Marco Antonio Campo Cabana

209 Pages·2008·4.55 MB·English

Marco Antonio Campo Cabana
PhD Dissertation
NUMERICAL ANALYSIS AND SIMULATIONS
OF SOME PROBLEMS WITH DAMAGE
IN SOLID MECHANICS
Departamento de Matema(cid:19)tica Aplicada
Facultad de Matema(cid:19)ticas
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Esta tesis doctoral fue presentada por Don Marco Antonio Campo Cabana en el
DepartamentodeMatema(cid:19)ticaAplicadadelaUniversidadedeSantiagodeCompostela
para la obtencio(cid:19)n del grado de Doctor en ciencias Matema(cid:19)ticas.
Fue dirigida por el Profesor Doctor Don Juan Manuel Vian~o Rey y el Profesor Doctor
Don Jos(cid:19)e Ramo(cid:19)n Ferna(cid:19)ndez Garc(cid:19)(cid:16)a, de la Universidade de Santiago de Compostela.
La defensa pu(cid:19)blica de la tesis tuvo lugar el d(cid:19)(cid:16)a 19 de octubre de 2007 ante el Tribunal
constituido por:
PRESIDENTE:
Dr. D. Alfredo Bermu(cid:19)dez de Castro Lo(cid:19)pez-Varela, Departamento
de Matema(cid:19)tica Aplicada, Universidade de Santiago de Compostela.
VOCALES:
Dr. Ingeniero D. Tod Laursen, Departament of Civil and
Environmental Engineering, Duke University.
Dr. Ingeniero D. Georgios Stavroulakis, DepartamentofProduction
Engineering and Management, Technical University of Crete.
Dr. D. Eduardo Casas Renter(cid:19)(cid:16)a, Departamento de Matema(cid:19)tica
Aplicada y Ciencias de la Computacio(cid:19)n, Universidad de Cantabria.
SECRETARIO:
Dr. D. Carlos V(cid:19)azquez Cendo(cid:19)n, Departamento de Matema(cid:19)tica
Aplicada, Universidade da Corun~a.
Obtuvo por unanimidad la cali(cid:12)cacio(cid:19)n de Sobresaliente cum laude
Octubre,2007.
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Acknowledgements
It is di(cid:14)cult to resume in a few lines all the grattitude I feel for the people who has
helped me in di(cid:11)erent ways along the research which has derived to this thesis.
First of all I have to strongly thank Professor Juan Vian~o for giving me the oportunity
of joininghis researchgroup andtherefore o(cid:11)er me the chanceto begina research work
in the (cid:12)eld of applied mathematics. No less grattitude deserves the other advisor of
this thesis, Jose R. Ferna(cid:19)ndez, whose decission and absolute dedication have been
decissive in the development of this work. For both of them, many thanks, not only
for their constant support and wisdom advises, but also for having passed through the
pro(cid:11)esional to the personal relation, which made everything easier and I hope I have
been able to correspond.
Many thanks also for all the collaborators I have had the luck to work with, specially
for Meir Shillor, Mircea Sofonea and Georgios Stavroulakis, who have always provided
me with helpful and motivate comments in a very pleasant enviroment.
I cannot forget now those friends who shared with me those years of research life as
(cid:13)atmates, supporting me in good and bad moments inconditionally. Thank you very
much for C(cid:19)esar, Visconti, Tere, Jesu(cid:19)s and Rafa.
It is a doubt now to thank all the research fellows of the faculty of Mathematics for all
those sport matches, chess games, co(cid:11)ee breaks and daily support in the battle (cid:12)eld.
Many thanks to Teresa, Julio, Carlitos, Brozos, Mar(cid:19)(cid:16)a, Andr(cid:19)es, Rosa... with special
(cid:19)
grattitude for Angel, my (cid:12)rst o(cid:14)cemate, who was always ready to sacri(cid:12)ce his time
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to solve hundreds of my doubts in my beginnings.
And last, but not least, I have to thank my parents, Jose Mar(cid:19)(cid:16)a and Maribel, and
my sister Ana for their inconditional love and never end e(cid:11)orts making everything
possible, and my life easier and happier.
Resumen 11
Introduction 21
1 Preliminaries 27
1.1 Material behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.2 Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.2.1 Modi(cid:12)ed constitutive laws . . . . . . . . . . . . . . . . . . . . 35
1.3 Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.4 Theoretical results and notation . . . . . . . . . . . . . . . . . . . . . 43
2 Quasistatic problems with damage 47
2.1 Quasistatic elastic damage problem . . . . . . . . . . . . . . . . . . . 49
2.1.1 An existence and uniqueness result . . . . . . . . . . . . . . . 52
2.1.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . 53
2.1.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 60
2.2 A frictional contact problem in viscoelasticity . . . . . . . . . . . . . 68
2.2.1 An existence and uniqueness result . . . . . . . . . . . . . . . 72
2.2.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . 74
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2.2.3 Numerical resolution of two-dimensional problems . . . . . . . 83
2.2.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 87
2.3 A contact problem in viscoelasticity with long memory . . . . . . . . 93
2.3.1 An existence and uniqueness result . . . . . . . . . . . . . . . 96
2.3.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . 100
2.3.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 106
2.4 A frictional contact problem in elasto-
viscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.4.1 An existence and uniqueness result . . . . . . . . . . . . . . . 117
2.4.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . 119
2.4.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 128
3 Dynamic viscoelastic contact problems with damage 135
3.1 A dynamic frictionless contact problem in viscoelasticity . . . . . . . 139
3.1.1 An existence and uniqueness result . . . . . . . . . . . . . . . 141
3.1.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . 145
3.1.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 153
3.2 A dynamic frictional contact problem in viscoelasticity . . . . . . . . 159
3.2.1 An existence and uniqueness result . . . . . . . . . . . . . . . 161
3.2.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . 169
3.2.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 176
3.3 A frictionless dynamic fully damageable contact problem . . . . . . . 181
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3.3.1 An existence and uniqueness result . . . . . . . . . . . . . . . 183
3.3.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . 187
3.3.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 194
Conclusions 201
Bibliography 202
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