = 1 ; w(1) = 1]; or
[f(0) = ?1 ; w(?1) = 1]
:
S(x;1) = 0 ) w[f(1)x] = 0 )
8
<
:
[f(1) = 0 ; w(0) = 0]; or
[f(1) = 1 ; w(1) = 0]; or
[f(1) = ?1 ; w(?1) = 0]
:
If w(x) = exp(Dx) we get the model of independence.
The structure of the function H reveals its above mentioned associative and commutative
character.
92 5.8. APPLICATIONS TO SCIENCE AND ENGINEERING
We can solve (5.78) in a much easier way if we observe that the variable z plays the role of
one parameter, i.e., for any flxed value of z, Equation (5.78) can be written in the form
S(x+ y) = H[S(x); S(y)];
and due to the associative character of H we can write
H(x; y) = w[w?1(x) + w?1(y)];
and its substitution into (5.78) leads to
S(x+ y; z) = wfw?1[S(x; z)] + w?1[S(y; z)]g )
) G(x+ y; z) = G(x; z) +G(y; z);
with G(x; z) = w?1S(x; z), which, for z held constant, is Cauchy?s Equation (4.7) and then:
G(x; z) = f(z)x ) S(x; z) = w[f(z)x]:
Model 2
Member 2 in the team wants to start from the following result : [26] based on some experi-
mental results of [108], suggest the following model for the survivor function
S(x; z) = S(y; z)N(y;x); (5.83)
where S(x; z) and S(y; z) are the survivor functions associated with two elements of lengths x
and y, respectively, and N(y; x) is an unknown function.
30
60
90
Element length
Figure 5.9: Experimental and theoretical survivor functions for lengths 30, 60 and 90 cm. (from
[26]).
flgure 5.9 shows the experimental survivor functions and those obtained using model (5.83)
(see [26]).
Note that (5.83) is an implicit function of S(x; z), or in other words, it is a functional
equation. Thus, it must be solved to know what the Bogdanofi and Kozin proposal is. [34]
showed that the only compatible functions for N(y; x) are those of the form N(y; x) = q(x)q(y) [46].
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 93
We know
S(x; z) = p(z)q(x); N(y; x) = q(x)q(y) : (5.84)
For S(x; z) to be a survivor function it must be non-increasing in z and we must have
S(x; 0) = 1 ) p(0)q(x) = 1 ) p(0) = 1;
S(x;1) = 0 ) p(1)q(x) = 0 ) p(1) = 0:
If q(x) = x we get the model of independence.
The hazard function associated with S(x; z) is
h(x; z) = ?p
0(z)
p(z) q(x) = ?[log p(z)]
0q(x) = s(z)q(x);
which shows that Model 2 is the Cox{proportional hazards model (see [54]). Thus, functional
Equation (5.83) characterizes the proportional hazards Cox-model.
Model 3
Member 3, based on expression (5.83), assumes that the survivor function of one element
of length x can be obtained from the survivor function of one element of length y and a given,
but unknown, function of x and y. In other words he assumes that the survivor function must
satisfy the functional equation
S(x; z) = K[S(y; z); N(x; y)]; (5.85)
which is a particular case of
G(x; y) = K[M(x; z); N(y; z)];
with S(x; y) = G(y; x) = M(y; x).
If we choose F and N to be invertible with respect to their flrst argument and K invertible
with respect to its flrst argument for a flxed value of the second, then the regularity conditions
hold and the general solution of the last equation is:
G(x; y) = f?1[p(x) + q(y)]; K(x; y) = f?1[l(x) + n(y)];
M(x; y) = l?1[p(x) + r(y)]; N(x; y) = n?1[q(x)? r(y)]; (5.86)
and then, for (5.85) we must have
S(x; z) = f?1[p(z) + q(x)]
= l?1[p(z) + r(x)];
K(x; y) = f?1[l(x) + n(y)];
N(y; z) = n?1[q(y)? r(z)];
(5.87)
implies
p(z) = cp(z) + a ) c = 1; a = 0;
f?1(x) = l?1
x? a? b
c
?
= l?1(x? b);
q(x) = cr(x) + b = r(x) + b;
and then, from Expression (5.87), model 3 becomes
94 5.8. APPLICATIONS TO SCIENCE AND ENGINEERING
Model 3 : S(x; z) = l?1[p(z) + r(x)];
K(x; y) = l?1[l(x) +m(y)];
N(x; y) = m?1[r(x)? r(y)];
(5.88)
where we have made m(x) = n(x)? b.
For S(x; z) to be a survivor function it must be non-increasing in z and we must have
S(x; 0) = l?1[p(0) + r(x)] = 1 )
8
<
:
[l(1) = p(0) = ?1]; or
[l(1) = p(0) = 1]; or
[r(x) = l(1)? p(0)]
:
S(x;1) = l?1[p(1) + r(x)] = 0 )
8
<
:
[l(0) = p(1) = 1]; or
[l(0) = p(1) = ?1]; or
[r(x) = l(0)? p(1)]
:
If l?1(x) = exp[D exp(Cx)] we get the model of independence.
One important aspect to point out here is that Equation (5.85) is more than a simple
generalization of Equation (5.83). In fact, it includes some extra compatibility conditions, in
the sense that no arbitrary N(y; x) is admissible in Model 3, even though, initially, the function
N(y; x) seems to be arbitrary. In order to prove this, we show that the only admissible N
functions are those appearing in Model 2 (Expression (5.84)) [46].
Let us assume that function K is that implied from Equation (5.88), that is
K(x; y) = xy = l?1[l(x) +m(y)] ) l(xy) = l(x) +m(y);
which, by making the change of variable u = log(x), can be written
l[exp(uy)] = l[exp(u)] +m(y):
This is a Pexider functional equation with solution
l(x) = c log[fi log(x)]; m(y) = c log(y):
Thus, we flnally get
N(x; y) = m?1[r(x)? r(y)] = exp
?r(x)? r(y)
c
?
= q(x)q(y) ;
q(x) = exp
?r(x)
c
?
;
which is model 2.
Reaching a consensus
In the second and flnal step the team is required to join and reach a consensus.
Normally, a consensus solution is understood as a linear combination of the quantitative
judgments of several individuals. However, in many cases the consensual solution reached does
not satisfy many of the properties that were satisfled by the solutions in the proposals given by
the difierent individuals (see [72]). This fact is irrelevant when one tries to use the consensus
model to make some evaluations, such as to calculate some probabilities, for example, but
becomes a very serious inconvenience when one tries to model a physical system. In fact, the
functional equations (5.78), (5.83) and (5.85) state some properties, which the difierent members
CHAPTER 5. USE OF FUNCTIONAL EQUATIONS 95
understand the physical system must satisfy. Thus, any member would not accept models
violating his/her associated functional equation. Thus, in the following, we shall understand
consensus as the intersection of the three families of models, if it exists, i.e., as models satisfying
all the requirements. We start by analyzing the common part of Models 1 and 2 (see the
corresponding equations).
S(x; z) = w[f(z)x] = p(z)q(x);
which implies
log[S(x; z)] = log fw[f(z)x]g = q(x) log[p(z)];
and making the change of variable u = f(z) we get
logfw[ux]g = q(x) logfp[f?1(u)]g;
which is Pexider?s functional Equation with the general continuous-at-a-point solution (see [46],
chapter 4)
logf[w(x)]g = ABxC ; q(x) = AxC ; logfp[f?1(x)]g = BxC :
Thus,
S(x; z) = exp
n
AB[f(z)x]C
o
; w(x) = exp
?
ABxC
?
;
p(z) = exp[BfC(z)]; q(x) = AxC ; (5.89)
which shows that Models 1 and 2 are not coincident but they share the common model
S(x; z) = exp
n
AB[f(z)x]C
o
? S(x; z) = exp[f(z)x]C = fl(z)xC ; (5.90)
where fl(z) is an arbitrary positive function.
Model (5.90) for C = 1 becomes the model of independence.
The hazard function for this model is
h(x; z) =
dfl(z)
dz
fl(z) x
c = ?dflog[fl(z)]gdz x
c = s(z)xc:
If now we look for the common part of Models 1 and 3 we get the functional equation
S(x; z) = l?1[p(z) + r(x)] = w[f(z)x];
which, by making the change of variable u = f(z), becomes Pexider?s Equation (5.37)
l[w(ux)] = p[f?1(u)] + r(x);
with the general continuous-at-a-point solution
l[w(x)] = A log(BCx); p[f?1(x)] = A log(Bx); r(x) = A log(Cx);
and then we flnally get
w(x) = l?1[A log(BCx)]; f?1(x) = p?1[A log(Bx)]; r(x) = A log(Cx);
which shows that Model 1 is a particular case of Model 3.
Finally, we compare Models 2 and 3. For the coincidence we must have
S(x; z) = l?1[p1(z) + r(x)] = p(z)q(x);
96 5.8. APPLICATIONS TO SCIENCE AND ENGINEERING
and taking logarithms we get
q(x) log p(z) = logfl?1[p1(z) + r(x)]g
which implies
lfexp[q(x) log p(z)]g = p1(z) + r(x);
which is Pexider?s Equation. Thus, we have
l[exp(x)] = A log(BCx);
r[q?1(x)] = A log(Bx);
p1
'p?1[exp(x)]? = A log(Cx);
and then
l?1(x) = exp [exp(x=A)=(BC)] ;
q(x) = exp (r(x)=A) =B;
p(z) = exp [exp (p1(z)=A) =C] ;
which shows that Model 2 is a particular case of Model 3.
Thus, we can conclude that a consensus model could be model (5.90), which is the family of
models common to all three members of the team.
flgure 5.10 shows the required separate and consensus proposals as well as the common
proposals associated with all three groups of only two members.
3
21
p(t)
q(x)
w[f(t)x]
(t)? xc
l [p(t)+r(x)]-1
Figure 5.10: Illustration of separate and consensus proposals. Image from [46]
As a flnal conclusion, we can add that functional equations can prove themselves to be a
very powerful tool to be used in model design. As a matter of fact, the engineer can state all
the conditions to be satisfled by the desired model in terms of functional equations. Then, by
flrst solving the resulting system and then in terms of its general solution, one can make the
selection by playing with the remaining degrees of freedom.
Part III
Theoretical Contributions. Proposed
Model
97
Chapter 6
The Weibull and Gumbel S{N Field
Stress Based Fatigue Models
In this Chapter, the model which gives title to this doctoral thesis is presented.
Its derivation is based on the application of functional equations, studied in the previous
chapter, and in the analysis of compatibility, statistical and physical conditions.
6.1 Introduction
The limitations of the models currently used in the fatigue design based on Wo?hler curves (see
Section 4.2 and table 4.1) are derived by the problem of characterizing to fatigue the material
carrying out tests under constant stress levels.
These models are applicable for a given constant stress level but fail to predict the fatigue
behavior when difierent stress levels are considered. Empirical mean stress equations, as those
of Goodmann [75], modifled Goodman (see Smith [122]), Soderberg [125], Morrow [96] and
[98], Haibach ([77], Smith-Watson-Topper [123] and Walker [129], are currently applied for
this transformation [59]. All of them refer lastly to the fatigue material properties related to
tests performed for a reference stress range or stress level, and difierent quality flttings to the
experimental results have been achieved according to the material and the state, notched or
un-notched, of the structural member.
The author of this doctoral thesis and her supervisors, based on the compatibility conditions
of the Wo?hler fleld together with statistical and physical conditions and solving a system of
functional equations, proposed for the flrst time a general fatigue regression model that includes
the consideration of the mean efiect, without the need of resorting to empirical relations, flrstly in
a simple 6-parameter Weibull model version [36], and later in an extended 9-parameter Weibull
model version [42], which parameter evaluation required a relatively simple test strategy. Even if
these two models represented a relevant advance in understanding the efiect of the mean stress,
its application was limited only to positive stress ranges R, a case frequently present in some
structural elements. However, since load spectra, to which mechanical and structural elements
are generally subject in practice, can include a signiflcant portion of cycles under compressive
99
100 6.1. INTRODUCTION
minimum stress, it implies an important limitation for the practical application of this model.
Particularly, the consideration of testing under fully reversed stress, a standard choice for many
laboratories, remained excluded.
In this work, a Weibull regression model for statistical analysis of stress life data for any
possible loading situations in tension and compression is developed that is later transformed in
a Gumbel model, as a revision of the previous version of the fatigue model, thus facilitating its
application to real loading spectra. The model enforces the compatibility condition of the Wo?hler
flelds associated with constant min and constant max, that leads to a system of functional
equations, which solution provides a model with the desired requirements.
The model depends on 8 parameters that can be estimated by maximum likelihood or non-
linear regression methods, and supplies all the material basic probabilistic fatigue information
to be used in a damage accumulation assessment for fatigue life prediction using practical load
spectra.
The main achievements of the model proposed are:
? According to the Buckingham?s theorem [31], only dimensionless variables are used in the
model and the corresponding regression equation. This implies on one hand less variables
involved in the problem, i.e., a simpler but not less powerful model and, on the other hand,
that the parameters or constants resulting in the model are also dimensionless, that is,
their values are independent of the units being used.
? The model is not based on arbitrary assumptions, but on sound physical and statistical
properties exigeable to any fatigue model. Thus the model is the only one resulting from
the selected constraints.
? The model provides useful statistical information including not only mean values but also
variability of the model, and permits calculating probabilities.
Concerning the testing strategy, in the experimental common practice, material fatigue char-
acterization consists of selecting several characteristic stress ratios R, ranging from ?1 to 0:50
or even higher positive ratios, covering the region of interest. This ensures the applicability of
the empirical stress level equation fltted to the data, that is obtained, for instance, by regression
[2]. The model proposed enables us the consideration of other possible testing strategies due to
its capability of evaluating data resulting from tests carried out for any constant max, min,
mean or stress ratio R. In fact, tests can be repeated for a certain set of flxed stress levels, the
only requirement being to cover a su?cient broad range of stress levels in order to guarantee
reliability in the parameter estimation.
Once the basic material fatigue information related to the Wo?hler fleld is supplied for any
possible combinations of max and min, a model is needed to perform a damage assessment
allowing for life prediction. In this case, the model provides probabilistic bases for calculating
the damage accumulation for any type of loading being considered. In fact, due to the possible
identiflcation of the probability of failure, represented by the percentile curves in the Wo?hler
fleld for any max and min with the damage state [42] the model can be used in cumulative
damage calculations for fatigue life prediction of components subject to even complex loading
histories. This issue is being the main objective of a joint research program launched presently
at the Empa - Du?bendorf, Switzerland. The experimental validation of the model (see Chapter
7) is based on data obtained from a set of experiments realized in this research institute.
The model is based in the works developed by Castillo (see [34] and [44]), Fern?andez{Canteli
(see [64], [65] and [66]) and Castillo and Fern?andez{Canteli (see [37], [35], [39] and [36]).
CHAPTER 6. THE WEIBULL AND GUMBEL S{N FIELD STRESS BASED FATIGUE
MODELS 101
6.2 Derivation of the model
Consider a fatigue test conducted at constant minimum and maximum stresses denoted for sim-
plicity m and M , respectively, and let N be the random number of cycles to failure associated
with the test. Then, one is interested in determining the probability of failure p of a randomly
chosen specimen when subject to such a test.
In this section a formula for p in terms of N; m and M is derived, using as few arbitrary
assumptions as possible. To this end, we use the Buckingham ? theorem [31], some knowledge
from fatigue and extreme value theory, and some compatibility assumptions.
The Buckingham ? theorem states that if we have a physically meaningful equation involving
a certain number, n, of physical variables, and these variables are expressible in terms of k
independent fundamental physical quantities, then the original expression is equivalent to an
equation involving a set of v = n ? k dimensionless variables constructed from the original
variables. This theorem states that in fact a set of less variables is su?cient to analyze the
problem under consideration.
From fatigue knowledge, we introduce two new variables: a stress 0, which can be the
fatigue limit or any other equivalent variable, and N0 (cycles), which are used to obtain the
required dimensionless variables. Thus, we conclude that our problem depends on the following
6 variables:
p; m; M ; 0; N;N0:
If we assume that there is a relationship among these variables
r(p; m; M ; 0; N;N0) = 0; (6.1)
using the Buckingham ? theorem, we can select the dimensionless variables ?m = m= 0,
?M = M= 0 and N? = N=N0 and p, already dimensionless, and then the relationship
g(p;N?; ?m; ?M ) = 0; (6.2)
or
p = h(N?; ?m; ?M ); (6.3)
is equivalent to (6.1). So, one of our aims is to obtain the function h(N?; ?m; ?M ).
With this purpose in mind, we proceed as follows: First, we apply the fatigue model presented
by Castillo et al. [39] for constant maximum stress M to the dimensionless variable
? ? = ?M ? ?m =
M ? m
0 : (6.4)
Natural scale instead of a logarithmic scale for the stress amplitude arises as a natural
requirement from the model. Otherwise inconsistencies arise in the solution of the functional
equation when null or negative values for the stress range, i.e., compression in the min, are
considered.
According to these authors ([37], [43]) , the only possible model satisfying several physical
conditions (weakest link principle and limited range), compatibility conditions (of life and stress
range), statistical conditions (stability, limit behavior) and extreme value analysis properties
([33], [39], [36] and [70]) is the model
p = 1? exp
(
?
?(logN? ?B) ( ?M ? ?m ? C)? E
D
?A)
: (6.5)
The physical meanings of the parameters are:
102 6.2. DERIVATION OF THE MODEL
A: Weibull shape parameter of the cumulative distribution function (cdf) in the S{N fleld.
B: Threshold value of log-lifetime (see flgure 6.1).
C: Endurance limit (? ?) (see flgure 6.1).
E: Parameter deflning the position of the corresponding zero-percentile hyperbola.
D: Weibull scale factor.
Log N
??
C
B
Figure 6.1: Schematic representation of the physical meanings of B and C model parameters.
We note that this model has not been arbitrarily chosen, but derived from a set of conditions
that a reasonable model must satisfy. Finally, the use of functional equations ([11], [46] and [49])
guarantees that the resulting model in (6.6) is the unique solution.
An interesting limiting case of this model for A !1 is the Gumbel model
p = 1? exp
?
? exp
?(logN? ?B) ( ?M ? ?m ? C)? E
D
?
; (6.6)
which also satisfles the previous conditions and will be used in this paper.
Assuming that the model is valid for any flxed values of m and M , and since for difierent
constant values of M one must have difierent models of the form (6.6), the parameters A;B;C;D
and E must be functions of M [42].
Similarly, if the constant load fatigue tests are run for constant values of m, one has
another family of models, where now the parameters A;B;C;D and E are functions of m.
Our next goal is to obtain these functions using the following compatibility condition:
If we run a constant load fatigue test oscillating from m to M , we can derive the model
as a particular case of (a) constant m or (b) constant M , but both models must be the same
(compatibility condition), that is:
?
(logN? ?Bm( ?m))(? ? ? Cm( ?m))? Em( ?m)
Dm( ?m)
?Am
=
?
(logN? ?BM ( ?M ))(? ? ? CM ( ?M ))? EM ( ?M )
DM ( ?M )
?AM
:
(6.7)
CHAPTER 6. THE WEIBULL AND GUMBEL S{N FIELD STRESS BASED FATIGUE
MODELS 103
?s
?max=1
?max=1.5
Log N
?min =0.4
?min =0.8
(a) (b)
?s
Log N
Figure 6.2: Wo?hler curves for difierent percentiles: (a) for constant max, (b) for constant min.
Image from [42]
The compatibility condition is represented at the intersection of both difierent families of
curves ( m; M ). Consider a situation, in which R? and N of failure is known. If the load
ratio R? = Rf is flxed, the probability of failure, deflned by the area inside the probability
density function (pdf), for a number of cycles Nf , will be p(N;R?) = pf1. On other hand, if
the number of cycles is flxed N = Nf , the new probability of failure for a stress ratio R? will
be p(R?; N) = pf2. The compatibility condition deflne that these two difierent probabilities pf1
and pf2 are equal (see flgure 6.3).
pf1
pf2
Nf Nf
f
???
N N
f
???
??? ???
Figure 6.3: Schematic representation about the compatibility in Castillo?s models (see [42], [40]).
This compatibility was shown in flgure 8.1, where the compatibility states that the set of
percentiles must intersect at horizontally aligned points.
Equation (6.7) is a functional equation, in which the unknowns are the 8 functions Bm( ?m),
Cm( ?m), Dm( ?m), Em( ?m), BM ( ?M ), CM ( ?M ), DM ( ?M ) and EM ( ?M ). The beauty of func-
tional equations is that a single equation allows us determining the solution for all the unknown
functions involved. The complete derivation of the model is made in Appendix A: Model
104 6.2. DERIVATION OF THE MODEL
Log N
??
?
?max=1.5
?min=0.8
*
*
?max=1
*
?min=0.4
*
Figure 6.4: Schematic Wo?hler curves for percentiles f0:01; 0:05; 0:5; 0:95; 0:99g for ?max = 1 and
?max = 1:5, and ?min = 0:4 and ?min = 0:8, illustrating the compatibility condition. Dashed
lines refer to Wo?hler curves for constant ?min, and continuous lines refer to Wo?hler curves for
constant ?max. Image from [42].
Derivation.
For the functional equation (6.7) to be satisfled for any N?, ?m and ?M , both models musthave the same parameters. Writing the model in (6.7) as
2
664
logN? ?
?
Bm( ?m) + Em(
?
m)
? ? ? Cm( ?m)
?
Dm( ?m)
? ? ? Cm( ?m)
3
775
Am
=
2
664
logN? ?
?
BM ( ?M ) +
EM ( ?M )
? ? ? CM ( ?M )
?
DM ( ?M )
? ? ? CM ( ?M )
3
775
AM
; 8N? (6.8)
and forcing the Weibull parameters to coincide one gets:
Am( ?m) = AM ( ?M ); 8 ?m; ?M (6.9)
Dm( ?m)
DM ( ?M )
= ?
? ? Cm( ?m)
? ? ? CM ( ?M )
= (
?
M ? ?m)? Cm( ?m)
( ?M ? ?m)? CM ( ?M )
; 8 ?m; ?M (6.10)
BM ( ?M ) = Bm( ?m)?
EM ( ?M )
? ? ? CM ( ?M )
+ Em(
?
m)
? ? ? Cm( ?m)
= Bm( ?m)?
EM ( ?M )
( ?M ? ?m)? CM ( ?M )
+ Em(
?
m)
( ?M ? ?m)? Cm( ?m)
; 8 ?m; ?M : (6.11)
The system of functional equations (6.9) to (6.11) deserve a careful attention because they
involve a deep knowledge about our problem. In particular, they are not simple equalities, but
each a full collection of equalities, because they must hold for any feasible pair ?m; ?M .Solving the system of functional equations (6.9) to (6.11) (see appendix) one gets the Weibull
model:
p = 1?exp
n
? [C0 + C1 ?m + C2 ?M + C3 ?m ?M + (C4 + C5 ?m + C6 ?M + C7 ?m ?M ) logN?]A
o
; (6.12)
CHAPTER 6. THE WEIBULL AND GUMBEL S{N FIELD STRESS BASED FATIGUE
MODELS 105
which depends on nine parameters supplying a complete probabilistic information for any
Wo?hler curves of the material related to whichever given stress level, and where p =
F (N ; ?m; ?M ). We note that (6.12) is the function h(N?; ?m; ?M ) in (6.3) sought after.We note that an important limiting case of the Weibull model is the Gumbel model which
results for A ! 1 [37]. In addition, if all the arguments used to obtain the Weibull model are
derived for the Gumbel model we obtain exactly the same functional equations. This implies
that a valid fatigue model is the following Gumbel model:
p = 1? exp f? exp [C0 + C1 ?m + C2 ?M + C3 ?m ?M + (C4 + C5 ?m + C6 ?M + C7 ?m ?M ) logN?]g ;
(6.13)
which has the advantage of having one parameter less, and even more important, that the range
of deflnition for logN includes the range (?1;1). This avoids deciding whether or not we are
in the allowable region.
6.3 Some properties of the model
The graphs (logN;? ) of the percentiles for flxed max or min are hyperbolas. We note that
the hyperbolas arise not because of a reasonable, though nevertheless arbitrary assumption, but
as the only possible solution to the functional equation (see Appendix A).
The two asymptotes of the hyperbolas can be calculated as follows.
? The asymptotic value of ? for large logN keeping min = m constant is
? m0 = limN!1? = ?
C4 + m(C5 + C6 + C7 m)
C6 + C7 m (6.14)
and the asymptotic value of ? for constant max = M is
? M0 = limN!1? =
C4 + M (C5 + C6 + C7 M )
C5 + C7 M : (6.15)
? Similarly, the asymptotic value of logN for large ? keeping min = m constant is
logNm0 = lim? !1 logN = ?
C2 + C3 m
C6 + C7 m ; (6.16)
and the asymptotic value of logN for constant max = M is
logNM0 = lim? !1 logN = ?
C1 + C3 M
C5 + C7 M : (6.17)
It is interesting to see, that the general model allows the asymptotes to be dependent on the
constant min and max levels being considered.
6.3.1 The regression equation for max-logN for difierent stress ratios R
Since the common practice consists of using a regression equation to flt the max-logN fleld, in
this subsection the regression model resulting from the Weibull and Gumbel models are derived.
As an illustrative example, in the evaluation of fatigue results for difierent materials proposed
by the MIL-HDBK-5G [2], a regression model of the form
logN = A1 +A2 log10(Smax(1?R)A3 ?A4); (6.18)
106 6.4. RESTRICTIONS
is considered, where A1; A2; A3 and A4 are constants with dimensions, except the dimensionless
constant A3. However, the bases for selecting this model are not given.
Alternatively, a regression model based on physical and statistical grounds is presented below.
According to the model (6.12), the p percentile of logN for a given max and stress ratio R
can be derived from expression (6.12) by replacing min by max ?? , to obtain
logN = C0 + C2 max + C1R max + C3R
2max ? (? log[1? p])1=A
C4 + C6 max + C5R max + C7R 2max
; (6.19)
and the regression equation for max-logN for difierent stress ratios R, taking into account
that the mean is a percentile dependent on A, can be obtained replacing (? log(1? p))A by
? (1 + 1=A), where ? is the gamma function, which is the mean for the Weibull W (0; 1; A)
model, leading to the expression
logN =
C0 + C2 max + C1R max + C3R 2max ? ?
?
1 + 1A
?
C4 + C6 max + C5R max + C7R 2max
; (6.20)
which for the Gumbel model becomes
logN = C0 + C2 max + C1R max + C3R
2max +
C4 + C6 max + C5R max + C7R 2max
; (6.21)
where = 0:57772 is the Euler-Mascheroni number.
The relevant issue of Equations (6.20) and (6.21) is that they have been derived from all the
indicated properties, and not arbitrarily chosen. Thus, this regression model will be selected to
flt the experimental data.
6.4 Restrictions
For the model to be physically and statistically valid its parameters must satisfy the following
constraints:
6.4.1 Physical restrictions
1. The asymptotic value ? m0 must be non-negative, i.e.
? m0 = ?C4 + m(C5 + C6 + C7 m)C6 + C7 m ? 0: (6.22)
2. The asymptotic value ? m0, due to physical reasons, must be non-increasing in m, that
is,
(C6 + C7 m)2 + C5C6 ? C4C7 ? 0: (6.23)
3. The asymptotic value ? M0 must be non-negative, i.e.
? M0 = C4 + M (C5 + C6 + C7 M )C5 + C7 M ? 0: (6.24)
4. The asymptotic value ? M0 must also be non-increasing in M , that is,
(C5 + C7 M )2 + C5C6 ? C4C7 ? 0: (6.25)
CHAPTER 6. THE WEIBULL AND GUMBEL S{N FIELD STRESS BASED FATIGUE
MODELS 107
5. The asymptotic value logNm0 must be non-increasing in m:
C3C6 ? C2C7 ? 0: (6.26)
6. The asymptotic value of logNM0 must be non-increasing in M :
C3C5 ? C1C7 ? 0: (6.27)
6.4.2 Statistical restrictions
1. The cdf in (6.12) must be non-decreasing in logN :
C4 + C5 m + C6 M + C7 m M > 0; m0 ? m ? M ? M0; (6.28)
which implies:
C4 + C5 m0 + C6 M0 + C7 m0 M0 > 0 (6.29)
C4 + C5 m0 + C6 m0 + C7 m0 m0 > 0 (6.30)
C4 + C5 M0 + C6 M0 + C7 M0 M0 > 0: (6.31)
2. The cdf in (6.12) must be non-increasing in m:
C1 + C3 M + (C5 + C7 M ) logN ? 0; N0 ? N ; m0 ? M ? M0; (6.32)
3. The cdf in (6.12) must be non-decreasing in M :
C2 + C3 m + (C6 + C7 m) logN ? 0; N0 ? N ; m0 ? m ? M0; (6.33)
4. The curvature of the zero-percentile of (logN;? ) for constant min must be non-negative,
that is, @
2?
@(logN)2 ? 0, which leads to:
(C6 + C7 m)
? m(C1C6 ? C3C4) + (C1C7 ? C3C5) 2m ? C2(C4 + C5 m)
?
+(C0 ? (? log(1? p))1=A)(C6 + C7 m)2 ? 0 (6.34)
It is su?cient to force the positivity at one point, as for example, p = 0:5.
5. The curvature of the zero-percentile of (logN;? ) for constant max must be non-negative,
that is, @
2?
@(logN)2 ? 0, leading to:
(C5 + C7 M )
? M (C2C5 ? C3C4) + (C2C7 ? C3C6) 2M ? C1(C4 + C6 M )
?
+(C0 ? (? log(1? p))1=A)(C5 + C7 M )2 ? 0; (6.35)
Inclusion of these constraints into the estimation method leads to valid models. This is an
important fact to be taken into consideration, because alternative methods do not take this into
account su?ciently, and lack generality.
108 6.4. RESTRICTIONS
6.4.3 Range of the problem. Simpliflcations of constraints
In order to make the application of these restriction (Equations (6.28)-(6.35)) easy, an analysis
of the ranges of application of the model can be made. The ranges are:
? The ranges of ?min and ?max are
?min 2 [?1; 1]; ?max 2 [0; 1] 8 min ? max; (6.36)
so, the range of ? is
? ? = ?max ? ?min ! 2 [?1; 2]; (6.37)
But, knowing that max must be greater than min, the range of ? is flnally ? 2 [0; 2]
(see flgure 6.5).
2
1
0
-1
?M=1
?M=0
?m=1
?m=-1
?M ?m
??
N
Figure 6.5: Schematic representation of the range of the problem.
? The range for the number of cycles, N , goes from N = 1 ! logN = 0, to N = 1 !
logN = 1.
But this last hypothesi about the range of number of cycles, doesn?t give a useful information
about the simpliflcation of the constraints because almost all these constraints are deflned in
the range of load. So, according to this, the flnal constraints are:
1. The cdf constraints (Equations (6.29) to (6.31)) are transformed in
C4 ? 0 (6.38)
C4 ? C5 ? 0 (6.39)
C4 + C5 ? 0 (6.40)
C4 ? C5 + C6 ? C7 ? 0 (6.41)
C4 + C5 + C6 + C7 ? 0: (6.42)
2. The curvature of the zero-percentile of (logN;? ) for constant min (Equation (6.34))
leads to:
(C6 ? C7)[(C0 + )(C6 ? C7) + C3(C4 ? C5) + C1(C7 ? C6) + C2(C5 ? C4)] ? 0 (6.43)
(C6 + C7)[(C0 + )(C6 + C7) + C1(C6 ?+C7)? (C2 + C3)(C4 + C5)] ? 0: (6.44)
CHAPTER 6. THE WEIBULL AND GUMBEL S{N FIELD STRESS BASED FATIGUE
MODELS 109
3. The curvature of the zero-percentile of (logN;? ) for constant max (Equaton (6.35)) is
now:
C5[C5(C0 + )? C1C4] ? 0 (6.45)
(C5 + C7)[(C0 + C2 + )(C5 + C7)? (C4 + C6)(C1 + C3)] ? 0: (6.46)
4. The asymptotic value ? m0 (Equation (6.23)) is:
(C6 ? C7)2 + C5C6 ? C4C7 ? 0 (6.47)
(C6 + C7)2 + C5C6 ? C4C7 ? 0: (6.48)
5. The asymptotic value ? M0 (Equation (6.25)) is transformed now to:
C25 + C5C6 ? C4C7 ? 0 (6.49)
(C5 + C7)2 + C5C6 ? C4C7 ? 0: (6.50)
6.5 Resulting models and submodels
In this section, we discuss some particular and interesting submodels of the general Gumbel
model (6.13).
6.5.1 General model
The general model with ? and logN asymptotes dependent on m and M was deflned in
Equation (6.13):
p = 1? exp f? exp [C0 + C1 ?m + C2 ?M + C3 ?m ?M + (C4 + C5 ?m + C6 ?M + C7 ?m ?M ) logN?]g ;
(6.51)
subject to constraints (6.22) to (6.35), Equations (6.38)-(6.50).
6.5.2 Submodel Nr.1
The flrst submodel is the linear model of (6.13). To have a linear submodel it is necessary to have
no asymptotes and any product between variables (load and number of cycles), because these
products transform the problem from a linear to a nonlinear problem. The variables afiected
are C5; C6 and C7 that must be equal to zero.
The simplest model is:
p = 1? exp f? exp [C0 + C1 ?m + C2 ?M + C3 ?m ?M + C4 logN?]g
C4 ? 0; (6.52)
Note that this submodel in a semilog scale leads to a Wo?hler fleld made of straight lines (see
flgure 6.6 (a)).
6.5.3 Submodel Nr. 2
This submodel has a flxed vertical asymptote. The model with logN asymptote independent
on m and M :
p = 1? exp
n
? exp
h
C0 + C2(C5C6 ?m + ?M +
C7
C6 ?m ?M ) + (C4 + C5 ?m + C6 ?M + C7 ?m ?M ) logN?
io
;
(6.53)
110 6.5. RESULTING MODELS AND SUBMODELS
which is obtained for C1 = C2C5C6 ;C3 =
C2C7
C6 , subject to constraints (6.22) to (6.35). ReplacingC3 = C7 = 0 into these constraints, and taking into account the range of load (Equations
(6.38)-(6.50)) one concludes that they are equivalent to the set of constraints:
C4 ? 0
C4 ? C5 ? 0
C4 + C5 ? 0
C4 ? C5 + C6 ? C7 ? 0
C4 + C5 + C6 + C7 ? 0
(C6 ? C7)[(C0 + )(C6 ? C7) + C2(C4 ? C5 + 1C6 (C4C7 ? C5C6 ? 2C5C7))] ? 0
(C6 + C7)[(C0 + )(C6 + C7)? C2C4(1 + C7C6 )] ? 0
C25 [C0 + ?
C2C4
C6 ] ? 0
(C5 + C7)[(C0 + )(C5 + C7) + C2(C5 ? C7 ? 2C5C4C6 ? 2C2C5)] ? 0
(C6 ? C7)2 + C5C6 ? C4C7 ? 0
(C6 + C7)2 + C5C6 ? C4C7 ? 0
C25 + C5C6 ? C4C7 ? 0
(C5 + C7)2 + C5C6 ? C4C7 ? 0: (6.54)
This model is represented in flgure 6.6 (b).
6.5.4 Submodel Nr. 3
The third submodel has a flxed horizontal asymptote. The model with ? asymptote indepen-
dent on m and M (flgure 6.6 (c)) is:
p = 1? exp
n
? exp
h
C0 + C1 ?m + C2 ?M + C3 ?m ?M + logN?
?
C4 + C5 ?m + C6 ?M + C5?C6 ?m? ?M
?
m ?M )
?io
;
(6.55)
which is obtained for C7 = C5C6 ?m? ?M , that taking into account the range of load transform C5 = C6and C7 = 0. The flnal constraints of this submodel are:
C4 ? 0
C4 ? C5 ? 0
C4 + C5 ? 0
C4 ? 2C5 ? 0
C4 + 2C5 ? 0
C5[C5(C0 + ? C1 + C2 ? C3) + C4(C3 ? C2)] ? 0
C5[C5(C0 + + C1 ? C2 + C3) + C4(C3 ? C2)] ? 0
C5[C5(C0 + )? C1C4] ? 0
C5[C5(C0 + + C2 ? C1 ? C3)? C4(C1 + C3)] ? 0: (6.56)
But, from Equations (6.47)-(6.50), the model leads to C5 = C6 = 0 that, in addition with C7 = 0
transform this submodel in the linear submodel.
CHAPTER 6. THE WEIBULL AND GUMBEL S{N FIELD STRESS BASED FATIGUE
MODELS 111
6.5.5 Submodel Nr. 4
This submodel with both flxed asymptotes, is derived from the two previous models, Equations
(6.53) and (6.55). So, with C5 = C6 = C7 = 0 the model leads to C3 = 0 and C1 = C2. Finally,
the model with ? and logN asymptotes independent on m and M (see flgure 6.6 (d)) is a
model with only three parameters:
p = 1? exp f? exp [C0 + C1( ?m + ?M ) + C4 logN?]g ;
C4 ? 0: (6.57)
??
?? ??
??
log N
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
(a)
(c) (d)
(b)
m2
M2
M1
m1
log N
log Nlog N
M1
M1M1
M2
M2
M2
m1
m1m1
m2
m2m2
Figure 6.6: Schematic representation of difierent sub-models.
6.5.6 Submodel Nr. 5
The last submodel presented in this section is obtained analyzing the situation of the asymptotes
and the cdf function:
p = 1? exp f? exp [C0 + C1 ?m + C2 ?M + logN?[C5( ?m ? ?M ]))]g ; (6.58)
which is obtained for C3 = C4 = C7 = 0, that operating become in C6 = ?C5. that taking
into account the range of load transform C5 = C6 and C7 = 0. The flnal constraints of this
112 6.6. PARAMETER ESTIMATION
sub-model are:
C1 ? 0
C5 ? 0
logN ? ?C1C5 : (6.59)
This submodel is used in the appendix D, where an informatic tool (SISIFO program) is pre-
sented.
6.6 Parameter estimation
The parameter estimation of the model can be done by several methods. The two most useful
are described below.
6.6.1 Maximum likelihood estimation
The most well known method for estimating the parameters of a statistical model is the maximum
likelihood method, which shows good statistical properties. Thus, it is one of the flrst possibilities
to be considered.
The log-likelihood function of the Weibull model (6.13) is
L =
X
i2I1
?
log(A) + (A? 1) log(H(N?i )) + log
?
C4 + C5 ?mi + C6 ?Mi + C7 ?mi ?Mi
?
? log(N?i )
?
?
X
i2I1[I0
HA(N?i );
(6.60)
where I1 and I0 are the set of non-runouts and runouts, respectively, Ni refers to the actual
value of the fatigue life in number of cycles, or the limit number of cycles for runouts, and
H(Ni) = C0+C1 ?mi+C2 ?Mi+C3 ?mi ?Mi+
?
C4 + C5 ?mi + C6 ?Mi + C7 ?mi ?Mi
?
logN?i : (6.61)
Similarly, for the Gumbel model (6.13), the log-likelihood becomes:
L =
X
i2I1
?H(N?i ) + log
?C4 + C5 ?mi + C6 ?Mi + C7 ?mi ?Mi
?? log(N?i )
??
X
i2I1[I0
exp(H(N?i )): (6.62)
Thus, to estimate the parameters of the model we can maximize (6.60) or (6.62) with respect
to the parameters, but subject to the set of constraints (6.38) to (6.50) or the equivalent ones
for simpler models. For the Weibull models one must add the condition H(Ni) ? 0;8i, which
is a very disturbing set of constraints, because the C?s are unknown. Thus, when possible, it is
recommendable to use the Gumbel model instead of the Weibull model because of its simplicity
and due to the fact that estimation is much easier. If the values of the A parameter are high,
as it happens to be with many materials, the Gumbel model is the most convenient option to
choose.
If the optimum value is not on the boundary of the feasible region, the asymptotic covariance
matrix of the estimates C0; C1; C2; C3; C4; C5; C6; C7 can be calculated using the well known
formula
Covar =
?
? @L
2
@Ci@Cj
!flflflflfl
?1
C
(6.63)
where C are the maximum likelihood parameter estimates. This matrix is the basic tool to
determine confldence intervals of other related variables, as percentiles for example.
If the optimum value is on the boundary of the feasible region, the covariance matrix must
be obtained by the bootstrap method (see [63], [99]).
CHAPTER 6. THE WEIBULL AND GUMBEL S{N FIELD STRESS BASED FATIGUE
MODELS 113
6.6.2 Parameter estimation by regression
Another possibility consists of using a regression model, i.e., minimize the following sum of
squares for the Weibull model
Q =
nX
i=1
0
@logN +
C0 + C2 ?max + C1R ?max + C3R ?max2 ? ?
?
1 + 1A
?
C4 + C6 ?max + C5R ?max + C7R ?max2
1
A
2
; (6.64)
based on (6.20), where the parameter A must be estimated using other methods, or for the
Gumbel model
Q =
nX
i=1
?
logN + C0 + C2
?max + C1R ?max + C3R ?max2 +
C4 + C6 ?max + C5R ?max + C7R ?max2
!2
; (6.65)
subject to the constraints (6.38) to (6.50).
To estimate the shape Weibull parameter A one can use a sample with constant max, min or
R = min= max, because the corresponding distribution is Weibull with the same A parameter.
Once this has been estimated, one can minimize (6.64) to estimate the remaining parameters.
For large sample sizes one could avoid the constraints, and assume that the data already
contain the necessary information about the constraints. However, this is very risky, and one
can face problems depending on the posterior use of the model
There are other methods to estimate the parameters of the model (see [38] and [44]).
6.7 Use of the model in practise
The general expression of the model is presented in Equation (6.13), based on the three variables
deflned in Section 6.2, ?m = m= 0, ?M = M= 0 and N? = N=N0. These equations show that
the model in (6.13) really depends on 8 parameters, and that 0 and N0 can be chosen arbitrarily
to obtain dimensionless stress ratios close to one to avoid numerical di?culties.
The proceeding steps in practice are:
Step 1: Design of the testing strategy. A set of testing cases encompassing several stress level
conditions, i.e., varying M and m is selected, for example, the set of pairs
f( mi ; Mi) j i = 1; 2; : : : ; ng;
which cover the desired region, where the regression equation is to be used.
Step 2: Choose the normalizing variables logN0 and 0. Choose logN0 and 0 and normalize
the data to dimensionless form.
Step 3: Estimate the model parameters. Use one of the estimation methods discussed in
Section 6.6 to estimate the parameters C0; C1; C2; C3; C4; C5; C6; C7 in (6.13).
Step 4: Extrapolate to other testing conditions. Use the model (6.13) and the parameters
C1; C2; C3; C4; C5; C7 and logN0 and 0 for any other testing condition (see Section 6.7.1).
114 6.8. EXAMPLE OF APPLICATION
6.7.1 Some difierent representations of the Gumbel model
There are, difierent parametric forms for the Wo?hler fleld, as required by the user. In the next
lines a representation of the most useful forms are deflned.
1. ? -logN for constant M :
logN = ?C0 + C1? ? C1 M ? C2 M + C3? M ? C3
2
M + log(? log(1? p))
C4 ? C5? + C5 M + C6 M ? C7? M + C7 2M
(6.66)
2. ? -logN for constant m:
logN = ?C0 ? C2? ? C1 m ? C2 m ? C3? m ? C3
2m + log(? log(1? p))
C4 + C6? + C5 m + C6 m + C7? m + C7 2m
(6.67)
3. ? -logN for constant mean:
logN = ?4C0 + 2C1? ? 2C2? + C3?
2 ? 4 mean(C1 + C2 + C3 mean) + 4 log(? log(1? p))
4C4 ? 2C5? + 2C6? ? C7? 2 + 4 mean(C5 + C6 + C7 mean)
(6.68)
4. ? -logN for constant R:
logN = ?C0(R? 1)
2 ?? (C2(1?R) +R(C1 + C3? ? C1R)) + (R? 1)2 log(? log(1? p))
C4(R? 1)2 +? (C6(1?R) +R(C5 + C7? ? C5R))
(6.69)
6.8 Example of application
6.8.1 Validation using data in the existing literature
To check the model performance and to illustrate its application to practical cases, the model
is applied in this section to a real example [40].
This example, uses some data published in the existing literature. The data correspond to
fatigue results from the MIL-HDBK-5G [2]. In particular, fatigue sample data from specimens
made of notched Inconel 718 bars including three stress ratios (R = ?0:50; 0:10; 0:50).
Since in Section 6.7 a methodology has been proposed to deal with practical cases, it is
applied here step by step, as follows:
Step 1: Design of the testing strategy. In this example, data is derived from published data,
so there isn?t a testing strategy. In Chapter 7 a real strategy will be presented.
Step 2: Choose the normalizing variables logN0 and 0. As indicated, we have chosen
logN0 = 0 (N0 = 1 cycle) and 0 = 1000 MPa. and normalized the data to dimensionless form
by dividing the lifetime and stresses by 1 cycle and 1000 MPa, respectively.
Step 3: Estimate the model parameters. We have used the maximum likelihood method for the
Gumbel models as discussed in Section 6.6.1 to estimate the parameters C0; C1; C2; C3; C4; C5; C6
and C7 in the Gumbel model (6.13) and the resulting parameter estimates are shown in table 6.1.
The following cases have been considered:
CHAPTER 6. THE WEIBULL AND GUMBEL S{N FIELD STRESS BASED FATIGUE
MODELS 115
Table 6.1: Parameter estimates cases 1 to 4.
Case Parameters
C0 C1 C2 C3 C4 C5 C6 C7
1 -46748.8 11631.7 -20502.9 -10430.8 2457.73 -5134. 5737.14 3070.4
2 -10.1959 44.7721 -39.1862 0 0 -5.66951 5.66951 0
3 -10.8552 42.8311 -36.9141 0 0 -5.56015 5.56015 0
4 -12.1504 49.975 -42.9144 0 0 -6.34962 6.34962 0
Case 0. It corresponds to the regression model (6.18) without constraints fltted to all data
excluding runouts. The data and the corresponding curves provided in the MIL-HDBK-5G
are plotted in the top graph of flgure 6.7. The flt is reasonably good, but the quality of
extrapolations based on this model is not guaranteed by a physically justifled regression
equation.
Case 1. It corresponds to the general 8 parameter regression model (6.13) without constraints
fltted to all data excluding runouts. The intermediate graph of flgure 6.7 shows the real
data classifled by R values, together with the estimated (regression) curves. The flt is
better than in the previous case, especially in the lower region. In adition, since the
regression model has been derived based on physical and statistical bases, the extrapolation
can be done with a higher reliability. However, taking into account that the constraints
were not imposed and normally not all will be satisfled, extrapolation must be done with
care.
Case 2. It corresponds to the Gumbel version of the model (6.57) with flxed asymptotes
including all the constraints and fltted with all data excluding the runouts. The corre-
sponding model is plotted in the lower part of flgure 6.7. Since the model used has been
constrained by a high number of constraints or conditions, the flt of the model to the data
is not as good as in the previous case. However, this is not a shortcoming but on the
contrary can be considered as an advantage. In fact, the plot reveals that the data point
with the smallest number of cycles to failure appears to be an outlier, and in fact it cor-
responds to the low cycle fatigue region. In addition, the curvature of the data points for
R = ?0:5 points out to the possibility of a plastic failure. Note that the outlier character
of this point was hidden in the two previous cases.
Case 3. It corresponds to the Gumbel version of the model (6.13) with flxed asymptotes
including all the constraints and fltted with all data including the runouts. The model
appears in the upper part of flgure 6.8. In this case the runouts were not removed, but
taken into consideration in the estimation process, using expression (6.62). Note that the
runouts do not contain exact information about the lifetime, but contain some information,
which is also valuable, and must not be neglected. A comparison of the plots of cases 2
and 3 show that they are very similar, and that the main difierences occur in the lower
right region, as expected. Note that again the data point with the least lifetime appears
as a clear outlier, which was not the case for cases 0 and 1.
Case 4. It corresponds to the Gumbel version of the model (6.13) with flxed asymptotes
including all the constraints and fltted with all data but the outlier and including the
runouts. The model appears in the lower part of flgure 6.8. Since the data point with the
smallest lifetime appears as an outlier and there are physical reasons to justify it, in this
116 6.8. EXAMPLE OF APPLICATION
S-N field
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
1000 10000 100000 1000000 10000000
Log N
R=-0.5
R=0.1
R=0.5
R=-0.5
R=0.1
R=0,5
S-N field
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
1000 10000 100000 1000000 10000000
Log N
?
R=-0.5
R=0.1
R=0.5
R=-0.5
R=0.1
R=0,5
S-N field
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
1000 10000 100000 1000000 10000000
Log N
R=-0.5
R=0.1
R=0.5
R=-0.5
R=0.1
R=0,5
?
m
ax
?
m
ax
m
ax
Figure 6.7: S{N curves for notched Kt = 3:3, AISI 4340 Alloy steel bar fltted by three difierent
methods. The upper corresponds to the MIL-HDBK-5G, the intermediate to the proposed model
without constraints and the lower to the proposed model including all the constraints.
case this data point has been removed and the model re-estimated. A comparison with
the plots of case 3 reveals that the resulting models are very similar, and then one can
conclude that the outlier has a negligible in uence due to the strong constraints imposed
to the model. So, the robustness of the proposed estimation regression method has been
CHAPTER 6. THE WEIBULL AND GUMBEL S{N FIELD STRESS BASED FATIGUE
MODELS 117
S- N field
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
1000 10000 100000 1000000 10000000
Log N
R=-0.5
R=0.1
R=0.5
R=-0.5
R=0.1
R=0,5
S- N field
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
1000 10000 100000 1000000 10000000
Log N
R=-0.5
R=0.1
R=0.5
R=-0.5
R=0.1
R=0,5
?
m
ax
?
m
ax
Figure 6.8: S{N curves for notched Kt = 3:3, AISI 4340 Alloy steel bar fltted using the proposed
regression model with constraints and including the runouts. In the lower flgure the outlier has
been removed.
shown comparing the results obtained when removing one outlier suspected to belong
to low-cycle fatigue, as shown in flgure 6.8, which shows practically the same resulting
evaluation results.
It seems reasonable to use this model as the most adequate to represent the material
fatigue strength corresponding to the given data. Then, the variance matrix of the Gumbel
parameter estimates C0; C1; C2; C5; C6) has been calculated using formula (6.63) and the
following matrix has been obtained
0
BBBBB@
3:302 ?9:247 8:892 1:296 ?1:446
?9:247 122:453 ?41:208 ?12:863 5:636
8:892 ?41:208 33:956 5:071 ?4:851
1:296 ?12:863 5:071 1:403 ?0:723
?1:446 5:636 ?4:851 ?0:723 0:726
1
CCCCCA
Step 4: Extrapolate to other testing conditions. Once the parameter estimates are available,
the model (6.12) can be used to extrapolate to other testing conditions. For example, one
can predict the expected lifetimes associated with other R values, plot the percentiles curves,
etc. For example, in table 6.2 the estimated percentile values associated with the difierent data
118 6.8. EXAMPLE OF APPLICATION
Data p Data p Data p Data p Data p Data p
1 0.004 6 0.161 11 0.734 16 0.959 21 0.860 26 0.312
2 0.356 7 0.664 12 0.638 17 0.639 22 0.259 27 0.369
3 0.632 8 0.249 13 0.912 18 0.874 23 0.876 28 0.144
4 0.512 9 0.082 14 0.618 19 0.220 24 0.385 29 0.092
5 0.933 10 0.049 15 0.516 20 0.223 25 0.547 30 0.376
Table 6.2: Estimated percentile values associated with the difierent data points using the Gumbel
fltted model.
points in flgure 6.9 are shown. They have been determined using the Gumbel fltted model. It
is interesting to see that the flrst data point has an associated value of 0:004, which reveals its
outlier character.
Further, the model can supply all the desired information corresponding to any possible
testing condition alternative. For example, in flgure 6.9 the S/N curves for constant R =
0:5; 0:1;?0:5 and ?1 (From top to bottom and left to right) including the percentiles 0:01, 0:05,
0:50, 0:95 and 0:99 are represented.
S-N field
30
29
28
2726
25
24
2322
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Log N
?
m
ax
R=0.5
p=0.01
p=0.05
p=0.5
p=0.95
p=0.99
S-N field
2120
19
1817
16
1514
13
12
11
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Log N
? m
a
x
R=0.1
p=0.01
p=0.05
p=0.5
p=0.95
p=0.99
S-N field
10
1
2
3
4
5
6
7
8
9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Log N
?
m
ax
R=-0.5
p=0.01
p=0.05
p=0.5
p=0.95
p=0.99
S-N field
R=-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Log N
?
m
ax
p=0.01
p=0.05
p=0.5
p=0.95
p=0.99
Figure 6.9: S{N curves for constant R = 0:5; 0:1;?0:5 and ?1 (From top to bottom and left to
right). The percentiles 0:01, 0:05, 0:50, 0:95 and 0:99 are represented.
CHAPTER 6. THE WEIBULL AND GUMBEL S{N FIELD STRESS BASED FATIGUE
MODELS 119
Similarly, in flgure 6.10 the ? -logN Wo?hler flelds for M = 0:8, m = 0, R = ?1 and
mean = ?0:20 are shown. They have been plotted using equations (6.66) to (6.69), respectively.
S-N field
?m ax=0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Log N
p=0.01
p=0.05
p=0.5
p=0.95
p=0.99
S-N field
? min=0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Log N
??
p=0.01
p=0.05
p=0.5
p=0.95
p=0.99
S-N field
R=-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Log N
p=0.01
p=0.05
p=0.5
p=0.95
p=0.99
??
??
S-N field
mean=-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Log N
p=0.01
p=0.05
p=0.5
p=0.95
p=0.99
??
?
Figure 6.10: S{N curves for constant M = 0:8, m = 0, mean = 0 and R = ?1 (From top to
bottom and left to right). The percentiles 0:01, 0:05, 0:50, 0:95 and 0:99 are represented.
6.9 Conclusions
The main conclusions obtained from this chapter are:
? A general Weibull regression model for statistical analysis of stress life data for the case
of max being tension has been developed. The model is based on statistical and physical
considerations, in particular, on compatibility conditions in the Wo?hler fleld that leads to
a system of functional equations.
? The model depends on 8 parameters that can be estimated by maximum likelihood
and also by non-linear regression. It supplies all the material basic probabilistic fatigue
information to be used in a damage accumulation assessment for fatigue life prediction of
structural and mechanical components under real loading spectra.
120 6.9. CONCLUSIONS
? The model was satisfactorily applied to the evaluation of fatigue results from an external
experimental program. In particular, in the example presented in Section 6.8.1, the
method allows us detecting some data points corresponding presumably to low-cycle
fatigue, so that when removed, the resulting model remains practically unaltered as the
original one. This is due to the large number of constraints that have been observed in
order to obtain only physically and statistically valid models. This is not the case of other
models commonly used in practice.
? Once the parameters of the model have been estimated, the model allows us obtaining any
kind of Wo?hler fleld according to the testing condition chosen, as it has been demonstrated
in the example of application.
? Finally, it is worthwhile mentioning that the model is the basic tool to develop a damage
accumulation tool involving any load spectrum.
Part IV
Experimental Validation of the
Models
121
Chapter 7
Experimental Validation of the
Model
In this Chapter, the experimental validation of the model proposed in chapter 6 is made.
Two difierent materials, low-alloy steel and aluminum alloy are tested.
The aim of this Chapter is to show that the results seem to conflrm the applicability
of the model showing its capability to provide linear and nonlinear trends for the S{N
curves and, furthermore the P{S{N fleld.
7.1 Introduction
Some of the problems faced when searching for a model able to predict adequately the material
lifetime have been shown in the previous chapters (see chapters 2 and 4). Real structures, equip-
ment, vehicles and machines are subjected to fatigue load histories involving complex varying
stress ranges and levels, which can cause their failure after a certain time period. Due to the
recent advances in mechanical and structural design methods, as for example, in flnite element
techniques, and to market pressure in reducing material consumption, in order to optimize pro-
duction costs, fatigue is becoming a determinant failure criterion for an increasing number of
structural members and mechanical components in real practice. Fatigue design demands a
cumulative damage model that requires a basic material fatigue characterization with consid-
eration of stress range and mean stress level. These are considered the main and secondary
parameters, respectively, governing the fatigue process.
The fundamental information is provided by means of laboratory tests, usually carried out
under difierent constant stress ranges, while the stress ratio is maintained throughout unchanged,
(S{N curves) or under a particular accelerated load history (ASTM E606 [5]). Thus, the eval-
uation of both (a) the standard fatigue data results, and (b) the damage model to be applied,
become crucial to the lifetime prediction.
Concerning the testing strategy, the common practice in material fatigue characterization
consists of selecting several data series with a reference constant stress level covering the region of
interest: commonly, constant M or constant m in structural design and constant mean or stress
ratio R = m= M in mechanical design. This information is insu?cient for the calculation of the
damage accumulation in real load cases in which varying mean stress levels are present. For this
reason, empirical transformation rules, not always su?ciently supported by experimentation,
123
124 7.2. MATERIALS
are currently applied. Researchers try to transform stress ranges at a certain mean stress level,
corresponding to the real load, to presumably \equivalent" stress ranges related to those provided
by the Wo?hler curves obtained in the laboratory.
In this chapter, the general Gumbel fatigue model for assessment of the Wo?hler fleld devel-
oped by the authors in the previous Chapter, that includes the consideration of the mean stress
efiect, is applied to two difierent materials to show that it is valid for some practical applications.
Continuing with the research in this doctoral thesis, the next step consists of making the
experimental validation of the proposed model (one example to validate the model theoretically
was presented in Chapter 6, Section 6.8.1). The experimental validation was made with two sets
of fatigue data corresponding to two difierent materials, low-alloy steel 42CrMo4 and aluminum
alloy AlMgSi1 [110].
The chapter is organized as follows: In section 7.2 the material is presented, then in section
7.3 a description of specimens is made, section 7.4 and 7.5 deflne the experimental program and
the testing strategy, respectively. Section 7.6 gives the parameter estimates and uses difierent
methods for checking the validity of the model (Section 7.7). Finally, Section 7.8 provides some
conclusions about the model and the materials.
7.2 Materials
For the validation of the model two difierent materials will be analyzed. These materials were
considered in an experimental fatigue program launched at Empa (Swiss Federal Laboratories
for Testing and Research at Du?bendorf (Switzerland)).
The materials chosen (a low alloy steel and an aluminum alloy) have a very difierent behavior
under cyclic and static load, so we can characterize the model for difierent states of load:
? A low{alloy steel 42CrMo4 (material number coded DIN-1.7225) with a nominal value of
the ultimate strength Rm = 1067MPa and of the yield strength, Ry = 975:3MPa.
? An aluminium alloy AlMgSi1 (material number coded DIN-3.2315) with a nominal value
of the ultimate strength Rm = 391:7MPa and of the yield strength, Rp0:2 = 364:3MPa.
Both materials were tested to know the static properties based on the ASTM E8M-04 norm
[7]. The results are shown in table 7.1 and in Appendix B. Furthermore, a metallographic
test (following the standard ASTM E7-03 [4]) was realized to each material to validate that the
chemical properties are the corrects one (see table 7.2 and Appendix B). More properties of
these materials are presented in [27] and [28].
These materials are used in structural construction because of its satisfactory service me-
chanical behavior and fatigue life. In addition, the AlMgSi1 shows good weather resistance,
lightweight and minimum thermal expansion properties.
Table 7.1: Static characteristics of the metallic alloys used.
Material Ry Rp0:2 Rm E modulus
[MPa] [MPa] [MPa] [MPa]
42CrMo4 975.3 967.3 1067 204870
AlMgSi1 { 364.3 391.7 72080
CHAPTER 7. EXPERIMENTAL VALIDATION OF THE MODEL 125
Table 7.2: Chemical compositions of 42CrMo4 steel according to DIN EN 10083-3:2007-01 and
AlMgSi1 alloy according with DIN EN 573-3 respectively.
42CrMo4 C Si Mn S P Cr Mo Fe
g/100g 0.38-0.45 ? 0.40 0.60-0.90 ? 0.035 ?0.025 0.90-1.20 0.15-0.30 remains
AlMgSi1 Si Fe Cu Mn Mg Cr Ni Zn various Ti Al
g/100g 0.70-1.30 0.50 0.10 0.40-1.0 0.60-1.20 0.25 - 0.20 0.15 0.10 remains
7.3 Specimens
The deflnition of the geometric characteristics of the specimens depends on the test to be run.
All these specimens are cylindric, but with difierent useful lengths. For example, Wolf, Fleck and
Ei er [133], deflne the geometric characteristics (see flgure 7.1 (b)) for temperature measures
in the fatigue behavior. Starke, Walther and Ei er [127], deflne the geometric characteristics of
the specimens to analyze fatigue behavior based on strain, temperature and electrical measures
(see flgure 7.1 (a)).
(b)
Figure 7.1: Difierent geometries for fatigue tests: (a) Starke et al. specimen [127], (b) Wolf et
al. specimen [133].
One aspect to be taken into account is the length efiect that occurs in fatigue tests. Castillo
et al. [47] deflnes a new model of fatigue, that can analyze the lifetime of a component based on
test specimens shorter than the real components. This model is in accordance with the weakest
link concept in the way as: an element of length L can be considered to be divided into a series
of n flctitious sub-elements of small length L0, i.e. L = nL0, and the failure occurs when the
weakest subelement fails [35].
The flnal specimens shown in flgure 7.2, are cylindrical and the test length of the specimens
was L2 mm, with 8 mm in diameter. The total length was L1 mm and the radius of transition
to the test section of the specimen was 55 mm. The difierent lengths used for each material are
shown in table 7.3.
Table 7.3: Specimen?s dimensions for each material.
Material L1 L2 d r
(mm) (mm) (mm) (mm)
42CrMo4 130 30 8 55
AlMgSi1 110 10 8 55
126 7.4. EXPERIMENTAL PROCEDURE
L1
r
d
L2
Figure 7.2: Geometry of the testing specimen.
7.4 Experimental procedure
The testing machines used for the validation were two: a servo hydraulic testing machine, 160
kN load capacity with a steel alloy grip based on ASTM E606 [5] (see flgure 7.3), and a high
frequency vibrophore machine with capacity of 150 Hz of frequency (see flgure 7.4).
(a)
(b)
Figure 7.3: Schenk Machine, used for 42CrMo4 tests. (a) General sight of the machine, (b)
detail of the machine, specimen?s grips.
In the case of the 42CrMo4 steel, all the tests were conducted using the servo hydraulic testing
machine at frequencies ranging from 1 to 10 Hz. For the AlMgSi1 alloy, the high frequency
vibrophore machine was be run at a maximum frequency of 80 Hz. In this way, only 34.7 hours
were needed to complete a test with a limit number of cycles of 10 millions.
In the case of 42CrMo4 steel a machine slower than that used for aluminum alloy was used
CHAPTER 7. EXPERIMENTAL VALIDATION OF THE MODEL 127
(a)
(b)
Figure 7.4: Rumul Machine, used for AlMgSi1 tests. (a) General sight of the machine, (b) detail
of the machine, specimen testing.
to avoid the temperature efiect created by the high frequencies. The specimen temperature was
continuously monitored to keep it su?ciently low in order not to in uence the failure mechanism
during testing. All the tests were run at 25?C degrees.
7.5 Type of load and testing strategy
In this section the testing strategy and the type of load are analyzed. The aim of any testing
strategy is to generate the most adequate test samples to estimate the model parameters in
(6.13).
The type of load proposed consists of testing specimens subject to alternating fatigue loads
from a value of ?m to a value ?M , which can change from test to test. Normally, a minimum,
of 8 to 12 tests with the same ?M and difierent ?m are su?cient to predict Wo?hler curves for
the associated constant ?M , but not for other possible combinations of ?m and ?M . Since we
aim at predicting fatigue life for a stress history of any combination of stresses, more tests are
needed by combining groups of tests run for constant ?M and varying ?m with groups of tests
run for constant ?m and varying ?M .
In our case, a seemingly e?cient testing strategy would consist of carrying out four groups
of tests, in which ?M are constant for each group of test. Then, difierent levels of ?m are chosen
to test a large range of load. In flgure 6.7 the distribution of these tests is shown schematically.
To this goal, in both materials all tests were conducted under four constant ?M levels corre-
sponding to given percentages of the yield strength and difierent values of ?m [110]. In the case
of the 42CrMo4 steel, the values of these levels correspond to the 0.98, 0.9, 0.8 and 0.7 of the
yield strength. For the AlMgSi1 alloy, the values of these levels correspond to the 0.9, 0.8, 0.7
and 0.6 of the yield strength (See flgure 7.5).
128 7.5. TYPE OF LOAD AND TESTING STRATEGY
Due to the lack of an exact knowledge of the fatigue limit for both materials, the minimum
testing amplitudes of these materials were estimated from the literature (see Boller and Seeger
[27, 28]) 450 MPa for the steel alloy and 80 MPa for the aluminium alloy. The difierent levels
were chosen to optimize the test times trying to avoid run-outs.
The distributions of the difierent test loads are shown in flgure 7.5. In addition, flve and ten
million of cycles were flxed as run-out values, respectively, for the 42CrMo4 and AlMgSi1 alloys.
The resulting lifetimes are shown in table 7.4.
AlMgS i10.6?R?0.2
0.7?R?0.2
0.8?R?0.2
0.9?R?0.2
200
250
300
350
-300 -200 -100 0 100 200
?
M
(M
Pa
)
42C rMo 4
0.7?Ry
0.98?Ry
0.8?Ry
0.9?Ry
600
700
800
900
1000
-800 -600 -400 -200 0 200 400
?
M
(M
Pa
)
?m (MPa)?m (MPa)
Figure 7.5: Distribution of the difierent tests loads. The left flgure corresponds to the 42CrMo4
steel, and the right flgure to the AlMgSi1 alloy.
42C rMo4
0.7?Ry
0.8?Ry
0.9?Ry
0.98?Ry
650
700
750
800
850
900
950
1000
-700 -500 -300 -100 100 300
?min (MPa)
?
m
ax
(M
Pa
)
AlMgSi1
0.6?Ry
0.7?Ry
0.8?Ry
0.9?Ry
150
200
250
300
350
400
-400 -300 -200 -100 0 100 200 300
?min (MPa)
?
m
a
x
(M
Pa
)
(a) (a)
(b)(b)
Figure 7.6: Schematic representation of the difierent between both distributions of the difierent
tests loads. The left flgure corresponds to the 42CrMo4 steel, and the right flgure to the AlMgSi1
alloy.
In flgure 7.6 we can observe that there is a big difierence between both distributions. Fur-
thermore, in this flgure we can observe the limits in the test distributions: Line (a) represents
the lower bound, deflned by the endurance limit; line (b) represents the higher bound, deflned by
the yield stress. Almost all the test results are between both lines, only four points are outside
this region. The aim of this is to know exactly the exact situation of the bounds.
CHAPTER 7. EXPERIMENTAL VALIDATION OF THE MODEL 129
Table 7.4: Resulting lifetimes for 42CrMo4 and AlMgSi1.
Material 42CrMo4 AlMgSi1
Nr. Test min max Ncycles min max Ncycles
1 17.51 955.50 65277 -182.15 327.87 19100
2 35.24 955.50 23700 -109.29 327.87 34000
3 55.27 955.50 52700 -72.86 327.87 42800
4 78.43 955.50 40900 36.43 327.87 153100
5 119.11 955.50 85900 -18.22 327.87 63800
6 108.91 955.50 124300 91.08 327.87 360400
7 124.21 955.50 222900 -182.15 291.44 28700
8 98.71 955.50 93500 -72.86 291.44 71700
9 -250.00 877.80 17281 -109.29 291.44 59900
10 -190.00 877.80 48787 -18.22 291.44 143500
11 -175.00 877.80 81244 36.43 291.44 326400
12 -360.00 877.80 3373 -182.15 255.01 48000
13 -305.00 877.80 21812 -145.72 255.01 57900
14 -332.50 877.80 7265 -109.29 255.01 113100
15 -167.50 877.80 125800 -18.22 255.01 348300
16 -500.00 780.23 11439 -63.75 255.01 172500
17 -437.50 780.23 14973 -71.42 218.58 526500
18 -375.00 780.23 32055 -281.42 218.58 37100
19 -312.50 780.23 483000 -246.42 218.58 54400
20 -343.75 780.23 36708 -211.42 218.58 80300
21 -265.63 780.23 532200 -176.42 218.58 96300
22 -281.25 780.23 123100 -141.42 218.58 175500
23 -550.00 682.73 183024 -106.42 218.58 172800
24 -620.00 682.73 19331
25 -525.00 682.73 347102
26 -585.00 682.73 33925
27 -500.00 682.73 381543
7.6 Parameter estimation
The parameter estimation is made in this section. For this, we cannot use simple software
tools, such Excel, but exists other specialized optimization softwares, as Mathematica or GAMS
(CONOPT solver), that make easier the job, specially with the implementation of the con-
straints.
7.6.1 General information
In Section 6.7 the steps proposed for the parameter estimation [40] were presented. A set
f( ?mi ; ?Mi) j i = 1; 2; : : : ; ng, of testing cases encompassing several stress level conditionsvarying ?M and ?m was selected to cover the desired region (Step 1).
Step 2: Choose the normalizing variables logN?0 and ?0. Choose logN0 and 0 and normalize
the data to dimensionless form. In both cases, the maximum stress was chosen as 0, 0 = 0:98Ry
MPa for 42CrMo4 and ?0 = 0:9Rp0:2 MPa for AlMgSi1. To deflne N0, the maximum number
130 7.6. PARAMETER ESTIMATION
of cycles were selected, that is, N0 = 532000 and N0 = 526500 cycles, respectively (see table 7.4).
Step 3: Estimate the model parameters. Use one of the estimation methods discussed in Section
6.6 to estimate the parameters C0 to C7 in the Gumbel model (6.13).
The implementation of the proposed estimation methods such as those described in Section
6.6 is not di?cult but cannot be easily implemented in simple software tools, such as Excel. The
use of a specialized optimization software, as Mathematica or GAMS for example, facilitates the
estimation job, and specially the implementation of constraints presented in Equations (6.22){
(6.35).
We have used the two difierent estimation methods, maximization of the loglikelihood
(6.62) and minimization of the regression equation (6.65). The parameter estimation will be
developed in the next subsections.
Step 3: Extrapolate to other testing conditions. Use the model (6.13) and the parameters C0
to C7 and logN0 and 0 for any other testing condition. This is illustrated in Section 7.7.
7.6.2 Parameter estimation for the 42CrMo4 steel
The parameters of all the cases considered have been estimated and the results are shown in
tables 7.5 for the 42CrMo4 steel.
Table 7.5: Parameter estimates for difierent estimation methods for the 42CrMo4 steel.
Case C0 C1 C2 C3 C4 C5 C6 C7
Max.Lik. -79.066 -63.141 85.309 38.297 0.000 0.000 2.394 0.000
L.Squares -77.338 -53.530 83.913 26.824 0.000 0.000 2.570 0.000
Figure 7.7 shows the experimental data and the resulting median curves according to the
fltted model. We can observe a reasonable flt [110].
S-N field
L.S.
0.80
0.95
1.10
1.25
1.40
0.001 0 .01 0 .1 1 10 100
Log N/N0
(M
Pa
)
Da ta 0.98? Ry
Da ta 0.9?Ry
Da ta 0.8?Ry
Da ta 0.7?Ry
0,98? Ry
0,9?Ry
0,8?Ry
0,7?Ry
S-N field
M.L.
0.80
0.95
1.10
1.25
1.40
0.001 0 .01 0 .1 1 10 100
Log N/N0
(M
Pa
)
Da ta 0.98? Ry
Da ta 0.9?Ry
Da ta 0.8?Ry
Da ta 0.7?Ry
0,98? Ry
0,9?Ry
0,8?Ry
0,7?Ry
??/
? 0
??/
? 0
Figure 7.7: S{N curves for constant ?M for Gumbel model with constraints for the 42CrMo4
steel using difierent methods: least squares (left side) and maximum likelihood (right side).
The variance-covariance matrix of the Gumbel parameter estimates C0 to C7 has been cal-
CHAPTER 7. EXPERIMENTAL VALIDATION OF THE MODEL 131
culated using formula (6.63) and the following matrix has been obtained for the 42CrMo4 steel:
0
BBBBBBBBBBBB@
423:723 ?386:651 ?334:520 ?5:221 1:930 ?0:627 0:602 367:882
?386:651 435:239 268:816 2:983 ?0:227 5:301 ?2:083 ?411:034
?334:520 268:816 283:325 4:445 ?1:889 ?0:974 0:451 ?256:652
?5:221 2:983 4:445 0:676 ?0:028 ?0:695 0:041 ?2:945
1:930 ?0:227 ?1:889 ?0:028 0:487 0:073 ?0:415 0:298
?0:627 5:301 ?0:974 ?0:695 0:073 0:966 ?0:042 ?4:726
0:602 ?2:083 0:451 0:041 ?0:415 ?0:042 0:672 2:012
367:882 ?411:034 ?256:652 ?2:945 0:298 ?4:726 2:012 388:523
1
CCCCCCCCCCCCA
:
7.6.3 Parameter estimation for the AlMgSi1 alloy
Similarly at 42CrMo4, the parameters of all the cases considered have been estimated and
presented in table 7.6 for the AlMgSi1 alloy. The graphical results are shown in flgure 7.8,
where we can observe a better flt than chromo [110].
Table 7.6: Parameter estimates for difierent estimation methods for the AlMgSi1 alloy.
Case C0 C1 C2 C3 C4 C5 C6 C7
Max. Lik. -78.507 -31.357 101.460 -34.960 0.000 0.000 13.302 -8.642
L.Squares -24.737 -16.091 31.803 -4.569 1.619 -1.619 2.837 -0.377
S-N fie ld
L.S .
0.70
0.85
1.00
1.15
1.30
1.45
1.60
0.01 0.1 1 10
Log N /N0
(M
Pa
)
S-N fie ld
M.L.
0.70
0.85
1.00
1.15
1.30
1.45
1.60
0.01 0.1 1 10
Log N /N0
(M
Pa
)
??/
? 0
Da ta 0.9?R?0.2
Da ta 0.8?R?0.2
Da ta 0.7?R?0.2
Da ta 0.6?R?0.2
0,9?R?0.2
0,8?R?0.2
0,7?R?0.2
0,6?R?0.2
Da ta 0.9?R?0.2
Da ta 0.8?R?0.2
Da ta 0.7?R?0.2
Da ta 0.6?R?0.2
0,9?R?0.2
0,8?R?0.2
0,7?R?0.2
0,6?R?0.2??/
? 0
Figure 7.8: S{N curves for constant ?M for Gumbel model with constraints for the AlMgSi1
alloy using difierent methods: least squares (left side) and maximum likelihood (right side).
Finally, the variance-covariance matrix of the Gumbel parameter estimates C0 to C7 has been
calculated using formula (6.63) and the following matrix has been obtained for the AlMgSi1 alloy:
0
BBBBBBBBBBBB@
206:358 ?247:932 ?69:555 ?17:021 12:080 ?10:425 8:345 196:913
?247:932 648:066 ?152:432 6:427 ?1:471 75:313 ?50:212 ?505:132
?69:555 ?152:432 195:600 13:421 ?11:664 ?36:919 27:165 115:082
?17:021 6:427 13:421 4:309 ?1:204 ?3:470 1:069 ?4:833
12:080 ?1:471 ?11:664 ?1:204 3:408 1:236 ?3:365 1:391
?10:425 75:313 ?36:919 ?3:470 1:236 13:511 ?6:889 ?58:325
8:345 ?50:212 27:165 1:069 ?3:365 ?6:889 10:118 39:604
196:913 ?505:132 115:082 ?4:833 1:391 ?58:325 39:604 394:455
1
CCCCCCCCCCCCA
:
132 7.7. PARAMETER VALIDATION
7.7 Parameter validation
After performing the parameter estimation, the quality of the fltted models were assessed. Some
methods for checking whether a fltted model is in agreement with the data were used.
This validation was done using the PP{plot and QQ{plot and the Kolmogorov-Smirnov and
Chi-square tests. For other treatment of the validation problem, see Drees, de Haan and Li [61]
and Fei, Lu and Xu [80].
Let x1; x2; :::; xn be a sample from a given population with cdf F (x). Let x1:n; x2:n; :::; xn:n
be the corresponding order statistics and p1:n; p2:n; :::; pn:n be plotting positions such as pi:n =
i=(n + 1). If F^ (x) is an estimate of F (x) based on x1; x2; :::; xn, then F^ (xi:n) is the estimated
probability corresponding to xi:n. The difierence between F^ (xi:n) and pi:n is an indication of the
quality of flt. The scatter plot of F^ (xi:n) versus pi:n; i = 1; 2; :::;n; is called a PP{plot, which is
very useful for model validation. If the model flts the data well, the graph will be close to the
45o line. Note that all the points in the PP{plot are inside the unit square [0; 1] ? [0; 1]. An
alternative and good complement to PP{plots are the QQ{plots, which represent the scatter plot
of the points xi:n versus F^ (xi:n). Figures 7.9 and 7.10 show these plots and the good agreement
between data and model.
A quantitative alternative to this model validation methods is the Kolmogorov-Smirnov test,
which can be used to test the log-Gumbel model 6.13, and uses the well known statistic:
D = max
1?i?n
F (H(N?i ))?
i? 1
n ;
1
n ? F (H(N
?
i ))
?
; (7.1)
where F is the Gumbel cdf and H(N?i ) is the sample order statistics (see Equation (6.61)). A
large value of D indicates that the model must be rejected as a valid model. We have calculated
the statistic in Equation (7.1) for the real sample and the corresponding estimated model (see
tables 7.5 and 7.6). In addition, to obtain a more precise critical value, we have simulated
1000 samples from the estimated model, and calculated the statistic in (7.1) using the simulated
samples and their resulting parameters estimates, obtaining a simulated sample of the statistic
in Equation (6.61), and the corresponding critical values, which are shown in tables 7.7 and 7.8,
where the signiflcance levels are also shown in parentheses. These tests reveal that the models
are far from being rejected at a signiflcant level of 0.05 for both materials.
The qualities of the fltted models (least squares and maximum likelihood) were also tested
using the Chi-square goodness of flt test, using 10 cells of the same probability. Since some
parameters were estimated, we have obtained the corresponding critical points for the Chi-
square tests by simulating 1000 samples and estimating the model parameters with the proposed
methods, following exactly the same process as the one indicated for the Kolmogoronov-Smirnov
test. The values of the Chi-square statistics and the corresponding critical values are indicated
in tables 7.7 and 7.8. Again, the models are far from being rejected at a signiflcance level of
0.05 for both materials.
In summary, the PP{plot and QQ{plot in flgures 7.9 and 7.10 and the Kolmogoronov-
Smirnov tests together with the Chi-Square tests indicate that the maximum likelihood and the
least squares based model provide from a practical point of view good and similar flts for the
42CrMo4 steel alloy and the AlMgSi1 aluminum alloy.
Thus, from all the above tests we can conclude that the log-Gumbel model is a good model
for both materials (42CrMo4 and AlMgSi1) [110].
One flnal conclusion that can be drawn from this example is that the constraints really
help when the sample size is small, because they reduce the efiect of outliers. The data in this
example really indicate that the sample sizes used are not as small as they could appear as a flrst
CHAPTER 7. EXPERIMENTAL VALIDATION OF THE MODEL 133
Table 7.7: Goodness of flt tests for the 42CrMo4 steel (in parenthesis the p value).
Case Nr. Kolmogoronov-Smirnov test Chi-Square test
tests Max. Difi. Critical Value Uniformity? ?2 Critical Value Ok?
M.L. 27 0.114 (0.36) 0.161 Yes 5.222 (0.86) 14.111 Yes
L.S. 27 0.099 (0.70) 0.556 Yes 5.222 (0.92) 56.333 Yes
Max.Lik.
0
0.2
0.4
0.6
0.8
1
0 0 .2 0.4 0 .6 0.8 1
P
P
L.Squares
0
0.2
0.4
0.6
0.8
1
0 0 .2 0.4 0 .6 0.8 1
P
P
P-P PlotP-P Plot
L.Squa re s
-5,5
-4,5
-3,5
-2,5
-1,5
-0,5
0,5
-5,5 -4,5 -3,5 -2,5 -1,5 -0,5 0,5
Q
Q
M ax. Lik.
-5,5
-4,5
-3,5
-2,5
-1,5
-0,5
0,5
-5,5 -4,5 -3,5 -2,5 -1,5 -0,5 0,5
Q
Q
Q-Q PlotQ-Q Plot
Figure 7.9: PP{plot for difierent analysis cases of the 42CrMo4 steel: least squares (left side),
maximum likelihood (right side).
look. The main reason is that important physical conditions and knowledge about the fatigue
problem has been used in deriving the model. As indicated, the model was not based on an
arbitrary selection of its functional form, but based on constraints of physical nature.
7.7.1 Validation of the theoretical example
Here, the validation of theoretical example presented in Section 6.8.1 is made. The methods and
steps used are the same than those for the experimental validation of chromo and aluminum
alloy.
134 7.7. PARAMETER VALIDATION
Table 7.8: Goodness of flt tests for the AlMgSi1 alloy (in parenthesis the p value).
Case Nr. Kolmogoronov-Smirnov test Chi-Square test
tests Max. Difi. Critical Value Uniformity? ?2 Critical Value Ok?
M.L. 23 0.111 (0.54) 0.183 Yes 7.000 (0.52) 14.826 Yes
L.S. 23 0.157 (0.51) 0.522 Yes 16.565 (0.37) 46.478 Yes
L.Square s
0
0.2
0.4
0.6
0.8
1
0 0 .2 0.4 0 .6 0.8 1
P
P
Max.Lik.
0
0.2
0.4
0.6
0.8
1
0 0 .2 0.4 0 .6 0.8 1
P
P
P-P Plot P-P Plot
L.Squa re s
-4
-3
-2
-1
0
1
-4 -3 -2 -1 0 1
Q
Q
Q-Q Plot
Max. Lik.
-4
-3
-2
-1
0
1
-4 -3 -2 -1 0 1
Q
Q
Q-Q Plot
Figure 7.10: PP{plot for difierent analysis cases of the AlMgSi1 alloy: least squares (left side),
maximum likelihood (right side).
The analysis of PP{plot (see flgure 7.11 (right side)) shows that relationship between model
probability and theoretical probability is almost a 45? line, so the flt of the model, with the
parameter estimated in section 6.8.1 were good.
The results of the Kolmogoronov-Smirnov and Chi-Square are shown in table 7.9. Note that
the results are better than those for the experimental validation model.
CHAPTER 7. EXPERIMENTAL VALIDATION OF THE MODEL 135
S-N field
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
1000 10000 100000 1000000 10000000
Log N
?
m
ax
R=-0.5
R=0.1
R=0.5
R=-0.5
R=0.1
R=0,5
0
0,2
0,4
0,6
0,8
1
0 0,2 0,4 0,6 0,8 1
P
P
PP plot
Figure 7.11: Representation of the Gumbel Model obtained in the example analyzed in section
6.8.1 (left side) and its PP{plot (right side).
Table 7.9: Goodness of flt tests for the theoretical example shown in section 6.8.1.
Nr. Kolmogoronov-Smirnov test Chi-Square test
tests Max. Difi. Critical Value Uniformity? ?2 Critical Value Ok?
30 0.067 0.242 Yes 1.333 11.071 Yes
7.7.2 Extrapolation
Once the parameter estimates are available, the model (6.13) can be used to extrapolate to other
testing conditions. For example, one can predict the expected lifetimes associated with other
constant values of ?M , plot the percentile curves, etc.
In table 7.10 the estimated percentile values associated with the difierent data points in flgure
7.12 for the 42CrMo4 steel are shown. Similarly, in table 7.11, the percentile values associated
with the difierent data points in flgure 7.13 for the AlMgSi1 alloy are shown [110].
Table 7.10: Estimated percentile values associated with the difierent data points using the
Gumbel fltted model for the 42CrMo4 steel.
Data p Data p Data p Data p
1 0.883 8 0.921 15 0.940 22 0.441
2 0.113 9 0.241 16 0.547 23 0.433
3 0.383 10 0.485 17 0.154 24 0.330
4 0.134 11 0.964 18 0.087 25 0.855
5 0.459 12 0.329 19 0.041 26 0.896
6 0.606 13 0.492 20 0.899 27 0.647
7 0.256 14 0.738 21 0.054
But, how can we know that really the model can be extrapolated to other load conditions?
We can answer to this question easily. If we know the parameter of one material, estimated
136 7.7. PARAMETER VALIDATION
S-N fie ld
=0.98?Ry
5
8
1
2
3
6
4
7
0.80
0.85
0.90
0.95
1.00
0.00 1 0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Da ta
S-N fie ld
0.9?Ry
15
12
13
14
11
9
10
1.00
1.10
1.20
1.30
1.40
0.00 1 0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Da ta
S-N fie ld
0.8?Ry
21
16
17
18
19 20
22
1.00
1.10
1.20
1.30
1.40
0.00 1 0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Da ta
S-N fie ld
0.7?Ry
25
23
27
26
24
1.20
1.25
1.30
1.35
1.40
0.00 1 0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Da ta
??
/?
0
??
/?
0
??
/?
0
??
/?
0
?M
?M=?M=
?M=
Figure 7.12: S{N curves representing constant ?M = 0:98; 0:9; 0:8 and 0:7Ry for the 42CrMo4
steel (from top to bottom and left to right). The percentiles 0:01, 0:05, 0:50, 0:95 and 0:99 are
represented.
Table 7.11: Estimated percentile values associated with the difierent data points using the
Gumbel fltted model for the AlMgSi1 alloy.
Data p Data p Data p Data p
1 0.544 7 0.472 13 0.048 19 0.873
2 0.872 8 0.967 14 0.828 20 0.155
3 0.451 9 0.214 15 0.371 21 0.720
4 0.083 10 0.369 16 0.258 22 0.006
5 0.754 11 0.454 17 0.427 23 0.941
6 0.774 12 0.366 18 0.728
with difierent series, and using the Gumbel model (6.13), we can delete one of these series and
estimate again the parameter for the model. Then, make the extrapolation of the model to the
deletes series and validate the flt of the curve.
All these processes are shown in the next flgures. The extrapolation has been done for the
AlMgSi1 alloy, described in Section 7.2, with the estimated parameter in Section 7.6.3. Figures
7.14, 7.15, 7.16 and 7.17 represent the re-estimation of the model without one of the initial
series. The parameters obtained after these new estimations are presented in table 7.12, all of
CHAPTER 7. EXPERIMENTAL VALIDATION OF THE MODEL 137
6
5
4
3
2
1
0.65
0.85
1.05
1.25
1.45
1.65
0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Dat a
S-N fie ld
=0.8?R?0.2
7
9
8
11
10
0.65
0.85
1.05
1.25
1.45
1.65
0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Dat a
S-N fie ld
=0.7?R?0.2
12 13
14
15
16
0.65
0.85
1.05
1.25
1.45
1.65
0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Dat a
S-N fie ld
=0.6?R?0.2
23
17
18
19
21
20
22
0.65
0.85
1.05
1.25
1.45
1.65
0.01 0.1 1 10
Log N/N0
(M
Pa
)
p= 0.01
p= 0.05
p= 0.5
p= 0.95
p= 0.99
Dat a
??
/?
0
?M
?M?M
??
/?
0
??
/?
0
??
/?
0
S-N f ield
0.9?R?0.2?M=
Figure 7.13: S{N curves representing constant ?M = 0:9; 0:8; 0:7 and 0:6Rp0:2 for the AlMgSi1
alloy (from top to bottom and left to right). The percentiles 0:01, 0:05, 0:50, 0:95 and 0:99 are
represented.
them have been estimated with maximum likelihood method (see Section 6.7.1). The PP{plot
of each new estimation (see flgure 7.18) shows the goodness of each estimation and, furthermore,
in table 7.13 uniformity test results are analyzed.
138 7.7. PARAMETER VALIDATION
S-N field
?max=0.9?Ry
6
5
4
3
2
1
0.65
0.85
1.05
1.25
1.45
1.65
1.85
0.01 0.1 1 10
Log N/N0
??
/s0
(M
Pa
)
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
?max=0.8?Ry
10
11
8
9
7
0.65
0.85
1.05
1.25
1.45
1.65
1.85
0.01 0,1 1 10
Log N/N0
/
(M
Pa
)
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
?max=0.7?Ry
12 13
14
15
16
0.65
0.85
1.05
1.25
1.45
1.65
1.85
0.01 0.1 1 10
Log N/N0
/
(M
Pa
)
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
?max=0.6?Ry
23
17
18
19
21
20
22
0.65
0.85
1.05
1.25
1.45
1.65
1.85
0.01 0.1 1 10
Log N/N0
/
(M
Pa
)
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
??
?
0
??
?
0
??
?
0
Figure 7.14: S{N fleld extrapolated to M = 0:9Ry (blue lines). The other S{N flelds represent
the curves obtained with the 0:8; 0:7 and 0:6Ry series data.
S-N field
ax=0.9?Ry
6
5
4
3
2
1
10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
?max=0.8?Ry
7
9
8
11
10
1 1 10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
ax=0.7?Ry
12
13
14
15
16
1 1 10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
ax=0.6?Ry
23
17
18
19
21
20
22
1 1 10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
?m
?m?m
0.65
0.85
1.05
1.25
1.45
1.65
0.01 0.1 1
Log N/N0
0.01 0.
Log N/N0
0.01 0.
Log N/N0
0.01 0.
Log N/N0
0.65
0.85
1.05
1.25
1.45
1.65
0.65
0.85
1.05
1.25
1.45
1.65
0.65
0.85
1.05
1.25
1.45
1.65
??
/?
0 (M
Pa
)
??
/?
0 (M
Pa
)
??
/?
0 (M
Pa
)
??
/?
0 (M
Pa
)
Figure 7.15: S{N fleld extrapolated to M = 0:8Ry (blue lines). The other S{N flelds represent
the curves obtained with the 0:9; 0:7 and 0:6Ry series data.
CHAPTER 7. EXPERIMENTAL VALIDATION OF THE MODEL 139
S-N field
?max=0.9?Ry
6
5
4
3
2
1
1 1 10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
ax=0.8?Ry
7
9
8
11
10
10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
ax=0.7?Ry
12 13
14
15
16
10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
ax=0.6?Ry
23
17
18
19
21
20
22
10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
?m
?m
?m
0.01 0.
Log N/N0
0.65
0.85
1.05
1.25
1.45
1.65
1 10.01 0.
Log N/N0
0.65
0.85
1.05
1.25
1.45
1.65
1 10.01 0.
Log N/N0
0.65
0.85
1.05
1.25
1.45
1.65
1 10.01 0.
Log N/N0
0.65
0.85
1.05
1.25
1.45
1.65
??
/?
0 (M
Pa
)
??
/?
0 (M
Pa
)
??
/?
0 (M
Pa
)
??
/?
0 (M
Pa
)
Figure 7.16: S{N fleld extrapolated to M = 0:7Ry (blue lines). The other S{N flelds represent
the curves obtained with the 0:9; 0:8 and 0:6Ry series data.
S-N field
?max=0.9?Ry
6
5
4
3
2
1
10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
ax=0.8?Ry
7
9
8
11
10
10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
ax=0.7?Ry
12 13
14
15
16
10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
S-N field
ax=0.6?Ry
23
17
18
19
21
20
22
10
p=0.01
p=0.05
p=0.50
p=0.95
p=0.99
Data
?m
?m
?m
1 10.01 0.
Log N/N0
0.65
0.85
1.05
1.25
1.45
1.65
1 10.01 0.
Log N/N0
0.65
0.85
1.05
1.25
1.45
1.65
1 10.01 0.
Log N/N0
0.65
0.85
1.05
1.25
1.45
1.65
1 10.01 0.
Log N/N0
??
/?
0 (M
Pa
)
0.65
0.85
1.05
1.25
1.45
1.65
??
/?
0 (M
Pa
)
??
/?
0 (M
Pa
)
??
/?
0 (M
Pa
)
Figure 7.17: S{N fleld extrapolated to M = 0:6Ry (blue lines). The other S{N flelds represent
the curves obtained with the 0:9; 0:8 and 0:7Ry series data.
140 7.7. PARAMETER VALIDATION
Table 7.12: Estimated parameter for difierent cases of study.
Serie Parameters
deleted C0 C1 C2 C3 C4 C5 C6 C7
Original -78.5072 -31.3573 101.4600 -34.9600 0.0000 0.0000 13.3021 -8.6420
0:9Ry -76.1105 -29.1199 87.8549 -21.9324 0.0000 0.0000 11.4951 -5.8264
0:8Ry -78.4884 -31.6460 101.1503 -35.2517 0.0000 0.0000 13.0518 -9.0942
0:7Ry -74.4392 -40.7260 94.5836 -22.3054 1.8062 0.0000 9.3937 -10.5329
0:7Ry -85.0225 -50.3607 105.2599 -20.9113 5.5017 0.0000 9.6941 -9.4481
Table 7.13: Goodness of flt test for the difierent cases of study.
Serie Nr. Kolmogoronov-Smirnov Chi-Square Test
deleted Test Max. Dif. Limit Uniformity? ?2 Critical Value ok?
Original 23 0.141 0.275 Yes 5.696 7.815 Ok
0:9Ry 17 0.074 0.318 Yes 1.588 7.815 Ok
0:8Ry 18 0.167 0.309 Yes 2.444 7.815 Ok
0:7Ry 18 0.194 0.309 Yes 4.222 7.815 Ok
0:6Ry 16 0.125 0.327 Yes 3.500 9.488 Ok
Withou t
0.9Ry
0
0,2
0,4
0,6
0,8
1
0 0,2 0,4 0,6 0,8 1
P
P
Without
0.8R y
0
0,2
0,4
0,6
0,8
1
0 0 ,2 0,4 0 ,6 0,8 1
P
P
Without
0.7R y
0
0,2
0,4
0,6
0,8
1
0 0 ,2 0,4 0 ,6 0,8 1
P
P
Without
0.6R y
0
0,2
0,4
0,6
0,8
1
0 0 ,2 0,4 0 ,6 0,8 1
P
P
Figure 7.18: Representation of the PP{plot obtained for difierent estimations. From the top
to bottom, and left to right: without 0:9Ry, without 0:8Ry, without 0:7Ry and 0:6Ry original
series data.
CHAPTER 7. EXPERIMENTAL VALIDATION OF THE MODEL 141
7.8 Conclusions
The main conclusions drawn from this chapter are:
? Two difierent materials (a low{alloy steel and an aluminum alloy) have been analyzed
and their model parameters obtained. The behaviors of these materials are difierent. The
results show less dispersion for the AlMgSi1 alloy than for the 42CrMo4 steel.
? The 42CrMo4 material shows linear or almost linear trend of the Wo?hler fleld, while the
AlMgSi1 material shows the typical curvature of S{N flelds.
? Difierent models have been obtained without parameter flxed. The model has the capacity
of choosing the best parameter for each material.
? The distribution of the points doesn?t afiect the parameter estimation, but a wide range
of data is necessary for a good estimation.
? Model validation have been made for both materials and, furthermore, for the theoretical
example presented. The results show the goodness flt of the models for difierent constant
ratios, i.e. R ratio or M .
? The capacity of the model to be extrapolated has been presented. Two difierent extrapo-
lation have been made: flrst, the P-S{N fleld for both materials, second re-estimation of
the model without some of the original data to validate the extrapolation.
? The extrapolation of the model can be done for any range of load, but better results are
obtained if the extrapolation is made when one of the middle series is deleted (see flgure
7.14 to flgure 7.17).
142 7.8. CONCLUSIONS
Chapter 8
Damage Measures and Damage
Accumulation
This chapter deals the problem of accumulating damage due to any fatigue load history.
First, some desirable properties for a damage model are discussed and difierent possible
alternatives fulfllling these properties are analyzed. Generally, current damage measures,
such as the Palmgrem-Miner?s rule, do not satisfy these properties. Next, a statistical fati-
gue regression model, able to predict the Wo?hler fleld for any combination of min, max
or R = min= max is presented.
This model is based on physical, statistical and compatibility conditions rather than on
arbitrary functions. The probability of failure is assumed to be the most suitable option
to assess damage measure. According to this, the procedure for obtaining the basic infor-
mation for constant load, i. e. the probabilistic Wo?hler fleld, is discussed, and formulas
for calculating the associated damage are given.
8.1 Introduction
The evaluation of fatigue damage is basic and very important in design of structures, because
fatigue is becoming determinant and the cause of a high number of failures of mechanical com-
ponents and structural elements in real practice.
Fatigue design of structures subject to varying loading is not possible without some basic
material fatigue characterization related to the stress range and stress levels, as the primary and
secondary parameters, respectively. This information allows one the evaluation of the damage
caused by the loading cycles associated with any given load spectrum. Generally, this important
information is obtained from tests conducted at constant stress level, either max, min, mean
or R = min= max, though Wo?hler curves obtained from tests run at mean = 0, i.e. for
completely reverse stress, are often preferred. The speed of the damage process depends on the
microstructure characteristics of the material and increases with the number of cycles, the stress
range and the stress level in every cycle, three factors to be taken into account when analyzing
damage.
143
144 8.2. DAMAGE MEASURES
The main objective of this chapter is to check the validity of the model developed in the
chapters below (chapter 6: Model derivation and chapter 7: Experimental validation) for fatigue
damage due to any stress history.
In most structural dynamic situations the loading process and hence the structural response
is assumed to be Gaussian [116], and so, very frequently, the damage accumulation is calculated
by simulating a Gaussian time history from a random process deflned by a power spectra density
(PSD) curve. Later, the rain- ow counting algorithm, which is assumed to be the best counting
method, and the Palmgrem-Miner rule ([95], [105]), frequently used due to its simplicity, are
applied to obtain an estimate of the structural fatigue life. The problem arises because in many
realistic applications the Gaussian hypothesis is not correct, since observed structural responses
are non-Gaussian ([81], [90]), due to either non-Gaussian external excitations (e.g. wave or
wind loads), or to the structural non-linearities. Difierent authors have pointed out that the
non-normal load can be responsible of an increase of the rate of fatigue damage accumulation
([83], [116] and [132]), so all the spectral methods valid for Gaussian loadings may provide non-
conservative estimates when applied to non-Gaussian loads [24]. In summary, constant loads
and lineal variable loads were used in order to illustrate damage evolution.
An important question was risen in Castillo et al. [41] what is damage? Or, equivalently
how is damage related to failure? A physical concept of failure, such as crack size, seems to be
adequate to deflne a service limit state, but fatigue failure, as an ultimate limit state, requires
a probabilistic framework allowing relating damage levels to probabilities of occurrence. In this
chapter we try to answer this question, furthermore, several methods to measure damage will
be analyzed and discuss to identify which is the best one and which of them are inadequate for
measuring damage ([41], [115]).
8.2 Damage measures
The problem of deflning and selecting damage measures must be done with care if one desires
they to be useful in practice. This problem is discussed in this section, with the main ideas
taken from Castillo et al. [41].
8.2.1 Some requirements for a damage measures
Consider a piece subject to a fatigue load. This piece has an initial damage, which is random.
The piece fails when the accumulated damage reaches a given threshold (critical) value. In
order to make sense and be a useful tool, a damage measure must be carefully selected to be
an indicator of the damage a certain piece of material has sufiered during its past life, including
fabrication.
Some properties to get a valid damage measure are:
Property 1.- Increasing with damage: The damage increases when increases the damage
measure.
Property 2.- Interpretability: The damage measures must provide a clear information of
the associated damage level.
Property 3.- Dimensionless measures: To avoid the problem of estimate the lifetime with
non-dimensionless variables, dimensionless variables should be chosen.
Property 4.- Known and flxed range: The range of variation of the damage measure must
be flxed and known, independently of the type of load and, if possible, of the material.
CHAPTER 8. DAMAGE MEASURES AND DAMAGE ACCUMULATION 145
Property 5.- Of known distribution: To know the probability of failure of a piece chosen
at random, its damage must have a known distribution.
8.2.2 Some damage measures
Several proposals of damage measures will be analyzed below. The aim of this study is to
illustrate the appropriateness and importance of the above properties [115].
Measures based on the number of cycles
As fatigue damage increases with the number of cycles N , the number of cycles to failure or any
increasing function of it are possible candidates for damage measures. Some possibilities are:
1. The number of cycles: The damage measure is the number of cycles N . This measure
does not satisfy Property 3 above (is not dimensionless), because if we use thousands or
dozens of cycles, instead of cycles, we have N/1000 or N/12, respectively.
According with the model (6.13), we have:
logNi ? G
B ? Elog? ? ? C ;
D
log? ? ? C
?
; (8.1)
and the range
0
B@e
B?
E
log? ? ? C ;1
1
CA depends on the stress range ? ?i , so it satisfles
Property 5 but not Property 4.
2. The logarithm of the number of cycles: Other alternative is the logarithm of the number
of cycles N. Unfortunately, this index satisfles neither Property 3 nor Property 4.
If logN is a Gumbel distribution (Equation (8.1)), and the range depends on the stress
level, this alternative has the same problems and limitations as the previous one.
3. The normalized logarithm of the number of cycles: A third alternative is the normalized
logarithm of the number of cycles to failure.
Di = logN
?
i
?i ; (8.2)
where ?i is the mean of logN?i when the stress range is ? ?i . Because this equation (8.2)
is a linear transformation, we have:
Di ? G
?i
?i ;
?i
?i
?
; (8.3)
but this measure it is not easily interpretable, so it does not satisfy Property 2. Further-
more Di does not satisfy Property 4.
4. The standardized logarithm of the number of cycles: The last option analyzed, is the
standardized logarithm of the number of cycles:
N?i =
logN?i ? ?i
i ; (8.4)
146 8.2. DAMAGE MEASURES
so that
N?i ? G
B log? ? + E
log? ? ? C ;
D
log? ? ? C
?
: (8.5)
This alternative does not depend on the stress range and material, so we can conclude
that N?i , N? and does not satisfy all the properties.
Based on the Palmgren-Miner number
1. The Palmgren-Miner number:
Mi = Ni=?0i; (8.6)
where ?0i is the mean value of the number of cycles to failure Ni for a stress level ? i.
The Palmgren-Miner number is dimensionless (because the random variable is divided by
its mean) so it satisfles Property 3. Taking logarithms one gets:
logMi = logNi ? log?0i (8.7)
and then
Mi ? logG(?i ? log?0i; ?i): (8.8)
The range of Mi is (e?i=?0i ;1) so it does not satisfy Property 4.
2. The logarithm the Palmgren-Miner number: The damage measure logMi has a Gumbel
distribution too (Equation (8.8)), but its range is (?i ? log?0i;1), and then, it does not
satisfy Property 4.
Based on the Gumbel variable
A very convenient candidate for a damage measure is the normalized variable Z of the Gumbel
model.
Normalization of the Gumbel model Normalization will be applied with the aim of estab-
lishing a relation among the fatigue data pertaining to difierent load levels, thus enabling us not
only a pooled parameter estimation, but the statistical interpretation of the damage measure to
be done.
We remind the reader that the cumulative distribution function of the two parameter Gumbel
family is given by:
F (x;?; ?) = 1? exp
?
? exp
?x? ?
?
?
;x ? ?;?1 < ? < 1; ? > 0; (8.9)
where ? and ? are the location and the scale parameters, respectively. When X follows a Gumbel
distribution G(x;?; ?), we write X ? G(x;?; ?). Its mean ? and variance 2 are:
? = ?? 0:57772?; (8.10)
2 = ?
2?2
6 : (8.11)
One important property of the Gumbel family is that it is stable with respect to location and
scale transformations, and also with respoect to minimum operations (see [33] and [46]). More
precisely:
X ? G(?; ?) , X ? ab ? G
?? a
b ;
?
b
?
(8.12)
CHAPTER 8. DAMAGE MEASURES AND DAMAGE ACCUMULATION 147
and
Xi ? G(?; ?) , Min(X1; X2; : : : ; Xn) ? G(?? ?n; ?): (8.13)
Let X be a random sample value coming from a given population. It is well known that,
no matter what statistical distribution X belongs to, it can be normalized by means of the
one-to-one transformation:
U = X ? ?X X ; (8.14)
where ?X and X are the mean value and the standard deviation of X, respectively, of the orig-
inal distribution, and U is the corresponding normalized value. The normalized distribution has
mean ?U = 0 and standard deviation U = 1. In addition, the new variable U is dimensionless.
According to Equation (8.12), if X follows a Gumbel distribution, as it is generally accepted
in the case of fatigue lives, U also follows a Gumbel distribution with parameters ?? and ??,
given by (see Equation 8.12):
?? = ?? ? =
?+ 0:57772? ? ?
??=p6 =
0:57772p6
? ; (8.15)
?? = ? =
?p6
?? =
p6
? : (8.16)
This demonstrates that the Gumbel parameters of the normalized distribution do not depend
on the parameters of the original distribution. Thus, it follows, that all distributions sharing
common parameters ? and ? transform, after normalization, to the same distribution.
An alternative normalization consists of using the random variable
Z = (logN
? ?B)(log? ? ? C)? E
D ; (8.17)
that is,
Z = C0 + C1 mi + C2 Mi + C3 mi Mi + (C4 + C5 mi + C6 Mi + C7 mi Mi) logNi: (8.18)
The normalized variable Z presented, satisfles all the Properties: increasing with damage,
interpretable, non-dimensional, with flxed range, and Z ? G(0; 1). Thus, we propose Z as a
convenient measure for the cumulative fatigue damage associated with a given stress level or
load history.
Based on the failure probability
In this subsection the failure probability as a damage measure is studied. According to our
Gumbel model (6.13), any number of cycles N below the zero-percentile curve leads to no
failure and, above this curve, there is a probability of failure dependent on N .
All specimens have maximum crack sizes below the failure size, in other words, below the zero
percentile curve, the crack grows but cannot reach failure for any possible initial crack siz. On
the contrary, above the zero percentiles, some specimens have cracks that have already reached
the failure size. Thus, above the zero-percentile curve, we can deflne our damage measure as
the failure probability (see [41], [115]):
PF = FG(B??i;?i)(logNi) = FD(D); (8.19)
148 8.3. CUMULATE DAMAGE ASSOCIATED WITH A GENERAL LOAD HISTORY
where PF is non-dimensional and has a uniform distribution (PF ? U(0; 1)). This measure
satisfles all desired Properties 1-5.
To deflne a damage measure below the zero-percentile, in the non-failure zone, a propor-
tionality criterion is used. This criteria was deflned by Castillo et al. [41], damage in this zone
is proportional to the number of cycles if the stress level is held constant. So, the probability
damage measure is:
PF =
8
<
:
Ni
N0;i ? 1 if Ni ? N0;i
FG(?i;?i)(logNi) if Ni > N0;i;
(8.20)
where N0;i is the zero-percentile number of cycles for a stress level ? i.
Finally we can conclude that a piece in the non-failure region has damage measure in the
range [?1; 0], where ?1 corresponds to zero cycles and 0 to the critical number of cycles Ni;0 in
the zero-percentile line. A piece in the failure region has a damage measure in the range (0; 1),
where 1 is the maximum damage measure, that corresponds to an inflnite life, only attainable
for the strongest possible piece.
Critical comparison
In this section, a comparison between all the cases studied is made. table 8.1 summarizes the
main characteristics and properties of several proposals of damage measures.
Some conclusions derived from this table are:
1. All measures increase with damage, as required.
2. Some measures are easier to evaluate than other.
3. All the measures are dimensionless except the measures based on Ni and logNi.
4. Some ranges are not independent of the material, load history or both. Only Z and Pf
satisfy this material independent property.
5. The measures Z and PF are the only ones that have a flxed statistical distribution, that
is, independent of the stress level.
6. Only measures Z and PF satisfy all the properties, so these measures are the best, followed
by N?.
8.3 Cumulate damage associated with a general load history
This section aims at studying the cumulative damage associated with a general load history
taking into account the in uence of the mean stress in damage accumulation (see Conway [53]
and Dowling [60]).
In flgure 8.1 we can see directly the number of cycles under a constant stress load with
the corresponding probability of failure, but it gives no direct information of the probability of
failure associated with a varying stress level history. An accumulation rule becomes necessary
to evaluate the damage produced by an arbitrary load history.
flgure 8.1 represents isodamage curves in the Wo?hler fleld, which are the curves corresponding
to the same damage level for difierent stress ranges. In this flgure two difierent load histories
leading to the same damage are shown. The flrst one is if the experiment is started (point
CHAPTER 8. DAMAGE MEASURES AND DAMAGE ACCUMULATION 149
Table 8.1: Properties and characteristics of difierent damage measures for the case of constant
stress levels (Legend: ? ? ?? very good, ? ? ? good, ?? medium and ? bad) [115].
Damage Property
measure Increasing Interpretability Dimesionless Range Distribution
Ni Yes ** No (e?i ;1) logG(B ? ?i; ?i)
logNi Yes * No (?i;1) G(B ? ?i; ?i)
Di = logNi?i Yes * Yes (?i=?i;1) G(?i=?i; ?i=?i)
N? = logNi ? ?i i Yes *** Yes (?
?
i ;1) G(??i ; ??i )
Mi = Ni?i Yes ** Yes (e
?i=?i ;1) logG(?i ? log?0i; ?i)
logMi = logNi ? log?i Yes * Yes (?i ? log ?0i;1) G(?i ? log ?0i; ?i)
Z = logNi ?B ? ?i?i Yes *** Yes (?1;1) G(0; 1)PF = FG(B+?i;?i) logNi Yes **** Yes (0; 1) U(0; 1)
?? = 0:57772
p6
? ; ?
? =
p6
?
A0) at stress level ? 1 for a number of cycles (until reaching point A1) and then changed to
another stress level ? 2 (point A2) for a number of cycles (until reaching point A3) and so on,
until the flnal curve is reached (point A7). In the second load history, everything occurs at
the reference level ? ref , where the points B1; :::; B7 are equivalent (in damage) to the points
A1; :::; A7, respectively. In other words, the four steps load history, moving from ? 1 and ? 4,
and the load history at the reference level ? ref lead to the same damage level. The important
consequence is that iso-damage curves permit the reduction of a load history to a probability of
failure, independently of the previous history of stress level or mean stress [41].
The following two rules permit damage evaluation of any stress history:
1. Iso-probability rule: Two load histories produce the same damage if the corresponding
probabilities of failure coincide.
2. Proportionality rule: The damage produced under the zero percentile curve is propor-
tional to the number of cycles, with a total maximum damage in that area being one unit.
8.3.1 Procedure to perform a damage analysis
Model (6.13) provides probabilistic bases for calculating the damage accumulation for any type
of loading being considered. In fact, due to the possible identiflcation of the probability of
failure P , represented by the percentile curves in the Wo?hler fleld for any min and max with
the damage state, the model can be used in cumulative damage calculations for fatigue life
prediction of components subject to complex loading histories.
To evaluate the accumulated damage, we proceed as follows:
1. The initial damage is set to 0 (no damage)
2. The damage p after the flrst cycle is calculated with the stress levels considered in this
cycle.
150 8.3. CUMULATE DAMAGE ASSOCIATED WITH A GENERAL LOAD HISTORY
??
3
Zero-percentile
No fatigue failure zone
N
4
2
1
I s o d a m a g e c u r v e s
ref
A4
A2
A3
A1
A5
A6
A7
B1 = B2 B7B3 = B4 B5 = B6
N1
N2
N3
N4
??
??
??
??
??
Figure 8.1: Illustration of the isodamage curves. flgure from [36].
3. The equivalent number of cycles associated with this damage level p for the stress levels
( ?m = ?m(N) and ?M = ?M (N)), for N? = 2, using the inverse function (quantile) of
(6.13), is calculated:
logN?eq =
log(? log(1? p))? (C0 + C1 ?m + C2 ?M + C3 ?m ?M )
C4 + C5 ?m + C6 ?M + C7 ?m ?M
(8.21)
4. The accumulated damage, represented by the probability of failure, is calculated by the
recursive formula:
PN+?N = F (N?eq +?N; ?m(N); ?M (N)); (8.22)
which gives the accumulated damage after N +?N cycles when the unit is subject to the
stress history given by ?m(N) and ?M (N).
5. After this point, the damage can be calculated based on the percentiles, only by repeating
steps 3 and 4 successively, until the damage level required is reached.
Figure 8.2 represent an theoretical example of application of this procedure:
1. The initial damage is equal to zero.
2. Knowing the flrst range of load, deflned by m1, M1 and N1, we obtain the probability
p1.
3. With the second range of load ( m2, M2, N2), we cannot obtain the probability p2 directly.
First we calculate the number equivalent of cycles (Neq1) in function of p1, to locate us
in the correct position inside the S{N fleld (flgure 8.2, (3) signal A). Then, with this
number of cycles and the increment between ?N = N2 ? N1 we obtain the probability
p2 = F (Neq1 +?N; m2; M2) (flgure 8.2, (3) signal B).
CHAPTER 8. DAMAGE MEASURES AND DAMAGE ACCUMULATION 151
??
Zero-percentile
N
p0
??
N
1??
(1)
p0
(2)
p1
N1
??
N
1??
p0
(3)
p1
N1 N2
?? 2
Neq1
A
B
??
N
1??
(4)
p2
N2
?? 2
Neq2
A
B?? 3
N3
??
N
1??
(5)
?? 2
?? 3
??
n
pn=1
Nn
N1 N3N2
??
3
??
2
??
1
?
(?m1,?M1,?1)
(?m2,?M2,?2)
(?m3,?M3,?3)
Figure 8.2: Theoretical example of application of the procedure for damage analysis.
152 8.4. EXAMPLE OF APPLICATIONS. VALIDATION OF DAMAGE ACCUMULATION
4. Like in the step nr. 3, we repeat all the procedure, this time with the load ( m3, M3,
N3), calculating flrst Neq2 and then p3.
5. Finally our specimen will be broken when the probability of failure will be equal to one
pn = 1.
8.4 Example of applications. Validation of damage accumula-
tion
As indicated, this model provides probabilistic bases for calculating the damage accumulation for
any type of loading being considered. In fact, due to the possible identiflcation of the probability
of failure (p), represented by the percentile curves in the Wo?hler fleld, with any damage state, the
model can be used in cumulative damage calculations for fatigue life prediction of components
subject to complex loading histories.
In this section difierent types of load are analyzed to validate the capacity of the model
described in chapter 6. These groups of loads are divided into two types:
1. Constant load, in which the specimen is subjected to a constant ? .
2. Variable load, in which the specimen is subjected to a non constant ? , but in which
there is a lineal relation between load and lifetime.
In all cases, the material used for this validation was 42CrMo4, and the parameter used are
those in chapter 7, section 7.6.2.
8.4.1 Constant loading
In this subsection a damage analysis of three difierent load histories is performed. All cases have
a constant ? (N) = 1130MPa, that correspond with ? ? = 1:099 ( 0 = 955:5 MPa), but each
one has difierent values of M , m and mean. The characteristics of each load are shown in
table 8.2. flgure 8.3 shows the difierent load histories.
Table 8.2: Characteristics of the three difierent load histories analyzed. Case of constant ? .
Case ? ? ?m ?M ?mean
(a) 1.099 -0.500 0.599 0.045
(b) 1.099 -0.399 0.700 0.151
(c) 1.099 (odd cycles) -0.399 0.700 (mean value)
(even cycles) -0.500 0.599 0.095
The results are presented in table 8.3 and flgure 8.4. In table 8.3 the difierent values of
the number of cycles for certain probabilities of failure (p = 0:01; 0:1; 0:5; 0:9 and p = 0:99) are
presented. The efiect of mean in the damage accumulation can be appreciated , i.e. higher
values of mean correspond with higher probabilities of failure.
mean?(a) < mean?(c) < mean?(b) ! N
?
(a) > N?(c) > N?(b)
CHAPTER 8. DAMAGE MEASURES AND DAMAGE ACCUMULATION 153
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 10 20 30 40 50
N*
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 10 20 30 40 50
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 10 20 30 40 50
?
*
(a) (b) (c)
N*N*
Figure 8.3: Load histories for the damage analysis, case of constant ? = 1050 MPa ! ? ? =
1:099 ( 0 = 955:5 MPa). (a) ?m = ?0:500, ?M = 0:599; (b) ?m = ?0:399, ?M = 0:700; (c)
?m = ?0:500, ?M = 0:599 for odd cycles and ?m = ?0:399, ?M = 0:700 for even cycles.
Table 8.3: Values of the number of cycles N? for the probabilities of p = 0:01; 0:1; 0:5; 0:9; 0:99
for constant ? and cases (a), (b) and (c).
Case Number of cycles N?
p = 0:01 p = 0:1 p = 0:5 p = 0:9 p = 0:99
(a) 10 51 188 434 704
(b) 1 5 15 30 45
(c) 3 8 27 55 85
Efiect the existence of a discontinuity on the damage accumulation
Here, the efiect of the existence of a discontinuity on the damage accumulation is analyzed. For
this, three difierent cases are studied (see flgure 8.5):
? Case a: When the discontinuity is situated at the beginning of the sequence (N? = 10).
? Case b: When the discontinuity is situated at the end of the sequence (N? = 100).
? Case c: When the discontinuities are situated at N? = 10 and N? = 30.
Table 8.4: Values of number of cycles N? for the probabilities of p = 0:01; 0:1; 0:5; 0:9; 0:99 when
there exits punctual high cycles in the load sequence.
Case Number of cycles N?
p = 0:01 p = 0:1 p = 0:5 p = 0:9 p = 0:99
Original 7 32 103 217 335
(a) 7 9 42 156 274
(b) 7 32 99 147 264
(c) 7 9 29 87 204
For all the cases the ? = 1000 MPa, corresponding with ? ? = 1:047 ( 0 = 955:5)MPa.
The punctual cycle increase the amplitude a 10%. The results shown in table 8.4 that the
154 8.4. EXAMPLE OF APPLICATIONS. VALIDATION OF DAMAGE ACCUMULATION
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000 100000
N*
Pr
ob.
(p)
)
(a) ?M*=0.598; ?m*=-0.5
(b) ?M*=0.7; ?m*=-0.399
(c) mix between (a) and (b)
Figure 8.4: Load histories for the damage analysis, case of constant ? = 1050 MPa ! ? ? =
1:099 ( 0 = 955:5 MPa). (a) ?m = ?0:500, ?M = 0:599; (b) ?m = ?0:399, ?M = 0:700; (c)
?m = ?0:500, ?M = 0:599 for odd cycles and ?m = ?0:399, ?M = 0:700 for even cycles.
probability of failure changes in function of where the discontinuity is situated. In all cases,
the specimen will break before than the original one (without discontinuity inside of the load
sequence). Otherwise, the worst case is the one corresponding to discontinuities at N? = 10 and
N? = 30 (case (c), flgure 8.5), because the mean is bigger and consequently the damage too.
Finally, when the situation of the discontinuities are difierent, the distribution of probabilities of
failure (cdf, flgure 8.6) is difierent. The increment of the probability of failure is function of the
equivalent number of cycles (N?eq) that depends also of the probability in the previous cycle (see
section 8.3.1 and Equations (8.21) and (8.22)), that is, the evolution of probabilities is difierent
although the increment of amplitude is the same in all the cases.
8.4.2 Variable loading
In this section difierent load histories are analyzed. All the sequences have a lineal amplitude
slope, but difierent m and M expressions (see flgure 8.7). Three difierent groups are studied:
1. Load history with constant ?m and variable ?M .
2. Load history with constant ?M and variable ?m.
3. Load history with variable ?M and ?m.
The general form of each expression is ? ? = m?N?+n = ?M? ?m, where ?M = m1 ?N?+n1
and ?m = m2 ? N? + n2. The values of the parameters m1;m2;m; n1; n2 and n are deflned in
table 8.5.
CHAPTER 8. DAMAGE MEASURES AND DAMAGE ACCUMULATION 155
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 10 20 30 40 50
N*
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
75 85 95 105 115 125
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
?
*
Original (a) Discontinuity at N*=10
(b) Discontinuity at N*=100 (c) Discontinuities at N*=10 and N*=30
N* N*
N*
Figure 8.5: Load sequences used for the analysis of damage accumulation when a discontinuity
appear in the sequence. From top to the bottom, and left to right: original sequence (without
discontinuity), discontinuity situated at N? = 10, discontinuity situated at N? = 100 and
punctual cycles situated at N? = 10 and N? = 30 (for N0 = 532000 cycles).
The aim of analyzing a two spectra in each group of load histories is to discover the efiect of
symmetrical spectra in the damage accumulation. Figure 8.8 represent a scheme of this analysis,
the aim is discover if the probability pA and the probability pB will be equal.
The cdfs obtained for each group of load histories are shown in flgure 8.9. It can be appreci-
ated how the second spectra of each group (designed by a2; b2 and c2 grow faster than the flrst
ones. This occurs because the second spectra damage begins to grow with a bigger value than in
the flrst one. Furthermore, the damage always grows, that is, if in the beginning the probability
is bigger, the failure will produce before than with the flrst spectra (designed by a1; b1 and c1).
Finally, if a comparison between all the difierent cdf?s is made, we can conclude that the
damage increase with the value of mean. In this example, we have analyzed six difierent load
156 8.4. EXAMPLE OF APPLICATIONS. VALIDATION OF DAMAGE ACCUMULATION
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000
N*
Pr
ob.
(p)
)
Original
(a) N*=10
(b) N*=100
(c) N*=10, and N*=30
Figure 8.6: Load sequences used for the analysis of damage accumulation when a discontinuity
appear in the sequence. From top to the bottom, and left to right: original sequence (without
discontinuities), discontinuity situated at N? = 10, discontinuity situated at N? = 100 and
discontinuities situated at N? = 10 and N? = 30 (for N0 = 532000 cycles).
Table 8.5: Parameter for the deflnition of the load histories expressions.
?M ?m ? ?
Case m1 n1 m2 n2 m n
a1 4:5 ? 10?4 0.50 0.00 -0.50 2:75 ? 10?4 0.50
a2 ?4:5 ? 10?4 0.66 0.00 -0.50 ?2:75 ? 10?4 0.58
b1 0.00 0.75 4:5 ? 10?4 0.10 2:75 ? 10?4 0.33
b2 0.00 0.75 ?4:5 ? 10?4 -0.25 ?2:75 ? 10?4 0.50
c1 1:8 ? 10?4 0.25 9 ? 10?5 -0.45 1:35 ? 10?4 0.33
c2 ?1:8 ? 10?4 0.59 ?9 ? 10?5 -0.62 ?1:35 ? 10?4 0.61
histories with the following ?mean expressions:
? a1: ?mean = 2:75 ? 10?4 ?N? + 0:50.
? a2: ?mean = ?2:75 ? 10?4 ?N? + 0:58.
? b1: ?mean = 2:75 ? 10?4 ?N? + 0:33.
? b2: ?mean = ?2:75 ? 10?4 ?N? + 0:50.
? c1: ?mean = 1:35 ? 10?4 ?N? + 0:33.
? c2: ?mean = ?1:35 ? 10?4 ?N? + 0:61.
CHAPTER 8. DAMAGE MEASURES AND DAMAGE ACCUMULATION 157
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 50 100 150 200
Ncycles
?
*
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
Ncycles
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 50 100 150 200
Ncycles
?
*
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 50 100 150 200
Ncycles
?
*
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 50 100 150 200
Ncycles
?
*
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 50 100 150 200
Ncycles
?
*
(a1) (b1) (c1)
(a2) (b2) (c2)
Figure 8.7: Variable load histories analyzed in the damage accumulation: (a1); (a2) constant ?m
and variable ?M , (b1; b2) constant ?M and variable ?m, (c1; c2) variable ?M and ?m.
N
?
??1 ??2
?N
m1
prob = pA
N
?
??1??2
?N
m1
prob = pB
(a) (b)
Figure 8.8: Schematic representation of the symmetrical problem.
If we choose a number of cycles, i.e. N? = 1 the ?mean becomes in:
a1 ! ?mean = 0:5001; a2 ! ?mean = 0:5798; b1 ! ?mean = 0:3252;
b2 ! ?mean = 0:4999; c1 ! ?mean = 0:3251; c2 ! ?mean = 0:6080
where it can be appreciated the failure sort of the difierent load histories:
meanc2 > meana2 > meana1 > meanb2 > meanb1 > meanc1
158 8.5. CONCLUSIONS
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000 100000
Pr
ob.
(p)
)
?M*=4.5?10 ?N +0.50;
?m*=-0.50;
?M*=-4.5?10 ?N +0.66;
?m*=-0.50;
-4
-4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000 100000
N*
Pr
ob.
(p
))
?M*=0.75;
?m*=-4.5?10 ?N +0.10;
?M*=0.75;
?m*=4.5?10 ?N -0.25;
-4
-4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000 100000
Pr
ob.
(p)
)
?M*=1.8?10 ?N +0.25;
?m*=-9?10 ?N -0.45;
?M*=1.8?10 ?N +0.59;
?m*=9?10 ?N -0.62
-4
-5
-5
-4
(b) (c)
N* N*
(c2)
(c1)
(a2)
(a1) (b1)
(b2)
Figure 8.9: Variable load histories analyzed in the damage accumulation: (a1); (a2) constant ?m
and variable ?M , (b1; b2) constant ?M and variable ?m, (c1; c2) variable ?M and ?m.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000 100000
N*
Pr
ob.
(p)
)
a1
a2
b1
b2
c1
c2
Figure 8.10: Variable load histories analyzed in the damage accumulation: (a1); (a2) constant
?m and variable ?M , (b1; b2) constant ?M and variable ?m, (c1; c2) variable ?M and ?m.
corresponding with
pfailurec2 > pfailurea2 > pfailurea1 > pfailureb2 > pfailureb1 > pfailurec1
and with
N?failurec2 < N
?
failurea2 < N
?
failurea1 < N
?
failureb2 < N
?
failureb1 > N
?
failurec1
Table 8.6 shows the difierent number of cycles N? for difierent probabilities of failure (p =
0:01; 0:1; 0:5; 0:9 and 0:99).
8.5 Conclusions
The main conclusions from this chapter are the following:
CHAPTER 8. DAMAGE MEASURES AND DAMAGE ACCUMULATION 159
Table 8.6: Values of number of cycles N? for the probabilities of p = 0:01; 0:1; 0:5; 0:9; 0:99 for
difierent variable load histories.
Case Number of cycles N?
p = 0:01 p = 0:1 p = 0:5 p = 0:9 p = 0:99
(a1) 161 229 283 317 337
(a2) 1 3 9 24 47
(b1) 355 501 623 700 744
(b2) 3 10 31 71 130
(c1) 1472 1607 1716 1785 1825
(c2) 1 2 7 20 34
? The normalized random variable in (8.17) described in Section 8.2.2 appears to be a
useful tool to facilitate the comparison of the cumulative fatigue damage produced by
difierent load histories involving constant or changing stress levels.
? A wide range of possible alternatives for selecting damage indices, including the logarithm
of the number of cycles, the number of cycles to failure, the Palmgren{Miner number, its
logarithm, its normalized or standardized form, the reference Weibull variable and the
failure probability, are possible, and have been described in this paper. However, some
of them are more convenient than others in the sense that they satisfy some desirable
properties, such as non{dimensionality, flxed range, interpretability, known distribution,
invariance with respect to load histories, etc.
? The probability of failure has been demonstrated to be the most convenient damage
measure for engineering design, due to the fact that it satisfles all desirable properties
and permits evaluating the probability of failure directly.
? The probability of failure is a very reasonable criterion for deflning cumulative damage
associated with difierent load histories. In fact, the Wo?hler percentile curves allow us an
easy interpretation of damage.
? The new model (6.13) is a good method for evaluating the damage in all ranges and load
histories: constant or variables.
? The mean stress is the most important factor in the damage accumulation process and it
gives to us a precious information. Understanding the variation of the mean stress in our
load history the damage accumulation will be obtained easily.
? The existence of discontinuities in a load sequence transform the mean value of the mean
stress, and this afiect in the damaage accumulation process as indicated in the previous
conclusion.
160 8.5. CONCLUSIONS
Part V
Conclusions
161
Chapter 9
Conclusions
This Chapter is the flnal chapter of the doctoral thesis presented in this document
and its objective is to summarize the conclusions drwan during the elaboration of the
doctoral thesis, and to present the future lines of research.
The Chapter is organized as follow: First, the general conclusions for each Chapter are
presented. Second, a summary of the original contribution is given. Finally, the future
lines of research are described.
9.1 Conclusions
The aim of this section is to present the most important conclusions obtained during the
elaboration of this doctoral thesis:
The new Gumbel fatigue model
? The model is based on statistical and physical considerations, in particular, on compati-
bility conditions in the Wo?hler fleld that leads to a system of functional equations.
? The model depends on 8 parameters that can be estimated by maximum likeli-
hood and also by non{linear regression methods. It supplies all the material basic
probabilistic fatigue information to be used in a damage accumulation assessment for
fatigue life prediction of structural and mechanical components under real loading spectra.
? The large number of constraints of the model provide us only physically and statistically
valid models, resulting good estimators of fatigue lifetime in the low-cycle range of fatigue
(see the example presented in Section 6.8.1).
? Once the parameters of the model have been estimated, the model allows us obtaining any
kind of Wo?hler fleld according to the testing condition chosen.
163
164 9.1. CONCLUSIONS
Experimental validation of the model
? Two difierent materials (a low{alloy steel and an aluminum alloy) have been analyzed
and their model parameters obtained. The behaviors of these materials are difierent. The
results show less dispersion for the AlMgSi1 alloy than for the 42CrMo4 steel.
? Difierent models have been obtained without flxed parameters. The model has the
capacity of choosing the best parameters for each material.
? The distribution of the points doesn?t greatly afiect the parameter estimation, but a wide
range of data is necessary for a good estimation.
? Model validation have been made for both materials and, furthermore, for the theoretical
example presented. The results show the goodness flt of the models for constant ratio: R
ratio or stress M .
? The capacity of the model to be extrapolated has been presented. Two difierent extrapo-
lation have been made: flrst, the P-S-N fleld for both materials, second re-estimation of
the model without some of the original data to validate the extrapolation.
? The extrapolation of the model can be done for any range of load, but better results are
obtained if the extrapolation is made when one of the middle series is deleted (see Figures
7.14 to 7.17).
Analysis of the damage accumulation
? The normalized random variable in Equation (8.17) described in Section 8.2.2 appears to
be a useful tool to facilitate the comparison of the cumulative fatigue damage produced
by difierent load histories involving constant or changing stress levels.
? A wide range of possible alternatives for selecting damage indices, including the logarithm
of the number of cycles, the number of cycles to failure, the Palmgren- Miner number,
its logarithm, its normalized or standardized form, the reference Weibull variable and the
failure probability, are possible, and have been described in this paper. However, some
of them are more convenient than others in the sense that they satisfy some desirable
properties, such as non-dimensionality, flxed range, interpretability, known distribution,
invariance with respect to load histories, etc.
? The probability of failure has been demonstrated to be the most convenient damage
measure for engineering design, due to the fact that it satisfles all desirable properties
and permits evaluating the probability of failure directly.
CHAPTER 9. CONCLUSIONS 165
? The probability of failure is a very reasonable criterion for deflning cumulative damage
associated with difierent load histories. In fact, the Wo?hler percentile curves allow us an
easy interpretation of damage.
? The model can be used for all range of load and for any type of load: constant or variable.
? The mean stress is the most important factor in the damage accumulation process and it
gives us a precious information. Understanding the variation of the mean stress in our
load history the damage accumulation will be obtained easily.
? The existence of discontinuities in a load sequence transform the mean value of the mean
stress, and this afiect in the damaage accumulation process as indicated in the previous
conclusion.
9.2 Summary of Contributions
The principal contributions provided in this doctoral thesis are:
? A new fatigue model useful for all range of load is deflned. The model can be used for the
analysis of material subjected to fatigue load in the range of tension-tension or tension-
compression test.
? The new fatigue model is a good tool to know the damage accumulation. The model
provides with a easy methodology good results in the analysis of the damage accumulation
with constant and variable load histories. In the case of random load histories, using a
second normalization, the model gives us a useful information of the damage process.
? A probabilistic useful model is developed. The advantages of this statistical approach to
the fatigue of materials are diverse, as the possibility of knowing the probability of failure
on a certain moment, or other statistical characteristics as the mean value of failure,
lifetime, cutofi of a material subjected to fatigue loads.
9.3 Future work
Finally, after the elaboration of this doctoral thesis, in which a new fatigue model is presented,
deflned and developed, a continuation of this research is planned. The principal ideas/subjects
that need more analysis in this research are:
1. Realize constant fatigue test with diverse metallic material with the objective of validate
the new Gumbel fatigue model and to know the characteristic parameter of more
materials. The methodology for this is explained in chapter 6 of this doctoral thesis.
2. Validate the new model for the low{cycle and for the high{cycle range of fatigue. This
doctoral thesis has validated the model for the central region of number of cycles, but for
166 9.3. FUTURE WORK
diverse application of the industrial engineering the materials are subjected to fatigue load
belonging in the other two ranges of fatigue lifetime.
3. Elaborate a research in which the objective is to flnd and deflne an optimum testing
strategy.
4. Study the efiect of other parameters in the damage accumulation process using the
Gumbel fatigue model (6.13). With respect to the in uence of the frequency, normally
is assumed that it has not in uence in the lifetime on the fatigue tests [102], but this is
not true when factors such as corrosion, high temperatures or random loads are analyzed.
This in uence depend of the material tested, and the type of test realized. Normally
this factor is analyzed when the fatigue cracks growth are studied (see [14], [16], [102] or
[119]). The efiect of high temperatures is also a good subject for a new line of research.
Its well known the use of metallic material in the aerospace industry, in which the ma-
terial is subjected to random fatigue load histories in combination with high temperatures.
5. Validate the model for other type of materials, such as composite materials, heterogenous
material (concrete), or new materials in which with diverse techniques the material is
modifled to obtain higher lifetimes.
Appendix A
Derivation of the Model
The functional equation (6.10) can be written as:
( ?M ? CM ( ?M ))Dm( ?m)?Dm( ?m) ?m ?DM ( ?M ) ?M +DM ( ?M )(Cm( ?m) + ?m) = 0; (A.1)
and solved as follows (see Acz?el [11] and Castillo et al. [49] and [46]):
0
BBB@
?M ? CM ( ?M )
1
DM ( ?M ) ?M
DM ( ?M )
1
CCCA =
0
BBB@
1 0
0 1
a0 b0
c0 d0
1
CCCA
?
?M ? CM ( ?M )
1
!
(A.2)
0
BBB@
Dm( ?M )
?Dm( ?m) ?m
?1
Cm( ?m) + ?m
1
CCCA =
0
BBB@
m0 n0
p0 q0
?1 0
0 1
1
CCCA
?
1
Cm( ?m) + ?m
!
(A.3)
with
?
1 0 a0 c0
0 1 b0 d0
!
0
BBB@
m0 n0
p0 q0
?1 0
0 1
1
CCCA =
?
0 0
0 0
!
(A.4)
from which one gets:
m0 = a0; n0 = ?c0; p0 = b0; q0 = ?d0: (A.5)
and replacing (A.5) into (A.2) and (A.3) and operating, one gets the solution of (6.10):
DM ( ?M ) =
a0d0 ? b0c0
a0 ? c0 ?M
(A.6)
CM ( ?M ) =
b0 ? ?M (d0 ? a0 + c0 ?M )
a0 ? c0 ?M
(A.7)
Dm( ?m) =
a0d0 ? b0c0
d0 + c0 ?m
(A.8)
Cm( ?m) =
b0 ? ?m(d0 ? a0 + c0 ?m)
d0 + c0 ?m
: (A.9)
where a0; b0; c0 and d0 are arbitrary constants.
167
168
Similarly, the functional equation (6.11) can be written as
ATB = 0; (A.10)
where
A =
0
BBBBBBBBBBBB@
BM ( ?M )CM ( ?M ) ?M ? EM ( M?) ?M ?BM ( ?M )( ?M )2
BM ( ?M )CM ( ?M )? EM ( ?M )
BM ( ?M ) ?M
CM ( ?M ) M ? ?( ?M )2
CM ( ?M )
BM ( ?M )
?M
1
1
CCCCCCCCCCCCA
(A.11)
and
B =
0
BBBBBBBBBBBB@
1
?Cm( ?m)? ?m
Cm( ?m) + 2 ?m
?Bm( ?m)
Bm( ?m)Cm( ?m)? Em( ?m) +Bm( ?m) ?m
?Cm( ?m) ?m ? ( ?m)2
?Bm( ?m)Cm( ?m) + Em( ?m)? 2Bm( ?m) ?m
Bm( ?m)Cm( ?m) ?m ?Em( ?m) ?m +Bm( ?m)( ?m)2
1
CCCCCCCCCCCCA
: (A.12)
To solve this functional equation, we write
0
BBBBBBBBBBBB@
BM ( ?M )CM ( ?M ) ?M ? EM ( M?) ?M ?BM ( ?M )( ?M )2
BM ( ?M )CM ( ?M )? EM ( ?M )
BM ( ?M ) ?M
CM ( ?M ) M ? ?( ?M )2
CM ( ?M )
BM ( ?M )
?M
1
1
CCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
fi fl ?
? ` ? ?
m1 n1 p1 q1
r1 s1 t1 u1
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
1
CCCCCCCCCCCCA
0
BBB@
CM ( ?M )
BM ( ?M )
?M
1
1
CCCA
(A.13)
0
BBBBBBBBBBBB@
1
?Cm( ?m)? ?m
Cm( ?m) + 2 ?m
?Bm( ?m)
Bm( ?m)Cm( ?m)? Em( ?m) +Bm( ?m) ?m
?Cm( ?m) ?m ? ( ?m)2
?Bm( ?m)Cm( ?m) + Em( ?m)? 2Bm( ?m) ?m
Bm( ?m)Cm( ?m) ?m ?Em( ?m) ?m +Bm( ?m)( ?m)2
1
CCCCCCCCCCCCA
=
0
BBBBBBBBBBBB@
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 ?1
a b c d
e f g h
m n p q
r s t u
1
CCCCCCCCCCCCA
0
BBB@
1
?Cm( ?m)? ?m
Cm( ?m) + 2 ?m
Bm( ?m)
1
CCCA
(A.14)
APPENDIX A. DERIVATION OF THE MODEL 169
with
0
BBB@
fi ? m1 r1 1 0 0 0
fl ` n1 s1 0 1 0 0
? p1 t1 0 0 1 0
? ? q1 u1 0 0 0 1
1
CCCA
0
BBBBBBBBBBBB@
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 ?1
a b c d
e f g h
m n p q
r s t u
1
CCCCCCCCCCCCA
=
0
BBB@
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1
CCCA (A.15)
from which we get:
a = ?fi; b = ??; c = ?m1; d = r1; m = ? ; n = ??; p = ?p1; q = t1; (A.16)
e = ?fl; f = ?`; g = ?n1; h = s1; r = ??; s = ??; t = ?q1; u = u1; (A.17)
From the fourth row of (A.13) and using CM ( ?m) from (A.7) one deduces that BM ( ?M )
must be of the following form:
BM ( ?M ) =
m2 + n2 ?M + p2( ?M )2
a0 ? c0 ?M
: (A.18)
From the flrst and second rows of (A.13) and using again CM ( ?m) from (A.7) one deduces
that EM ( ?M ) must be of the following for:
EM ( ?M ) =
m3 + n3 ?M + p3( ?M )2 + q3( ?M )3
( ?M ? n1)(a0 ? c0 ?M )2
: (A.19)
Similarly, from the flfth and sixth rows of (A.14) and using Cm( ?m) from (A.9) one can
derive the following two expressions for Bm( ?m) and Em( ?m), respectively:
Bm( ?m) =
m4 + n4 ?m + p4( ?m)2
d0 + c0 ?m
(A.20)
Em( ?m) =
m5 + n5 ?m + p5( ?m)2
(d0 + c0 ?m)2
: (A.21)
We note that in order to deduce that the numerator of Em( ?m) is a second degree polynomial
equation, one has also to use the seventh and eighth rows of (A.14) and compare the resulting
expressions for Bm( ?m) and Em( ?m) with those in (A.20) and (A.21).
Replacing (A.18), (A.19), (A.20) and (A.21) into (6.11) one obtains a polynomial in ?m and ?M , which must be identically equal to zero, that is, all its coe?cients must be null. This leadsto p2 = 0 and p4 = 0, which replaced into (6.7) provides the model:
2
664
logN? ?
?
Bm( ?m) + Em(
?
m)
? ? ? Cm( ?m)
?
Dm( ?m)
? ? ? Cm( ?m)
3
775
Am
=
2
664
logN? ?
?
BM ( ?M ) +
EM ( ?M )
? ? ? CM ( ?M )
?
DM ( ?M )
? ? ? CM ( ?M )
3
775
AM
= (A.22)
= 1? exp
'
? [C0 + C1 ?m + C2 ?M + C3 ?m ?M + (C4 + C5 ?m + C6 ?M + C7 ?m ?M ) logN?]A
?
(A.23)
in which the parameters have been redeflned.
170
Appendix B
Specimen Characterization
B.1 Material characterization
B.1.1 Metallographic test
Standard
The metallographic test is based on the ASTM E7-03 [4]. This standard describes the aim of
the test, terminology, and test procedure.
Procedure
Two solid rectangular specimens are necessary (dimensions 1 ? 1 ? 0:5 cm). Furthermore, to
analyze the C/S contents some grams of steel fllings are necessary. The material surface is flled.
The WD-XRF test measures the wave lengths of difierent chemical components (in percent-
age). Then, to know the C/S content in the material, a IR-Detection test is made. This test
deflnes the percentage of CO2 and SO2 that exists in the specimen.
Results: 42CrMo4
Table B.1: Result of metallographic test for the 42CrMo4 material
Element C Si Mn S P Cu Ni Cr Mo Fe
Content 0.40 0.24 0.76 0.033 0.012 0.18 0.11 1.2 0.23 Rest
DIN 1.7225 0.38-0.45 ?0.40 0.60-0.90 ?0.035 ?0.035 - - 0.90-1.20 0.15-0.30 Rest
B.1.2 Static tests
Standard
The tension test is based on the ASTM EM8-08 [7]. This standard describes the aim of the test,
terminology, and test procedure.
Procedure
Machines used for tension testing shall conform to the Practices E4 [6]. To transmit the measured
force applied by the testing machine to the test specimens various types of gripping devices may
171
172 B.1. MATERIAL CHARACTERIZATION
be used. To ensure axial tensile stress within the gage length, the axis of the test specimens
should coincide with the center line of the heads of the testing machine. In flgure B.1 some
pictures of the testing machine are shown.
(a) (c)(b)
Figure B.1: Tension testing machine. (a) machine, (b) detail of specimens in the machine, (c)
detail of the broken specimen.
Figure B.2 and flgure B.3 present the dimensions of the specimens used in these tests for the
42CrMo4 steel and the AlMgSi1 alloy respectively.
Figure B.2: Geometric deflnition of the tension test specimen for the 42CrMo4 steel.
APPENDIX B. SPECIMEN CHARACTERIZATION 173
Figure B.3: Geometric deflnition of the tension test specimen for the AlMgSi1 alloy.
Results
Tables B.2 B.3 present the static parameters of the 42CrMo4 and the AlMgSi1 materials.
Figures B.4 to B.6 show the ?{ curves obtained in the tension tests. Figures B.7 to B.9
show the corresponding curves for the AlMgSi1 material.
Table B.2: Static tests results for the 42CrMo4 steel.
Nr. Date T d Ry Rp0.2 Fm Rm Ag A E modul
Test [C] [mm] [MPa] [MPa] [kN] [MPa] [%] [%] [MPa]
1 13=06=07 21.70 9.97 978.00 966.00 83.13 1065 6.00 14.10 201820
2 14=06=07 21.70 9.97 976.00 969.00 83.40 1068 6.00 14.00 206610
3 15=06=07 21.70 9.97 972.00 967.00 83.40 1068 5.90 14.10 206180
? 21.70 9.97 975.33 967.33 83.31 1067 5.97 14.13 204870
2 0.00 0.00 2.16 1.08 0.11 1.23 0.04 0.04 1873.91
Table B.3: Static tests results for the AlMgSi1 alloy.
Nr. Date T d Rp0.2 Fm Rm Ag A E modul
Test [C] [mm] [MPa] [kN] [MPa] [%] [%] [MPa]
1 31=07=07 21.50 7.77 359.00 18416 388.00 6.00 10.50 72308
2 31=07=07 21.60 7.77 369.00 18789 396.00 6.00 10.60 72258
3 31=07=07 21.70 7.77 365.00 18557 391.00 5.80 10.40 71681
? 21.60 7.77 364.33 18587 391.7 5.93 10.5 72082.33
2 0.07 0.00 3.56 1.33 2.86 0.08 0.07 246.40
174 B.1. MATERIAL CHARACTERIZATION
TEST Nr.1
Rm=1065 MPa
Ry = 973 MPa
Rp0.2=966M Pa
E=201820 MPa
0
200
400
600
800
1000
1200
0 2 4 6 8 10 12 14 16
? (% )
?
(M
Pa
)
Figure B.4: Static test result for the 42CrMo4 steel. Specimen Nr. 1.
TEST Nr.2
Rm=1068 MPa
Ry = 977 MPa
Rp0.2=969M Pa
E=206610 MPa
0
200
400
600
800
1000
1200
0 2 4 6 8 10 12 14 16
? (%)
?
(M
Pa
)
Figure B.5: Static test result for the 42CrMo4 steel. Specimen Nr. 2.
APPENDIX B. SPECIMEN CHARACTERIZATION 175
TEST Nr.3
Rm=1068 MPa
Ry = 978 MPa
Rp0.2=967M Pa
E=206180 MPa
0
200
400
600
800
1000
1200
0 2 4 6 8 10 12 14 16
? (% )
?
(M
Pa
)
Figure B.6: Static test result for the 42CrMo4 steel. Specimen Nr. 3.
TEST Nr.1
Rp0.2=359 MPa
E=72308MPa
0
100
200
300
400
500
0 2 4 6 8 10 12
? (% )
?
(M
Pa
)
Figure B.7: Static test result for the AlMgSi1 alloy. Specimen Nr. 1.
176 B.1. MATERIAL CHARACTERIZATION
TEST Nr.2
Rp0.2=369 MPa
E=72258MPa
0
100
200
300
400
500
0 2 4 6 8 10 12
? (% )
?
(M
Pa
)
Figure B.8: Static test result for the AlMgSi1 alloy. Specimen Nr. 2.
TEST Nr.3
Rp0.2=364.3 MPa
E=71681 MPa
0
100
200
300
400
500
0 2 4 6 8 10 12
? (% )
?
(M
Pa
)
Figure B.9: Static test result for the AlMgSi1 alloy. Specimen Nr. 3.
APPENDIX B. SPECIMEN CHARACTERIZATION 177
B.2 Geometric specimens deflnition
B.2.1 42CrMo4
The geometrical specimen deflnition for the 42CrMo4 steel is shown in flgure B.10.
Figure B.10: Geometric fatigue test deflnition for the 42CrMo4 steel.
B.2.2 AlMgSi1
The geometrical specimen deflnition for the AlMgSi1 alloy is shown in flgure B.11.
Figure B.11: Geometric fatigue test deflnition for the AlMgSi1 alloy.
178 B.2. GEOMETRIC SPECIMENS DEFINITION
Appendix C
Experimental Protocols
C.1 Constant load tests
Next tables present the experimental protocol made in the constant load test for the materials
analyzed.The table shown by difierent columns the number of test (Nr.), the diameter of the
specimen (`), the maximum stress ( M ), the static and dynamic force (FStat and FDyn), the
machine use, the frequency (f), the start and flnish date and the number of cycles to failure
obtained in each test point.
C.1.1 42CrMo4 steel
Table C.1: Experimental protocol for the 42CrMo4 steel. Constant load tests
Nr. ` M FStat FDyn Machine f Date Cycles
[mm] [MPa] [kN ] [kN ] [Hz] Start End
1 8.00 955.50 24.46 23.58 Rumul 101.36 25/06/2007 25/06/2007 65277
2 8.00 955.50 24.90 23.13 Rumul 88.90 25/06/2007 25/06/2007 23700
3 8.00 955.50 25.41 22.63 Rumul 90.00 26/06/2007 26/06/2007 52700
4 8.00 955.50 25.97 22.03 Rumul 90.10 26/06/2007 26/06/2007 40900
5 8.00 955.50 27.03 21.04 Rumul 90.29 24/08/2007 25/08/2007 85900
6 8.00 955.50 26.75 21.28 Rumul 90.50 25/08/2007 25/08/2007 124300
7 8.00 955.50 27.14 20.90 Rumul 90.70 26/08/2007 26/08/2007 222900
8 8.00 955.50 26.51 21.54 Rumul 89.34 29/06/2007 29/06/2007 93500
9 8.01 877.80 15.80 28.38 Torsion-Tension 3.00 23/07/2007 24/07/2007 17281
10 8.01 877.80 17.33 26.90 Torsion-Tension 5.00 24/07/2007 24/07/2007 48787
11 8.00 877.80 17.66 26.46 Torsion-Tension 5.00 08/08/2007 13/08/2007 81244
12 8.00 877.80 13.01 31.11 Torsion-Tension 2.00 08/08/2007 08/08/2007 3373
13 8.01 877.80 14.41 29.76 Torsion-Tension 3.00 16/08/2007 16/08/2007 21812
14 8.01 877.80 13.72 30.46 Torsion-Tension 3.00 16/08/2007 16/08/2007 7265
15 8.00 877.80 17.85 26.27 Rumul 90.77 15/08/2007 15/08/2007 125800
16 8.00 780.27 7.05 32.18 Torsion-Tension 3.00 14/08/2007 14/08/2007 11439
17 8.00 780.27 8.61 30.60 Torsion-Tension 4.00 14/08/2007 14/08/2007 14973
18 8.00 780.27 10.19 29.04 Torsion-Tension 7.00 14/08/2007 14/08/2007 32055
19 8.00 780.27 11.76 27.46 Rumul 90.78 08/08/2007 08/08/2007 483000
20 8.00 780.27 10.97 28.24 Torsion-Tension 3.00 15/08/2007 15/08/2007 36708
179
180 C.1. CONSTANT LOAD TESTS
Nr. ` M FStat FDyn Machine f Date Cycles
[mm] [MPa] [kN ] [kN ] [Hz] Start End
21 8.00 780.27 12.94 26.30 Rumul 90.82 15/08/2007 15/08/2007 532200
22 8.00 780.27 12.54 26.69 Rumul 90.74 14/08/2007 14/08/2007 123100
23 8.00 682.73 3.34 30.99 Torsion-Tension 3.00 30/07/2007 31/07/2007 183024
24 8.00 682.73 1.58 32.77 Torsion-Tension 3.00 31/07/2007 31/07/2007 19331
25 8.00 682.73 3.97 30.36 Torsion-Tension 6.00 13/08/2007 14/08/2007 347102
26 8.00 682.73 2.46 31.86 Torsion-Tension 1.00 15/08/2007 16/08/2007 33925
27 8.00 682.73 4.59 29.74 Torsion-Tension 7.00 14/08/2007 15/08/2007 381543
C.1.2 AlMgSi1 alloy
Table C.2: Experimental protocol for the AlMgSi1 alloy. Constant load tests
Nr. ` M FStat FDyn Machine f Date Cycles
[mm] [MPa] kN kN [Hz] Start End
1 8.00 327.87 3.66 12.82 Rumul 78.02 17/08/2007 17/08/2007 19100
2 8.00 327.87 5.49 10.99 Rumul 78.90 17/08/2007 17/08/2007 34000
3 8.00 327.87 6.41 10.07 Rumul 78.27 26/08/2007 26/08/2007 42800
4 8.00 327.87 9.16 7.32 Rumul 78.23 25/08/2007 25/08/2007 153100
5 8.00 327.87 7.78 8.70 Rumul 78.21 27/08/2007 27/08/2007 63800
6 8.00 327.87 10.53 5.95 Rumul 78.29 27/08/2007 27/08/2007 360400
7 8.00 291.44 2.75 11.90 Rumul 78.20 17/08/2007 17/08/2007 28700
8 8.00 291.44 5.49 9.15 Rumul 78.27 27/08/2007 27/08/2007 71700
9 8.00 291.44 4.58 10.07 Rumul 78.26 20/08/2007 20/08/2007 59900
10 8.00 291.44 6.87 7.78 Rumul 78.25 27/08/2007 27/08/2007 143500
11 8.00 291.44 8.24 6.41 Rumul 78.12 27/08/2007 27/08/2007 326400
12 8.00 255.01 1.83 10.98 Rumul 78.35 20/08/2007 20/08/2007 48000
13 8.00 255.01 2.75 10.07 Rumul 78.19 06/09/2007 06/09/2007 57900
14 8.00 255.01 3.66 9.15 Rumul 78.40 20/08/2007 20/08/2007 113100
15 8.00 255.01 5.95 6.86 Rumul 78.33 11/09/2007 11/09/2007 348300
16 8.00 255.01 4.81 8.01 Rumul 78.35 11/09/2007 11/09/2007 172500
17 8.00 218.58 -1.58 12.57 Rumul 78.28 12/09/2007 12/09/2007 37100
18 8.00 218.58 -0.70 11.68 Rumul 78.30 12/09/2007 12/09/2007 54400
19 8.00 218.58 0.18 10.81 Rumul 78.23 12/09/2007 12/09/2007 80300
20 8.00 218.58 1.06 9.92 Rumul 78.33 13/09/2007 13/09/2007 96300
21 8.00 218.58 1.94 9.06 Rumul 78.42 13/09/2007 13/09/2007 175500
22 8.00 218.58 2.82 8.17 Rumul 78.44 13/09/2007 13/09/2007 172800
23 8.00 218.58 3.70 7.29 Rumul 78.42 13/09/2007 13/09/2007 526500
Appendix D
SISIFO program
D.1 Introduction
This appendix summarizes the work done by the author during her stay at the Empa Reseach
Institute.
The principal objectives are:
1. Solve the problem by using difierent programs to flrst estimate the model parameters and,
then, extend the analysis of difierent situations that appear in the daily practice of fatigue
of materials.
2. Develop a compiler tool (Matlab) for the new fatigue model together with an a user
interface (SISIFO) to facilitate the use of the program.
3. Present the difierent parts of the program and explaining the objectives of each one.
4. Describe all the components of the program.
5. Show by means of simple examples how the program works.
The report is organized as follows:
1. The fatigue model considerations are presented in section D.2 together with the required
background and how the problem can be solved.
2. In section D.3 the program (SISIFO) is presented. Difierent sections and subsections
describe its components and the steps needed to use it.
3. Section D.4 describes a general example of application in order to facilitate the under-
standing of the complete process of the parameter estimation and the damage analysis.
D.2 Background of SISIFO program: the new fatigue Castillo?s
model
In this section the most important considerations related to program codes, and the main sim-
pliflcations made are described.
181
182 D.3. PROGAM SISIFO
D.2.1 Simpliflcation of Castillo?s model
In this work one of the submodels presented in section 6.5.6 is implemented as a computer
program. This model depends on 4 parameters (C0, C1, C2 and C5), the rest of parameters are
assumed null, that is, C3 = C4 = C7 = 0 and C6 = ?C5. The flnal expression of the model is:
p = 1? exp f? exp [C0 + C1 ?m + C2 ?M ? (C5( ?M ? ?m)) logN?]g ; (D.1)
where p = F (N?; ?m; ?M ), which supplies a complete probabilistic information for any Wo?hler
curves of the material related to whichever given stress level.
Finally, the restrictions of this submodel stated in Equation (6.13) are:
C0 ? 0;
C5 ? 0;
min(logN?) ? ?C1=C5;
min(logN?) ? C2=C5; (D.2)
D.2.2 Parameter estimation
The parameter estimation of the model was given in section 6.6, and the program is prepared
to estimate the parameter by two methods (maximum likelihood and least{squares).
D.2.3 Validation of the model
The methods used to validate the model were shown in section 7.7.
D.2.4 Damage accumulation
The damage accumulation is based on the methodology presented in section 8.3.1.
D.3 Progam SISIFO
D.3.1 Introduction and general organization
The interface SISIFO has been created to solve the problems of estimation and analysis of
the new fatigue model deflned and described by Castillo et al. [40]. The principal objective of
the program is to reproduce all the necessary steps (described in section 6.7) for the complete
analysis of a material subjected to a certain load history. Other secondary objectives of the
program are:
1. Provide the necessary information for the correct use of the difierent components in the
program, such as type of data, flles, etc.
2. Work with data series from experimental tests. Make the normalization of data following
strategy described in chapter 6.
3. Estimate the model parameters, characteristic of each material.
4. Validate the model resulting after the estimation process.
5. Obtain the S{N curves and the P{S{N curves.
APPENDIX D. SISIFO PROGRAM 183
6. Deflne the method of extrapolation to other load conditions.
7. Analyze the damage accumulation of a specimen subject to a non constant load history.
8. Perform the rain ow analysis and flltering of the rain ow matrix depending on the en-
durance limit of the material.
SISIFO has been created using the GUIDE tool of the MATLAB program. GUIDE
(Graphical User Interface Development Environment) is a set of tools that can extend all the
options of MATLAB, designing easily interfaces (GUIs). A GUI is a graphical user interface
that contains components, that enable a user to perform interactive tasks. SISIFO is a set of
menus, toolbars, push buttons, radio buttons, pop-up menus etc, in which additional plots are
displayed easily.
The flrst screen of SISIFO represents the start of the process. The program is divided in
two basic parts: parameter estimation and damage analysis (see flgure D.1 and flgure D.2). The
parameter estimation menu permits the user to analyze series of experimental data and estimate
the set of parameters of the Castillo model. Furthermore the user can plot the S{N curves, the
P{S{N curves, validate the model and extrapolate the problem to other load conditions. The
second part of the program permits deflning a load history (given by the user) and analyzes the
damage accumulation of this load history. In addition a rain ow analysis can be performed.
Figure D.1: Start screen of the SISIFO program.
The methodology of work is presented in flgure D.3 where the steps mentioned in section 6.7
are represented.
The chapter is organized in two sections and diverse subsections in which all the components
of the program are described. In section D.4 some examples of application will be explained.
184 D.3. PROGAM SISIFO
Figure D.2: Main windows of the program: the parameter estimation window (top flgure) and
the damage analysis window (bottom flgure).
APPENDIX D. SISIFO PROGRAM 185
Step 1:
Testing Strategy
Step 4:
Extrapolation
Step 3:
Parameter Estimation
Step 2:
Normalization of
the variables
Figure D.3: Representation of the difierent steps to use the Castillo model in the SISIFO pro-
gram.
D.3.2 Program installation
For the use of the SISIFO program no installation process is necessary, because the executable
flle called sisifoV22.exe can be directly used.
If MATLAB is not installed, we need to install a MATLAB component called mcrIn-
staller.exe. The steps to be followed are:
1. Install the flle called mcrInstaller.exe from the program CD. With this installation the
computer is prepared to execute MATLAB codes. For the complete installation the user
must follow the steps described in flgures D.4 and D.5.
2. Execute the flle sisifoV22.exe.
3. Use the program.
186 D.3. PROGAM SISIFO
(1) (2)
(4)(3)
Figure D.4: Representation of the difierent steps to be followed for the installation of the SISIFO
program. Steps 1 to 4.
D.3.3 Menu 1: parameter estimation
In this section the parameter estimation menu is described. To access this part of the program,
the option Parameter estimation has to be chosen from the Option menu (see flgure D.6).
The parameter estimation window is formed by four panels: Data acquisition, Parameter
estimation, Model validation and Extrapolation (see flgure D.7).
Data acquisition
The objective of this panel is to load data from experimental tests and prepare them for the
Parameter estimation in the next panel. The homogenization of the data is also realized inside
this panel.
The methodology is the following:
APPENDIX D. SISIFO PROGRAM 187
(5)
(8)(7)
(6)
Figure D.5: Representation of the difierent steps to be followed for the installation of the SISIFO
program. Steps 5 to 8.
1. Deflne the maximum number of cycles NRunOut.
2. Choose the testing conditions case ( M = cte, R = cte).
3. Load data from an external flle.
4. Plot data and homogenize data.
188 D.3. PROGAM SISIFO
Figure D.6: Entering the Parameter estimation menu of SISIFO.
If the user specifles an incorrect number of cycles, i.e. with ortographic mistakes, a screen
with an error dialog box will appear, informing about this error and how to solve it. One example
is shown in flgure D.8.
There are two difierent cases of test conditions that SISIFO is able to analyze: test with
M = cte or test with R = cte, where M is the maximum stress and R is the stress ratio,
deflned as R = m= M . The user can choose the corresponding option from the pop-up menu
(see flgure D.9).
The data to be analyzed is read by the program by using the button Load Data (see flgure
D.10). The data flle is an excel flle .xls. The structure of this flle is shown in flgure D.11:
? First column: Number of cycles N . It represents the abscissas of the S{N curve.
? Second column: ? in the case of M = cte or M in the case of R = cte. It represents
the ordenate of the S{N curves.
? Third column: The constant value chosen in the pop-up menu ( M or R).
? Fourth column: Outliers are deflned by a value of 1.
Before plotting data, the user has to specify the difierent data labels such as the name of the
data set and axes (a panel will appear on the screen after loading of the data flle). In this panel
the following information appears (see flgure D.12):
? Number of data sets: a real number. A maximum of four data set is permitted. Normally
three or four sets are recommended.
? Label of each data sets.
? Axes information, the label of each axis.
If the user wants to enter a Greek symbol, i.e. ;? , MATLAB is prepared to accept
LaTeX code. In table D.1 some of the most useful codes are presented:
APPENDIX D. SISIFO PROGRAM 189
Panel 1
:
Data adquisitio
n
Panel 4:
Extrapolatio
n
Panel 3
:
Model validatio
n
Panel 2
:
Parameter estimatio
n
Figure D.7: Panels that form the Parameter estimation window of SISIFO.
190 D.3. PROGAM SISIFO
Figure D.8: Deflnition of the maximum number of cycles (run outs). Left side, a correct deflni-
tion; right side, bad input.
Figure D.9: Example of the testing option chosen by the user for the data analysis.
Table D.1: Useful symbols in MATLAB code.
Symbol Code Symbol Code
nsigma ? nDelta n sigma
ab (subindex) a b ab (superindex) ab^
logN nlog N exp a n exp a
APPENDIX D. SISIFO PROGRAM 191
Figure D.10: Example for loading data.
Figure D.11: Example of data flle.
192 D.3. PROGAM SISIFO
Figure D.12: Deflnition of the data sets labels.
The data sets are plotted on two difierent charts: flrst without normalization (see flgure
D.13). For the normalization process SISIFO uses by default N0 = 1 and 0 = 1000 MPa.
There are other two difierent buttons in this panel: Clear all and Save Plot. The flrst is
used to clear all the data and the plots, useful when the user wants to change the data at the
beginning of the parameter estimation process. The second is used to save the plot, in this case
a new window will appear on the screen, in which the user can choose difierent options, such as
change the symbol of the series, the colour, name, etc (see flgure D.14).
Parameter estimation
The objective of this panel is to estimate the parameters of the model. To this end the Castillo
fatigue model is used [40]. For the estimation process SISIFO uses the normalizated data in the
previous panel.
The steps to use this panel are:
1. Choose the estimation method. The user can choose between \maximum likelihood" or
\least-squares regression".
2. Choose the data to be used with or without run-outs and with or without outliers.
3. Estimate the parameters. The user can choose some estimation methods.
4. Plot the results. The user can choose from two difierent plot options: S{N curves or P{S{N
curves.
The estimation methods were presented in section 6.6: maximum likelihood (section 6.6.1)
and least-squares (section 6.6.2). The user may choose the method in the pop-up menu (see
flgure D.15).
The user can estimate the parameters of the model for difierent cases (see flgure D.16). The
difierences between these cases refer to the existence of run-outs or outliers. If the user checks the
APPENDIX D. SISIFO PROGRAM 193
Figure D.13: The two plots obtained with SISIFO: upper flgure, plot of the initial data, and
lower flgure, plot of the normalized data.
Without run-out or Without outliers check boxes, the program modifles the data sets deleting
the corresponding, run-out and outlier data respectively. Furthermore, when one of these check
boxes is selected, the data plot of the flrst panel changes and shows the data selected for the
estimation.
The data fllter works in two difierent ways: in the case that the user chooses estimation
without run-outs, the program reviews all the test data and deletes data in which the number
of cycles to failure is larger than N0 deflned in the flrst panel; if the user chooses estimation
without outliers, the program analyzes the fourth column of the data flle, if the value in the
cell is equal to one, an outlier is deflned and the program deletes this row for the parameter
estimation.
When the user presses the button Calculate a new window appears on the screen of SISIFO
(see flgure D.17). In this window some estimation options may be selected:
194 D.3. PROGAM SISIFO
Figure D.14: Plot options in the Save plot window.
Figure D.15: Parameter estimation methods represented in the corresponding pop-up menu in
SISIFO.
APPENDIX D. SISIFO PROGRAM 195
Figure D.16: SISIFO Check boxes to choose the flnal data for the parameter estimation, with
or without run-outs and/or outliers.
Figure D.17: Deflnition of the parameter estimation options.
196 D.3. PROGAM SISIFO
? Initial parameters, x0: It is necessary to deflne a set of initial parameters for the MATLAB
estimations (always a local optimum). With these parameters the program tries to obtain
the best estimates. These initial parameters must satisfy the constraints of the model
(deflned in section 6.4). If these initial parameters don?t satisfy the constraints, a dialog
box will appear on the screen indicating us the problem (see flgure D.18).
? Tolerance, Tol: The user has the option of increasing the tolerance value of the parameter
estimation algorithm. In the case that the user doesn?t specify a value, SISIFO chooses a
default value of Tol = 10?4.
? Number of iterations, Nr:iter: The user has the option of increasing the maximum number
of iterations to be used in the optimization process. In the case that the user doesn?t specify
as value, the program chooses a default value Nr:Iter = 400.
Figure D.18: Dialog box: Error on initial parameters.
If the calculated parameters don?t satisfy the constraints of the model, a screen similar to
flgure D.18 will appear indicating us the problem, otherwise the resulting parameters will appear
on this panel (see flgure D.19).
Figure D.19: Parameters calculated by the SISIFO program.
APPENDIX D. SISIFO PROGRAM 197
The results can be plotted in two difierent ways: the S{N curves or the P{S{N curves. The
user has to choose the type of graphic in the pop-up menu shown in flgure D.20.
Figure D.20: Difierent plot options of the estimation results.
In both cases, having chosen one option, a new window will appear (see flgure D.21). In this
window the user deflnes the ranges of the plots. If the probability stress fleld plot (P{S{N fleld)
is chosen, the user may specify the difierent probabilities (between zero and one 2 [0; 1]). These
will appear in the graphic (see flgure D.22). Note that the program is prepared for the analysis
and estimation of a maximum of four difierent sets of data. In order to display the results the
user needs to press the Plot Model button.
Figure D.21: Plot ranges of the chart.
One example of the type of graphics obtained with the program is shown in flgure D.23. The
experimental data is plotted by dots and the model results as continuous lines.
Finally, this panel has also the buttons Save Plot and Clear all. They work as explained in
the flrst panel.
The button Save result of this panel saves the obtained parameters estimates in an excel flle
(.xls). The program saves the flle with the default path, and the name Param.xls. When the
flle has been saved a dialog box will appear on the SISIFO screen conflrming the creation of the
flle Param.xls (see flgure D.24).
198 D.3. PROGAM SISIFO
Figure D.22: Probability deflnition of the P{S{N curves.
(a) (b)
Figure D.23: Example of graphics obtained after the parameter estimation: (a) S{N curves; (b)
P{S{N curves.
Model validation
The objective of this panel is to validate the model with the obtained parameters of the previous
panel: Parameter estimation, based on the concepts and methods explained in section 7.7. From
a user point of view it is easy to work with panels in the SISIFO program because the user only
needs to press buttons to obtain the results, without the necessity of loading flles or deflning
certain values.
For the evaluation of the PP{plot, the user has to press the button near to the PP{plot
graphic called Calculate and SISIFO will generate the corresponding PP{plot (see flgure D.25).
For the evaluation of the Kolmogoronov-Smirnov Test (K.S.T.) and the ?2 Test, the user has to
press the button called Calculate at the bottom of the panel (see flgure D.26).
The button Save plot works similar to the flrst panel 4.2. The button called Save Result
APPENDIX D. SISIFO PROGRAM 199
Figure D.24: Dialog box conflrming the creation of the flle Param.xls.
Figure D.25: PP{plot obtained by the SISIFO program.
Figure D.26: Resulting of the model validation performed by the SISIFO program.
saves the result in an excel flle (.xls) called ValidationTest.xls. In flgure D.27 the conflrmation
of the flle creation is shown.
In flgure D.28 an example of the created flle is shown. The flrst row corresponds to the
results obtained in the ?2 test, and the second row to the Kolmogoronov-Smirnov test (KST
test). The flrst column represents the value obtained with the estimated parameters, the second
with the critical value for each validation test and the last column corresponds to the difierence
between the flrst two columns. If this difierence is negative the set is not uniform.
200 D.3. PROGAM SISIFO
Figure D.27: Conflrmation after creating the validation flle.
Figure D.28: Example of the ValidationTest.xls flle.
Extrapolation
Finally the last panel in this menu is the panel called Extrapolation. Its objective is to show a
certain S{N or P{S{N curve for a certain constant value ( M or R ratio).
The steps required to obtain these curves are:
1. Choose the case of study, M or R constant values.
2. Choose certain probabilities for the P{S{N curves or any probability to plot the S{N
curves.
The case study is chosen from a Pop-up menu (see flgure D.29). After having chosen the
case, the user deflnes a constant value corresponding to each case ( M or R respectively).
APPENDIX D. SISIFO PROGRAM 201
Figure D.29: Deflnition of the case study in the extrapolation panel.
The S{N curves are plotted when the text boxes with labels p1; p2; p3; p4 and p5, that deflne
the difierent probabilities are not fllled, otherwise the P{S{N fleld will appear in the Figure. In
the flgure D.30 an example of the extrapolation is shown, the left side represents the S{N curves
for the case and value chosen and the right side represents the P{S{N curves for the same value
and case study.
Figure D.30: Example of extrapolation: left side, S{N curves for a M = 900MPa; right side,
P{S{N curves for the same value and case of study.
The buttons Clear All and Save Plot work similar to those in the flrst panels (section D.3.3).
D.3.4 Menu 2: damage analysis
In this section, the Damage analysis menu is described. To access this menu the option Damage
Analysis has to be chosen in the Option menu (see flgure D.31).
This menu consists of three panels: Load deflnition, Rain ow analysis and Damage analysis
(see flgure D.33).
202 D.3. PROGAM SISIFO
Figure D.31: Entering the Damage analysis menu of SISIFO.
Figure D.32: Loading a load sequence in the SISIFO program.
Load deflnition
The objective of this panel is to load or to deflne a load history that will be used for the Damage
analysis.
The methodology of using is as follows:
1. Deflne a load history.
2. Plot the spectrum and save the load sequence.
APPENDIX D. SISIFO PROGRAM 203
Panel 1
:
Load definitio
n
Panel 2
:
Rainflow analysi
s
Panel 3
:
Damage analysi
s
Figure D.33: Difierent panels in the Damage analysis menu of the SISIFO program.
204 D.3. PROGAM SISIFO
(a)
(c)
(b)
Figure D.34: Difierent types of load histories: (a) ? = 900 MPa,(b) ? = 0:1?N+900 MPa,(c)
? = 0:4 ?N + 600 MPa.
APPENDIX D. SISIFO PROGRAM 205
Figure D.35: Representation of a quasi-random load history in SISIFO.
To deflne the load history the user has three difierent options:
1. Load a load history flle with all the information about the sequence of loads: The user will
press the button called Load, and choose the correct flle (see flgure D.32). This flle (.xls
flle) will be formed by three columns, the flrst column represents the number of cycles, the
second the minimum stress m and the third column represents the maximum stress M .
2. Deflne a constant load history or a variable linear load history: The user creates the load
history (? = M ? m = m ?N + n) deflning the expressions of M = m1 ?N + n1 and
m = m2 ?N + n2. The constants m1, m2, n1 and n2 are chosen by the user. Finally, the
parameter m corresponds to the slope of the load history, how the load increase/decrease
with the number of cycles and n corresponds to the value of ? when N = 0. The meaning
of m1;m2 and n1 and n2 is the same than m and n. Three examples of load histories are
shown in flgure D.34.
In the sequence plot the program shows three difierent lines, two of them correspond to
the deflnition of m and M (red lines) and the third one represents the load sequence (in
blue).
206 D.3. PROGAM SISIFO
3. Load a quasi-random load spectrum: First the user needs to load the spectrum using the
button called Load (as in the flrst point). After this, the user deflnes the stresses that
deflne the load sequence (because normally the spectra are dimensionless and based on
classes) and deflnes the position of the zero load Zero Class (see flgure D.35).
The sequence shown in this case is the result of the algorithm of four points. The sequence
represents the real cycles of the load history.
In order to plot the difierent load histories the user presses the button called Plot Load.
The button Save Plot and Clear All work similar as in section D.3.2. The button called Save
sequence creates an excel flle in the current directory with all the information of the load history
in three columns (flrst, number of cycles; second, minimum stress and third column maximum
stress in the cycle), but it is implemented only for quasi-random spectra.
Figure D.36: Error dialog box used to tell the user that the sequence flle has not been created.
If the user presses the button called Save results when the load history is difierent, a new
Dialog box will appear in SISIFO indicating us that the flle has not been created (see flgure
D.36, in which a variable load history is presented but not a quasi-random one, so SISIFO tells
APPENDIX D. SISIFO PROGRAM 207
us that the flle has not been created). The aim of this is to obtain a sequence flle that can be
applicated to a fatigue test machine.
Rain ow analysis
The objective of this panel is to calculate the Rain ow matrix for that load history.
For the rain ow analysis the program needs only the number of classes (normally less than
64). After pressing the button Calculate a plot with the rain ow analysis in a color scale is
obtained. The user can save the results in an excel flle (.xls) by pressing the button Save
Results. In flgure D.37 a rain ow plot is shown.
Figure D.37: Rain ow matrix in color scale. The color scale of the right side represents the
frequence of cycles in the rain ow matrix.
The second part of this panel corresponds to a fllter that the user may apply to the load
sequence. The objective is to delete cycles lower than the endurance limit of the material. This
is useful to reproduce the damage calculations and time on fatigue test machines. In this case,
a new window will appear in the program and the user specifles in it the necessary data to fllter
the rain ow matrix: Endurance limit, R-ratio and a security factor (between zero and one) to
use in the fllter process.
There are two options for the R-ratio, equal to minus one or equal to zero. In the flrst
case, the endurance limit doubled in order to get the endurance range instead of the amplitude
parameter deflnition window (see flgure D.38). In the case of R = 0 the endurance limit is equal
to the limit deflned by the user. The security factor modifles the flnal value of the endurance
limit.
Finally, the buttons Save Plot and Save Results work similar to the buttons explained in the
previous sections.
Damage analysis
The objective of the last panel of the SISIFO program is to analyze the damage accumulation
(deflned by the probability of failure p) of a material subjected to a certain load history.
The methodology is the following:
208 D.3. PROGAM SISIFO
Figure D.38: Parameter deflnition for the Rain ow fllter analysis.
1. Load the model parameters. If they don?t exist the program sends the user to the flrst
menu of SISIFO in order to calculate the parameters (see flgure D.39).
Figure D.39: Dialog box indicating the absence of the parameter flle.
2. Deflne the point of study, that is, number of cycles (N), minimum stress ( m) and maxi-
mum stress ( M ).
3. Press Calculate. The program calculates the probability of failure (p) for the point deflned
in the previous step and plot the cumulative distribution function (cdf) for this load and
parameters (see flgure D.40).
It is very important that the user specifles a good value for m, M and N .
Finally, the buttons Save Plot and Save Results work similar to the buttons explained in the
previous sections.
APPENDIX D. SISIFO PROGRAM 209
Figure D.40: Representation of the probability of failure calculated with the model parameters
for a certain point of study. The red point corresponds to the point of study, the blue line
corresponds to the cdf of the model for all number of cycles with this load.
D.4 Example of application
In this section, some illustrative examples of applications are given with the aim of showing how
the SISIFO program works. The section is organized in three subsections: flrst, an example of
how the parameters of a material can be estimated is given, second, it is shown how the damage
accumulation can be analyzed and flnally, an example of how a rain ow matrix can be estimated
and flltered.
D.4.1 Estimation of the Castillo model parameters
Presentation of the problem
Consider a metallic material. We want to know the lifetime of these material subjected to a
certain variable load history, using the new fatigue model deflned by Castillo et al. [40].
The data of the problem are:
? A set of experimental data tests have been obtained from fatigue tests. The results are
shown in table D.2. The data consists of four difierent series for constant M .
? The number of cycles deflned for runouts are N = 10000000 cycles.
Following the methodology deflned by Castillo et al. [40] (see section 6.7) the steps in the
damage accumulation analysis will be:
1. Design of the testing strategy: Presented in table D.2.
2. Choose the normalizing variables N0 and 0: Deflned by SISIFO (see section D.3.2).
3. Estimate the model parameters: Using the SISIFO program (see section D.4.2).
4. Extrapolate to other testing conditions: Calculate the damage accumulation for the load
history described before and estimate the trend of the S{N curves in the case of i.e. R = ?1.
210 D.4. EXAMPLE OF APPLICATION
Table D.2: Resulting lifetimes from the laboratory tests.
Nr. Test min max Ncycles Nr. Test min max Ncycles
1 -182.15 327.87 19100 18 -281.42 218.58 37100
2 -109.29 327.87 34000 19 -246.42 218.58 54400
3 -72.86 327.87 42800 20 -211.42 218.58 80300
4 36.43 327.87 153100 21 -176.42 218.58 96300
5 109.29 327.87 10000000 22 -141.42 218.58 175500
6 -18.22 327.87 63800 23 -106.42 218.58 172800
7 91.08 327.87 360400 24 -71.42 218.58 526500
8 182.15 327.87 10000000 25 -36.42 218.58 8639500
9 72.86 327.87 81600 26 -182.15 255.01 48000
10 -182.15 291.44 28700 27 -145.72 255.01 57900
11 -72.86 291.44 71700 28 -109.29 255.01 113100
12 -109.29 291.44 59900 29 27.32 255.01 7500000
13 -18.22 291.44 143500 30 -18.22 255.01 348300
14 36.43 291.44 326400 31 -63.75 255.01 172500
15 109.29 291.44 10000000 32 55.01 255.01 10000000
16 91.08 291.44 10000000 33 72.86 255.01 238200
17 72.86 291.44 207700
Preparation of the data
Before estimating the model parameters, the normalization of the data is performed. For this
we need to load our data in SISIFO and show the normalization of the data.
The flle example1.xls contains all the test data and also deflnes which of these points are
outliers (indicated by a value equal to one in the fourth column). flgure D.41 shows how the
data looks like.
Figure D.42 presents the plots obtained by SISIFO, the upper flgure corresponds to the
original data and the lower one to the normalized data.
Parameter estimation
After the data normalization the parameter estimation is carried out. The maximum likelihood
method is chosen for the estimation and for this example the estimation is done without runouts
and outliers (note that in flgure D.43 the check boxes RunOut and Outliers are checked).
The initial parameters correspond to
Cinitial = (?10:19593; 44:77213;?39:18622; 0:0; 0:0;?5:66951; 5:66951; 0:0)
which satisfy all the model constraints. The flnal parameters are shown in flgure D.43. The
parameters obtained have been saved in the current directory (see flgure D.44).
APPENDIX D. SISIFO PROGRAM 211
Figure D.41: Deflnition of the data flle example1.xls.
212 D.4. EXAMPLE OF APPLICATION
Figure D.42: Graphics representing the original and normalized data.
Figure D.43: Parameter estimation with all the data, using the maximum likelihood method.
APPENDIX D. SISIFO PROGRAM 213
Figure D.44: File saved by SISIFO with the estimated parameters.
The S{N and P{S{N curves are shown in flgure D.45. We can observe that with these
parameters the curves flt well with the experimental data.
(a) (b)
Figure D.45: S{N curves and P{S{N curves obtained with the parameter estimation.
Model validation
For the model validation the last parameters obtained are chosen. Pressing the two buttons
called Calculate (flrst one close to PP{plot and second one on the validation uniformity test) in
the model validation panel we obtain the results shown in flgure D.46.
Extrapolation to other load conditions
We want the model to extrapolate and plot a new P{S{N curve corresponding to the value
R = ?1. For this we choose the case R = cte in the pop-up menu and then we deflne the value
to minus one. Pressing the button Calculate the new curve is shown (see flgure D.47).
214 D.4. EXAMPLE OF APPLICATION
Figure D.46: Obtained results for the model validation.
(a) (b)
Figure D.47: P{S{N curve obtained for R = ?1, extrapolated from the original data.
Before going to the second menu for the damage analysis we want to show how the flrst
menu looks like when all the panels are in use. This is presented in flgure D.48.
APPENDIX D. SISIFO PROGRAM 215
Figure D.48: Representation of the Menu 1: Parameter estimation after using all the panels.
216 D.4. EXAMPLE OF APPLICATION
D.4.2 Analysis of the damage accumulation
Presentation of the problem
Consider a metallic material. We want to analyze the damage accumulation of a material
subjected to a certain variable load history, using the new fatigue model deflned by Castillo et
al. [40].
The data of the problem is:
? A set of experimental data tests have been obtained from fatigue tests. The results are
shown in table D.3. The data consists of four difierent series for constant M .
? The number of cycles deflned for runout are N = 10000000 cycles.
? Endurance limit is 420 MPa for R = 0.
? The damage accumulation will be analyzed for the following load history: ? = 1250 MPa
.
Table D.3: Resulting lifetimes from the laboratory tests for the Example Nr.2.
Nr. Test min max Ncycles Nr. Test min max Ncycles
1 17.502 955.50 65277 15 -167.50 877.80 125800
2 35.24 955.50 23700 16 -500.00 780.30 11439
3 55.27 955.50 52700 17 -437.50 780.30 14973
4 78.43 955.50 40900 18 -375.00 780.30 32055
5 119.11 955.50 85900 19 -312.50 780.30 483000
6 108.91 955.50 124300 20 -343.75 780.30 36708
7 124.21 955.50 222900 21 -265.63 780.30 532200
8 98.71 955.50 93500 22 -281.25 780.30 123100
9 -250.00 877.80 17281 23 -550.00 682.74 183024
10 -190.00 877.80 48787 24 -620.00 682.74 19331
11 -175.00 877.80 81244 25 -525.00 682.74 347102
12 -360.00 877.80 3373 26 -585.00 682.74 33925
13 -305.00 877.80 21812 27 -500.00 682.74 381543
14 -332.50 877.80 7265
In this section the parameter estimation is made as in the flrst example. Since the results
of the parameter estimates are presented in table D.4, we go directly to the analysis of damage
accumulation for the load history deflned above.
Table D.4: Parameters estimated with SISIFO for Example Nr.2.
C0 C1 C2 C3 C4 C5 C6 C7
-46.29 -3.77 33.464 0.00 0.00 -1.43 1.43 0.00
Deflning the variable load history
In this section the load history is created. The load history corresponds to the expression
? = 1250 MPa, that is m = 0 and n = 1250 MPa (constant load history), with m = ?250
MPa and M = 1000 MPa.
APPENDIX D. SISIFO PROGRAM 217
(a) (b)
Figure D.49: (a) Load history deflnition: ? = 1250 MPa (constant load history), with
m = ?250 MPa and M = 1000 MPa. (b) Error dialog box created by SISIFO when the
flle sequence.xls can not be created.
The sequence is deflned by pressing the button Deflne load. Pressing Plot Results we obtain
the graphic N cycles vs. . flgure D.49 (a) shows the error dialog box created by SISIFO when
the load history deflned is not a random load history and the program is not prepared to create
sequences of this type of loads.
The program is not prepared to make the rain ow analysis to this type of load sequences.
flgure D.49 (b) shows the Dialog box that SISIFO creates when we press the button Calculate
on the Rain ow analysis panel.
Damage analysis
Finally, the damage is analyzed. First we need to load the parameter values of our material by
pressing the button called Load Parameters.
We want to know the probability of failure when the material is subjected to the load deflned
in section D.4.3, or, more precisely, when the number of cycles is equal to N = 4000, m = ?250
MPa and M = 1000 MPa. flgure D.51 shows the results of this analysis when we press the
Calculate button.
Concluding with this section, flgure D.52 shows how the second menu looks like when all the
218 D.4. EXAMPLE OF APPLICATION
Figure D.50: Error dialog box created by SISIFO when the rain ow analysis cannot be carried
out.
procedures have been used.
D.4.3 Rain ow analysis in a quasi-random load history
Rain ow analysis is normally made to work easily with complex load histories, as quasi-random
load histories. In this section an example of rain ow analysis is made: flrst obtaining the rain ow
matrix and then flltering the resulting sequence.
The rain ow matrix represents a certain number of classes (less than 64) containing the
frequency of cycles between a certain minimum stress (from) to a maximum stress (to). The
diagonal of this matrix is equal to zero because these cycles would have an amplitude equal to
zero.
The steps for obtaining the rain ow matrix (RFM) are:
1. Load a sequence. For this we choose the option Load data from the panel Load deflnition.
We choose the flle example2.xls.
2. Deflnition of the maximum and minimum stresses, and the position of the zero load. We
APPENDIX D. SISIFO PROGRAM 219
Figure D.51: Damage results obtained with SISIFO, for the load history ? = 1250 MPa in the
point N = 4000, m = ?250 MPa and M = 1000 MPa
choose the values of M = 900 MPa, m = ?200MPa, and zero class in 7 (in a sequence
with 12 classes). flgure D.53 shows the resulting spectrum.
3. Select the number of classes in our RFM equal to 32. Press the button called Calculate.
The resulting RFM is shown in flgure D.54 (a). Note that the diagonal is equal to zero and
there is a quasi symmetric distribution of the points. The results are saved by pressing
the button Save Results.
To optimize the time in our fatigue test with random loads and/or in the calculation of the
damage accumulation we decide to do a flltering of the rain ow matrix obtained before with the
aim of deleting the cycles with an amplitude ? ? ? lim (limit of endurance). In the section
D.4.2 the values for the R ratio, security factor (FS) and the ? limit are deflned. Pressing the
button Calculate, fllling the difierent text boxes and flnally pressing Ok! we obtain the flltered
rain ow matrix (see flgure D.54 (b)).
It can be appreciated that both rain ow matrices are equal. But the width of the empty
diagonal of the flltered matrix is bigger, because all the small non damaging cycles have been
deleted (see flgure D.54). Pressing the button Save results the flle RFMflltered.xls is created and
the new sequence (without the smallest cycles) is also created (flle called sequenceFiltered.xls).
220 D.4. EXAMPLE OF APPLICATION
Figure D.52: Representation of Menu 2: Damage Analysis after using it.
APPENDIX D. SISIFO PROGRAM 221
Figure D.53: Random spectrum used for the example of rain ow analysis.
(a) (b)
Figure D.54: Comparison between the rain ow matrix (a) and the flltered rain ow matrix (b).
222 D.4. EXAMPLE OF APPLICATION
Appendix E
Nomenclature
Symbol Nomenclature
Chapter 3 Basic concepts
Stress of a cycle
min, m Minimum stress of a cycle
max, M Maximum stress of a cycle
mean Mean stress in a cycle
? Stress range of a cycle
a Stress amplitude of a cycle
R Stress ratio
A Amplitude ratio
Chapter 3 Fatigue under fracture mechanics point of view
a Crack length
F Dimensionless function used in the fracture mechanics approach
K Stress intensity factor
?K Stress intensity factor range
fi Relative crack length
b Width dimension of a member
da
dN Crack growth rate
C Constant used in the fatigue crack growth
m Constant used in the fatigue crack growth
?Kth Fatigue crack growth intensity factor threshold
Kc Fatigue crack growth intensity factor critic (failure)??K Equivalent zero-to-tension (R = 0) stress intensity factor
Constant of the material used in the fatigue crack growth theories
da
dt , _a Time based growth rate, crack growth velocity
A, n Constant of the material used in the crack velocity formulation
C1, C2 Material parameter used by Broek & Schijve [30] in the crack growth formulation
P , Q Material parameter used by Frost & Dugdale [69] in the crack growth formulation
f? Strength applied in the Yokobori [134] crack growth formulation
fi Image strength in the Yokobori [134] crack growth formulation
? Distance from the head of crack to the dislocation. Used by the Yokobori [134]
crack growth formulation
` Crack angle. Used by the Yokobori [134] crack growth formulation
223
224
Symbol Nomenclature
Chapter 3 Stress based approach to fatigue
e Equivalent stress
u Ultimate strength
y Yield stress
0f Reversal fatigue strength
M Mean stress sensitivity factor
Chapter 3 Strain based approach to fatigue
? Strain
l Instantaneous length
l0 Original length
?l Longitudinal deformation
A Cross-sectional area
A0 Original cross-sectional area
S Engineering stress
e Engineering strain
%EI Percentage of elongation
%RA Reduction of area
E Modulus of elasticity
K Monotonic strength coe?cient. Used by Ramberg and Osgood [112]
n Monotonic strain hardening exponent. Used by Ramberg and Osgood [112]
f True fracture strength
?f True fracture ductility
?e Elastic strain
?p Plastic strain
K 0 Cyclic strength coe?cient. Used by Ramberg and Osgood [112]
n0 Cyclic strain hardening exponent. Used by Ramberg and Osgood [112]
?? Strain range
0 Fatigue strength coe?cient
?0 Fatigue ductility coe?cient
b Fatigue strength exponent
c Fatigue ductility exponet
?a Strain amplitude
Nf Number of cycles to failure
Chapter 4 Models used in fatigue
a, b Parameters of the Basquin function [20]
1 Fatigue limit used by Stromeyer [128]
a, b, B Parameters of the Palmgren function [105]
Ni
Nfi
Fraction of lifetime used in the Palmgren-Miner rule [95]
? i Stress ranges used by Dixon & Mood [57]
pi Probability of failure
?(X) Mean value of the X variable
(X) Standard deviation of the X variable
?(N) Bastenaire?s auxiliary function [21]
A, E, c Bastenaire?s function parameters [21]
SA Fatigue strength used by Spindel & Haibach [126]
APPENDIX E. NOMENCLATURE 225
Symbol Nomenclature
k Slope of the S{N curves [126]
fi Spindel parameter [126]
NE Cutofi of the S{N curves by Spindel [126]
SE Endurance limit [126]
Y Lifetime in the Pascual & Meeker model [107]
fl0, fl1 Pascual-Meeker model parameters [107]
Fatigue limit of the specimen [107]
? Error term in the Pascual-Meeker equation [107]
F (x;?; ?; fl) Weibull cumulative distribution function
? Location parameter in the Weibull distribution
? Scale parameter in the Weibull distribution
fl Shape parameter in the Weibull distribution
B Threshold value of lifetime used by Castillo et al. [36]
C Endurance limit used by Castillo et al. [36]
E Position of the corresponding zero-percentile hyperbola
[36]
D Scale factor [36]
A Weibull shape parameter of the whole cdf in the S{N
fleld [36]
N? Normalized number of cycles [36]
? ? Normalized stress level [36]
Nref Number of cycles of reference used for the normalization
[36]
? ref Stress level of reference used in the normalization [36]
Chapter 6 Derivation of the Gumbel fatigue model
N0 Equivalent number of cycles used in the normalization [40]
0 Equivalent stress used in the normalization [40]
N? Normalized number of cycles [40]
? Normalized stress [40]
?m Normalized minimum stress [40]
?M Normalized maximum stress [40]
C1; C2; C3; C4; C5; C6 and C7 Parameters of the Gumbel fatigue mode [40]
? m0 Asymptotic value of ? for constant m[40]
? M0 Asymptotic value of ? for constant M [40]
logNm0 Asymptotic value of logN for constant m [40]
logNM0 Asymptotic value of logN for constant M [40]
Euler-Mascheroni number
L Log-Likelihood function
I1 Set of non-runouts
I0 Set of runouts
H(Ni) Auxiliary function of the log-likelihood formulation
Covar Covariance matrix of the estimation parameter process
C Parameters estimated maximum likelihood
Q Regression Weibull-Gumbel equation
226
Symbol Nomenclature
Chapter 7 Experimental validation of the model
L1 Total length of the specimen
L2 Useful length of the specimen
d Diameter of the specimen
r Curvature radius of the specimen
D Value of the Kolmogorov-Smirnov test
F (H(N?i )) Sample order statistics
i Position of the sample
n Total number of specimens (sample size)
?2 Value of the Chi2 test
Chapter 8 Damage measures and damage accumulation
Mi Palmgren-Miner number
Z Normalized variable related with the Castillo fatigue model [40]
PF Probability of failure
N?eq Equivalent number of cycles used in the damage accumulation formulation
of the Castillo fatigue model [40]
PN+?N Accumulated damage [40]
Appendix B Specimen characterization
T Temperature (C)
Ry Yield strength (MPa)
Rp0:2 Yield strength at ? = 0:2% of strain
Rm Ultimate strength
? Mean value
Standard deviation
Bibliography
[1] ASTM Standard E1049. Standard Practices for Cycle Counting in Fatigue Analysis.
Philadelphia, 1985.
[2] MIL-HDBK-5G: Application of Fracture Mechanics for Selection of Metallic Structural
Materials. Mlitary Handbook, 1994.
[3] ASTM Standard E739. Standard Practice for Statistical Analysis of Linear or Linearized
stress{life (S{N) and strain{life (?{N) fatigue data. Philadelphia, 1998 (reapproved).
[4] Standard Terminology Relating to Metallography. Annual Book of ASTM Standards, vol.
03.01 edition, 2003.
[5] Annual Book of ASTM Standards. ASTM E606-92. Vol. 03.01.2005 edition, 2005.
[6] Practices for Force Veriflcation to Testing Machines. Annual Book of ASTM Standards,
vol. 03.01 edition, 2007.
[7] Standart Test Methods for Tension Testing of Metallic Materiasls [Metric]. Annual Book
of ASTM Standards, vol. 03.01 edition, 2008.
[8] N. H. Abel. M?ethode g?enerale pour trouver des functions d?une seule quantit?e variable
lorsqu?une propiet?e de ces fonctions est exprim?ee par une equation entre deux variables.
Mag. Naturvidenskab, 1:1{10, 1823.
[9] N. H. Abel. Unterschungen uber die Reihe 1 + (m=1)x+ (m(m? 1))(1:2)x2 + :::. Journal
Reine Angewandte Mathematical, 1:311{319, 1826.
[10] N. H. Abel. Untersuchungen der Functionen zweier unabhangigen veranderlichen Gro?ssen
x und y, wie f(x; y) welche die Eigenschaft haben, dass f [z; f(x; y)] eine Symmetrische
Function von x; y und z ist. Journal Reine Angewandte Mathematical, 1:5{11, 1826.
[11] J. Acz?el. Lectures in functional equations and their applications. Mathematics in science
and Engineering. Academic Press, 19, 1966.
[12] J. Acz?el. Functional equations: History, applications and theory. Dordrecht, 1984.
[13] J. Acz?el and J. Dhombres. Functional Equations in several variables. Cambridge University
Press,, 1989.
[14] H. Alawi and M. Shabans. Fatigue crack growth under random loading. Engineering
Fracture Mechanics, 32(5):845{854, 1989.
[15] A. Anatolij. Linear functional equations. Russian Estate University, 1996.
227
228 BIBLIOGRAPHY
[16] Y.M. Baik and K.S. Kim. The combined efiect of frequency and load level on fatigue crack
growth in stainless steel 304. International Journal of Fatigue, 23:417{425, 2001.
[17] R. Balasubrahmayan and K.S. Lau. Functional equations in probability theory. Boston
Academic Press, 1991.
[18] J. A. Bannantine, J. J. Comer, and J. L. Handrock. Fundamentals of Metal Fatigue
Analysis. Prentice Hall, New York, 1990.
[19] J.M. Barson and S.T. Rolfe. Fracture and Fatigue Control in Structures. Prentice Hall,
Upper Saddle River, New Jersey, 2nd edition, 1987.
[20] O. H. Basquin. The exponential law of endurance tests. American Society for Testing and
Materials Proceedings, 10:625{630, 1910.
[21] F. A. Bastenaire. Et?ude Statistique et Physique de la Dispersion des R?esistances et des
Endurances a la Fatigue. PhD thesis, Faculty of Sciences, University of Paris, Paris,
France, 1960.
[22] F. A. Bastenaire. New method for the statistica1 evaluation of constant stress amplitude
fatigue-test results. Probabilistic Aspects of Fatigue, American Society for Testing and
Materials, ASTM STP 511:3{28, 1972.
[23] C. Bathias and R.M. Pelloux. Et?ude de la zone plastifl?ee a fond de flssure: application a
la propagation des flssures de fatigue dans les aciers maraging et les aciers aust?enitiques.
Saclay, 1972.
[24] D. Benasciutti and R. Tovo. Fatigue life assessment in non-Gaussian random loadings.
International Journal of Fatigue, 28:733{746, 2006.
[25] B.A. Bilby, A. H. Cottrell, and K. H. Swiden. Analysis of stresses and strains near the
end of a crack transversing a plate. Proceedings of the Royal Society, A272:304, 1963.
[26] J. Bogdanofi and F. Kozin. Efiect of length on fatigue life of cables. Journal of Engineering
Mechanics, 113:925{940, 1987.
[27] C. Boller and T. Seeger. Materials Data for Cycling Loading. Part B: Low-Alloy Steels.
Elsevier, ISBN: 0-444-42871-2 edition, 1987.
[28] C. Boller and T. Seeger. Materials Data for Cycling Loading. Part D: Aluminium and
Titanium Alloys. Elsevier, ISBN: 0-444-42873-9 edition, 1987.
[29] P. W. Bridgman. Stress distribution at the neck of tension specimen. American Society
for Testing and Materials Proceedings, 32:553{572, 1944.
[30] D. Broek. Elementary Engineering Fracture Mechanics. Kluwer Academic Publications,
Dordrecht, The Netherlands, 4th edition, 1986.
[31] E. Buckingham. On physically similar systems; illustrations of the use of dimensional
equations. Physical Review, 4:345{376, 1914.
[32] R.G. Carlson. Fatigue Studies of Inconel, BMI-1335, UC-25 Metallurgy and Ceramics.
Ohio, 1959.
BIBLIOGRAPHY 229
[33] E. Castillo. Extreme value theory in engineering. Academic Press, 1988.
[34] E. Castillo, A. Fern?andez, J. R. Ruiz, and J. M. Sarabia. Statistical models for analysis of
fatigue life of longs elements. Journal of Mechanical Engineering, ASCE, 116:1036{1049,
1990.
[35] E. Castillo and A. Fern?andez-Canteli. A general regression model for lifetime evaluation
and prediction. International Journal of Fracture, 107:117{137, 2001.
[36] E. Castillo and A. Fern?andez-Canteli. A parametric lifetime model for the prediction
of high-cycle fatigue based on stress level and amplitude. Fatigue Fracture Engineering
Materrial Structure, 29:1031{1038, 2006.
[37] E. Castillo, A. Fern?andez-Canteli, V. Esslinger, and B. Thu?rlimann. Statistical model for
fatigue analysis of wires, strands and cables. International Association for Bridge and
Structural Engineering, P-82/85:1{40, 1985.
[38] E. Castillo, A. Fern?andez-Canteli, and A.S. Hadi. On fltting a fatigue model to data.
International Journal of Fatigue, 21:97{106, 1999.
[39] E. Castillo, A. Fern?andez-Canteli, A.S. Hadi, and M. L?opez-Aenlle. A fatigue model with
local sensitivity analysis. International Journal of Fatigue, 30:149{168, 2007.
[40] E. Castillo, A. Fern?andez-Canteli, R. Koller, M.L. Ruiz-Ripoll, and A. Garc??a. A sta-
tistical fatigue model covering the tension and compression Wo?hler flelds. Probabilistic
Engineering Mechanics, doi:10.1016/j.probengmech.2008.06.003, 2007.
[41] E. Castillo, A. Fern?andez-Canteli, M. L?opez-Aenlle, and M.L. Ruiz-Ripoll. Some fatigue
damage measures for longitudinal elements based on the Wo?hler fleld. Fatigue & Fracture
of Engineering Materials & Structures, 30:1063{1075, 2007.
[42] E. Castillo, A. Fern?andez-Canteli, and M.L. Ruiz-Ripoll. A general model for fatigue
damage due to any stress history. International Journal of Fatigue, 30:150{164, 2008.
[43] E. Castillo and J. Galambos. Lifetime regression models based on a functional equation
of physical nature. Journal of Applied Probability, 24:160{169, 1987.
[44] E. Castillo and A.S. Hadi. Modeling lifetime data with applicaiton to fatigue models.
Journal of the American Statistical Association, 90, No. 431:1041{1054, 1995.
[45] E. Castillo, A. Iglesias, and M.R. Ruiz-Cobo. Functional equations in applied sciences.
Amsterdam, 2005.
[46] E. Castillo, A. Iglesias, and R. Ruiz-Cobo. Functional equation in applied science. Elsevier,
1st. edition, 2004.
[47] E. Castillo, M. L?opez-Aenlle, A. Ramos, A. Fern?andez-Canteli, R. Kieselbach, and
V. Esslinger. Specimen length efiect on parameter estimation in modelling fatigue strength
by Weibull distribution. International Journal of Fatigue, 28:1047{1058, 2006.
[48] E. Castillo, A. Ramos, R. Koller, M. L?opez-Aenlle, and A. Fern?andez-Canteli. A critical
comparison of two models for assessment of fatigue data. International Journal of Fatigue,
30, No. 1:45{57, 2008.
230 BIBLIOGRAPHY
[49] E. Castillo and M.R. Ruiz-Cobo. Functional equations in science and engineering. Marcel
Dekker, New York, 1992.
[50] L. F. Jr. Co?n. A study of the efiect of cyclic thermal stresses on a ductile metal.
Transactions of ASME, 76:931{950, 1954.
[51] V.J. Colangelo and F.A. Heiser. Analysis of Metallurgical Failures. Wiley, New York,
1974.
[52] J.A. Collins. Failure of Materials in Mechanical Design. Wiley, New York, 1993.
[53] J. B. Conway and L. H. Sjodahlo. Analysis and representation of fatigue data. ASM
International, Cincinatti, 1991.
[54] D. R. Cox. Regression models and life-tables. Journal of the Royal Statistical Society,
2:187{202, 1972.
[55] J. G. Dhombres and R. Ger. Conditional cauchy equations. Glasnik Matematicki Series
III, 13:39{62, 1975.
[56] G.E. Dieter. Mechanical Metallurgy. McGraw-Hill, New York, 1976.
[57] W. J. Dixon and A. M. Mood. A method for obtaining and analyzing sensitivity data.
Journal of the American Statistical Association, 43:109{126, 1948.
[58] N. E. Dowling. Fatigue at notches and the local strain and fracture mechanics approaches.
Fracture Mechanics, American Society of Testing and Materials, ASTM STP 677, 1979.
[59] N.E. Dowling. Mechanical Behavior of Materials. Engineering Method for Deformations,
Fracture and Fatigue. Pearson, New Jersey, 3nd edition, 1998.
[60] N.E. Dowling and S. Thangjitham. An overview and discussion of basic methodology for
fatigue. Fatigue and Fracture Mechanics. ASTM STP 1389, 31:3{36, 2000.
[61] H. Drees, L. Haan, and D. Li. On large deviation for extremes. Statistical Probabilistic
Letters, 64(1):51{62, 2003.
[62] S. Durham, J. Lynch, and W.J. Padgett. TP 2-ordering and the IFR property with appli-
cations. Probability in the Engineering and Informational Science, 4:73{88, 2000.
[63] B. Efron and R. J. Tibshirani. An Introduction to the Bootstrap. Chapman & Hall/CRC,
1993.
[64] A. Fern?andez-Canteli. Criterios para la normalizaci?on de ensayos de fatiga en armaduras
activas y pasivas de hormig?on armado y pretensado. PhD thesis, Universidad Polit?ecnica
de Madrid, Madrid, 1981.
[65] A. Fern?andez-Canteli. Statistical interpretation of the miner-number using an index of
probability of total damage. In ABSE Colloquium Lausanne. pp. 309{319, 1982.
[66] A. Fern?andez-Canteli, V. Esslinger, and B. Thu?rlimann. Ermu?dungsfestigkeit von
Bewehrungs{und Spannsta?hlen. Bericht Nr. 8002-1, IBK. ETH Zrich, Birkha?user Verlag,
ISBN:3-7643-1613-6 edition, 1984.
[67] A.G. Forman. Journal of Basic Engineering, 89:459{464, 1967.
BIBLIOGRAPHY 231
[68] A. M. Freudenthal. Statistical aspects ol fatigue. American Society for Testing and Ma-
terials, ASTM STP 121:3, 1952.
[69] N. E. Frost and D.S. Dugdale. The propagation of fatigue craks in sheet specimens. Journal
of the Mechanics and Physics of Solids, 6, No. 2:92{110, 1965.
[70] J. Galambos. The Asymptotic Theory of Extreme Order Statistics. Krieger Publ., Mel-
bourne, Florida, second edition edition, 1987.
[71] G. Galileo. Discorsi e dimostrazioni intorno a due nouve scienze. Leyden, 1638.
[72] G. Genest and J. V. Zidek. Combining probability distributions: a critique and a annoted
bibliography. Statistical Science, 1:114{148, 1986.
[73] W.Z. Gerber. Calculation of the allowable stresses in iron structures. Z. Bayer Archit Ing
Ver, 6:101{110, 1874.
[74] J. Goodman. Journal of Mechanics Applied to Engineering. Longmans, Green, New York,
1st edition, 1899.
[75] J. Goodmann. Mechanics applied to Engineering. Longmans, Green and Co., London,
1919.
[76] E. Haibach. Modifled linear damage accumulation hypothesis. In Proceedings of the Con-
ference on Fatigue of welded Structures, pp. xx-xxii. Brighton, England, 1970.
[77] E. Haibach. Betriebsfestigkeit. Verfahren und Daten zur Bauteilberechnung. Du?sseldorf:
VDI-Verlag, 1989.
[78] E. Haibach and B. Atzori. Ein statistisches Verfahren fr das erneute Auswerten von er
anregende. Aluminium, 51, No. 4:267{272, 1975.
[79] B.P. Haigh. Experiments on the fatigue of brasses. Journal of the Institute of Metals,
18:55{86, 1917.
[80] Fei H.L. and Xu X.L. Lu X.W. Procedures for testing outlying observations with aweibull
or extreme-value distribution. Acta Math Appl Sinica, 21(4):549{61, 1998.
[81] S.L.J Hu and L.D. Lutes. Non-normal description of morrison-type wave forces. Journal
of Engineering Mecchanics, ASCE, 113(3):196{209, 1987.
[82] J. Kepler. Chilias logarithmorum ad totidem numeros rotundos. Marburg, 1624.
[83] D.P. Kihl, S. Sarkani, and J.E. Beach. Stochastic fatigue damage accumulation under
broadband loadings. International Journal of Fatigue, 17(5):321{9, 1995.
[84] J. Kohout and S. Vechet. New functions for description of fatigue curves and their ad-
vantages. In Proceedings of the Seventh International Fatigue Congress, Vol. II, pp.783{8.
Beijing, 1999.
[85] J. Kohout and S. Vechet. Regression of fatigue curves using functions with graphs close
to broken line. In Proceedings of Sixth National Conference Degradation of Properties of
Structural Materials by Fatigue, pp. 29{33 (ISBN 80-7100-629-7). Zilina, Slovakia, 1999.
232 BIBLIOGRAPHY
[86] J. Kohout and S. Vechet. A new function for fatigue curves characterization and its
multiple merits. International Journal of Fatigue. Technical note, 23:175{183, 2001.
[87] M. Kuczma. Functional equations on restricted domains. Aequationes Mathematicae,
18:1{34, 1978.
[88] T. E. Langlais and J. H. Vogel. Overcoming limitations of the conventional strain-life
fatigue damage model. Journal of Engineering Materials and Technology, 117:103{108,
1995.
[89] A. M. Legendre. Elements de geometrie. Note II edition, 1791.
[90] L.D. Lutes, M. Corazao, S.L.J. Hu, and J.J. Zimmermann. Stochastic fatigue damage
accumulation. Journal of Structural Engineering ASCE, 110(ST11):2585{2601, 1984.
[91] J.Y. Mann. The historical development of reseach on the fatigue of materials and struc-
tures. The Journal of the Australian Institute of Metals, pages 222{241, 1958.
[92] S. S. Manson. Behavior of materials under conditions of thermal stress. University of
Michigan Engineering Research Institute, MI, 1953.
[93] M. Matsuishi and T. Endo. Fatigue of metals subjected to varying stress. Fukuoka, Japan,
1968.
[94] I. Milne. The importance of the managment of structural integrity. Engineering Failure
Analysis, 1, No. 3:171 { 181, 1958.
[95] M.A. Miner. Cumulative damage in fatigue. Journal of Applied Mechanics, 12:159{164,
1945.
[96] J. Morrow. Fatigue properties of metals. Section 3.2 of Fatigue Design Handbook. Society
for Automotive Engineers, Warrendale, AE-4 edition, 1964.
[97] J. Morrow. Fatigue design handbook. section 3.2. Advances in Engineering, SAE, War-
rendale, PA, 4:21{29, 1968.
[98] J. D. Morrow. Cyclic plastic strain energy and fatigue of metals. in internal friction,
damping and cyclic plasticity. ASTM, West Conshohocken, PA, pages 45{86, 1965.
[99] A. Naess and B. Hungness. Estimating confldence intervals of long period design by
bootstrapping. Journal of Ofishore Mechanics and Arctic Engineering-Transactions of the
ASME, 124 (1):2{5, 2002.
[100] J. Napier. Cannon of Logarithms. 1614.
[101] W. Nelson. Analysis of residuals from censored data. Technometrics, 15:697{715, 1973.
[102] S. Nirbhay, R. Khelawan, and G.N. Mathur. Efiect of stress ratio and frequency on fatigue
crack growth rate of 2618 aluminium alloy silicon carbide metal matrix composite. Bulletin
of Materials Science, 24(2):169{171, 2001.
[103] N. Oresme. Questiones super geometriam Euclidis. Manuscript, Paris, 1347.
[104] N. Oresme. Tractatus de conflgurationibus qualitatum et motuum. Manuscript, Paris, 1352.
BIBLIOGRAPHY 233
[105] A. Palmgrem. Die Lebendauer von Kugellagern. Ver. Deut. Ingr., 68:339{341, 1924.
[106] P. C. Paris and F. Erdogan. A critical analysis of crack propagation law. Transactions of
the ASME Journal Basic Engineering, 85, 4:528{538, 1963.
[107] F.G. Pascual and W.Q. Meeker. Estimating fatigue curves with the random fatigue-limit
model. Technometrics, 41, No. 4:277{290, 1999.
[108] R. Picciotto. Tensile fatigue characteristics of sized polyester/viscose yarn and their efiect
on weaving performance. Masters Thesis. PhD thesis, North Carolina State Univ., North
Carolina, 1970.
[109] L.P. Pook and N.E. Frost. A fatigue crack growth theory. International Journal of Fracture,
9:53{61, 1973.
[110] Koller R., Ruiz-Ripoll M.L., Garc??a A., Fern?andez-Canteli A., and Castillo E. Experimen-
tal validation of a statistical model for the Wo?hler fleld corresponding to any stress level
and amplitude. International Journal of Fatigue, doi:10.1016/j.ijfatigue.2008.09.003.
[111] D. Radaj and C.M. Sonsino. Fatigue Assessment of Welded Joints by Local Approaches.
Abington Publishing, Cambridge, 1998.
[112] W. Ramberg and W. R. Osgood. Description of stress-strain curves by three parameters.
NACA Technical Note, 902, 1943.
[113] T.M. Rassias. Functional equations and inequalities. Dordrecht: Lluwer Academic Pub-
lisher, 2000.
[114] R. C. Rice et al. Fatigue Design Handbook. Society of Automotive Engineers, Warrendale,
PA, 3rd edition, 1997.
[115] M.L. Ruiz-Ripoll, A. Garc??a, E. Castillo, A. Fern?andez-Canteli, and R. Koller. Fatigue
damage measures with a statistical model for the whler fleld. In Proceding for the IAL-
CEE?08. First International Conference in Life Cycling Engineering. pp. 221{227. Taylor
& Francis Group, London, 2008.
[116] S. Sarkani, G. Michaelov, D.P. Kihl, and J.E. Beach. Fatigue of welded steel joints under
wideband loadings. Probabilistic Engineering Mechanics, 11:221{227, 1996.
[117] J. Schijve. NLR-TR-101-361. Amsterdam, 1962.
[118] W. Schu?tz. View points of material selection for fatigue loaded structures. Darmstadt,
1968.
[119] M. Shankar and J.B. Ahna. Frequency efiects on fatigue behavior of nextel720TM/alumina
at room temperature. Journal of the European Ceramic Society, 28:2783{2789, 2008.
[120] C.G. Small. Functional equations, how to solve them. Springer. Boston Academic Press,
New York, 2007.
[121] J. Smital. On functions and functional equations. Hilger, New York, 1988.
[122] J.O. Smith. The efiect of range of stress on the fatigue strength of metals. Urbana, Il,
1942.
234 BIBLIOGRAPHY
[123] K.N. Smith, P. Watson, and T.H. Topper. A stress-strain function for the fatigue of metals.
Journal of Materials, 5:767{778, 1970.
[124] C. R. Soderberg. Fatigue of safety and working stress. Transactions of the American
Society of Mechanical Engineers, 52 (Part APM-52-2):13{28, 1930.
[125] C.R. Soderberg. Factor of safety and working stress. Transactions of the ASTM, 52:13{28,
1939.
[126] J.E. Spindel and E. Haibach. Some consideration in the statistical determination of the
shape of S{N curves. Statistical Analysus of Fatigue Data, ASTM, ASTM STP 744:89{113,
1981.
[127] P. Starke, F. Walther, and D. Ei er. PHYBAL-A new method for lifetime prediction based
on strain, temperature and electrical measurements. International Journal of Fatigue,
28(9):1028{1036, 2006.
[128] C. E. Stromeyer. The determination of fatigue limits under alternating stress conditions.
Proceedings of the Royal Society of London. Series A, 90, No. 620:411{425, 1914.
[129] E.K. Walker. The efiect of the stress ratio during crack propagation and fatigue for 2024-T3
and 7075-T6 aluminium. ASTM-SPT, 462:1{14, 1970.
[130] W. Weibull. Fatigue testing and analysis of results. Pergamon, Oxford, 1st edition, 1961.
[131] J. S. Wilson and B. P. Haigh. Stresses in bridges. Engineering (London), 116:411{413,
1923.
[132] S.R. Winterstein. Nonlinear vibration models for extremes and fatigue. Journal of Engi-
neering Mechanic ASCE, 114(10):772{90, 1988.
[133] B. Wolf, C. Fleck, and D. Ei er. Characterization of the fatigue behaviour of the mag-
nesium alloy AZ91D by means of mechanical hysteresis and temperature measurements.
Technical Note. International Journal of Fatigue, 26:1357{1363, 2004.
[134] T. Yokobori, S. Konosu, and A.T.Jr. Yokobori. Micro and Macro Fracture Mechanics
Approach to Brittle Fracture and Fatigue Crack Growth. advances in Research on the
Strength and Fracture of Materials. Master?s thesis, Engineering Science and Mechanics
Departament Virginia Polytechnic Institute and State University, Waterloo, Canada, 1977.
[135] L. Yung-Li, J. Pan, R. Hathaway, and M. Barkey. Fatigue Testing and Analysis. Theory
and Practice. Elsevier. Butterworth Heinemann, Oxford, 1st edition, 2004.
[136] J. Zhang and D.B. Kececioglu. New approaches to determine the endurance strength
distribution. In 4th ISSAT International Conference on Reliability & Quality in Design.
Seattle, WA, 1998.