Variational approaches in the study of nonlinear problems arising in Contact Mechanics Contents Abstract v Rezumat ix Publications of the Thesis xiii… [610092]
HABILITATION THESIS
Specialization: Mathematics
Andaluzia Cristina Matei
Variational approaches in the study of nonlinear
problems arising in Contact Mechanics
Contents
Abstract v
Rezumat ix
Publications of the Thesis xiii
SCIENTIFIC RESULTS 1
Preface 3
The most frequent notation 11
I A variational approach via Lagrange multipliers 13
1 Slip-independent frictional contact problems 14
1.1 An antiplane problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.1 The model and its weak solvability . . . . . . . . . . . . . . . . . . . . . 14
1.1.2 Nonconforming discretization and optimal a priori error estimates . . . . 18
1.2 An elasto-piezoelectric problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.1 An abstract auxiliary result . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.2 The model and its weak solvability . . . . . . . . . . . . . . . . . . . . . 24
1.2.3 Discretization and an optimal a priori error estimate . . . . . . . . . . . 26
2 Slip-dependent frictional contact problems 30
2.1 An abstract result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 An antiplane problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 A 3D slip-dependent frictional contact problem . . . . . . . . . . . . . . . . . . 35
3 Contact problems for nonlinearly elastic materials 39
3.1 Problems governed by strongly monotone and Lipschitz continuous operators . . 39
3.1.1 Abstract results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
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3.1.2 Contact models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Problems governed by proper convex l.s.c functionals . . . . . . . . . . . . . . . 45
3.2.1 An abstract result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.2 3D contact models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Problems governed by a nonlinear, hemicontinuous, generalized monoton operator 53
3.3.1 An abstract result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.2 An antiplane frictional contact problem . . . . . . . . . . . . . . . . . . . 56
4 Viscoelastic frictional contact problems 58
4.1 The case of viscoelasticity with long-memory term . . . . . . . . . . . . . . . . . 58
4.1.1 An abstract result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.2 A mechanical model and its weak solvability . . . . . . . . . . . . . . . . 60
4.2 The case of viscoelasticity with short-memory term . . . . . . . . . . . . . . . . 64
4.2.1 An abstract result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.2 A mechanical model and its weak solvability . . . . . . . . . . . . . . . . 67
5 Frictionless contact problems 71
5.1 The case of electro-elastic materials . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.1 The case of nonconductive foundation . . . . . . . . . . . . . . . . . . . . 72
5.1.2 The case of conductive foundation . . . . . . . . . . . . . . . . . . . . . . 75
5.2 The case of viscoplastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2.1 The model and its weak solvability . . . . . . . . . . . . . . . . . . . . . 81
5.2.2 A convergence result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Contact problems involving multi-contact zones 87
6.1 The case of linear elastic operators . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.1 Abstract results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.2 3D contact models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 The case of nonlinear elastic operators . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.1 Abstract results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.2 3D contact models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7 Unilateral frictional contact problems 101
7.1 Abstract results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 A 3D contact model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
II A variational approach via bipotentials 105
8 Unilateral frictionless contact problems 106
8.1 The model and its weak solvability via bipotentials . . . . . . . . . . . . . . . . 106
8.2 New approach versus previous approaches . . . . . . . . . . . . . . . . . . . . . 108
9 Frictional contact problems 111
9.1 The model and its weak solvability via bipotentials . . . . . . . . . . . . . . . . 111
9.2 New approach versus previous approaches . . . . . . . . . . . . . . . . . . . . . 114
III A variational approach via history-dependent quasivariational
inequalities on unbounded time interval 115
10 Preliminaries 116
10.1 A fixed point result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.2 An abstract history-dependent quasivariational inequality . . . . . . . . . . . . . 118
11 Viscoplastic problems 121
11.1 Mechanical models and their weak solvability . . . . . . . . . . . . . . . . . . . . 121
11.2 A convergence result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
12 Electro-elasto-viscoplastic contact problems 129
12.1 The mechanical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
12.2 Weak formulation and main results . . . . . . . . . . . . . . . . . . . . . . . . . 131
CAREER EVOLUTION AND DEVELOPMENT PLANS 137
13 Further research directions 139
14 Further plans 149
14.1 On the scientific and professional career . . . . . . . . . . . . . . . . . . . . . . . 149
14.2 On the academical career . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
BIBLIOGRAPHY 151
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v
Abstract
The present work is a collection of results in the study of nonlinear problems by means of three
variational approaches: a variational approach via Lagrange multipliers, a variational approach
via bipotentials and a variational approach via history-dependent quasivariational inequalities
on unbounded time intervals.
The study we present in this thesis has an interdisciplinary character and a strong appli-
cability feature, combining mathematical areas as PDEs, Nonlinear Analysis, Convex Analysis
and Calculus of Variations with Mechanics of Continua, Mechanics of Materials and Contact
Mechanics.
All the problems we discuss in this thesis are related to models in Contact Mechanics for
several types of deformable solid materials. The scientific results mentioned in the present thesis
represent a part of the scientific results of the candidate, published after obtaining the Ph.D.
degree in Mathematics. The results we focus on are presented without proofs; details can be
found in the papers mentioned in the Publications of the Thesis (just after the Abstract in
Romanian).
The thesis comprises SCIENTIFIC RESULTS, CAREER EVOLUTION AND DEVELOP-
MENT PLANS and BIBLIOGRAPHY.
The presentation of the SCIENTIFIC RESULTS is organized into three parts.
Part I, devoted to the study of a class of contact models by a variational approach with
Lagrange multipliers , is a collection of new mixed variational problems. The variational for-
mulations via Lagrange multipliers in non-smooth mechanics are suitable formulations to effi-
ciently approximate the weak solutions; this motivates the research in this direction. Structured
in seven chapters, Part I discusses slip-independent frictional contact problems/slip-dependent
frictional contact problems, contact problems for several types of nonlinearly elastic materials,
frictional contact viscoelastic problems, frictionless contact problems involving electro-elastic or
viscoplastic materials, contact problems involving multi-contact zones, unilateral frictional con-
tact problems, focusing on their weak solvability. Presenting new abstract results as nonlinear
analysis tools is also under attention. The abstract problems we consider herein are new abstract
variational systems. In order to solve them, the main techniques we use rely on saddle point or
fixed point techniques.
Part II adopts a variational approach via bipotentials in the weak solvability of a class of
nonlinearly elastic contact problems. This second part comprises two chapters, Chapters 8-9. In
Chapter 8 it is analyzed a unilateral frictionless contact model while in Chapter 9 it is analyzed a
frictional contact model, both models leading to new variational systems. In order to solve these
systems, the main technique we use is a minimization technique. Using a separated bipotential
we get existence and uniqueness results suitable to discuss a simultaneous computation of the
displacement field and the Cauchy stress tensor. The results presented in Part II represent first
vi
steps in a new research direction, more complex models related to non-separated bipotentials
being also envisaged.
Part III discusses the variational analysis via history-dependent quasivariational inequalities
for a class of viscoplastic or electro-elasto-viscoplastic contact problems on unbounded time inter-
val. This third part comprises three chapters, Chapters 10-12. Some preliminaries are presented
in Chapter 10: a fixed point result and an existence and uniqueness result for an auxiliary
problem consisting of an abstract history-dependent quasivariational inequality formulated on
unbounded time interval. Using these preliminaries, we analyze in Chapter 11 two viscoplastic
problems and in Chapter 12 an electro-elasto-viscoplastic problem. The weak formulations we
deliver are new variational problems. Working on the interval [0 ,∞), a continuation of the
research going to the Asymptotic Analysis in Contact Mechanics is envisaged.
The main contributions:
Ithe statement and the solution of three new classes of abstract problems
•stationary mixed variational problems governed by nonlinear maps
•evolutionary mixed variational problems (with short-memory term)
•time-dependent mixed variational problems (with long-memory term)
Ithe weakly solvability of contact models by new variational techniques
•fornonlinearly elastic, viscoelastic, viscoplastic or electro-elastic materials via a variational
approach with Lagrange multipliers
•fornonlinearly elastic materials governed by possibly set valued elastic operators by means
ofa variational approach via bipotentials theory
•onunbounded time interval .
We end this thesis by presenting some CAREER EVOLUTION AND DEVELOPMENT
PLANS. The presentation is structured in two chapters, Chapters 13-14. Chapter 13 presents
further research directions such us: qualitative and numerical analysis in the study of mixed
variational problems or in the study of variational systems via bipotentials; variational for-
mulations in contact mechanics/ weak solutions via weighted Sobolev spaces or via Lebesgue
spaces with variable exponent; optimal control problems in contact mechanics; mathematical
techniques in the study of dissipative dynamic contact problems; asymptotic analysis in contact
mechanics, regularity results; convergence results; viscoelastic problems via fractional differential
operators/fractional calculus of variations. Chapter 14 presents further plans on the scientific,
professional and academic career such as: to do a research activity allowing to continue to pub-
lish in international journals of hight level, to participate to international meetings in order to
disseminate the best results, to organize scientific meetings, to extend the editorial activities
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for scientific journals, to continue the collaborations started in the past and to establish new
contacts, to apply for national/international/interdisciplinary research projects as manager or
member, to publish Lecture Notes and new monographs addressed to the students or researchers,
to extend the advising activity to Ph.D. theses.
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ix
Rezumat
Prezenta lucrare este o colect ¸ie de rezultate ˆ ın studiul unor probleme neliniare, studiu realizat
prin intermediul a trei abordˇ ari variat ¸ionale: o abordare variat ¸ionalˇ a cu multiplicatori Lagrange,
o abordare variat ¸ionalˇ a via bipotent ¸iali ¸ si a abordare variat ¸ionalˇ a bazatˇ a pe teoria inegalitˇ at ¸ilor
cvasivariat ¸ionale cu termen istoric-dependent formulate pe interval de timp nemˇ arginit.
Studiul prezentat prin intermediul acestei teze are atˆ at caracter interdisciplinar cˆ at ¸ si o
puternicˇ a trˇ asˇ aturˇ a aplicativˇ a, ˆ ımbinˆ and domenii de matematicˇ a aplicatˇ a cum ar fi Ecuat ¸ii cu
derivate part ¸iale, Analizˇ a neliniarˇ a, Analizˇ a convexˇ a ¸ si Calcul variat ¸ional cu Mecanica mediilor
continue, Mecanica materialelor ¸ si Mecanica contactului.
Toate problemele discutate ˆ ın aceastˇ a tezˇ a sunt in legˇ aturˇ a cu modele ˆ ın mecanica contac-
tului pentru mai multe tipuri de materiale solide deformabile. Rezultatele ¸ stiint ¸ifice ment ¸ionate
ˆ ın prezenta tezˇ a reprezintˇ a o parte dintre rezultatele ¸ stiint ¸ifice ale candidatei, publicate dupˇ a
obt ¸inerea titlului de Doctorˆ ın Matematicˇ a. Rezultatele focalizate sunt prezentate fˇ arˇ a demonstra-
t ¸ii; detalii pot fi gˇ asite ˆ ın lista de lucrˇ ari intitulatˇ a ”Publicat ¸iile tezei”, listˇ a ce apare ˆ ın prezentul
manuscris imediat dupa rezumatul tezei in limba romˆ ana.
Teza cuprinde REZULTATE S ¸TIINT ¸IFICE, PLANURI DE DEZVOLTARE SI EVOLUT ¸IE
A CARIEREI ¸ si BIBLIOGRAFIE.
Prezentarea REZULTATELOR S ¸TIINT ¸IFICE este organizatˇ a ˆ ın trei pˇ art ¸i.
Partea I, dedicatˇ a studiului unei clase de modele ˆ ın mecanica contactului prin intermediul
unei abordˇ ari variat ¸ionale via multiplicatori Lagrange, este o colect ¸ie de noi probleme variat ¸ionale
mixte. Formulˇ arile variat ¸ionale via multiplicatori Lagrange ˆ ın mecanica nenetedˇ a sunt formulˇ ari
care permit o eficientˇ a aproximare a solut ¸iilor slabe; aceasta motiveazˇ a cercetarea ˆ ın aceastˇ a
direct ¸ie. Structuratˇ a in ¸ sapte capitole, Partea I analizeazˇ a variat ¸ional probleme de contact cu
frecare independentˇ a sau dependentˇ a de alunecare, probleme de contact pentru diferite tipuri de
materiale neliniar elastice, probleme vˆ ascoelastice de contact cu frecare, probleme de contact cu
frecare neglijabilˇ a pentru materiale electro-elastice sau vˆ ascoplastice, probleme ce implicˇ a mai
multe zone de contact, probleme de contact unilateral cu frecare. Se are ˆ ın vedere de asemenea
prezentarea unor noi rezultate abstracte care pot fi considerate unelte utile de analizˇ a neliniarˇ a.
Problemele abstracte discutate ˆ ın aceastˇ a parte a lucrˇ arii sunt noi sisteme variat ¸ionale abstracte.
Principalele tehnici utilizate ˆ ın rezolvarea lor sunt tehnici de punct ¸ sa ¸ si tehnici de punct fix.
Partea a-II-a adoptˇ a o abordare variat ¸ionalˇ a via bipotent ¸iali ˆ ın vederea rezolvˇ arii ˆ ın sens slab
a unei clase de probleme de contact pentru materiale neliniar elastice. Aceastˇ a a doua parte
a lucrˇ arii are douˇ a capitole, Capitolele 8-9. ˆIn Capitolul 8 se analizeazˇ a un model de contact
unilateral fˇ arˇ a frecare ˆ ın timp ce ˆ ın Capitolul 9 se analizeazˇ a un model de contact cu frecare,
ambele modele conducˆ and la noi sisteme variat ¸ionale. Principala tehnicˇ a utilizatˇ a ˆ ın vederea
rezolvˇ arii acestor sisteme este o tehnicˇ a de minimizare. Utilizˆ andu-se un bipotent ¸ial separat se
obt ¸in rezultate de existent ¸ˇ a ¸ si unicitate care permit un calcul simultan al cˆ ampului deplasare ¸ si
al tensorului tensiune Cauchy. Rezultatele prezentate ˆ ın aceastˇ a parte a tezei reprezintˇ a primi
x
pa¸ si ˆ ıntr-o nouˇ a direct ¸ie de cercetare, fiind vizate de asemenea modele mai complexe care implicˇ a
bipotent ¸iali neseparat ¸i.
Partea a-III-a prezintˇ a rezultate ˆ ın analiza variat ¸ionalˇ a, via inegalitˇ at ¸i cvasivariat ¸ionale cu
termen istoric-dependent, pentru o clasˇ a de probleme de contact vˆ ascoplastice sau electro-elasto-
vˆ ascoplastice, formulate pe interval de timp nemˇ arginit. Aceastˇ a a treia parte are trei capitole,
Capitolele 10-12. ˆIn Capitolul 10 sunt prezentate cˆ ateva preliminarii: un rezultat de punct
fix ¸ si un rezultat de existent ¸ˇ a ¸ si unicitate pentru o problemˇ a auxiliarˇ a ce constˇ a dintr-o ine-
galitate cvasivariat ¸ionalˇ a abstractˇ a cu termen istoric-dependent, formulatˇ a pe interval de timp
nemˇ arginit. Utilizˆ and aceste preliminarii, ˆ ın Capitolul 11 analizˇ am douˇ a probleme vˆ ascoplastice
¸ si ˆ ın Capitolul 12 o problemˇ a electro-elasto-vˆ ascoplasticˇ a. Formulˇ arile variat ¸ionale obt ¸inute sunt
noi probleme variat ¸ionale. Lucrˆ and pe intervalul [0 ,∞), se are ˆ ın vedere o continuare a cercetarii
in direct ¸ia Analizei Asimptotice.
Principalele contribut ¸ii:
Iformularea ¸ si rezolvarea a trei noi tipuri de probleme abstracte:
•probleme variat ¸ionale mixte stat ¸ionare guvernate de aplicat ¸ii neliniare
•probleme variat ¸ionale mixte de evolut ¸ie (cu termen memorie scurtˇ a)
•probleme variat ¸ionale mixte dependente de timp (cu termen memorie lungˇ a)
Istudiul solut ¸iilor slabe, prin intermediul unor noi tehnici de calcul variat ¸ional, al unor
modele ˆ ın mecanica contactului
•pentru materiale neliniar elastice, vˆ ascoelastice, vˆ ascoplastice sau electro-elastice prin in-
termediul unei abordˇ ari variat ¸ionale cu multiplicatori Lagrange
•pentru materiale neliniar elastice guvernate de operatori elastici posibil multivoci , prin
intermediul unei abordˇ ari variat ¸ionale via bipotent ¸iali
•peinterval de timp nemˇ arginit .
Prezenta tezˇ a se ˆ ıncheie cu prezentarea unor PLANURI DE DEZVOLTARE S ¸I EVOLUT ¸IE
A CARIEREI. Aceastˇ a prezentare este structuratˇ a ˆ ın douˇ a capitole, Capitolele 13-14. ˆIn
Capitolul 13 sunt indicate direct ¸ii de cercetare pe care candidata le are ˆ ın vedere pentru pe-
rioada urmˇ atoare, direct ¸ii precum: analizˇ a calitativˇ a ¸ si numericˇ a ˆ ın studiul unor probleme
variat ¸ionale mixte sau ˆ ın studiul unor sisteme variat ¸ionale via bipotent ¸iali; formulˇ ari variat ¸ionale
ˆ ın mecanica contactului /solut ¸ii slabe prin intermediul spat ¸iilor Sobolev cu pondere sau prin in-
termediul spat ¸iilor Lebesgue cu exponent variabil; probleme de control optimal ˆ ın mecanica con-
tactului; un studiu matematic pentru probleme de contact dinamice disipative, analizˇ a asimp-
toticˇ a ˆ ın mecanica contactului, rezultate de regularitate, rezultate de convergent ¸ˇ a, probleme
vˆ ascoelastice via operatori diferent ¸iali fract ¸ionari/calcul variational fract ¸ionar. ˆIn Capitolul 14
xi
sunt prezentate planuri viitoare ˆ ın carierˇ a, atˆ at din punct de vedere ¸ stiint ¸ific ¸ si profesional
cˆ at ¸ si din punct de vedere academic, precum: desfˇ a¸ surarea unei activitˇ at ¸i de cercetare de
calitate care sˇ a conducˇ a la publicat ¸ii ˆ ın jurnale internat ¸ionale de ˆ ınalt nivel, participarea la
evenimente ¸ stiint ¸ifice ˆ ın cadrul cˇ arora sˇ a fie diseminate principalele rezultate obt ¸inute, organi-
zarea de evenimente ¸ stiint ¸ifice, extinderea activitˇ at ¸ii editoriale pentru jurnale ¸ stiint ¸ifice, con-
tinuarea colaborˇ arilor ˆ ıncepute ˆ ın trecut ¸ si stabilirea de noi contacte, aplicarea pentru proiecte
nat ¸ionale/internat ¸ionale/interdisciplinare ca director sau membru de echipˇ a, publicarea de note
de curs sau monografii adresate student ¸ilor sau cercetˇ atorilor, extinderea activitˇ at ¸ii de coor-
donare ¸ stiint ¸ificˇ a, de la lucrˇ ari de licent ¸ˇ a sau disertat ¸ii, la teze de doctorat.
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Publications of the Thesis
The Publications of the Thesis, specifying them in the order of their appearance in the present
manuscript, are the following.
[1] S. H¨ ueber, A. Matei and B. Wohlmuth, Efficient algorithms for problems with friction,
SIAM Journal on Scientific Computing , DOI: 10.1137/050634141, 29(1) (2007), 70-92.
[2] S. H¨ ueber, A. Matei and B. Wohlmuth, A mixed variational formulation and an optimal
a priori error estimate for a frictional contact problem in elasto-piezoelectricity, Bull. Math.
Soc. Math. Roumanie ,48(96)(2) (2005), 209-232.
[3]A. Matei , On the solvability of mixed variational problems with solution-dependent sets
of Lagrange multipliers, Proceedings of The Royal Society of Edinburgh, Section: A Mathematics ,
143(05) (2013), 1047-1059, http://dx.doi.org/10.1017/S0308210512000637.
[4]A. Matei , A mixed variational formulation for a slip-dependent frictional contact model,
Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on
Engineering 2014 , 2-4 July, 2014, London, U.K., pp 750-754 (ISBN: 978-988-19253-5-0, ISSN:
2078-0958).
[5]A. Matei , Weak Solutions via Lagrange Multipliers for a Slip-dependent Frictional Con-
tact Model, IAENG International Journal of Applied Mathematics ,44(3) (2014), 151-156.
[6]A. Matei and R. Ciurcea, Contact problems for nonlinearly elastic materials: weak solv-
ability involving dual Lagrange multipliers, Australian and New Zealand Industrial and Applied
Mathematics Journal (ANZIAM JOURNAL) , DOI:10.1017/S1446181111000629, 52(2010), 160-
178.
[7]A. Matei and R. Ciurcea, Weak solvability for a class of contact problems, Annals of
the Academy of Romanian Scientists Series on Mathematics and its Applications ,2(1) (2010),
25-44.
[8]A. Matei , Weak solvability via Lagrange multipliers for two frictional contact mod-
els, Proceedings of ”11-` eme Colloque Franco-Roumain de Math´ ematiques Appliqu´ ees”, 2012,
Bucharest, Annals of the University of Bucharest (mathematical series) ,4(LXII) (2013), 179-
191.
[9]A. Matei , An existence result for a mixed variational problem arising from Contact
Mechanics, Nonlinear Analysis Series B: Real World Application ,20(2014), 74-81,
DOI: 10.1016/j.nonrwa.2014.01.010.
[10]A. Matei and R. Ciurcea, Weak solutions for contact problems involving viscoelastic ma-
terials with long memory, Mathematics and Mechanics of Solids , DOI: 10.1177/1081286511400515,
16(4) (2011), 393-405.
[11]A. Matei , An evolutionary mixed variational problem arising from frictional contact me-
chanics, Mathematics and Mechanics of Solids , DOI: 10.1177/1081286512462168, 19(3) (2014),
225-241.
xiv
[12]A. Matei , A variational approach for an electro-elastic unilateral contact problem,
Mathematical Modelling and Analysis ,14(3) (2009), 323-334.
[13] S. H¨ ueber, A. Matei , B. Wohlmuth, A contact problem for electro-elastic materials,
ZAMM-Journal of Applied Mathematics and Mechanics , Special Issue: Mathematical Model-
ing: Contact Mechanics, Phase Transition, Multiscale Problems, DOI:10.1002/zamm.201200235,
93(10-11) (2013), 789-800.
[14] M. Barboteu, A. Matei and M. Sofonea, On the behavior of the solution of a viscoplas-
tic contact problem, Quarterly of Applied Mathematics , DOI: http://dx.doi.org/10.1090/S0033-
569X-2014-01345-4, 72(4) (2014), 625-647.
[15]A. Matei , Weak solvability via Lagrange multipliers for contact problems involving
multi-contact zones, Mathematics and Mechanics of Solids , DOI: 10.1177/1081286514541577,
published online July 7, 2014.
[16]A. Matei , Two abstract mixed variational problems and applications in Contact Me-
chanics, Nonlinear Analysis Series B: Real World Application , published online October 13,
2014, http://dx.doi.org/10.1016/j.nonrwa.2014.09.014.
[17]A. Matei , A variational approach via bipotentials for unilateral contact problems, Jour-
nal of Mathematical Analysis and Applications , http://dx.doi.org/10.1016/j.jmaa.2012.07.065,
397(1) (2013), 371-380.
[18]A. Matei , A variational approach via bipotentials for a class of frictional contact prob-
lems, Acta Applicandae Mathematicae , DOI: 10.1007/s10440-014-9868-1, (2014), 134(1) (2014),
45-59.
[19] M. Barboteu, A. Matei and M. Sofonea, Analysis of Quasistatic Viscoplastic Contact
Problems with Normal Compliance, The Quarterly Journal of Mechanics and Applied Mathe-
matics ,65(4) (2012), 555-579, DOI: 10.1093/qjmam/hbs016.
[20] M. Boureanu, A. Matei and M. Sofonea, Analysis of a Contact Problem for Electro-
elastic-visco-plastic Materials, Communications on Pure and Applied Analysis ,11(3) (2012),
1185-1203, DOI: 10.3934/cpaa.2012.11.1185.
1
SCIENTIFIC RESULTS
2
3
Preface
”Each progress in mathematics is based on
the discovery of stronger tools and easier methods,
which at the same time makes it easier to
understand earlier methods. By making these stronger
tools and easier methods his own, it is possible for
the individual researcher to oriented
himself in the different branches of mathematics. ”
David Hilbert, 1900
The present thesis is a collection of results in the weak solvability of a class of nonlinear
problems by means of three variational approaches: a variational approach via Lagrange multi-
pliers, a variational approach via bipotentials and a variational approach via history-dependent
quasivariational inequalities on unbounded time intervals.
The research from this thesis has an interdisciplinary character. New trends in Advanced
Applied Mathematics are required, combining mathematical areas as PDEs, Nonlinear Analysis,
Convex Analysis and Calculus of Variations with Mechanics of Continua, Mechanics of Materials
and Contact Mechanics; see e.g. [1, 24, 50, 57, 83, 92, 151, 160] for important mathematical
tools, [131, 162] for numerical approximation techniques, [48, 49, 59, 81, 124, 128, 130, 138]
for applied mathematics in contact mechanics, [79, 88, 161, 166] for an engineering approach
in contact mechanics, [45, 73] for viscoplasticity and [76, 158] for piezoelectricity, to give just
a few examples of foundational books. It is worth to underline also the strong applicability
feature of the research from this thesis: all the problems we discuss are related to models in
Contact Mechanics for various kind of deformable solid materials. Solving contact problems for
nonlinearly materials is a challenging topic of non smooth mechanics. The contact models are
very complex. Most of them are analyzed by variational methods because of the difficulty of
finding strong solutions. After establishing the well-possedness of a contact model, the next
target is the approximation of the weak solution. Currently, obtaining variational formulations
which are suitable to an efficient approximation of the weak solutions is an issue of great interest.
The scientific results mentioned in the present thesis represent a part of the scientific results
of the candidate after obtaining the Ph.D. degree. The results we focus on are presented without
proofs; details can be found in the papers mentioned in the Publications of the Thesis, a list
placed just after the Abstract in Romanian.
The thesis comprises SCIENTIFIC RESULTS, CAREER EVOLUTION AND DEVELOP-
MENT PLANS and BIBLIOGRAPHY.
The presentation of the SCIENTIFIC RESULTS is organized into three parts.
PART I – A variational approach via Lagrange multipliers
PART II – A variational approach via bipotentials
4
PART III – A variational approach via history-dependent quasivariational inequalities on
unbounded time interval
Part I, devoted to the study of a class of contact models by a variational approach with La-
grange multipliers, is based on the papers [69, 68, 105, 109, 112, 99, 104, 100, 107, 101, 111, 98,
70, 11, 110, 113], specifying them in the order of their appearance in the present manuscript. In
this part of the thesis we discuss slip-independent or slip-dependent frictional contact problems,
contact problems for several types of nonlinearly elastic materials, frictional contact viscoelastic
problems, frictionless contact problems involving electro-elastic or viscoplastic materials, con-
tact problems involving multi-contact zones, unilateral frictional contact problems, focusing on
their weak solvability. Presenting new abstract results as nonlinear analysis tools is also under
attention. The abstract problems we consider herein are new abstract variational systems. After
presenting their solution, we show how these abstract results were used to solve contact problems
for different types of materials or different types of contact conditions, frictionless or frictional.
The main techniques we use rely on a saddle point technique and fixed point techniques. The
saddle point theory, who originates from Babuˇ sca-Brezzi works, was successfully developed and
applied in a large number of publications, see e.g. the books [22, 23, 61, 129] and the papers
[3, 63, 66, 67, 69, 132, 164] to give only a few examples. The first part of the thesis is a col-
lection of new mixed variational problems. The mixed variational formulations in non-smooth
mechanics are suitable formulations to efficiently approximate the weak solutions; this motivates
the research in this direction.
Part I is structured in seven chapters.
Chapter 1, which is concerned with the analysis of a class of slip-independent frictional
contact problems, comprises two sections: Section 1.1 based on the paper [69] and Section 1.2
based on the paper [68].
Section 1.1 focuses on an antiplane frictional contact model which is related to a saddle point
problem while Section 1.2 discusses an elasto-piezoelectric frictional contact problem whose vari-
ational formulation is related to a generalized saddle point problem with non-symmetric bilinear
forma(·,·).From the variational point of view both problems have the following form:
a(u,v) +b(v,λ) = (f, v)X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ.
Chapter 2, devoted to the analysis of a class of slip-dependent frictional contact problems,
comprises three sections: Section 2.1 and Section 2.2 are based on the paper [105] while Section
2.3 is based on the papers [109, 112].
Section 2.1 presents an abstract existence result in the study of the following mixed variational
problem with solution dependent-set of Lagrange multipliers, Λ = Λ( u).
Givenf∈X, f̸= 0X,find(u,λ)∈X×Ysuch thatλ∈Λ(u)⊂Yand
a(u,v) +b(v,λ) = (f, v)X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ(u).
5
The applicability of the abstract result we present is illustrated in Section 2.2 in the study of
an antiplane problem and in Section 2.3 in the study of a 3D slip-dependent frictional contact
problem.
Chapter 3, structured in three sections, is related to the analysis of a class of contact problems
for nonlinearly elastic materials leading to weak formulations governed, in Section 3.1, by a
strongly monoton and Lipschitz continuous operator, in Section 3.2 by a proper convex l.s.c
functional and in Section 3.3 by a nonlinear hemicontinuous generalized monotone operator.
Section 3.1 is based on the papers [99, 104], Section 3.2 is based on the papers [100, 104] and
Section 3.3 is based on the paper [107].
Section 3.1 analyzes the case of single-valued elastic operators; herein the mixed variational
formulation via Lagrange multipliers leads to a mathematical problem of the form below.
Givenf, h∈X,findu∈Xandλ∈Λ such that
(Au,v )X+b(v,λ) = (f, v)X for allv∈X,
b(u,µ−λ) ≤b(h,µ−λ) for allµ∈Λ,
whereAisa strongly monoton and Lipschitz continuous operator.
In Section 3.2 the constitutive law we use is expressed in a form of a subdifferential inclusion
governed by a proper convex lower semicontinuous functional. Thus, we focus on the case
of possibly multi-valued elastic operators. The mixed variational formulation via Lagrange
multipliers leads us to a mathematical problem having the following form.
Givenf∈X,findu∈Xandλ∈Λsuch that
J(v)−J(u) +b(v−u,λ)≥(f,v−u)Xfor allv∈X
b(u,µ−λ)≤0 for allµ∈Λ,
whereJisa proper convex lower semicontinuous functional.
In Section 3.3 we study the weak solvability via Lagrange multipliers of a class of nonlinearly
elastic contact models leading to a mixed variational problem governed by a nonlinear, hemicon-
tinuous, generalized monotone operator. Using a fixed point theorem for set valued mapping, we
analyze here the existence of the solution of the following abstract mixed variational problem.
Givenf∈X′,find(u,λ)∈X×Λsuch that
(Au,v )X′,X+b(v,λ) = (f, v)X′,X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ,
whereAisa nonlinear, hemicontinuous, generalized monotone operator. Then, we apply the
abstract result to the analysis of an antiplane contact problem involving a class of nonlinearly
elastic materials.
6
Chapter 4, devoted to a class of viscoelastic frictional contact problems has two sections,
treating the case of the viscoelasticity with long memory as well as the case of the viscoelasticity
with short memory. Section 4.1 is based on the paper [101] and Section 4.2 is based on the paper
[111].
In Section 4.1 we can see haw the viscoelastic model with long memory leads to a time-
dependent mixed variational problem involving an integral operator , which, in an abstract frame-
work, has the following form.
Givenf: [0,T]→X,findu: [0,T]→Xandλ: [0,T]→Ysuch that, for all t∈[0,T],we
haveλ(t)∈Λand
(Au(t),v)X+ (∫t
0B(t−s)u(s)ds,v)X+b(v,λ(t)) = (f(t), v)Xfor allv∈X
b(u(t),µ−λ(t))≤0 for allµ∈Λ.
In Section 4.2 we study a viscoelastic model with short memory leading to an evolutionary
mixed variational problem having the form below.
Givenf: [0,T]→X, g∈Wandu0∈X,findu: [0,T]→Xandλ: [0,T]→Λ(g)⊂Y
such that for all t∈(0,T),we have
a( ˙u(t),v) +e(u(t),v) +b(v,λ(t)) = (f(t),v)Xfor allv∈X,
b( ˙u(t),µ−λ(t))≤0 for allµ∈Λ(g),
u(0) =u0.
Chapter 5, who studies a class of frictionless contact problems, comprises two sections. Sec-
tion 5.1 is based on the papers [98, 70] and Section 5.2 is based on the paper [11].
Section 5.1 analyzes the case of electro-elastic materials, treating the case of nonconductive
foundation as well as the case of conductive foundation. Both weak formulations we deliver are
generalized saddle point problem . The variational formulation in the nonconductive case consists
of the following nonhomogeneous and nonsymmetric mixed variational problem. Givenf, g∈X,
g̸= 0X,findu∈Xandλ∈Λ such that
a(u,v)X+b(v,λ) = (f, v)X for allv∈X,
b(u,µ−λ) ≤b(g,µ−λ) for allµ∈Λ.
In the conductive case, the weak formulation consists of the following coupled variational system .
Givenf∈Xandq∈Y,find(u,φ,λ )∈X×Y×Λsuch that
a(u,v) +e(v,φ) +b(v,λ) = (f,v)Xfor allv∈X,
c(φ,ψ)−e(u,ψ) +j(λ,φ,ψ ) = (q,ψ)Yfor allψ∈Y,
b(u,µ−λ) ≤ 0 for all µ∈Λ.
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In Section 5.2, contact models involving viscoplastic materials are studied. The models lead
to a weak formulation via Lagrange multipliers which consists of a variational system coupled
with an integral equation; see the problem below.
Find a displacement field u: [0,T]→V, a viscoplastic stress field β: [0,T]→Qand a
Lagrange multiplier λ: [0,T]→Λsuch that, for all t∈[0,T],
(Lu(t),v)V+ (β(t),ε(v))Q+ (Pu(t),v)V+b(v,λ(t)) = (f(t),v)Vfor allv∈V,
b(u(t),µ−λ(t))≤b(geθ,µ−λ(t)) for allµ∈Λ,
β(t) =∫t
0G(Eε(u(s)) +β(s),ε(u(s)))ds+σ0− Eε(u0).
Chapter 6, divided in two sections, focuses on the weak solvability of a class of contact
problems involving two contact zones, for elastic materials. Section 6.1 is based on the paper
[110] and Section 6.2 is based on Sections 3 and 4 of the paper [113].
Section 6.1 focuses on the case of linearly elastic materials. To start, we present existence,
uniqueness and boundedness results for a class of abstract generalized saddle point problems, as
well as abstract convergence results for a class of regularized problems. The abstract problem
we analyze has the following form:
a(u,v−u) +b(v−u,λ) +j(v)−j(u)≥(f, v−u)X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ.
Based on the abstract results we get two models are investigated. Every model is mathemat-
ically described by a boundary value problem which consists of a system of partial differential
equations associated with a displacement condition, a traction condition, a frictional contact
condition and a frictionless unilateral contact condition. In both models the unilateral contact
is described by Signorini’s condition with non zero gap. The difference between the models is
given by the frictional condition we use. Thus, in the first model we use a frictional condi-
tion with prescribed normal stress, while in the second one we use a frictional bilateral contact
condition.
Section 6.2 focuses on the case of nonlinearly elastic materials. Firstly, we present abstract
results in the study of a generalized saddle point problem having the following form:
J(v)−J(u) +b(v−u,λ) +φ(v)−φ(u)≥(f, v−u)X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ.
Nextly, we apply the abstract results to the weak solvability of two contact models.
The study made in Chapter 7 goes to the weak solvability of an unilateral frictional contact
model. In Section 7.1 we study the existence and the uniqueness of the solution of an abstract
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mixed variational problem governed by a functional Jand a bifunctional jas follows:
J(v)−J(u) +b(v−u,λ) +j(λ,v)−j(λ,u)≥(f, v−u)X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ.
Next, in Section 7.2, we apply the abstract results to the weak solvability of an unilateral
frictional contact model.
Part II, devoted to a variational approach via bipotentials for a class of contact problems for
nonlinearly elastic materials, relies on the papers [106, 108]. The presence of the bipotentials
in mechanics of solid was noticed quite recently, but the literature covering this subject is
growing. According to Buliga-Saxc´ e-Vall´ ee, starting from an extension of Fenchel inequality, the
bipotentials were introduced as non-smooth mechanics tools used to model various multivalued
laws. Several bipotential functions are related to the Coulomb’s friction law [26], Cam-Clay
models in soil mechanics [134, 159], cyclic plasticity [16, 133], viscoplasticity of metals with non-
linear kinematical hardening rule [64], Lemaitre’s damage law [15], the coaxial laws [136, 155],
the elastic-plastic bipotential of soils [13]. For other important results related to the bipotential
theory we refer for instance to [25, 27, 28, 135]; see also the recent work [156].
Herein, two contact problems are focused: a unilateral frictionless contact problem and a fric-
tional contact problem with prescribed normal stress. In order to solve them, the main technique
we use is a minimization technique. Using a separated bipotential we investigate the existence
and the uniqueness of the solutions. The unknown is the pair consisting of the displacement
vector and the Cauchy stress tensor. The main advantage of this approach is that it allows to
compute simultaneously the displacement field and the Cauchy stress tensor. Also we discuss
the relevance of the approach reported to previous variational approaches: the primal varia-
tional formulation and the dual variational formulation. We recall that the primal variational
formulation is the weak formulation in displacements, and the dual variational formulation is
the weak formulation in terms of stress.
The problems we treat in this second part of the thesis lead to new variational systems
governed by bipotentials. The investigation on this direction can be extended to more complex
models governed by non-separated bipotentials attached to the constitutive map and its Fenchel
conjugate.
Part II comprises two chapters, Chapter 8 and Chapter 9.
In Chapter 8 we present the results in the study of a class of unilateral frictionless contact
problem obtained in the paper [106]. The variational approach we use leads us to a variational
problem having the following form.
Findu∈U0⊂Vandσ∈Λ⊂L2
s(Ω)3×3such that
b(v,σ)−b(u,σ)≥(f,v−u)V for allv∈U0
b(u,µ)−b(u,σ)≥0 for allµ∈Λ.
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In Chapter 9 we present the results in the study of a class of frictional contact problem obtained
in the paper [108]. In this case, the variational approach via bipotentials leads to a variational
problem governed by a functional jas follows.
Findu∈Vandσ∈Λ⊂L2
s(Ω)3×3such that
b(v,σ)−b(u,σ) +j(v)−j(u)≥(f,v−u)Vfor allv∈V
b(u,µ)−b(u,σ)≥0 for allµ∈Λ.
Part III discusses the variational analysis via history-dependent quasivariational inequalities
of a class of viscoplastic or electro-elasto-viscoplastic contact problem on unbounded time in-
terval. Part III focuses on results obtained in the papers [10, 20]. The first study of a contact
problem on the unbounded interval [0 ,∞) was made in [144]. The next important contribution
was the paper [146], followed by [10, 20], and more recently by [148].
In this third part of the thesis we focus on new contact models related to quasivariational
inequalities defined on unbounded time interval and governed by two nondifferentiable convex
functional in which one depends on the history of the solution,
u(t)∈K, (Au(t),v−u(t))X+φ(Su(t),v)−φ(Su(t),u(t))
+j(u(t),v)−j(u(t),u(t))≥(f(t),v−u(t))X for allv∈K.
These inequalities have a special structure, involving a history-dependent term. In addition,
working on the time interval [0 ,∞), a continuation of the research going to the Asymptotic
Analysis in Contact Mechanics is envisaged.
Part III comprises three chapters, Chapter 10-Chapter 12. Some auxiliary abstract results
are presented in Chapter 10: a fixed point result and an existence and uniqueness result.
In Chapter 11 we present results obtained in [10] in the study of two viscoplastic problems
and in Chapter 12 we present results obtained in [20] in the study of an electro-elasto-viscoplastic
problem.
The names of the three parts in the presentation of the SCIENTIFIC RESULTS indicate the
main research directions in which the candidate had original contributions. Let us nominee here
the main contributions :
Ithe statement and the solution of three new classes of abstract problems :
•abstract stationary mixed variational problems governed by nonlinear maps
•abstract evolutionary mixed variational problems (with short-memory term)
•abstract time-dependent mixed variational problems (with long-memory term)
Ithe weakly solvability of contact models ( by new variational techniques )
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•fornonlinearly elastic, viscoelastic, viscoplastic or electro-elastic materials via a variational
approach with Lagrange multipliers
•fornonlinearly elastic materials governed by possibly set valued elastic operators by means
ofa variational approach via bipotentials theory
•onunbounded time interval .
We end this thesis by presenting some CAREER EVOLUTION AND DEVELOPMENT
PLANS. The presentation is structured in two chapters, Chapters 13-14. Chapter 13 presents
further research directions such us: qualitative and numerical analysis in the study of mixed
variational problems or in the study of variational systems via bipotentials; variational for-
mulations in contact mechanics/ weak solutions via weighted Sobolev spaces or via Lebesgue
spaces with variable exponent; optimal control problems in contact mechanics; mathematical
techniques in the study of dissipative dynamic contact problems; asymptotic analysis in contact
mechanics, regularity results; convergence results; viscoelastic problems via fractional differential
operators/fractional calculus of variations. Chapter 14 presents further plans on the scientific,
professional and academic career such as: to do a research activity allowing to continue to pub-
lish in international journals of hight level, to participate to international meetings in order to
disseminate the best results, to organize scientific meetings, to extend the editorial activities
for scientific journals, to continue the collaborations started in the past and to establish new
contacts, to apply for national/international/interdisciplinary research projects as manager or
member, to publish Lecture Notes and new monographs addressed to the students or researchers,
to extend the advising activity to Ph.D. theses.
Acknowledgments The end of this thesis is a special moment for me. At this special moment
it is my pleasure to express my gratitude to my mentors Professor Vicent ¸iu Rˇ adulescu, Professor
Mircea Sofonea and Professor Barbara Wohlmuth. Special thanks equally go to Professor Sorin
Micu and Professor Constantin Niculescu for encouragement and stimulating discussions over
the years. I also express my thanks to my co-authors and my colleagues for a nice collaboration.
And many special thanks go to my family for constant support and understanding!
December 2014
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The most frequent notation
•S3denotes the space of second order symmetric tensors on R3.
•Every field in R3orS3is typeset in boldface.
•By·and∥ · ∥we denote the inner product and the Euclidean norm on R3andS3, respec-
tively.
•For eachu,v∈Rd,u·v=uivi,∥v∥= (v·v)1/2; for eachσ,τ∈Sd,σ·τ=σijτij,∥τ∥=
(τ·τ)1/2; the indices iandjrun between 1 and dand the summation convention over
repeated indices is applied.
•u= (ui) denotes the displacement field.
•un=u·ndenotes the normal displacement ( nbeing herein the outward normal vector).
•uτ=u−unndenotes the tangential component of the displacement field.
•σ= (σij) denotes the Cauchy stress tensor.
•σn= (σn)·ndenotes the normal component of the stress on the boundary.
•στ=σn−σnndenotes the tangential component of the stress on the boundary.
•Ω = Ω ∪∂Ω; Γ =∂Ω.
•H1(Ω)d(d∈ {2,3}) denotes the standard Sobolev space.
•Lp(Ω)d(d∈ {2,3}, p≥1) denotes the standard Lebesgue space.
•γ:H1(Ω)→L2(Ω) is Sobolev’s trace operator for scalar valued functions.
•γ:H1(Ω)d→L2(Ω)dis Sobolev’s trace operator for vector valued functions ( d∈ {2,3}).
•For eachw∈H1(Ω)d, wν=γw·νandwτ=γw−wννa.e. on Γ (d∈ {2,3}).
•Divdenotes the divergence operator for tensor valuated functions.
•divdenotes the divergence operator for vector valuated functions.
•E= (Eijkl) (orC= (Cijkl)) denotes a fourth order elastic tensor.
•ε=ε(u) is the infinitezimal strain tensor with components εij=1
2(
∂ui
∂xj+∂uj
∂xi)
for alli,j∈
{1,2,3}.
•IfE= (Eijl) is the piezoelectric tensor, E⊤denotes the transpose of the tensor Egiven by
Eσ·v=σ· E⊤v,σ∈Sd,v∈Rd,andE⊤= (E⊤
ijl) = (Elij) for alli,j,l∈ {1,…,d}.
•l.s.c = lower semicontinuous
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Part I
A variational approach via Lagrange
multipliers
13
Chapter 1
Slip-independent frictional contact
problems
This chapter is based on the papers [69, 68]. Firstly, we study the antiplane shear deformation
of two elastic bodies in frictional contact on their common boundary. To model the friction, we
use Tresca’s law. Our study is based on a mixed variational formulation with dual Lagrange
multipliers, the well-possednes of this weak formulation being guaranteed by arguments in the
saddle point theory. This approach results in an efficient iterative solver for the nonlinear problem
with a negligible additional effort compared to solving a linear problem. Nextly, we study the
frictional contact between an elasto-piezoelectric body and a rigid foundation. Our study is
based on a non-symmetric mixed variational formulation involving dual Lagrange multipliers.
The well-posedness of this variational problem is justified by combining a fixed point technique
with a saddle point technique.
1.1 An antiplane problem
This section is based on the paper [69]. The mechanical model used in this section involves the
particular type of deformation that a solid can undergo, the antiplane shear deformation . For
a cylindrical body subject to antiplane shear, the displacement is parallel to the generators of
the cylinder and is independent of the axial coordinate. The antiplane shear (or longitudinal
shear, generalized shear) may be viewed as complementary to the plane strain deformation , and
represents the Mode III, fracture mode for crack problems.
1.1.1 The model and its weak solvability
Let us consider two cylinders Bm,Bs⊂R3having the generators parallel to the x3-axis of a
rectangular cartesian coordinate system Ox 1x2x3.We use a superscript kto indicate that a
quantity is related to the cylinder Bk, k=m,s. We assume that the bodies are homogeneous,
14
15
isotropic and elastic media; more precisely, we shall use the constitutive law
σk=λktr(ε(uk))I+ 2µkε(uk) in Bk, (1.1)
whereσk= (σk
ij) denotes the stress field, ε(uk) = (εij(uk)) the linearized strain tensor, λk>0
andµk>0 are the Lam´ e coefficients, tr( ε(uk)) =∑3
i=1εii(uk) andIis the unit tensor in
R3.Moreover, we assume that the generators are sufficiently long so that end effects in the
axial direction are negligible. Let us denote by Ωka cross-section, which is a domain in R2.
Thus, Bk= Ωk×(−∞,+∞).For each domain Ωk, we assume that its boundary Γkis Lipschitz
continuous and is divided into three disjoint measurable parts Γk
1, Γk
2and Γk
3, with meas (Γk
1)>0.
We assume that the bodies are clamped on Γk
1, body forces of density fk
0act on Ωkand surface
tractions of density fk
2act on Γk
2.Moreover, we assume that the bodies in the initial configuration
are in contact on their common boundary part Γ 3= Γ1
3= Γ2
3and that Γ3is a compact subset
of∂Ωk\Γk
1, k=m, s. We load the solid in a special way, as follows,
fk
0= (0,0,fk
0), fk
0=fk
0(x1,x2) : Ωk→R, (1.2)
fk
2= (0,0,fk
2), fk
2=fk
2(x1,x2) : Γk
2→R. (1.3)
The unit outward normal on Γk×(−∞,+∞) is denoted by nk,nk= (nk
1,nk
2,0) and is defined
almost everywhere. We note that on Γ 3,
n=ns=−nm,σsns=−σmnm,στ=σs
τ=−σm
τ.
For a vectorvk,we denote by vk
nandvk
τitsnormal andtangential parts on the boundary, given by
vk
n=vk·nkandvk
τ=vk−vk
nnk,respectively. Furthermore, for a given stress field σk,we denote
byσk
nandσk
τthenormal and the tangential parts on the boundary, that is σk
n= (σknk)·nk
andσk
τ=σknk−σk
nnk,respectively.
The body forces (1.2) and the surface tractions (1.3) would be expected to give rise to a
deformation of the elastic cylinder Bk,such that the displacement ukis of the form
uk= (0,0,uk), uk=uk(x1,x2) : Ωk→R. (1.4)
The Cauchy stress vector on Γk×(−∞,+∞) is given by
σknk= (0,0,µk∂nuk),
where, as usual, ∂nuk=∇uk·nk.In addition,
Divσk= (0,0,µk∆uk). (1.5)
According to the physical setting, we have
Divσk+fk
0=0 in Bk, (1.6)
uk=0 on Γk
1×(−∞,+∞), (1.7)
σknk=fk
2 on Γk
2×(−∞,+∞). (1.8)
16
Finally, we have to describe the frictional contact condition on Γ 3×(−∞,+∞).Sinceus
n=
um
n= 0,on Γ 3×(−∞,+∞) the contact is bilateral. The friction was modeled by using Tresca’s
law,
∥στ∥ ≤g,
∥στ∥<g⇒us
τ−um
τ= 0,
∥στ∥=g⇒στ=−β(us
τ−um
τ), β > 0,on Γ 3×(−∞,+∞), (1.9)
wheregis the friction bound . We note that uk
τ= (0,0,uk) andσk
τ= (0,0,σk
τ) withσk
τ=µk∂nuk.
In addition, we have
∂nu=µs∂nus=−µm∂numon Γ 3. (1.10)
The mathematical description of the mechanical model is the following one.
Problem 1.1. Find the displacement fields uk: Ω→R, k=m, s, such that
µk∆uk+fk
0= 0 in Ωk, (1.11)
uk= 0 on Γk
1, (1.12)
µk∂nuk=fk
2 on Γk
2, (1.13)
|∂nu| ≤g,
|∂nu|<g⇒us−um= 0,
|∂nu|=g⇒∂nu=−β(us−um), β > 0,on Γ3. (1.14)
In the study of Problem 1.1 we made the following assumptions.
Assumption 1.1. fk
0∈L2(Ωk), fk
2∈L2(Γk
2).
Assumption 1.2. g∈L2(Γ3), g ≥0a.e. on Γ3.
In order to write a weak formulation for Problem 1.1, we need the Hilbert space
Vk={
vk∈H1(Ωk)|vk= 0 a. e. on Γk
1}
k∈ {m,s}(vk= 0 in the sense of the trace)
endowed with the inner product
(uk,vk)Vk=∫
Ωk∇uk· ∇vkdx, for alluk,vk∈Vk,
and the associated norm,
∥vk∥Vk=∥∇vk∥L2(Ωk)2, vk∈Vk.
17
We consider the product space V=Vm×Vsand leta:V×V→Rbe the bilinear form
a(u,v) =∑
k∈{m,s}µk(uk,vk)Vk.
This form is continuous with the continuity constant Ma=µm+µsandV-elliptic, with the
V-ellipticity constant ma= min {µm,µs}.
We definef∈Vsuch that
(f,v)V=∑
k∈{m,s}(∫
Ωkfk
0vkdx+∫
Γk
2fk
2vkds)
.
Next, we define a Lagrange multiplier λinD= (H1/2(Γ3))′,as follows:
⟨λ,w⟩Γ3=−∫
Γ3∂nuwds (1.15)
where ⟨·,·⟩Γ3denotes the duality pairing. Furthermore, we introduce a nonempty closed convex
set,
Λ ={
ζ∈D:⟨ζ, w⟩Γ3≤∫
Γ3g|w|ds, w ∈H1/2(Γ3)}
. (1.16)
We denote by [ v] the jump on Γ3:
[v] =vs−vm, v = (vm,vs)∈V.
Let us denote by b:V×D→R,the bilinear and continuous form
b(v, ζ) =⟨ζ,[v]⟩Γ3. (1.17)
The weak formulation of Problem 1.1 is as follows.
Problem 1.2. Findu∈Vandλ∈Λsuch that
a(u, v) +b(v,λ) = (f,v)Vfor allv∈V,
b(u, ζ−λ) ≤0 for allζ∈Λ.
The well-possedness of Problem 1.2 is given by the following theorem.
Theorem 1.1. [Theorem 2.1 in [69]] If Assumptions 1.1-1.2hold true, then there exists a unique
solution (u, λ)∈V×Λof Problem 1.2. Moreover, if f1, f2are elements in Vcorresponding to
the data (f0, f2)1,(f0, f2)2,respectively, then we have the stability result,
∥u1−u2∥V+∥λ1−λ2∥D≤α+ma+Ma
αm a∥f1−f2∥V, (1.18)
where (u1,λ1),(u2,λ2)are the solutions of Problem 1.2 corresponding to f1, f2∈V,respectively.
The proof of Theorem 1.1, given in [69], is based on the saddle point theory (see for instance
[50]).
18
1.1.2 Nonconforming discretization and optimal a priori error esti-
mates
In this subsection, we give the discretization of Problem 1.2, and we present an optimal a priori
error estimate for the discretization error. Let us assume that the bodies Ωk,k=m,s, are
polygonal domains. To approximate V, we use lowest order finite elements on simplicial or
quadrilateral triangulations. The finite element space associated with the shape regular trian-
gulation Th,Ωkis denoted by S1(Ωk,Th,Ωk).The meshsize his defined by the maximal diameter
of the elements in Th,ΩmandTh,Ωs.For simplicity, we assume that Γk
1,k=m,s, and Γ 3can be
written as union of edges, Γ 3both from the triangulation Th,Ωsof the slave side and from the
triangulation Th,Ωmof the master side. Before introducing the discrete spaces, we decompose
the set of all vertices into three disjoint sets S,MandN. By the first subset S, we denote
all vertices on Γ3of the triangulation Th,Ωson the slave side, by Mall vertices on Γ3of the
triangulation Th,Ωmon the master side. The set Ncontains all remaining ones. Then we have
for the discrete spaces Vk
h
Vk
h={
vk
h∈S1(
Ωk,Th,Ωk)
:vk
h= 0 on Γk
1}
⊂Vk,
and we define Vh=Vm
h×Vs
h.For the discretization of the Lagrange multiplier space we use
dual shape functions, introduced in [163]. In the case of linear or bilinear finite elements in
2D, the dual basis functions are associated with the vertices. We use discontinuous piecewise
linear functions having value two at the associated vertex and value minus one at the two
neighbor vertices as basis functions. We denote this discrete Lagrange multiplier space by
Mh= span {ψp, p∈ S} , whereψpdenotes the basis function associated with the vertex p. Then
the biorthogonality of the basis functions yields
⟨ψp,φq⟩Γ3=δpq∫
Γ3φqds, p,q ∈ S, (1.19)
whereφqare the standard nodal basis functions of S1(Ωs,Th,Ωs) associated with the vertex q.
The finite element space Vhcan be written in terms of the standard finite element basis φas
Vh= span {φp, p∈ S ∪ M ∪ N} . Additionally to the basis φ, we introduce the constrained
finite element basis ˆφ, see [165]. To introduce these basis functions, we define the entries of the
coupling matrices DandMbetween the finite element basis functions φpand the basis functions
for the Lagrange multiplier space ψpby
D[p,q] =⟨ψp,φq⟩Γ3, p, q ∈ S,
M[p,q] =⟨ψp,φq⟩Γ3, p∈ S, q∈ M.
19
Due to the biorthogonality (1.19), the matrix Dis diagonal. In terms of ˆM=D−1M, we obtain
the constrained basis ˆφofVhfrom the nodal basis φofVhby the transformation
ˆφ=
ˆφN
ˆφM
ˆφS
=
Id 0 0
0Id ˆM⊤
0 0Id
φN
φM
φS
=Qφ. (1.20)
We note that only basis functions associated with a node p∈ M are changed, and that by
definition
b( ˆφq,ψp) = 0, p∈ S, q∈ M.
For simplicity of notation, we use the same symbol for a function in VhandMhas for its
algebraic representation with respect to the nodal basis. Let vhbe the algebraic representation
of an element vh∈Vhwith respect to the basis φand let ˆvhbe the corresponding algebraic
representation with respect to the constrained basis ˆφ. Then we have the relation vh=Q⊤ˆvh.
Now, after an easy computation, taking into account the biorthogonality (1.19), we get
b(vh,µh) =⟨µh,∑
p∈Svpφp−∑
q∈Mvqφq⟩Γ3=⟨µh,∑
p∈Sˆvpˆφp⟩Γ3=⟨µh,∑
p∈Sˆvpφp⟩Γ3.
Before introducing the discrete set Λ hfor the admissible Lagrange multiplier, we define for
vh∈Vhthe restriction to the slave side of the interface Γ 3by
vh,S=∑
p∈Svpφp.
Now we define the discrete convex set Λ hby
Λh={
µh∈Mh:⟨µh,vh,S⟩Γ3≤∫
Γ3g|vh,S|hds, v h∈Vh}
, (1.21)
where the mesh dependent absolute value |vh,S|hof the function vh,Sis given by
|vh,S|h=∑
p∈S|vp|φp.
We remark that in general |vh,S|h̸=|vh,S|.Everywhere below in this subsection, we assume that
gis a strictly positive constant. In this case, the convex set Λ hcan be equivalently written as
Λh={
µh∈Mh:µh=∑
p∈Sγpψp,|γp| ≤g, p ∈ S}
. (1.22)
The discrete formulation of Problem 1.2 is the following.
20
Problem 1.3. Finduh∈Vhandλh∈Λhsuch that
a(uh,vh) +b(vh,λh) = (f,vh)V, v h∈Vh,
b(uh,ζh−λh) ≤0, ζh∈Λh.
Using the discrete inf-sup property for the spaces MhandVh,see, e.g., [163], we get the
existence and the uniqueness of the solution. In order to obtain an optimal a priori error
estimate, several lemmas will be proved. We note that Λ h̸⊂Λ.
Before presenting the first lemma, we have to consider for a function vh∈Vhthe discrete
jump ˆvh,Son the interface Γ 3in the constrained basis and its mesh dependent absolute value by
ˆvh,S=∑
p∈Sˆvpφp,|ˆvh,S|h=∑
p∈S|ˆvp|φp,
respectively.
Lemma 1.1. [Lemma 3.1 in [69]] Let (u,λ)be the solution of Problem 1.2 and (uh,λh)the
solution of Problem 1.3. Then we have
b(u,λ) =∫
Γ3g|[u]|ds, b (uh,λh) =∫
Γ3g|ˆuh,S|hds.
Furthermore, the following result holds.
Lemma 1.2. [Lemma 3.2 in [69]] Let (u,λ)∈V×Λbe the solution of Problem 3.18 and let
(uh,λh)∈Vh×Λhbe the solution of the discrete Problem 1.3. Then there exists a positive
constantCindependent of the meshsize h,such that for all vh∈Vh, µh∈Mh,
∥u−uh∥2
V+∥λ−λh∥2
M≤C{
∥u−vh∥2
V+∥λ−µh∥2
M}
+b(u,λ h−λ).
Let us denote γsl= supp [u]⊂Γ3, γst= Γ 3\γsl,and we introduce the sets
W∗=˚γst∩γsl,
W0={
w∈Γ3: [u](w) = 0 and ∃r>0 : [u](w−ε) [u](w+ε)<0,0<ϵ<r}
,
W=W∗∪ W0.
Everywhere below we will use the following assumption.
Assumption 1.3. The number of points in Wis finite.
The minimum distance between the elements in Wis denoted by a, i.e.,
a= inf{|wj−wk|: 1≤j̸=k≤Nw},
whereNwdenotes the number of points in W.By Assumption 1.3, Nw<∞and thusa >0.
Forh<a
2=:h0, we find between two neighbor points in Watleast two vertices inS. Then the
following lemma holds.
21
Lemma 1.3. [Lemma 3.3 in [69]] Let (u,λ)∈V×Λbe the solution of Problem 1.2 and let
(uh,λh)∈Vh×Λhbe the solution of Problem 1.3. Under Assumption 1.3 and the regularity
assumption uk∈H3
2+ν(Ωk),0<ν≤1
2,k=m,s, we then have the a priori error estimate
b(u,λ h−λ)≤Ch1
2+ν∥λ−λh∥M∑
k=m,s|uk|3
2+ν,Ωk
for a positive constant Cindependent of h<h 0.
Based on the results obtained in Lemma 1.2 and Lemma 1.3 and using the well known
approximation properties for the spaces VhandMh,by applying Young’s inequality, the following
theorem holds.
Theorem 1.2 (Theorem 3.4 in [69]) .Let(u,λ)∈V×Λbe the solution of Problem 1.2 and
let(uh,λh)∈Vh×Λhbe the solution of Problem 1.3. Under Assumption 1.3 and the regularity
assumption uk∈H3
2+ν(Ωk),0<ν≤1
2,k=m,s, we then have the a priori error estimate
∥u−uh∥V+∥λ−λh∥M≤Ch1
2+ν∑
k=m,s|uk|3
2+ν,Ωk
for a positive constant Cindependent of the meshsize h<h 0.
Remark 1.1. The discrete nonlinear problem was solved by using an inexact primal-dual active
set strategy in Section 4 of the paper [69]. Numerical examples validating the theoretical result
and illustrating the performance of the algorithm are also presented in [69], see Section 5.
1.2 An elasto-piezoelectric problem
This section is based on the paper [68]. The piezoelectricity is the ability of certain crystals to
produce a voltage when subjected to mechanical stress. The word is derived from the Greek
piezein, which means to squeeze or press. Piezoelectric materials also show the opposite ef-
fect, called converse piezoelectricity ; i.e., the application of an electrical field creates mechanical
stresses (distortion) in the crystal. Because the charges inside the crystal are separated, the
applied voltage affects different points within the crystal differently, resulting in the distor-
tion. Many materials exhibit the piezoelectric effect (e.g. ceramics: BaTiO 3, KNbO 3, LiNbO 3,
LiTaO 3, BiFeO 3). The first mathematical model of an elastic medium taking linear interac-
tion of electric and mechanical fields into account was constructed by W. Voigt, see [157], and
more refined models can be found for example in the works of R. Toupin [153, 154], R. Mindlin
[121, 122, 123], S. Kalinski and J. Petikiewicz [80] and T. Ikeda [76].
22
1.2.1 An abstract auxiliary result
In this subsection we present the results in the study of the following abstract problem.
Problem 1.4. Givenf∈X,findu∈Xandλ∈Ysuch thatλ∈Λ⊂Yand
a(u,v) +b(v,λ) = (f, v)X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ.
We underline that Problem 1.4 is not a classical saddle point problem , becausea(·,·) isnon-
symmetric . The study of this problem was made under the following assumptions.
Assumption 1.4. (X,(·,·)X,∥ · ∥ X)and(Y,(·,·)Y,∥ · ∥ Y)are two Hilbert spaces.
Assumption 1.5. a(·,·) :X×X→Ris a non-symmetric bilinear form such that
(i1)there exists Ma>0 :|a(u,v)| ≤Ma∥u∥X∥v∥Xfor allu,v∈X,
(i2)there exists ma>0 :a(v,v)≥ma∥v∥2
X for allv∈X.
Assumption 1.6. b(·,·) :X×Y→Ris a bilinear form such that
(j1)there exists Mb>0 :|b(v,µ)| ≤Mb∥v∥X∥µ∥Yfor allv∈X, µ∈Y,
(j2)there exists α>0 : inf
µ∈Y,µ̸=0Ysup
v∈X,v̸=0Xb(v,µ)
∥v∥X∥µ∥Y≥α.
Assumption 1.7. Λis a closed convex subset of Ysuch that 0Y∈Λ.
Leta0(u, v) andc(u, v) be the symmetric, respectively the antisymmetric part of a(u, v),
that is
a0(u, v) =1
2(a(u, v) +a(v, u)), c(u,v) =1
2(a(u, v)−a(v, u)).
For a given r∈[0,1],we introduce the following bilinear form
ar(u, v) =a0(u, v) +rc(u, v), u, v ∈X, (1.23)
as a ”perturbation” of a0(·,·).We underline that a1(u, v) =a(u, v) and for all r∈[0,1]ar(u, v)
isX-elliptic with the same ellipticity-constant ma.Moreover, the bilinear forms a0(·,·) andc(·,·)
are continuous with the same continuity-constant Ma.
Let us consider the following problem.
Problem 1.5. For a given f∈X,findu∈Xandλ∈Ysuch thatλ∈Λand
a0(u,v) +b(v,λ) = (f, v)X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ.
Lemma 1.4. [Lemma 3.4 in [68]] Assumptions 1.4–1.7hold true. Given f∈X,there exists a
unique solution of Problem 1.5, (u, λ)∈X×Λ.
23
LetL:X×Λ→Rbe the functional defined as follows:
L(v, µ) =1
2a(v,v)−(f,v)X+b(v,µ).
Using this definition, an equivalent formulation of Problem 1.5. is the following saddle point
problem: find u∈Xandλ∈Λ such that
L(u, µ)≤ L(u, λ)≤ L(v, λ)v∈X, µ∈Λ.
We consider now the following ”perturbate” problem.
Problem 1.6. For a given f∈X,findu∈Xandλ∈Ysuch thatλ∈Λ,and
ar(u,v) +b(v,λ) = (f, v)X for allv∈X, (1.24)
b(u,µ−λ) ≤0 for allµ∈Λ. (1.25)
We have the following lemmas.
Lemma 1.5. [Lemma 3.2 in [68]] Assume that for every f∈Xthere exists a unique solution of
Problem 1.6, (u, λ)∈X×Λ.If(u1, λ1)and(u2, λ2)are solutions of Problem 1.6 corresponding
to two given functions f1, f2∈X,then
∥u1−u2∥X+∥λ1−λ2∥Y≤α+ma+ 2Ma
αm a∥f1−f2∥X.
Lemma 1.6. [Lemma 3.3 in [68]] Let τ∈[0,1].Assume that for every f∈Xthere exists a
unique solution of Problem 1.6 with r=τ,(u, λ)∈X×Λ.Then, for every f∈Xthere exists
a unique solution (u, λ)of Problem 1.6 with r∈[τ,τ+t0],where
t0<αm a
Ma(α+ma+ 2Ma). (1.26)
Applying Lemma 1.6 and Lemma 1.5 we were led to the following result.
Theorem 1.3. [Theorem 3.1 in [68]] Let f∈X.If Assumptions 1.4–1.7hold true, then there
exists a unique solution of Problem 1.4, (u,λ)∈X×Λ.Moreover, if (u1, λ1)and(u2, λ2)are
two solutions of Problem 1.4, corresponding to two given functions f1, f2∈X,then
∥u1−u2∥X+∥λ1−λ2∥Y≤α+ma+ 2Ma
αm a∥f1−f2∥X.
The proof of Theorem 1.3 can be found in [68]. The main idea of this proof was to use the
results known in the saddle point theory, see, e.g., [22, 23, 50, 61], for the symmetric part of
a(·,·).The prove was completed by a fixed point technique. The reader can found a version of
this fixed point technique in [83], in the framework of the elliptic variational inequalities of the
first kind.
24
1.2.2 The model and its weak solvability
In this subsection we study the weak solvability of an elasto-piezoelectric model in the following
physical setting. An elasto-piezoelectric body which occupies the bounded domain Ω ⊂Rd
(d∈ {2,3},is in frictional contact with a rigid foundation. We consider two partitions of the
boundary Γ = ∂Ω : firstly, we consider a partition given by the measurable parts Γ 1, Γ2and Γ 3,
such that meas Γ 1>0 and Γ3is a compact subset of ∂Ω\Γ1; secondly, we consider a partition
given by the measurable parts Γ a,Γb,such that meas Γ a>0.The unit outward normal to Γ
is denoted by nand is assumed to be constant on Γ 3,i.e. Γ 3is a straight line or a face. We
associate the body with a rectangular cartesian coordinate system Ox 1x2x3such thate1=n 3.
We assume that the body is clamped on Γ 1, body forces of density f0act on Ω, a surface
traction of density f2acts on Γ 2,a surface electric charge of density q2acts on Γ b,and the
electric potential vanishes on Γ a.Moreover, we assume that on Γ 3the deformable body is in
bilateral contact with the rigid foundation. Herein φdenotes the electric potential.
The equilibrium equations are given by
Divσ+f0=0 in Ω, (1.27)
divD=q0 in Ω, (1.28)
whereD= (Di) is the electric displacement field, and q0is the volume density of free electric
charges.
To describe the behavior of the material, we use the following constitutive law:
σ=Cε(u) +E⊤∇φ in Ω, (1.29)
D=Eε(u)−β∇φ in Ω, (1.30)
where C= (Cijls) is the elastic tensor, E= (Eijl) is the piezoelectric tensor, and βis the
permittivity tensor. Note that (1.29) represents an electro-elastic constitutive law and (1.30)
describes a linear dependence of the electric displacement field on the strain and electric fields.
Such kind of electro-mechanic relations can be found in the literature, see, e.g., [157].
To complete the model, we have to prescribe the mechanic and electric boundary conditions.
According to the physical setting, we write
u=0 on Γ 1, (1.31)
σn =f2 on Γ 2, (1.32)
φ= 0 on Γ a, (1.33)
D·n=q2 on Γ b. (1.34)
25
Finally, we describe the frictional bilateral contact using Tresca’s law:
un= 0,∥στ∥ ≤g,
∥στ∥<g⇒uτ= 0,
∥στ∥=g⇒there exists α>0 such thatστ=−αuτon Γ 3, (1.35)
where the constant g≥0 represents the friction bound . When the strict inequality holds, the
material point is in the sticky zone; when the equality holds, the material point is in the slippy
zone. The boundary of these zones is unknown a priori.
To resume, we have the following problem:
Problem 1.7. Find the displacement field u: Ω→Rdand the electric potential field φ: Ω→R
such that (1.27)–(1.35)hold.
In the study of Problem 1.7, we made the following assumptions.
Assumption 1.8. C= (Cijls) : Ω×Sd→Sd;Cijls=Cijsl=Clsij∈L∞(Ω);
There exists mC>0such that Cijlsεijεls≥mC∥ε∥2,for allε∈Sd,a.e. on Ω.
Assumption 1.9. E= (Eijk) : Ω×Sd→Rd;Eijk=Eikj∈L∞(Ω).
Assumption 1.10. β= (βij) : Ω×Rd→Rd;βij=βji∈L∞(Ω);
There exists mβ>0such thatβij(x)DiDj≥mβ∥D∥2,D∈Rd,a.e.x∈Ω.
Assumption 1.11. f0∈L2(Ω)d,f2∈L2(Γ2)d.
Assumption 1.12. q0∈L2(Ω), q 2∈L2(Γb).
Let us introduce the following Hilbert spaces:
V ={
v∈H1(Ω)d|v= 0 on Γ 1}
,
Vn={
v∈V|vn= 0 on Γ 3}
,
Φ ={
θ∈H1(Ω)|θ= 0 on Γ a}
.
We introduce the functional space ˜V=V×Φ,that is a Hilbert space endowed with the
inner product
(˜u,˜v)˜V= (u,v)H1(Ω)d+ (φ,θ)H1(Ω),˜u= (u,φ),˜v= (v,θ)∈˜V;
the corresponding norm is denoted by ∥ · ∥ ˜V.Leta:˜VטV→Rbe the bilinear form given by:
a(˜u,˜v) =∫
ΩCε(u)·ε(v)dx+∫
ΩEε(v)· ∇φdx (1.36)
−∫
ΩEε(u)· ∇θdx+∫
Ωβ∇φ· ∇θdx.
26
Moreover, using Riesz’s representation theorem, we define ˜f∈˜Vsuch that for all ˜v∈˜V,
(˜f,˜v)˜V=∫
Ωf0vdx+∫
Γ2f2vdΓ−∫
Γbq2θdΓ +∫
Ωq0θdx.
LetDbe the dual space of the space S={w=v|Γ3v∈Vn}.
We define
={
µ∈D|⟨µ,v|Γ3⟩Γ3≤∫
Γ3g∥v∥dΓ,v∈Vn}
, (1.37)
where ⟨·,·⟩Γ3denotes the duality pairing between DandS.
We suppose that the stress σis a regular enough function to define λ∈Das follows
⟨λ,ζ⟩Γ3=−∫
Γ3στ·ζds,ζ∈S.
Furthermore, we introduce a bilinear and continuous form as follows:
b:˜V×D→R, b (˜v,µ) =⟨µ,v|Γ3⟩Γ3. (1.38)
The mechanical model leads us to the following variational formulation.
Problem 1.8. Find ˜u∈˜Vandλ∈such that
a(˜u,˜v) +b(˜v,λ) = ( ˜f,˜v)˜V,˜v∈˜V,
b(˜u,µ−λ) ≤0,µ∈.
Theorem 1.4. [Theorem 2.1 in [68]] If Assumptions 1.8–1.12hold true, then, Problem 1.8 has
a unique solution (˜u,λ)∈˜V×.Moreover, if (˜u1,λ1)and(˜u2,λ2)are two solutions of
Problem 1.8 for two functions ˜f1,˜f2∈˜V,corresponding to data {f0,f2, q0, q2}1,respectively
{f0,f2, q0, q2}2,then we have
∥˜u1−˜u2∥˜V+∥λ1−λ2∥D≤C∥˜f1−˜f2∥˜V,
whereC > 0is a constant that depends of C,Eandβ.
The proof of Theorem 1.4, given in [68], is based on the abstract result we presented before,
Theorem 1.3.
1.2.3 Discretization and an optimal a priori error estimate
In this subsection, we discuss the discrete 2D case. Let us assume that Ω ⊂R2is a polygonal
domain and that Γ 1,Γ3and Γ acan be written as union of edges of the triangulation. Further-
more, let us denote by τa unit vector such that n·τ= 0.We refer the body to a rectangular
cartesian coordinate system Ox 1x2such thate1=n 3ande2=τ 3.To simplify the writing,
27
everywhere below we will write nandτinstead ofn 3andτ 3,respectively. To approximate
˜V, we use standard conforming finite elements of lowest order on quasi-uniform simplicial tri-
angulations, and we denote by S1(Ω,Th,Ω) the finite element space associated with the shape
regular triangulation Th,Ω.The meshsize his defined by the maximal diameter of the elements
inTh,Ω.Let us consider the discrete spaces
Vh={
vh∈[S1(Ω,Th,Ω)]2:vh| 1=0}
⊂V,
(Vh)n={
vh∈Vh: (vh)n| 3= 0}
⊂Vn,
Φh={
θh∈S1(Ω,Th,Ω) :θh| a= 0}
⊂Φ.
Let us denote
˜Vh=Vh×Φh⊂˜V
and
Mh={
µh∈M|µh=NMh∑
i=1γiψin+NMh∑
i=1αiψiτ}
,
whereNMhis the number of vertices on Γ3and for every i= 1,…,N Mh,ψiis thei-th. scalar dual
basis function of the standard nodal Lagrange finite element basis function and γi, αiare real
coefficients. According to [163], we consider the dual basis such that the following biorthogonality
relation holds
⟨ψi,φj⟩Γ3=δij∫
Γ3φjds, i,j = 1,…,N Mh, (1.39)
whereφm, m= 1,…,N Mh,are the standard scalar nodal basis functions of S1(Ω,Th,Ω), re-
stricted to Γ 3.Furthermore, every element vhof (Vh)ncan be written on Γ 3as a combination
of standard basis functions φias follows
vh=NMh∑
j=1ζjφjτ, ζ j∈R, j= 1,…,N Mh.
Defining a mesh dependent absolute value of an element vh∈(Vh)nby
|vh|h=NMh∑
j=1|ζj|φj,
we set has follows
h={
µh∈Mh|⟨µh,vh⟩Γ3≤∫
Γ3g|vh|hds,vh∈(Vh)n}
.
We now consider the following discrete problem.
28
Problem 1.9. Find ˜uh∈˜Vhandλh∈hsuch that
a(˜uh,˜vh) +b(˜vh,λh) = (˜f,˜vh)˜V, ˜vh∈˜Vh
b(˜uh,µh−λh) ≤0, µh∈h.
Existence and uniqueness of a solution follows from a discrete inf-sup condition for the spaces
˜VhandMh, see, e.g., [163].
Let us denote by PC={pi: 1≤i≤NMh}the set of vertices on Γ3.
The following result takes place.
Lemma 1.7. [Lemma 4.2 in [68]] Let (˜u= (u,φ),λ)∈˜V×be the solution of Problem 1.8
and let (˜uh= (uh,φh),λh)∈˜Vh×hbe the solution of Problem 1.9. Then, the following
equalities hold
b(˜u,λ) =∫
Γ3g|u|ds, (1.40)
b(˜uh,λh) =∫
Γ3g|uh|hds. (1.41)
Using this lemma we have got the following result.
Lemma 1.8. [Lemma 4.3 in [68]] Let (˜u,λ)∈˜V×be the solution of Problem 1.8 and let
(˜uh,λh)∈˜Vh×hbe the solution of Problem 1.9. Then, there exists a positive constant C
independent of the meshsize h,such that for all vh∈˜Vh,µh∈Mh,
∥˜u−˜uh∥2
˜V+∥λ−λh∥2
−1
2,Γ3≤C{
∥˜u−˜vh∥2
˜V+∥λ−µh∥2
−1
2,Γ3}
+b(˜u,λh−λ).
Let us denote γsl= supp(u| 3·τ) andγst= Γ 3\γsl.
We made the following assumption.
Assumption 1.13.
•γstis a compact subset of Γ3such that the number of points in ˚γst∩γslis finite;
•˚γst=γst.
LetWC={wj: 1≤j≤Nw}be the set of points in ˚γst∩γsl. The minimum distance between
the elements in WCis denoted by a, i.e.,a= inf{|wj−wk|: 1≤j̸=k≤Nw}, where | · |
denotes the Euclidean norm. By Assumption 4.1, Nw<∞and thusa>0. Forh<a
2=:h0, we
find between two neighbor points in WCat least two vertices in PC.
29
Let us denote by Ihthe standard interpolation operator restricted on Γ 3,i.e.,
Ihu=NMh∑
i=1u(pi)φi,
and let us define the following modified interpolation operator by
(˜Ihu)(pi) =
u(pi) if supp φi⊂γsl,
0 else,
for eachi= 1,…,N Mh.
We underline that, under Assumption 1.13, we can write on Γ 3the following identities
|˜Ihu|h=|˜Ihu|, (1.42)
sgn(u·τ) = sgn( ˜Ihu·τ).
The following lemma holds.
Lemma 1.9. [Lemma 4.4 in [68]] Let (˜u,λ)∈˜V×be the solution of Problem 1.8 and let
(˜uh,λh)∈˜Vh×hbe the solution of Problem 1.9. Under the additional regularity assumption
u∈[
H3
2+ν(Ω)]2,0<ν≤1
2,and Assumption 1.13, we then have the estimate
b(˜u,λh−λ)≤Ch1
2+ν|u|3
2+ν,Ω∥λ−λh∥−1
2,Γ3
for a positive constant Cindependent of h<h 0.
A straightforward consequence of the results obtained in Lemmas 1.8-1.9 is the following
theorem.
Theorem 1.5. [Theorem 4.1 in [68]] Let (˜u,λ)∈˜V×be the solution of Problem 1.8 and let
(˜uh,λh)∈˜Vh×hbe the solution of Problem 1.9. Under the additional regularity assumption
˜u∈[H3
2+ν(Ω)]3,0< ν≤1
2and Assumption 1.13, we then have the following optimal a priori
error estimate
∥˜u−˜uh∥˜V+∥λ−λh∥−1
2,Γ3≤Ch1
2+ν|˜u|3
2+ν,Ω
for a positive constant Cthat is independent of the meshsize h<h 0.
Remark 1.2. The a priori results can be extended to the 3D case. The results also hold in
the multibody case with nonconforming meshes at the contact interface, see e.g. [62, 66] for the
necessary techniques. A numerical example was given in Section 5 of the paper [68].
Chapter 2
Slip-dependent frictional contact
problems
This chapter is based on the papers [105, 109, 112]. A slip-dependent frictional contact law is a
law in which the friction bound depends on the slip. The first mathematical results on contact
problem with slip displacements dependent friction in elastostatics were obtained in [72]. For
other mathematical results in the study of slip-dependent frictional contact models see, e.g.,
[36, 65, 97, 103, 145] for a treatment in the frame of quasivariational inequalities or, see e.g,
[115, 116, 117] for a treatment in the frame of hemivariational inequalities. In the present work,
the interest lies into a variational approach involving dual Lagrange multipliers which allows to
apply modern numerical techniques (see e.g. [162]) in order to approximate the weak solution.
2.1 An abstract result
This section presents the results obtained in Section 2 and Section 3 of the paper [105]. In this
section we consider an abstract mixed variational problem, the set of the Lagrange multipliers
being dependent on the solution.
Problem 2.1. Givenf∈X, f̸= 0X,find(u,λ)∈X×Ysuch thatλ∈Λ(u)⊂Yand
a(u,v) +b(v,λ) = (f, v)X for allv∈X, (2.1)
b(u,µ−λ) ≤0 for allµ∈Λ(u). (2.2)
We shall discuss the existence of the solution based on a fixed point technique for weakly
sequentially continuous maps.
Let us make the following assumptions.
Assumption 2.1. (X,(·,·)X,∥ · ∥ X)and(Y,(·,·)Y,∥ · ∥ Y)are two Hilbert spaces.
Assumption 2.2. a(·,·) :X×X→Ris a symmetric bilinear form such that
30
31
(i1)there exists Ma>0 :|a(u,v)| ≤Ma∥u∥X∥v∥Xfor allu,v∈X,
(i2)there exists ma>0 :a(v,v)≥ma∥v∥2
X for allv∈X.
Assumption 2.3. b(·,·) :X×Y→Ris a bilinear form such that
(j1)there exists Mb>0 :|b(v,µ)| ≤Mb∥v∥X∥µ∥Yfor allv∈X, µ∈Y,
(j2)there exists α>0 : inf
µ∈Y,µ̸=0Ysup
v∈X,v̸=0Xb(v,µ)
∥v∥X∥µ∥Y≥α.
Assumption 2.4. For eachφ∈X,Λ(φ)is a closed convex subset of Ysuch that 0Y∈Λ(φ).
Assumption 2.5. Let(ηn)n⊂Xand(un)n⊂Xbe two weakly convergent sequences, ηn⇀ η
inXandun⇀u inX,asn→ ∞.
(k1)For eachµ∈Λ(η),there exists a sequence (µn)n⊂Ysuch thatµn∈Λ(ηn)and
lim inf n→∞b(un,µn−µ)≥0.
(k2)For each subsequence (Λ(ηn′))n′of the sequence (Λ(ηn))n,if(µn′)n′⊂Ysuch that
µn′∈Λ(ηn′)andµn′⇀µ inYasn′→ ∞,thenµ∈Λ(η).
Theorem 2.1. [Theorem 2.1 in [105]] If Assumptions 2.1-2.5hold true, then Problem 2.1 has
a solution. In addition, if (u,λ)∈X×Λ(u)is a solution of Problem 2.1, then
(u,λ)∈K1×(
Λ(u)∩K2), (2.3)
where
K1={v∈X|∥v∥X≤1
ma∥f∥X}
and
K2={µ∈Y|∥µ∥Y≤ma+Ma
αm a∥f∥X}.
The proof of Theorem 2.1, which can be found in [105], is based on the saddle point theory,
see [50], and a fixed point result for weakly sequentially continuous maps, see [5].
Theorem 2.1 is a new result which improves and extends the existence results of solutions for
mixed problems which are equivalent to saddle point problems, see e.g. [61]. The main difficulty
here it was generated by the dependence Λ = Λ( u).A convergence of Mosco type for the convex
sets of Lagrange multipliers it was required; Assumption 2.5 it was crucial.
2.2 An antiplane problem
This section is based on Section 4 of the paper [105]. In this section we apply the abstract result
obtained in Section 2.1 to the weak solvability of a slip-dependent frictional antiplane contact
problem.
Let us consider the following mechanical model.
32
Problem 2.2. Find a displacement field u:¯Ω→Rsuch that
div (µ(x)∇u(x)) +f0(x) = 0 in Ω, (2.4)
u(x) = 0 on Γ 1, (2.5)
µ(x)∂νu(x) =f2(x) on Γ 2, (2.6)
|µ(x)∂νu(x)| ≤g(x,|u(x)|),
µ(x)∂νu(x) =−g(x,|u(x)|)u(x)
|u(x)|ifu(x)̸= 0on Γ 3. (2.7)
Herein Ω ⊂R2is a bounded domain with Lipschitz continuous boundary Γ partitioned in
three measurable parts Γ 1,Γ2,Γ3such that the Lebesgue measure of Γ iis strictly positive, for
everyi∈ {1,2,3}.Problem 2.2 models the antiplane shear deformation of an elastic, isotropic,
nonhomogeneous cylindrical body, in frictional contact on Γ 3with a rigid foundation. Referring
the body to a cartesian coordinate system Ox 1x2x3such that the generators of the cylinder are
parallel with the axis Ox 3, the domain Ω ⊂Ox 1x2denotes the cross section of the cylinder. The
functionµ=µ(x1,x2) :¯Ω→Rdenotes a coefficient of the material (one of Lam´ e’s coefficients ),
the functions f0=f0(x1,x2) : Ω→R, f2=f2(x1,x2) : Γ 2→Rare related to the density of
the volume forces and the density of the surface traction, respectively and g: Γ3×R+→R+
is a given function, the friction bound . Hereν= (ν1,ν2), νi=νi(x1,x2),for eachi∈ {1,2},
represents the outward unit normal vector to the boundary of Ω and ∂νu=∇u·ν.
The unknown of the problem is the function u=u(x1,x2) :¯Ω→Rthat represents the
third component of the displacement vector u.We recall that, in the antiplane physical setting,
the displacement vectorial field has the particular form u= (0,0,u(x1,x2)).Once the field uis
determined, the stress tensor σcan be computed:
σ=
0 0 µ∂u
∂x1
0 0 µ∂u
∂x2
µ∂u
∂x1µ∂u
∂x20
.
The mechanical problem has the following structure: (2.4) represents the equilibrium equa-
tion, (2.5) is the displacement boundary condition, (2.6) is the traction boundary condition and
(2.7) is a frictional contact condition. The condition (2.7) is Tresca’s law of dry friction with
slip-dependent friction bound g. To give an example of such a function gwe can consider
g(x,r) =k(1 +δe−r);k,δ> 0. (2.8)
The slip-dependent friction law (2.7) with the friction bound ggiven by (2.8) describes the slip
weakening phenomenon which appears in the study of geophysical problems, see for example
33
[29, 30, 74, 75, 137]. For details concerning the frictional antiplane model we send the reader to
[145] and to the references therein.
We are interested on the weak solvability of Problem 2.2 under the following assumptions.
Assumption 2.6. f0∈L2(Ω), f 2∈L2(Γ2).
Assumption 2.7. µ∈L∞(Ω), µ(x)≥µ∗>0a.e. in Ω.
Assumption 2.8. There exists Lg>0such that
|g(x,r1)−g(x,r2)| ≤Lg|r1−r2|r1,r2∈R+,a.e.x∈Γ3;
The mapping x7→g(x, r)is Lebesgue measurable on Γ3,for allr∈R;
The mapping x7→g(x,0)belongs toL2(Γ3).
Let us describe the functional setting. To start, we introduce the space
X={
v∈H1(Ω)|γv= 0 a.e. on Γ 1}
. (2.9)
The spaceXis a Hilbert space endowed with the inner product given by
(u,v)X=∫
Ω∇u· ∇vdx for allu,v∈X,
and the associated norm
∥v∥X=∥∇v∥L2(Ω)2for allv∈X.
Leta:X×X→Rbe the bilinear form
a(u,v) =∫
Ωµ∇u· ∇vdx for allu,v∈X (2.10)
andf∈Xdefined as follows
(f,v)X=∫
Ωf0vdx+∫
Γ3f2γvdΓ. (2.11)
Let Γ 3⊂Γ such that Γ 3∩Γ1=∅.We consider the space
S={ev=γv|Γ3v∈X} (2.12)
endowed with the Sobolev-Slobodeckii norm
∥ev∥Γ3=(∫
Γ3∫
Γ3(ev(x)−ev(y))2
∥x−y∥2dsxdsy)1/2
for allev∈S.
We can introduce now a second Hilbert space, the dual of the space S,
Y=S′. (2.13)
34
Also, we can define a second bilinear form b:X×Y→R,
b(v, ζ) =⟨ζ,γv|Γ3⟩, (2.14)
where ⟨·,·⟩denotes the duality pairing between the spaces YandS.
We define a Lagrange multiplier λ∈Y,
⟨λ,z⟩=−∫
Γ3µ∂νuzdΓ for allz∈S, (2.15)
where the space Yis defined in (2.13) and the space Sis defined in (2.12).
Furthermore, for each φ∈X,we introduce a subset of the space Y,
Λ(φ) ={
ζ∈Y:⟨ζ, γw|Γ3⟩ ≤∫
Γ3g(x,|γφ(x)|)|γw(x)|dΓ for allw∈X}
. (2.16)
Problem 2.2 has the following weak formulation.
Problem 2.3. Findu∈Xandλ∈Λ(u)⊂Ysuch that
a(u, v) +b(v,λ) = (f,v)X for allv∈X; (2.17)
b(u,ζ−λ)≤0 for allζ∈Λ(u). (2.18)
Each solution of Problem 2.3 is called weak solution of Problem 2.2.
Notice that for each µ∈Ywe have
∥µ∥Y= sup
γw| 3∈S, γw| 3̸=0S<µ,γw |Γ3>
∥γw|Γ3∥Γ3
≤csup
v∈X, v̸=0Xb(v,µ)
∥v∥X,
wherec>0.
Theorem 2.2. [Theorem 4.3 in [105]] If Assumptions 2.6-2.8hold true, then Problem 2.2has
a weak solution. In addition, if (u,λ)is a weak solution of Problem 2.2, then (u,λ)∈K1×(
Λ(u)∩K2), whereK1={v∈X|∥v∥X≤1
ma∥f∥X}, K 2={µ∈Y|∥µ∥Y≤ma+Ma
α ma∥f∥X}, X
given by (2.9),Ygiven by (2.13),fgiven by (2.11),Ma=∥µ∥L∞(Ω), ma=µ∗andα=1
c.
The proof of Theorem 2.2 was based on the previous abstract result, Theorem 2.1. The main
difficult part of the proof consists in the verification of Assumption 2.5. The crucial point was
the construction of an appropriate sequence ( µn)n⊂Y:
⟨µn,ζ⟩=∫
Γ3g(x,|γηn(x)|) sgnγun(x)ζ(x)dΓ
−∫
Γ3g(x,|γη(x)|)|γun(x)|dΓ
+⟨µ,γu n|Γ3⟩for allζ∈S.
For details, see [105].
35
2.3 A 3D slip-dependent frictional contact problem
This section, devoted to the weak solvability of a 3D slip-dependent frictional contact problem,
is based on the papers [109, 112]. The model we focus on was previously analyzed into the
framework of quasi-variational inequalities in [36]. The novelty herein consists in the variational
approach we use. Thus, we propose a mixed variational formulation in a form of a generalized
saddle point problem, the set of the Lagrange multipliers being solution-dependent.
The physical setting is as follows. We consider a deformable body that occupies the bounded
domain Ω ⊂R3with smooth (say Lipschitz continuous) boundary Γ partitioned into three
measurable parts, Γ 1,Γ2and Γ 3,such thatmeas (Γ1)>0.The unit outward normal vector to
Γ is denoted by νand is defined almost everywhere. The body is clamped on Γ 1, body forces
of densityf0act on Ω and surface traction of density f2acts on Γ 2.On Γ 3the body is in
slip-dependent frictional contact with a rigid foundation.
The 3D slip-dependent frictional contact model is mathematically described as follows.
Problem 2.4. Findu:Ω→R3andσ:Ω→S3such that
Divσ(x) +f0(x) =0 inΩ, (2.19)
σ(x) =Eε(u(x)) inΩ, (2.20)
u(x) =0 onΓ1, (2.21)
σν(x) =f2(x) onΓ2, (2.22)
uν(x) = 0 onΓ3, (2.23)
∥στ(x)∥ ≤g(x,∥uτ(x)∥),
στ(x) =−g(x,∥uτ(x)∥)uτ(x)
∥uτ(x)∥
ifuτ(x)̸=0 onΓ3. (2.24)
Problem 2.4 has the following structure: (2.19) represents the equilibrium equation, (2.20)
represents the constitutive law for linearly elastic materials, (2.21) represents the homogeneous
displacements boundary condition, (2.22) represents the traction boundary condition and (2.23)-
(2.24) model the bilateral contact with friction, the friction law involving a slip-dependent friction
boundg.
For more details on this model see e.g. [36] and the references therein.
In order to weakly solve Problem 2.4 we made the following assumptions.
Assumption 2.9. E= (Eijls) : Ω×S3→S3,
•Eijls=Eijsl=Elsij∈L∞(Ω),
•There exists mE>0such that Eijlsεijεls≥mE∥ε∥2,ε∈S3,a.e. in Ω.
Assumption 2.10. f0∈L2(Ω)3,f2∈L2(Γ2)3.
36
Assumption 2.11. g: Γ3×R+→R+,
•there exists Lg>0 :|g(x,r1)−g(x,r2)| ≤Lg|r1−r2|r1,r2∈R+,a.e.x∈Γ3;
•the mappingx7→g(x, r)is Lebesgue measurable on Γ3,for allr∈R+;
•the mappingx7→g(x,0)belongs toL2(Γ3).
Let us introduce the following Hilbert space.
V={v∈H1(Ω)3|γv= 0 on Γ 1, vν= 0 on Γ 3}. (2.25)
Definef∈Vusing Riesz’s representation theorem,
(f,v)V=∫
Ωf0(x)·v(x)dx+∫
Γ2f2(x)·γv(x)dΓ (2.26)
for allv∈V.
Also, we introduce the space
S={γw|Γ3w∈V}, (2.27)
whereγw|Γ3denotes the restriction of the trace of the element γw∈Vto Γ 3. Thus,S⊂
H1/2(Γ3;R3) whereH1/2(Γ3;R3) is the space of the restrictions on Γ 3of traces on Γ of functions
ofH1(Ω)3. It is known that Scan be organized as a real Hilbert space, see for instance [1, 92].
We use the Sobolev-Slobodeckii norm
∥ζ∥S=(∫
Γ3∫
Γ3∥ζ(x)−ζ(y)∥2
∥x−y∥3dsxdsy)1/2
.
Let us introduce now the following real Hilbert space,
D=S′(the dual of the space S). (2.28)
The duality paring between DandSwill be denoted by ⟨·,·⟩.
For eachφ∈Vwe define
(φ) ={µ∈D|⟨µ,γv|Γ3⟩ ≤∫
Γ3g(x,∥φτ(x)∥)∥vτ(x)∥dΓv∈V}.
Let us define a Lagrange multiplier λ∈D,
⟨λ,ζ⟩=−∫
Γ3στ(x)·ζ(x)dΓ (2.29)
for allζ∈S.
Notice thatλ∈Λ(u).
37
We also define
a:V×V→Ra(u,v) =∫
ΩEε(u)·ε(v)dx; (2.30)
b:V×D→Rb(v,µ) =⟨µ,γv|Γ3⟩. (2.31)
Therefore, Problem 2.4 has the following weak formulation.
Problem 2.5. Findu∈Vandλ∈Λ(u)⊂Dsuch that
a(u,v) +b(v,λ) = (f,v)Vfor allv∈V (2.32)
b(u,ζ−λ)≤0for allζ∈Λ(u). (2.33)
Each solution of Problem 2.5 is called weak solution of Problem 2.4.
Theorem 2.3. [An existence result (Theorem 2 in [112])] If Assumptions 2.9 -2.11 hold true,
then Problem 2.4 has a weak solution.
The idea of the proof was to use the abstract result, Theorem 2.1; for details of the proof
of Theorem 2.3 we send the reader to the paper [112] or to the conference paper [109] (the
paper [112] is an revised/extended version of the conference paper [109]). However, it is worth
to mention here the crucial point of the proof: to construct an appropriate sequence ( µn)nin
order to verify Assumption 2.5. Let us give an example: for each n≥1,
<µn,ζ>=∫
Γ3g(x,∥ηnτ(x)∥)ψ(unτ(x))·ζ(x)dΓ
−∫
Γ3g(x,∥ητ(x)∥)∥unτ(x)∥dΓ +⟨µ,γun|Γ3⟩,
for allζ∈S,where
ψ(r) =
r
∥r∥ifr̸=0;
0 ifr=0.
Notice that the form a(·,·) defined in (2.30) verifies Assumption 2.2 with
Ma=∥E∥∞andma=mE, (2.34)
where
∥E∥∞= max
0≤i,j,k,l≤d∥Eijkl∥L∞(Ω).
Also, we note that for each µ∈D,there exists c>0 such that
∥µ∥D≤csup
v∈V,v̸=0Vb(v,µ)
∥v∥V,
38
and we can take
α=1
c. (2.35)
Let us introduce now
K1={v∈V|∥v∥V≤1
ma∥f∥V}; (2.36)
K2={µ∈D|∥µ∥D≤ma+Ma
αm a∥f∥V}. (2.37)
Theorem 2.4. [A boundedness result (Theorem 3 in [112])] If (u,λ)is a weak solution of
Problem 2.4, then
(u,λ)∈K1×(
Λ(u)∩K2)
whereK1andK2are given by (2.36)-(2.37),Vgiven by (2.25),Dgiven by (2.28),fgiven by
(2.26),maandMabeing the constants in (2.34)andαbeing the constant in (2.35).
The proof of Theorem 2.4 uses the abstract result, Theorem 2.1.
Chapter 3
Contact problems for nonlinearly
elastic materials
This chapter is based on the papers [99, 100, 104, 107]. In this chapter we discuss a class of
problems which model the contact between nonlinearly elastic bodies and rigid foundations,
under the small deformation hypothesis, for static processes. The contact between the body and
the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem
we discuss a weak formulation consisting of a system of a nonlinear variational equation and a
variational inequality, involving dual Lagrange multipliers. The weak solvability of the models
is based on the saddle point theory and fixed point techniques.
3.1 Problems governed by strongly monotone and Lips-
chitz continuous operators
This section presents some results obtained in the papers [99, 104] drawing the attention to the
weak solvability via dual Lagrange multipliers for a class of contact problems leading to mixed
variational problems governed by strongly monotone and Lipschitz continuous operators.
3.1.1 Abstract results
In this subsection we present results obtained in Section 5 of the paper [99] and some results
obtained in Section 2 of the paper [104], focusing on the following abstract problem.
Problem 3.1. Givenf, h∈X,findu∈Xandλ∈Λsuch that
(Au,v )X+b(v,λ) = (f, v)X for allv∈X, (3.1)
b(u,µ−λ) ≤b(h,µ−λ) for allµ∈Λ. (3.2)
The study was made under the following hypotheses.
39
40
Assumption 3.1. (X,(·,·)X,∥ · ∥ X)and(Y,(·,·)Y,∥ · ∥ Y)are two Hilbert spaces.
Assumption 3.2. A:X→Xis a nonlinear operator such that:
there exists mA>0 : (Au−Av, u −v)X≥mA∥u−v∥2
Xfor allu,v∈X,
there exists LA>0 :∥Au−Av∥X≤LA∥u−v∥Xfor allu,v∈X.
Assumption 3.3. b:X×Y→Ris a bilinear form such that:
there exists Mb>0 :|b(v,µ)| ≤Mb∥v∥X∥µ∥Yfor allv∈X, µ∈Y,
there exists α>0 : inf
µ∈Y,µ̸=0Ysup
v∈X,v̸=0Xb(v,µ)
∥v∥X∥µ∥Y≥α.
Assumption 3.4. Λ⊂Yis a closed convex set such that 0Y∈Y.
Under these assumptions, Problem 3.1 is not a saddle point problem. This is a new variational
problem, a mixed variational problem governed by a nonlinear operator A.
The following existence and uniqueness result holds.
Theorem 3.1. If Assumptions 3.1–3.4hold true, then there exists a unique solution of Problem
3.1,(u,λ)∈X×Λ.
The proof of Theorem 3.1 is based on Theorem 5.2 in [99] if Λ is an unbounded set and on
Theorem 2.1 in [104] if Λ is a bounded set.
3.1.2 Contact models
This subsection, based on Section 3, Section 4 and Section 6 in [99] and, on a part of Section
3 in [104], presents results in the weak solvability of frictionless unilateral or frictional bilateral
contact problems, for nonlinearly elastic materials, by using a technique involving dual Lagrange
multipliers and applying the abstract results presented in Section 3.1.1.
Physical setting and mathematical description of the models
We consider a body that occupies the bounded domain Ω ⊂R3,with the boundary partitioned
into three measurable parts, Γ 1,Γ2and Γ 3,such thatmeas (Γ1)>0.The unit outward normal
vector to Γ is denoted by nand is defined almost everywhere. The body Ω is clamped on Γ 1,
body forces of density f0act on Ω and surface traction of density f2acts on Γ 2.On Γ 3the body
can be in contact with a rigid foundation.
In order to describe the behavior of the materials, we use the constitutive law,
σ=F(ε(u)) in Ω (3.3)
where Fdenotes a nonlinear elastic operator. This kind of constitutive law can be found in the
literature, see for example [59] and the references therein. As an example, we may consider
σ=λ0(trε)I3+ 2µ0ε+β(ε−PKε) (3.4)
41
whereλ0andµ0denote Lame’s constants, trε(u) =εkk, I3= (δij) is the unit in S3, Kdenotes
a closed convex subset of S3that contains the zero element 0 S3, PK:S3→Kis the projection
operator onto K,andβis a strictly positive constant. A second example is the following
constitutive law,
σ=k(trε)I3+ψ(∥εD∥2)εD, (3.5)
wherek > 0 is a coefficient of the material, ψ:R→Ris a constitutive function and εD=
ε−1
3(trε)I3,is the deviator of the tensor ε.
Assuming that on Γ 3the body is in frictional bilateral contact with a rigid foundation, we
use Tresca’s law to state the following mechanical problem.
Problem 3.2. Findu:¯Ω→R3andσ:¯Ω→S3such that
Divσ+f0=0 inΩ,
σ=F(ε(u)) inΩ,
u=0 onΓ1,
σn =f2 onΓ2,
(C)
un= 0,∥στ∥ ≤ζ,
if∥στ∥<ζthenuτ= 0,
if∥στ∥=ζthen there exists ψ>0 :στ=−ψuτonΓ3,
whereζ >0denotes the friction bound.
If we assume that on Γ 3,the body can be in frictionless unilateral contact with a rigid
foundation, we can model the contact by Signorini’s condition with zero gap, yielding the second
problem.
Problem 3.3. Findu:¯Ω→R3andσ:¯Ω→S3such that
Divσ+f0=0 inΩ, (3.6)
σ=F(ε(u)) inΩ, (3.7)
u=0 onΓ1, (3.8)
σn =f2 onΓ2, (3.9)
στ=0, σn≤0, un≤0, σnun= 0 onΓ3. (3.10)
Finally, if we model the contact on Γ 3by Signorini’s condition with non zero gap, we have
to replace (3.10) with the following contact condition,
στ=0, σn≤0, un−g≤0, σn(un−g) = 0 on Γ 3, (3.11)
42
whereg: Γ3→Ris the gap between the deformable body and the foundation, measured along
the outward normal n.Thus, we can formulate the third problem.
Problem 3.4. Findu:¯Ω→R3andσ:¯Ω→S3such that (3.6)-(3.9)and(3.11)hold.
Additional details on this subsection, including a description of the physical significance for
the contact conditions (C), (3.10) and (3.11), can be found for instance in [59].
Once the displacement field uis determined, the stress tensor σcan be obtained via relation
(3.3).
Hypotheses and weak formulations
Herein we state the hypotheses and present the weak formulations with dual Lagrange multipliers
for each of the models described in the previous section.
Assumption 3.5. F: Ω×S3→S3;
there exists M > 0such that ∥F(x,ε1)−F(x,ε2)∥ ≤M∥ε1−ε2∥for allε1,ε2∈S3,a.e. in Ω;
there exists m> 0such that for all ε1,ε2∈S3,and almost everywhere in Ω :
(F(x,ε1)− F(x,ε2))·(ε1−ε2)≥m∥ε1−ε2∥2;
for allε∈S3,x→ F (x,ε)is Lebesgue measurable in Ω;
x→ F (x,0S3)belongs toL2(Ω)3×3.
Referring to (3.4), we note that, using the property of the non-expansivity of the projection
map, it can be proved that the map
F: Ω×S3→S3;F(x,ε) =λ0(trε)I3+ 2µ0ε+β(ε−PKε)
satisfies Assumption 3.5. Moreover, referring to (3.5), under appropriate assumptions on the
constitutive function ψ,see [59] p.125, the map
F: Ω×S3→S3;F(x,ε) =k(trε)I3+ψ(∥εD∥2)εD,
satisfies Assumption 3.5.
Moreover, we made the following assumptions.
Assumption 3.6. f0∈L2(Ω)3,f2∈L2(Γ2)3.
Assumption 3.7. There exists gext: Ω→Rsuch thatgext∈H1(Ω), γg ext= 0 almost every-
where on Γ1, γg ext≥0almost everywhere on Γ\Γ1, g=γgextalmost everywhere on Γ3.
Assumption 3.8. The unit outward normal to Γ3,denoted byn3,is a constant vector.
43
Weak formulation of Problem 3.2. Let us introduce the space
V1={v∈H1(Ω)3|v= 0 a.e. on Γ 1, vν= 0 a.e. on Γ 3}.
We define an operator A:V1→V1such that, for each u∈V1, Auis the element of V1that
satisfies,
(Au,v)V1=∫
ΩFε(u)·ε(v)dx for allv∈V1. (3.12)
Also, we define f∈V1such that,
(f,v)V1=∫
Ωf0·vdx+∫
Γ2f2·γvda for allv∈V1.
LetDTbe the dual of the space γ(V1) ={γv v ∈V1}.We defineλ∈DTsuch that
⟨λ,γv⟩T=−∫
Γ3στ·vτda, for allγv∈γ(V1),
where ⟨·,·⟩Tdenotes the duality pairing between DTandγ(V1).Furthermore, we define a bilinear
form as follows,
b:V1×DT→R, b(v,µ) =⟨µ,γv⟩T,for allv∈V1,µ∈DT. (3.13)
Let us introduce the following subset of DT,
Λ ={
µ∈DT:⟨µ,γv⟩T≤∫
Γ3ζ∥vτ∥dΓ for allγv∈γ(V1)}
. (3.14)
We have the following weak formulation of Problem 3.2.
Problem 3.5. Findu∈V1andλ∈Λ,such that
(Au,v)V1+b(v,λ) = (f,v)V1 for allv∈V1,
b(u,µ−λ)≤0 for allµ∈Λ.
A solution of Problem 3.5 is called a weak solution to Problem 3.2.
Weak formulation of Problem 3.3. We introduce the space
V={v∈H1(Ω)3|v= 0 a.e. on Γ 1}.
We define an operator A:V→Vsuch that, for each u∈V, Auis the element of Vthat
satisfies,
(Au,v)V=∫
ΩFε(u)·ε(v)dx for allv∈V. (3.15)
44
Next, we define f∈Vsuch that
(f,v)V=∫
Ωf0·vdx+∫
Γ2f2·γvda for allv∈V. (3.16)
LetDSbe the dual of the space γ(V) and let us denote by ⟨·,·⟩Sthe duality pairing between
DSandγ(V).We defineλ∈DSsuch that
⟨λ,γv⟩S=−∫
Γ3σnvnda for allγv∈γ(V). (3.17)
In addition, we define a bilinear form as follows
b:V×DS→R, b(v,µ) =⟨µ,γv⟩S,for allv∈V,µ∈DS. (3.18)
We introduce the following subset of DS,
Λ ={
µ∈DS:⟨µ,γv⟩S≤0 for allγv∈ K}
, (3.19)
where
K={γv∈γ(V) :vn≤0 almost everywhere on Γ 3}. (3.20)
We arrive to the following weak formulation of Problem 3.3.
Problem 3.6. Findu∈Vandλ∈Λ,such that
(Au,v)V+b(v,λ) = (f,v)V for allv∈V,
b(u,µ−λ)≤0 for allµ∈Λ.
A solution of Problem 3.6 is called a weak solution to Problem 3.3.
Weak formulation of Problem 3.4. We can keep (3.15)-(3.20). Thus, we can write the
following weak formulation of Problem 3.4.
Problem 3.7. Findu∈Vandλ∈Λ,such that
(Au,v)V+b(v,λ) = (f,v)V for allv∈V,
b(u,µ−λ)≤b(gextn3,µ−λ)for allµ∈Λ.
A solution of Problem 3.7 is called a weak solution to Problem 3.4.
45
Weak solvability of the models
The well-posedness of Problem 3.5 is given by the following theorem.
Theorem 3.2. [Theorem 6.1 in [99]] If Assumptions 3.5-3.6and3.8hold true, then Problem
3.5has a unique solution (u,λ)∈V1×Λ.Moreover, if (u,λ)and(u∗,λ∗)are two solutions of
Problem 3.5corresponding to the data f∈V1andf∗∈V1,there exists CT>0such that
∥u−u∗∥V1+∥λ−λ∗∥DT≤CT∥f−f∗∥V1. (3.21)
The well-posedness of Problem 3.6 is given by the following theorem.
Theorem 3.3. [Theorem 6.2 in [99]]If Assumptions 3.5-3.6and3.8hold true, then Problem
3.6has a unique solution (u,λ)∈V×Λ.Moreover, if (u,λ)and(u∗,λ∗)are two solutions of
Problem 3.6corresponding to the data f∈Vandf∗∈V,there exists CS
1>0such that
∥u−u∗∥V+∥λ−λ∗∥DS≤CS
1∥f−f∗∥V. (3.22)
Finally, we discuss the well-posedness of Problem 3.7.
Theorem 3.4. [Theorem 6.3 in [99]]If Assumptions 3.5-3.8hold true, then Problem 3.7has a
unique solution (u,λ)∈V×Λ.Moreover, if (u,λ)and(u∗,λ∗)are two solutions of Problem
3.7corresponding to the data (f, gextn3)∈V×Vand(f∗, g∗
extn3)∈V×V,there exists CS
2>0
such that
∥u−u∗∥V+∥λ−λ∗∥DS≤CS
2(
∥f−f∗∥V+∥gextn3−g∗
extn3∥V)
. (3.23)
The proofs of Theorems 3.2, 3.3 and 3.4 are based on Theorem 3.1; for details see [99] and
[104].
3.2 Problems governed by proper convex l.s.c functionals
The results we present in this section were obtained in the papers [100, 104]. This section focuses
on the weak solvability of a class of contact models, under the small deformations hypothesis, for
static processes, for materials whose behavior is described by a constitutive law stated in a form
of a subdifferential inclusion. The weak solvability of the models is based on weak formulations
with dual Lagrange multipliers.
3.2.1 An abstract result
This subsection, based on Section 4 in [100] and on a part of Section 2 in [104], delivers abstract
results in the study of the following problem.
46
Problem 3.8. Findu∈Xandλ∈Λsuch that
J(v)−J(u) +b(v−u,λ)≥(f,v−u)Xfor allv∈X
b(u,µ−λ)≤0 for allµ∈Λ.
This is a new variational problem, a mixed variational problem governed by a functional J.
The analysis of this problem was made under the following hypotheses.
Assumption 3.9. (X,(·,·)X,∥ · ∥ X)and(Y,(·,·)Y,∥ · ∥ Y)are Hilbert spaces.
Assumption 3.10. J:X→[0,∞)is a convex and lower semicontinuous functional such that
there existm1, m 2>0 :m1∥v∥2
X≥J(v)≥m2∥v∥2
Xfor allv∈X.
Assumption 3.11. b:X×Y→Ris a bilinear form such that:
i) there exists Mb>0 :|b(v,µ)| ≤Mb∥v∥X∥µ∥Yfor allv∈X, µ∈Y;
ii) there exists α>0 : inf
µ∈Y,µ̸=0Ysup
v∈X,v̸=0Xb(v,µ)
∥v∥X∥µ∥Y≥α.
Assumption 3.12. Λis a closed convex subset of Ythat contains 0Y.
Theorem 3.5. If Assumptions 3.9-3.12hold true, then Problem 3.8has at least one solution.
For the proof of Theorem 3.5 see the proof of Theorem 3 in [100] if Λ is unbounded, and the
proof of Theorem 2.3 in [104] if Λ is bounded, respectively.
In order to study the uniqueness and the stability of the solution, the following additional
assumption it was used.
Assumption 3.13. J:X→[0,∞)is a Gˆ ateaux differentiable functional such that,
there exists L>0 :∥∇J(u)− ∇J(v)∥X≤L∥u−v∥Xfor allu,v∈X,
there exists m> 0 : (∇J(u)− ∇J(v),u−v)X≥m∥u−v∥2
Xfor allu,v∈X.
Notice that a pair ( u,λ) is a solution of Problem 3.8 if and only if
(P) :
(∇J(u),v)X+b(v,λ) = (f,v)X for allv∈X,
b(u,µ−λ)≤0 for allµ∈Λ.
Theorem 3.6. Under Assumptions 3.9-3.13, Problem 3.8has a unique solution, which depends
Lipschitz continuously on the data f.
For the proof of Theorem 3.6 see the proof of Theorem 4 in [100] if Λ is unbounded, and the
proof of Theorem 2.4 in [104] if Λ is bounded, respectively.
47
3.2.2 3D contact models
This subsection is based on Section 3 and Section 5 in [100] and, on a part of Section 3 in [104].
In this subsection we apply the abstract results we have presented in the previous subsection,
to the weak solvability of two classes of contact problems.
A frictionless unilateral contact model
This model was analyzed in the paper [100]. The physical setting is as follows. We consider
a body that occupies the bounded domain Ω ⊂R3,with the boundary partitioned into three
measurable parts, Γ 1,Γ2and Γ 3,such thatmeas (Γ1)>0.The unit outward normal to Γ is
denoted byνand is defined almost everywhere. The body Ω is clamped on Γ 1, body forces
of densityf0act on Ω and surface traction of density f2act on Γ 2.On Γ 3the body can
be in contact with a rigid foundation. In order to describe the behavior of the materials, we
use a nonlinear constitutive law expressed by the subdifferential of a proper, convex, lower
semicontinuous functional and the contact will be modelled using Signorini’s condition with zero
gap.
Problem 3.9. Findu:¯Ω→R3andσ:¯Ω→S3,such that
Divσ(x) +f0(x) =0 inΩ,
σ(x)∈∂ω(ε(u(x))) inΩ,
u(x) =0 onΓ1,
σν(x) =f2(x) onΓ2,
στ(x) =0, uν(x)≤0,σν(x)≤0,σν(x)uν(x) = 0 onΓ3.
The study was made under the following assumptions.
Assumption 3.14. f0∈L2(Ω)3;f2∈L2(Γ2)3.
Assumption 3.15. ω:S3→[0,∞)is a convex, lower semicontinuous functional such that
there existα1, α2>0 :α1∥ε∥2≥ω(ε)≥α2∥ε∥2for allε∈S3.
To give an example of such a function ωwe can consider
ω:S3→[0,∞), ω (ε) =1
2Aε·ε+β
2∥ε−PKε∥2(3.24)
where Ais a fourth order symmetric tensor satisfying the ellipticity condition, βis a strictly
positive constant, K⊂S3denotes a closed, convex set containing the element 0 S3andPK:S3→
Kis the projection operator; see e.g. [59].
48
Let us introduce the spaces
V={v∈H1(Ω)3:γv=0a.e. on Γ 1},
and
L2
s(Ω)3×3={µ= (µij) :µij∈L2(Ω), µij=µjifor alli,j∈ {1,2,3}}.
We define a functional as follows,
W:L2
s(Ω)3×3→[0,∞), W (τ) =∫
Ωω(τ(x))dx.
Next, we define the functional
J:V→[0,∞), J (v) =W(ε(v)). (3.25)
Also, we define f∈Vas follows,
(f,v)V=∫
Γ2f2·γvda+∫
Ωf0·vdx for allv∈V.
Let us denote by Dthe dual of the space γ(V).We define the following subset of D,
Λ ={µ∈D:⟨µ,γv⟩ ≤0,for allγv∈ K}, (3.26)
where
K={γv∈γ(V) :vν≤0 a.e. on Γ 3};
⟨·,·⟩denotes the duality pairing between Dandγ(V).
In addition, we define the bilinear form
b:V×D→R, b (v,µ) =⟨µ,γv⟩ for allv∈V,µ∈D (3.27)
and the Lagrange multiplier λ∈D,
⟨λ,γv⟩=−∫
Γ3σνvνdΓ, for allγv∈γ(V).
Problem 3.9 has the following weak formulation.
Problem 3.10. Findu∈Vandλ∈Λsuch that
J(v)−J(u) +b(v−u,λ)≥(f,v−u)Vfor allv∈V
b(u,µ−λ)≤0 for allµ∈Λ.
49
Based on the previous abstract result, the following theorems take place.
Theorem 3.7 (An existence result (Theorem 5 in [100])) .If Assumptions 3.14and3.15hold
true, then Problem 3.10has at least one solution.
Let us make now the following additional assumption.
Assumption 3.16. ωis a Gˆ ateaux differentiable functional such that:
there exists L>0:∥∇ω(ε)− ∇ω(τ)∥ ≤L∥ε−τ∥for allε,τ∈S3;
there exists m> 0:(∇ω(ε)− ∇ω(τ))·(ε−τ)≥m∥ε−τ∥2for allε,τ∈S3.
An example of such a function is ωin (3.24).
Theorem 3.8 (An existence, uniqueness and stability result, (Theorem 6 in [100])) .If As-
sumptions 3.14,3.15and3.16hold true, then Problem 3.10has a unique solution. Moreover, if
(u1,λ1)and(u2,λ2)are two solutions of Problem 3.10corresponding to the data f1,f2∈V,
then there exists C > 0such that
∥u1−u2∥V+∥λ1−λ2∥D≤C∥f1−f2∥V.
The proof of Theorem 3.8, gave in [100], is based on the previous abstract result, Theorem
3.6.
A frictional contact model
The model we discuss now was analyzed in the paper [104]. The physical setting is the follow-
ing. A body occupies the bounded domain Ω ⊂R3,with the boundary partitioned into three
measurable parts, Γ 1,Γ2and Γ 3,such thatmeas (Γ1)>0.The unit outward normal vector to Γ
is denoted by νand is defined almost everywhere. The body Ω is clamped on Γ 1, body forces of
densityf0act on Ω and surface traction of density f2acts on Γ 2.On Γ 3the body is in frictional
contact with a foundation.
According to the previous physical setting we state the following boundary value problem.
Problem 3.11. Findu:Ω→R3andσ:Ω→S3such that
Divσ+f0=0 inΩ, (3.28)
σ(x)∈∂ω(ε(u(x))) inΩ, (3.29)
u=0 onΓ1, (3.30)
σν=f2 onΓ2, (3.31)
−σν=F,∥στ∥ ≤k|σν|,στ=−k|σν|uτ
∥uτ∥ifuτ̸=0onΓ3, (3.32)
50
whereω:S3→[0,∞) is a constitutive function, F: Γ3→R+is the prescribed normal stress
andk: Γ3→R+is the coefficient of friction.
Problem 3.11 has the following structure: (3.28) represents the equilibrium equation, (3.29)
represents the constitutive law, (3.30) represents the displacements boundary condition, (3.31)
represents the traction boundary condition and (3.32) models the frictional contact with pre-
scribed normal stress. For details on this model we send the reader to, e.g., [147].
We made the following assumptions.
Assumption 3.17. ω:S3→[0,∞)is a convex, lower semicontinuous functional. In addition,
there existα1, α2>0such thatα1∥ε∥2≥ω(ε)≥α2∥ε∥2for allε∈S3.
Assumption 3.18. The density of the volume forces verifies f0∈L2(Ω)3and the density of
the tractions verifies f2∈L2(Γ2)3.
Assumption 3.19. The prescribed normal stress verifies F∈L2(Γ3)andF(x)≥0a.e.x∈Γ3.
Assumption 3.20. The coefficient of friction verifies k∈L∞(Γ3)andk(x)≥0a.e.x∈Γ3.
Let us replace now (3.32) with the following condition
uν= 0,∥στ∥ ≤ζ,στ=−ζuτ
∥uτ∥ifuτ̸=0. (3.33)
This condition is a frictional bilateral contact condition where ζ: Γ3→R+denotes the friction
bound.
Now, a second model can be formulated as follows.
Problem 3.12. Findu:Ω→R3andσ:Ω→S3such that (3.28)-(3.31)and(3.33)hold true.
We shall study Problem 3.12 under Assumptions 3.17-3.18, and in addition we shall make
the following assumption.
Assumption 3.21. The friction bound verifies ζ∈L2(Γ3)andζ(x)≥0a.e.x∈Γ3.
Weak solvability of Problem 3.11
Let us introduce two functional spaces
V={v∈H1(Ω)3:γv=0a.e. on Γ 1},
and
L2
s(Ω)3×3={µ= (µij) :µij∈L2(Ω), µij=µjifor alli,j∈ {1,2,3}}
We now introduce the functional
W:L2
s(Ω)3×3→[0,∞), W (τ) =∫
Ωω(τ(x))dx. (3.34)
51
Using the functional Wwe introduce a new one
J:V→[0,∞), J (v) =W(ε(v)). (3.35)
We definef∈Vsuch that, for all v∈V,
(f,v)V=∫
Ωf0(x)·v(x)dx+∫
Γ2f2(x)·γv(x)dΓ−∫
Γ3F(x)vν(x)dΓ.
Next, letDbe the dual of the Hilbert space
S={ev=γv| 3v∈V}.
We defineλ∈Dsuch that
⟨λ,w⟩=−∫
Γ3στ(x)·wτ(x)dΓ for all w∈S,
where ⟨·,·⟩denotes the duality pairing between DandS.Furthermore, we define a bilinear form
as follows,
b:V×D→R, b(v,µ) =⟨µ,γv| 3⟩,for allv∈V,µ∈D.
Let us introduce the following subset of D,
Λ ={
µ∈D:⟨µ,γv|Γ3⟩ ≤∫
Γ3kF∥vτ∥dΓv∈V}
. (3.36)
We are led to the following weak formulation of Problem 3.11.
Problem 3.13. Findu∈Vandλ∈Λ,such that
J(v)−J(u) +b(v−u,λ)≥(f,v−u)V for allv∈V,
b(u,µ−λ)≤0 for allµ∈Λ.
A solution of Problem 3.13 is called a weak solution of Problem 3.11.
Theorem 3.9. [Theorem 3.1 in [104]] If Assumptions 3.17-3.20hold true, then Problem 3.13has
at least one solution (u,λ)∈V×Λ.If, in addition, Assumption 3.16is fulfilled, then Problem
3.13has a unique solution; moreover, there exists C > 0such that
∥u−u∗∥V+∥λ−λ∗∥D≤C∥f−f∗∥V, (3.37)
where (u,λ)and(u∗,λ∗)are two solutions of Problem 3.13corresponding to the data f∈V
andf∗∈V.
52
The proof of Theorem 3.9, given in [104], is based on Theorem 3.6.
Weak solvability of Problem 3.12
Herein we use the Hilbert space
V1={
v∈H1(Ω)3|γv=0a.e. on Γ 1, vν= 0 a.e. on Γ 3}
.
We definef1∈V1such that, for all v∈V1,
(f1,v)V=∫
Ωf0(x)·v(x)dx+∫
Γ2f2(x)·γv(x)dΓ.
Next, we define the functional
J1:V1→[0,∞), J 1(v) =W(ε(v)),
whereWis the functional defined in (3.34).
LetD1be the dual of the Hilbert space
S1={ev=γv| 3v∈V1}.
We defineλ∈D1such that
⟨λ,w⟩=−∫
Γ3στ(x)·wτ(x)dΓ, for allw∈S1,
where ⟨·,·⟩denotes the duality pairing between D1andS1.Furthermore, we define a bilinear
form as follows,
b:V1×D1→R, b 1(v,µ) =⟨µ,γv| 3⟩,for allv∈V1,µ∈D1.
Let us introduce the following subset of D1,
Λ1={
µ∈D1:⟨µ,γv|Γ3⟩ ≤∫
Γ3ζ(x)∥vτ∥dΓv∈V1}
.
Clearly,λ∈Λ1.Furthermore,
b1(u,λ) =∫
Γ3ζ(x)∥uτ(x)∥dΓ,
b1(u,µ)≤∫
Γ3ζ(x)∥uτ(x)∥dΓ for all µ∈Λ1.
Consequently, we are led to the following weak formulation of Problem 3.12.
53
Problem 3.14. Findu∈V1andλ∈Λ1,such that
J1(v)−J1(u) +b1(v−u,λ)≥(f1,v−u)V for allv∈V1,
b1(u,µ−λ)≤0 for allµ∈Λ1.
Theorem 3.10. [Theorem 3.2 in [104]] If Assumptions 3.17-3.18and3.21hold true, then Prob-
lem3.14has a unique solution; moreover, there exists C > 0such that
∥u−u∗∥V1+∥λ−λ∗∥D1≤C∥f−f∗∥V1
where (u,λ)and(u∗,λ∗)are two solutions of Problem 3.14corresponding to the data f∈V1
andf∗∈V1.
A solution of Problem 3.14 is called a weak solution of Problem 3.12.
The proof of Theorem 3.10, given in [104], is based on Theorem 3.6.
3.3 Problems governed by a nonlinear, hemicontinuous,
generalized monoton operator
This section, based on the paper [107], focuses on a new theoretical result which will allow to
explore contact models for a class of nonlinearly elastic materials leading to mixed variational
problems governed by nonlinear, hemicontinuous, generalized monoton operators. The key herein
is not the saddle point theory; the key here is a fixed point theorem for set valued mapping.
3.3.1 An abstract result
This subsection is based on the Sections 2-4 of the paper [107]. In this subsection we focus on
the following mixed variational problem.
Problem 3.15. Givenf∈X′,find(u,λ)∈X×Λsuch that
(Au,v )X′,X+b(v,λ) = (f, v)X′,X for allv∈X, (3.38)
b(u,µ−λ) ≤0 for allµ∈Λ. (3.39)
Here and everywhere below X′denotes the dual of the space Xand Λ is a subset of a space Y.
Assumption 3.22. (X,(·,·)X,∥·∥ X)and(Y,(·,·)Y,∥·∥ Y)are two real reflexive Banach spaces.
Assumption 3.23. Λis a closed convex bounded subset of Ysuch that 0Y∈Λ.
Assumption 3.24. There exists a functional h:X→Rsuch that:
•(i1)h(tw) =trh(w)for allt>0, w∈Xandr>1;
•(i2) (Av−Au,v−u)X′,X≥h(v−u)for allu,v∈X;
•(i3)If(xn)n⊂Xis a sequence such that xn⇀x inXasn→ ∞,thenh(x)≤lim sup
n→∞h(xn).
54
Notice that ( i1) and (i2) in Assumption 3.24 express a generalized monotonicity property for
the operator A:X→X′.According to the literature, the operator Ais a relaxed h−monotone
operator, see for example [42] and the references therein.
Assumption 3.25. The operator A:X→X′is hemicontinuous, i.e., for all u,v∈X, the
mappingf:R→(−∞,+∞), f(t) = (A(u+tv),v)X′,Xis continuous at 0.
Assumption 3.26.(Au,u )X′,X
∥u∥X→ ∞ as∥u∥X→ ∞.
Assumption 3.27. The formb:X×Y→Ris bilinear. In addition,
•for each sequence (un)n⊂Xsuch thatun⇀u inXasn→ ∞ we haveb(un,µ)→b(u,µ)
asn→ ∞,for allµ∈Λ.
•for each sequence (λn)n⊂Ysuch thatλn⇀λ inYasn→ ∞,we haveb(v,λn)→b(v,λ)
asn→ ∞,for allv∈X.
Under Assumptions 3.22-3.27, Problem 3.15 has at least one solution. Assumptions 3.22-
3.27 impose a new technique in order to handle Problem 3.15, namely a fixed point technique
involving a set valued mapping, instead of a saddle point technique. Let us mention here the
main tool.
Theorem 3.11. LetK ̸=∅be a convex subset of a Hausdorff topological vector space E. Let
F:K → 2Kbe a set valued map such that
(h1)for eachu∈ K,F(u)is a nonempty convex subset of K;
(h2)for eachv∈ K,F−1(v) ={u∈ K :v∈F(u)}contains an open set Ovwhich may be
empty;
(h3)∪
v∈KOv=K;
(h4)there exists a nonempty set V0contained in a compact convex subset V1ofKsuch that
D=∩
v∈V0Oc
vis either empty or compact.
Then, there exists u0∈ K such thatu0∈F(u0).
We note that 2Kdenotes the family of all subsets of K,andOc
vis the complement of Ovin
K. For a proof of this theorem we refer to [152].
Let us construct a bounded convex closed nonempty subset of Xas follows,
Kn={v∈X:∥v∥X≤n}
wherenis an arbitrarily fixed positive integer. We consider the following auxiliary problem.
Problem 3.16. Givenf∈X′,find(un,λn)∈Kn×Λsuch that
(Aun,v−un)X′,X+b(v,λn)−b(un,µ)≥(f, v−un)X′,X (3.40)
for all (v,µ)∈Kn×Λ.
55
Lemma 3.1. [Lemma 1 in [107]] A pair (un,λn)∈Kn×Λis a solution of Problem 3.16if and
only if it verifies
(Av,v−un)X′,X+b(v,λn)−b(un,µ)≥(f,v−un)X′,X+h(v−un) (3.41)
for all (v,µ)∈Kn×Λ.
Let us define a set valued map F:Kn×Λ→2Kn×Λas follows,
F(u,λ) ={
(v,µ)∈Kn×Λ : (Au,v−u)X′,X+b(v,λ)−b(u,µ)<(f, v−u)X′,X}
.
Arguing by contradiction, using the map F(·,·) and Theorem 3.11, the following existence result
was delivered.
Theorem 3.12. [Theorem 2 in [107]] If Assumptions 3.22-3.25and Assumption 3.27hold true,
then Problem 3.16has at least one solution (un,λn)∈Kn×Λ.
Based on Theorem 3.12 we have got the following existence result.
Theorem 3.13. [Theorem 3 in [107]] If Assumptions 3.22-3.27hold true, then Problem 3.15
has at least one solution.
The proofs of Lemma 3.1, Theorems 3.12-3.13 can be found in [107].
Let us present an example of spaces X,Y, subset Λ,operatorAand formb(·,·) which verify
Assumptions 3.22-3.27.
Let Ω be a bounded domain in R2with smooth boundary Γ. Let pbe a real number such
that∞>p≥4.We define a subspace of W1,p(Ω) as follows,
X={v:v∈W1,p(Ω), γv= 0 a.e. on Γ D} (3.42)
where Γ Dis a part of Γ with positive Lebesgue measure and γ:W1,p(Ω)→Lp(Γ) is the Sobolev
trace operator. It is known that the space Xis a Banach space endowed with the norm
∥u∥X=∥∇u∥Lp(Ω)N.
Letp′be the conjugate exponent of p,i.e.1
p+1
p′= 1.We now consider Γ Ca part of Γ such
thatmeas (ΓC)>0 and Γ C∩ΓD=∅.Then, we can take
Y=Lp′(ΓC). (3.43)
Next, we define a subset of Yas follows,
Λ ={µ∈Y:⟨µ,γv | C⟩ ≤∫
ΓCg|γv(x)|dΓ for allv∈X}, (3.44)
wheregis a positive real number.
56
We defineA:X→X′as follows: for each u∈X, Au is the element of X′such that
(Au,v )X′,X=∫
Ωµ∥∇u(x)∥p−2∇u(x)· ∇v(x)dx for allv∈X (3.45)
whereµis a positive real number. The operator Ais hemicontinuous and relaxed h−monotone
withh≡0 being in fact Lipschitz continuous and monotone. Besides, for each u∈X, u̸= 0 X,
we have
(Au,u )X′,X
∥u∥X=µ∥u∥p−1
X.
Finally, we define b:X×Lp′(ΓC)→Ras follows
b(v,µ) =⟨µ,γv | C⟩, (3.46)
where ⟨·,·⟩is the duality pairing between Lp′(ΓC) andLp(ΓC).
3.3.2 An antiplane frictional contact problem
This subsection is based on Section 5 of the paper [107]. Herein we apply the abstract existence
result, Theorem 3.13, to the weak solvability of the following boundary value problem.
Problem 3.17. Findu:¯Ω→Rsuch that
div (µ∥∇u(x)∥p−2∇u(x)) +f0(x) = 0 in Ω, (3.47)
u(x) = 0 on Γ D, (3.48)
µ∥∇u(x)∥p−2∂νu(x) =f2(x) on Γ N, (3.49)
|µ∥∇u(x)∥p−2∂νu(x)| ≤g,
µ∥∇u(x)∥p−2∂νu(x) =−gu(x)
|u(x)|ifu(x)̸= 0on Γ C. (3.50)
This problem models the antiplane shear deformation of a nonlinearly elastic cylindrical
body, in frictional contact on Γ Cwith a rigid foundation. See [145] for details on the antiplane
contact models. We also refer to the works [114, 115, 116, 117] for a treatement of some antiplane
contact problems in a general setting of the hemivariational inequalities.
Herein Ω ⊂R2is a bounded domain with smooth boundary Γ partitioned in three measurable
parts Γ D,ΓN,ΓCwith positive Lebesgue measures. Referring the body to a cartesian coordinate
systemOx 1x2x3such that the generators of the cylinder are parallel with the axis Ox 3, the
domain Ω ⊂Ox 1x2denotes the cross section of the cylinder. The functions f0=f0(x1,x2) :
Ω→R,f2=f2(x1,x2) : Γ N→Rare related to the density of the volume forces and the density
of the surface traction, respectively, and g >0 is the friction bound. The vector ν= (ν1,ν2),
57
νi=νi(x1,x2),for eachi∈ {1,2}, represents the outward unit normal vector to the boundary
of Ω and∂νu=∇u·ν.The behavior of the nonlinearly elastic material is described by the
following constitutive law:
σ(x) =ktrε(u(x))I3+µ∥εD(u(x))∥p−2εD(u(x)) (3.51)
whereσis the Cauchy stress tensor, tr is the trace of a cartesian tensor of second order, εis the
infinitesimal strain tensor, uis the displacement vector, I3is the identity tensor, k,µ > 0 are
material parameters, pis a constant such that 4 ≤p<∞,εDdenotes the deviator of the tensor
ε, defined byεD=ε−1
3(trε)I3.The constitutive law (3.51) is a Hencky-type constitutive law.
The mechanical problem has the following structure: (3.47) represents the equilibrium equa-
tion, (3.48) is the displacement boundary condition, (3.49) is the traction boundary condition
and (3.50) is Tresca’s law of dry friction; see e.g. [145, 147] for more details on frictional laws.
We shall study Problem 3.17 assuming that
f0∈Lp′(Ω), f 2∈Lp′(ΓN). (3.52)
We definef∈X′as follows
(f,v)X′,X=∫
Ωf0(x)v(x)dx+∫
ΓDf2(x)γv(x)dΓ for allv∈X. (3.53)
Next, we define a Lagrange multiplier λ∈Yas follows
⟨λ,z⟩=−∫
ΓCµ∥∇u(x)∥p−2∂νu(x)z(x)dΓ for allz∈Lp(ΓC), (3.54)
whereYis the space defined in (3.43).
Problem 3.17 has the following weak formulation.
Problem 3.18. Findu∈Xandλ∈Λ⊂Ysuch that
(Au, v )X′,X+b(v,λ) = (f,v)X′,X for allv∈X. (3.55)
and
b(u,ζ−λ)≤0for allζ∈Λ. (3.56)
Theorem 3.14. [Theorem 4 in [107]]If 4≤p<∞, k,µ,g > 0, f0∈Lp′(Ω),andf2∈Lp′(ΓN),
then Problem 3.18has at least one solution.
The proof of Theorem 3.14, given in [107], is based on Theorem 3.13.
As each solution of Problem 3.18 is called weak solution of Problem 3.17, Theorem 3.14
ensures us that Problem 3.17 has at least one weak solution.
Chapter 4
Viscoelastic frictional contact problems
This chapter is based on the papers [101, 111]. We discuss antiplane models which describe
the contact between a deformable cylinder and a rigid foundation, under the small deformation
hypothesis, for quasistatic processes. The behavior of the material is modelled using viscoelastic
constitutive laws and the frictional contact is modelled using Tresca’s law. We draw the attention
to the weak solvability of the models based on a weak formulation with dual Lagrange multipliers
in the case of viscoelastic materials with long memory as well as in the case of viscoelastic
materials with short memory. The results we have got are based on new abstract results in the
study of new classes of mixed variational problems: a class of time dependent mixed variational
problems and a class of evolutionary mixed variational problems.
4.1 The case of viscoelasticity with long-memory term
In this section we present the results obtained in the paper [101], discussing the weak solvability
of a contact model for viscoelastic materials with long memory, by using arguments which involve
dual Lagrange multipliers; for a classical approach of such kind of models we refer to, e.g. [143].
The weak solvability of the proposed model through an approach with Lagrange multipliers is
related to the solvability of a new abstract variational problem.
4.1.1 An abstract result
In this subsection we shall present an abstract result obtained under the following assumptions.
Assumption 4.1. (X,(·,·)X,∥ · ∥ X)and(Y,(·,·)Y,∥ · ∥ Y)are two Hilbert spaces.
Assumption 4.2. A:X→Xis an operator such that:
there exists mA>0 : (Au−Av, u −v)X≥mA∥u−v∥2
Xfor allu,v∈X,
there exists LA>0 :∥Au−Av∥X≤LA∥u−v∥Xfor allu,v∈X.
58
59
Assumption 4.3. b:X×Y→Ris a bilinear form such that:
there exists Mb>0 :|b(v,µ)| ≤Mb∥v∥X∥µ∥Yfor allv∈X, µ∈Y,
there exists α>0 : inf
µ∈Y,µ̸=0Ysup
v∈X,v̸=0Xb(v,µ)
∥v∥X∥µ∥Y≥α.
Assumption 4.4. Λ⊂Yis a closed convex set such that 0Y∈Y.
Assumption 4.5. f∈C([0,T];X).
Assumption 4.6. B∈C([0,T];L(X)).
Problem 4.1. Givenf: [0,T]→X,findu: [0,T]→Xandλ: [0,T]→Ysuch that, for every
t∈[0,T],we haveλ(t)∈Λand
(Au(t),v)X+ (∫t
0B(t−s)u(s)ds,v)X+b(v,λ(t)) = (f(t), v)Xfor allv∈X (4.1)
b(u(t),µ−λ(t))≤0 for allµ∈Λ. (4.2)
This is a new variational problem, a time dependent mixed variational problem with long-
memory (a mixed variational problem governed by an integral term).
Letη∈C([0,T],X) andt∈[0,T].We consider the following auxiliary problem.
Problem 4.2. Finduη(t)∈Xandλη(t)∈Λsuch that
(Auη(t),v)X+b(v,λη(t)) = (f(t)−η(t),v)Xfor allv∈X
b(uη(t),µ−λη(t))≤ 0 for allµ∈Λ.
Problem 4.2 has a unique solution, ( uη(t),λη(t))∈X×Λ.In addition,
uη∈C([0,T],X) andλη∈C([0,T],Y). (4.3)
Let us define
T:C([0,T],X)→C([0,T],X), (Tη)(t) =∫t
0B(t−s)uη(s)ds.
The operator Tis a contraction.
Letη∗∈C([0,T];X) be the unique fixed point of the operator Tand (uη∗,λη∗) be the
solution of Problem 4.2 for η=η∗.Lett∈[0,T].Notice that the pair ( uη∗(t),λη∗(t)) verifies
(4.1) and (4.2). It was proved the following theorem.
Theorem 4.1. If Assumptions 4.1−4.6hold true, then there exists a unique solution of Problem
4.1,(u,λ),such that (u(t),λ(t))∈X×Λfor allt∈[0,T]and
u∈C([0,T];X), λ∈C([0,T];Y).
60
Moreover, given f1, f2∈C([0,T];X),there exists C > 0such that
∥u1−u2∥C([0,T];X)+∥λ1−λ2∥C([0,T];Y)≤C∥f1−f2∥C([0,T];X),
(u1,λ1)and(u2,λ2)being the solutions of Problem 4.1 corresponding to the data f1andf2,
respectively.
The proof of Theorem 4.1 can be found in [101] if Λ is unbounded (see Theorem 2 in [101]),
and it follows from Theorems 2.1 and 2.2 in [104], combined with the Banach’s fixed point
theorem, if Λ is a bounded set.
4.1.2 A mechanical model and its weak solvability
In this subsection we discuss the weak solvability of the following contact model.
Problem 4.3. Findu:Ω×[0,T]→Rsuch that, for every t∈[0,T],
div(
a(x,∇u(x,t)) +∫t
0θ(x,t−s)∇u(x,s)ds)
+f0(x,t) = 0 inΩ
u(x,t) = 0 onΓ1
a(x,∇u(x,t))·ν(x) +∫t
0θ(x,t−s)∂νu(x,s)ds=f2(x,t)onΓ2
|a(x,∇u(x,t))·ν(x) +∫t
0θ(x,t−s)∂νu(x,s)ds| ≤g(x);
|a(x,∇u(x,t))·ν(x) +∫t
0θ(x,t−s)∂νu(x,s)ds|<g(x)
⇒u(x,t) = 0;
|a(x,∇u(x,t))·ν+∫t
0θ(x,t−s)∂νu(x,s)ds|=g(x)
⇒there exists β >0such that
a(x,∇u(x,t))·ν(x) +∫t
0θ(x,t−s)∂νu(x,s)ds
=−βu(x,t)
onΓ3, (4.4)
where Ω ⊂R2is an open, bounded, connected subset, with Lipschitz continuous boundary Γ
partitioned in three measurable parts Γ 1,Γ2,Γ3such that the Lebesgue measure of Γ 1is positive.
This problem models the antiplane shear deformation of a cylindrical body in bilateral frictional
contact on Γ 3with a rigid foundation. The domain Ω denotes the cross section of the cylinder,
the unknown u=u(x1,x2) :¯Ω×[0,T]→Rrepresents the 3rdcomponent of the displacement
vector,a:¯Ω×R2→R2is a constitutive function, θ:¯Ω×[0,T]→Ris a coefficient of relaxation,
61
g: Γ3→Ris the friction bound and the functions f0: Ω×[0,T]→R, f2: Γ2×[0,T]→Rare
related to the density of the volume forces and the density of the surface traction, respectively.
Hereνis the unit outward normal vector on the boundary Γ ,defined almost everywhere, and
∂νu=∇u·ν.Notice thatν= (ν1,ν2), νi=νi(x1,x2),for eachi∈ {1,2}.
Assumption 4.7.
•there exists La>0 :∥a(x,ξ1)−a(x,ξ2)∥ ≤La∥ξ1−ξ2∥for allξ1,ξ2∈R2,a.e.x∈Ω;
•there exists Ma>0 : (a(x,ξ1)−a(x,ξ2))·(ξ1−ξ2)≥Ma∥ξ1−ξ2∥2,for allξ1,ξ2∈
R2,a.e.x∈Ω;
•For eachξ∈R2,x→a(x,ξ)is measurable in Ω;
•The mapping x→a(x,0)∈L2(Ω)2.
Let us give three examples of constitutive laws related to three examples of such functions a.
Example 4.1. We can describe the behavior of the material with the following constitutive law
σ(x,t) =λ(x)(trε(u(x,t)))IS3+ 2µ(x)ε(u(x,t)) +∫t
0θ(x,t−s)ε(u(x,s))ds
+∫t
0ζ(x,t−s)tr(ε(u(x,s)))IS3ds,
whereλandµare coefficients of the material, trε(u) =εkk(u),IS3is the unit tensor and θ, ζ
are coefficients of relaxation. In the antiplane context, the equilibrium equation reduces to
div(
µ(x)∇u(x,t) +∫t
0θ(x,t−s)∇u(x,s)ds)
+f0(x,t) = 0 inΩ×(0,T),
see for example [145].In this situation we define
a(x,ξ) =µ(x)ξ.
Assumingµ∈L∞(Ω), µ(x)≥µ∗>0a.e.x∈Ω,then Assumption 4.7 is fulfilled.
Example 4.2. Let us describe the behavior of the material with the viscoelastic constitutive law
σ(x,t) =λ(x)(trε(u(x,t)))IS3+ 2µ(x)ε(u(x,t))
+2β(x)(ε(u(x,t))−PKε(u(x,t)))
+∫t
0θ(x,t−s)ε(u(x,s))ds
+∫t
0ζ(x,t−s) tr(ε(u(x,s)))IS3ds,
62
whereλ, µ andβare coefficients of the material, Kis the non-empty, closed and convex von
Mises set
K={σ∈S3|1
2σD·σD≤k2, k> 0} (4.5)
andPK:S3→ K represents the projection operator on K. We recall that σDis the deviator of
σ,i.e.,σD=σ−1
3(trσ)IS3.
The equilibrium equation reduces to the following scalar equation
div(
(µ(x) +β(x))∇u(x,t)−2β(x)PeK1
2∇u(x,t) +∫t
0θ(x,t−s)∇u(x,s)ds)
+f0(x,t) = 0 in Ω×(0,T),
whereeK=B(0R2,k),(kgiven by (4.5)) andPeK:R2→eKis the projection operator on eK.
We define
a(x,ξ) = [µ(x) +β(x)]ξ−2β(x)PeK1
2ξ.
Let us assume that µ∈L∞(Ω), µ(x)≥µ∗>0a.e.x∈Ω,andβ∈L∞(Ω).Taking into account
the non-expansivity of the projection map PeK,Assumption 4.7 is verified.
Example 4.3. The behavior of the material is described now as follows,
σ(x,t) =k0(trε(u(x,t)))IS3+ψ(|εD(u(x,t)))|2)εD(u(x,t))
+∫t
0θ(x,t−s)ε(u(x,s))ds+∫t
0ζ(x,t−s)tr(ε(u(x,s)))IS3ds,
wherek0>0is a coefficient of the material, εD(u)is the deviatoric part of ε=ε(u)and
ψ:R→Ris a constitutive function. In the antiplane context the equilibrium equation reduces
to
div(1
2ψ(1
2|∇u(x,t)|2
R2)
∇u(x,t) +∫t
0θ(x,t−s)∇u(x,s)ds)
+f0(x,t) = 0 in Ω×(0,T).
Thus, we can consider
a(x,ξ) =1
2ψ(1
2|ξ|2
R2)
ξ.
Assume that ψ:R→Ris a piecewise continuously differentiable function such that there exist
positive constants c1, c2, d1andd2which verify the following inequalities
ψ(ξ2)≤d1,−c1≤ψ′(ξ2)≤0, c 2≤ψ(ξ2) + 2ψ′(ξ2)ξ≤d2;
see[59],page 125. In this case Assumption 4.7 is fulfilled, too.
In addition to Assumption 4.7, we made the following assumptions.
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Assumption 4.8. θ∈C([0,T];L∞(Ω)).
Assumption 4.9. f0∈C([0,T];L2(Ω));f2∈C([0,T];L2(Γ2)).
Assumption 4.10. g∈L2(Γ3), g(x)≥0a.e. on Γ3.
Let us introduce the space V={v∈H1(Ω)v= 0 a.e. on Γ 1}(v= 0 in the sense of the trace) .
We defineA:V→Vas follows: for each u∈V, Au is the unique element of Vsuch that
(Au,v )V=∫
Ωa(x,∇u(x,t))· ∇v(x)dx.
Besides, we define a function fas follows,
f: [0,T]→V,(f(t),v)V=∫
Ωf0(t)vdx+∫
Γ2f2(t)vdx, for allv∈V.
In addition, we define an operator B: [0,T]→ L(V) such that, for t∈[0,T] andu∈V, B(t)u
is the element of Vwhich verifies
(B(t)u,v)V=∫
Ωθ(t)∇u· ∇vdx for allv∈V.
We note that
f∈C([0,T];V)
B∈C([0,T];L(V)).
LetD= (γ(V))′be the dual of the space γ(V) ={w=v|Γv∈V}.For everyt∈[0,T] we
defineλ(t)∈Das follows
⟨λ(t),γw⟩=−∫
Γ3a(x,∇u(x,t))·ν(x)γw(x)dΓ
−∫
Γ3∫t
0θ(t−s)(x)∂νu(x,s)dsγw (x)dΓ for allγw∈γ(V),
where ⟨·,·⟩is the duality pairing between ( γ(V))′andγ(V).Also, we introduce Λ ⊂(γ(V))′as
follows,
Λ ={µ∈D|⟨µ,γw⟩ ≤∫
Γ3g(x)|γw(x)|dΓ for allγw∈γ(V)}. (4.6)
Next, we define
b:V×D→Rb(v,µ) =⟨µ,γv⟩. (4.7)
We have the following weak formulation of Problem 4.3.
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Problem 4.4. Givenf: [0,T]→V,findu: [0,T]→Vandλ: [0,T]→Dsuch that, for every
t∈[0,T],we haveλ(t)∈Λand
(Au(t),v)V+ (∫t
0B(t−s)u(s)ds,v)V+b(v,λ(t)) = (f(t),v)Vfor allv∈V
b(u(t),µ−λ(t))≤0 for allµ∈Λ.
Theorem 4.2. [Theorem 1 in [101]] If Assumptions 4.7-4.10are fulfilled, then Problem 4.4has
a unique solution (u,λ)with the regularity
u∈C([0,T];V), λ∈C([0,T];D).
Moreover, given f1, f2∈C([0,T];V),there exists C > 0such that
∥u1−u2∥C([0,T];V)+∥λ1−λ2∥C([0,T];D)≤C∥f1−f2∥C([0,T];V), (4.8)
(u1,λ1)and(u2,λ2)being the solutions of Problem 4.4corresponding to the data f1andf2,
respectively.
The proof of Theorem 4.2 is based on the abstract result, Theorem 4.1.
4.2 The case of viscoelasticity with short-memory term
This section is based on the paper [111]. We discuss herein an abstract mixed variational
problem which consists of a system of an evolutionary variational equation in a Hilbert space
Xand an evolutionary inequality in a subset of a second Hilbert space Y, associated with an
initial condition. The existence and the uniqueness of the solution is proved based on a fixed
point technique. The continuous dependence on the data was also investigated. The abstract
results we obtain can be applied to the mathematical treatment of a class of frictional contact
problems for viscoelastic materials with short memory. In this section we consider an antiplane
model for which we deliver a mixed variational formulation with friction bound dependent set of
Lagrange multipliers. After proving the existence and the uniqueness of the weak solution, we
study the continuous dependence on the initial data, on the densities of the volume forces and
surface tractions. Moreover, we prove the continuous dependence of the solution on the friction
bound.
4.2.1 An abstract result
LetTbe a positive real number. In this subsection we study the following abstract problem.
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Problem 4.5. Givenf: [0,T]→X, g∈Wandu0∈X,findu: [0,T]→Xandλ: [0,T]→
Λ(g)⊂Ysuch that for each t∈(0,T),we have
a( ˙u(t),v) +e(u(t),v) +b(v,λ(t)) = (f(t),v)X for allv∈X, (4.9)
b( ˙u(t),µ−λ(t))≤0 for allµ∈Λ(g), (4.10)
u(0) =u0. (4.11)
This is an evolutionary mixed variational problem with short-memory term .
Problem 4.5 was studied under the following assumptions.
Assumption 4.11. (X,(·,·)X,∥ · ∥ X),(Y,(·,·)Y,∥ · ∥ Y)and(W,(·,·)W,∥ · ∥ W)are three Hilbert
spaces.
Assumption 4.12. a(·,·) :X×X→Ris a symmetric bilinear form such that
(a1)there exists Ma>0 :|a(u,v)| ≤Ma∥u∥X∥v∥Xfor allu,v∈X;
(a2)there exists ma>0 :a(v,v)≥ma∥v∥2
Xfor allv∈X.
Assumption 4.13. e(·,·) :X×X→Ris a symmetric bilinear form such that
(e1)there exists Me>0 :|e(u,v)| ≤Me∥u∥X∥v∥Xfor allu,v∈X;
(e2)there exists me>0 :e(v,v)≥me∥v∥2
Xfor allv∈X.
Assumption 4.14. b(·,·) :X×Y→Ris a bilinear form such that
(b1)there exists Mb>0 :|b(v,µ)| ≤Mb∥v∥X∥µ∥Yfor allv∈X, µ∈Y;
(b2)there exists α>0 : inf µ∈Y,µ̸=0Ysupv∈X,v̸=0Xb(v,µ)
∥v∥X∥µ∥Y≥α.
Assumption 4.15. f∈C([0,T];X).
Assumption 4.16. For eachφ∈W,Λ(φ)is a closed convex subset of Ysuch that 0Y∈Λ(φ).
Assumption 4.17. If(ηn)n⊂Wand(wn)n⊂Xare two sequences such that ηn→ηinW
andwn⇀w inX,asn→ ∞,then:
(k1)for eachµ∈Λ(η)⊂Y,there exists a sequence (µn)n⊂Ysuch thatµn∈Λ(ηn)
for alln≥1,andlim sup
n→∞b(wn,µ−µn)≤0;
(k2)For each subsequence (Λ(ηn′))n′of the sequence (Λ(ηn))n,if(µn′)n′⊂Ysuch that
µn′∈Λ(ηn′)andµn′⇀µ inYasn′→ ∞,thenµ∈Λ(η).
Letη∈C([0,T];X) be given and let us consider the following intermediate problem.
Problem 4.6. Givenf: [0,T]→Xandg∈Wfindwη: [0,T]→Xandλη: [0,T]→Λ(g)⊂Y
so that, for each t∈[0,T],we have
a(wη(t),v) +e(η(t),v) +b(v,λη(t)) = (f(t),v)X for allv∈X, (4.12)
b(wη(t),µ−λη(t))≤0 for allµ∈Λ(g). (4.13)
66
The existence and the uniqueness of the solution of this problem is provided by the following
lemma.
Lemma 4.1. [Lemma 2 in [111]] Problem 4.6 has a unique solution with the regularity
wη∈C([0,T];X), λ η∈C([0,T];Y). (4.14)
Let us associate with Problem 4.6 the following functional.
Lt
η:X×Λ(g)→R,Lt
η(v, µ) =1
2a(v,v)−(fη(t),v)X+b(v,µ). (4.15)
It was proved that a pair ( wη(t),λη(t)) verifies (4.12) and (4.13) if and only if it is a solution
of the following saddle point problem.
Find (wη(t),λη(t))∈X×Λ(g) so that
Lt
η(wη(t), µ)≤ Lt
η(wη(t), λη(t))≤ Lt
η(v, λ η(t)) for all v∈X, µ∈Λ(g).
(4.16)
Following [50, 61], it was proved that the Problem 4.16 has a solution.
Let us consider the operator T:C([0,T];X)→C([0,T];X) defined as follows: for each
η∈C([0,T];X),
Tη(t) =∫t
0wη(s)ds+u0for allt∈[0,T]. (4.17)
Lemma 4.2. [Lemma 3 in [111]] The operator Thas a unique fixed point η∗∈C([0,T]; Λ(g)).
The main abstract result in this subsection is the following one.
Theorem 4.3. [Theorem 4 in [111]] If Assumptions 4.11-4.17hold true, then Problem 4.5has
a unique solution with the regularity
u∈C1([0,T];X), λ∈C([0,T]; Λ(g)).
The proof of Theorem 4.3, which can be found in [111], is based on the saddle point theory
and Banach’s fixed point theorem.
In addition to this theorem it is worth to mention the following three results.
Let us start with the following stability properties.
Proposition 4.1. [Proposition 5 in [111]] If Assumptions 4.11-4.17 hold true, then:
(p1)givenf∈C([0,T];X), g∈Wand two initial data u1
0, u2
0∈X, there exists c1>0such
that
∥u1−u2∥C1([0,T];X)≤c1∥u1
0−u2
0∥X, (4.18)
whereu1,u2are the corresponding solutions of Problem 4.5;
(p2)giveng∈W, f 1,f2∈Xand two initial data u1
0, u2
0∈X, there exists c2>0such that
∥u1−u2∥C1([0,T];X)≤c2(∥f1−f2∥X+∥u1
0−u2
0∥X), (4.19)
whereu1,u2are the corresponding solutions of Problem 4.5.
67
Next, we mention a boundedness property.
Proposition 4.2. [Proposition 6 in [111]] If Assumptions 4.11-4.17hold true, then there exist
two positive friction bound independent constants K1andK2such that
∥u∥C1([0,T];X)≤K1; (4.20)
∥λ∥C([0,T];Y)≤K2, (4.21)
where (u,λ)is the solution of Problem 4.5.
Finally, let us indicate some convergence properties.
Proposition 4.3. [Proposition 7 in [111]] If Assumptions 4.11-4.17hold true and (gn)n⊂W
is a sequence such that gn→ginWasn→ ∞,then for all t∈[0,T],
un(t)→u(t)inXasn→ ∞ ; (4.22)
˙un(t)→˙u(t)inXasn→ ∞ ; (4.23)
λn(t)→λ(t)inYasn→ ∞, (4.24)
where (u,λ)and(un,λn)denote the solutions of Problem 4.5associated with the data (f,g,u 0)∈
C([0,T];X)×W×Xand(f,gn,u0)∈C([0,T];X)×W×X, n≥1.
Propositions 4.1, 4.2 and 4.3 have been proved in [111].
4.2.2 A mechanical model and its weak solvability
In this subsection we discuss the weak solvability of the following model.
Problem 4.7. Find a displacement field u:¯Ω×[0,T]→Rsuch that, for all t∈(0,T),we have
div(θ(x)∇˙u(x,t) +µ(x)∇u(x,t)) +f0(x,t) = 0 in Ω , (4.25)
u(x,t) = 0 on Γ 1, (4.26)
θ(x)∂ν˙u(x,t) +µ(x)∂νu(x,t) =f2(x,t) on Γ 2, (4.27)
|θ(x)∂ν˙u(x,t) +µ(x)∂νu(x,t)| ≤g(x),
θ(x)∂ν˙u(x,t) +µ(x)∂νu(x,t) =−g(x)˙u(x,t)
|˙u(x,t)|if ˙u(x,t)̸= 0
on Γ 3, (4.28)
u(0) =u0in Ω. (4.29)
Herein [0,T] is the time interval and Ω ⊂R2is a bounded domain with Lipschitz continuous
boundary. The boundary will be denoted by Γ and will be partitioned in three measurable parts
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Γ1,Γ2,Γ3such that the Lebesgue measure of Γ 1is positive. Problem 4.7 models the antiplane
shear deformation of a viscoelastic, isotropic, nonhomogeneous cylindrical body in frictional
contact on Γ 3with a rigid foundation. Referring the body to a cartesian coordinate system
Ox 1x2x3such that the generators of the cylinder are parallel with the axis Ox 3, the domain
Ω⊂Ox 1x2denotes the cross section of the cylinder. The function θ=θ(x1,x2) :¯Ω→Ris the
viscoelastic coefficient, µ=µ(x1,x2) :¯Ω→Rdenotes a coefficient of the material (one of Lam´ e’s
coefficients ), the functions f0=f0(x1,x2,t) : Ω×(0,T)→R,f2=f2(x1,x2,t) : Γ 2×(0,T)→R
are related to the density of the volume forces and the density of the surface traction, respectively
andg: Γ3→R+is the friction bound, a given function. Here ν= (ν1,ν2) (νi=νi(x1,x2),for
eachi∈ {1,2}), represents the outward unit normal vector to the boundary Γ and ∂νu=∇u·ν.
The unknown of the problem is the function u=u(x1,x2,t) :¯Ω×[0,T]→Rwhich represents
the third component of the displacement vector u.
In the study of Problem 4.7 we assume that the elasticity and the viscosity coefficients fulfill
the following assumptions.
Assumption 4.18. µ∈L∞(Ω),and there exists µ∗>0such thatµ(x)≥µ∗a.e.x∈Ω.
Assumption 4.19. θ∈L∞(Ω),and there exists θ∗>0such thatθ(x)≥θ∗a.e.x∈Ω.
Assumption 4.20. f0∈C([0,T];L2(Ω)), f 2∈C([0,T];L2(Γ2)).
Assumption 4.21. g∈L2(Γ3)such thatg(x)≥0a.e.x∈Γ3.
Finally, we made the following assumption for the initial displacement.
Assumption 4.22. u0∈X.
Let us introduce the Hilbert space
X={v∈H1(Ω)|γv= 0 a.e. on Γ 1}. (4.30)
We define the bilinear forms a:X×X→Rande:X×X→Rby equalities
a(u,v) =∫
Ωθ∇u· ∇vdx for allu, v∈V; (4.31)
e(u,v) =∫
Ωµ∇u· ∇vdx for allu, v∈V. (4.32)
Lett∈[0,T].We definef(t)∈Xas follows
(f(t),v)X=∫
Ωf0(t)vdx+∫
Γ2f2(t)γvdΓ for allv∈X. (4.33)
We consider the space
S={ev=γv|Γ3v∈X} (4.34)
69
and we denote its dual by D.Also, we define a bilinear form b:V×D→Ras follows
b(v, ζ) =⟨ζ,γv|Γ3⟩, (4.35)
where ⟨·,·⟩denotes the duality pairing between the spaces DandS.
Next, for each φ∈L2(Γ3) we define Λ( φ) as follows:
Λ(φ) ={
ζ∈D:⟨ζ, γw|Γ3⟩ ≤∫
Γ3|φ(x)||γw(x)|dΓ for allw∈X}
. (4.36)
Let us define now a Lagrange multiplier λ,such that at each t∈[0,T], λ(t)∈Yand
⟨λ(t),z⟩=−∫
Γ3(θ∂ν˙u(t) +µ∂νu(t))zdΓ for allz∈S, (4.37)
whereSis defined in (4.34).
We delivered the following mixed variational formulation of Problem 4.7.
Problem 4.8. Findu: [0,T]→Xandλ: [0,T]→Λ(g)⊂Dsuch that, for all t∈(0,T),
a( ˙u(t), v) +e(u(t),v) +b(v,λ(t)) = (f(t),v)X for allv∈X,
b( ˙u(t), ζ−λ(t)) ≤0 for allζ∈Λ(g),
u(0) =u0.
Theorem 4.4. [Theorem 8 in [111]] If Assumptions 4.18–4.22hold true, then Problem 4.8has
a unique solution (u,λ)with the regularity
u∈C1([0,T];X), λ∈C([0,T]; Λ(g)).
The proof of Theorem 4.4 is based on the previous abstract result, Theorem 4.3, see [111]
for details. In addition, the following propositions hold true.
Proposition 4.4. [Proposition 9 in [111]] if Assumptions 4.18–4.22hold true, then:
(i1)givenf0∈C([0,T];L2(Ω)), f2∈C([0,T];L2(Γ2)), g∈L2(Γ3)and two initial data u1
0,
u2
0∈X,there exists c1>0such that
∥u1−u2∥C1([0,T];X)≤c1∥u1
0−u2
0∥X (4.38)
whereu1,u2are the corresponding solutions of Problem 4.8.
(i2)givenf1
0, f2
0∈L2(Ω), f1
2, f2
2∈L2(Γ2), g∈L2(Γ3)and two initial data u1
0, u2
0∈X,there
existsc2>0such that
∥u1−u2∥C1([0,T];X)≤c2(∥f1
0−f2
0∥L2(Ω)+∥f1
2−f2
2∥L2(Γ2)+∥u1
0−u2
0∥X) (4.39)
whereu1,u2are the corresponding solutions of Problem 4.8.
70
Besides, we have the following boundedness result.
Proposition 4.5. [Proposition 10 in [111]] If Assumptions 4.18–4.22hold true, then the solution
(u,λ)of Problem 4.8is bounded.
Finally, we have the continuous dependence of the weak solution on the friction bound.
Proposition 4.6. [Proposition 11 in [111]] If Assumptions 4.18–4.22hold true, then if (gn)n⊂
L2(Γ3)is a sequence of friction bounds such that gn≥0a.e. on Γ3,for alln≥1,andgn→g
inL2(Γ3)asn→ ∞,we have, for all t∈[0,T],
un(t)→u(t)inXasn→ ∞ ;
˙un(t)→˙u(t)inXasn→ ∞ ;
λn(t)→λ(t)inYasn→ ∞,
where (u,λ)and(un,λn)are solutions of Problem 4.8associated with the friction bounds gand
gn,for alln≥1.
The proof of these three propositions are based on the previous abstract results, Proposition
4.1, Proposition 4.2 and Proposition 4.3.
Chapter 5
Frictionless contact problems
This chapter is based on the papers [98, 70, 11]. Firstly, we focus on a mechanical model which
describes the frictionless unilateral contact between an electro-elastic body and a rigid electrically
nonconductive foundation. For this model, a mixed variational formulation is provided. Using
elements of the saddle point theory and a fixed point technique, an abstract result is proved.
Based on this abstract result, the existence of a unique weak solution of the mechanical problem
is established.
Next, we analyze the frictionless unilateral contact between an electro-elastic body and a
rigid electrically conductive foundation. On the potential contact zone, we use the Signorini
condition with non-zero gap and an electric contact condition with a conductivity depending on
the Cauchy vector. We provide a weak variationally consistent formulation and show existence,
uniqueness and stability of the solution. Our analysis is based on a fixed point theorem for
weakly sequentially continuous maps.
Finally, we consider a mathematical model which describes the frictionless contact between
a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic and
the contact is modeled with normal compliance and unilateral constraint. We provide a mixed
variational formulation of the model which involves dual Lagrange multipliers, then we prove its
unique weak solvability. We also prove an estimate which allows us to deduce the continuous
dependence of the weak solution with respect to both the normal compliance function and the
penetration bound.
5.1 The case of electro-elastic materials
In this section, based on the papers [98, 70], we discuss the weak solvability via dual Lagrange
multipliers of a class of electro-elastic contact models for linearly elastic materials.
71
72
5.1.1 The case of nonconductive foundation
In this subsection, devoted to the case of nonconductive foundation, we present results obtained
in the paper [98]. Let us start with an abstract auxiliary result.
Assumption 5.1. (X,(·,·)X,∥ · ∥ X)and(Y,(·,·)Y,∥ · ∥ Y)are two Hilbert spaces.
Assumption 5.2. a(·,·) :X×X→R,is a non-symmetric form such that
there exists Ma>0 :|a(u,v)| ≤Ma∥u∥X∥v∥Xfor allu,v∈X;
there exists ma>0such thata(v,v)≥ma∥v∥2
Xfor allv∈X.
Assumption 5.3. b(·,·) :X×Y→Ris a bilinear form such that
there exists Mb>0 :|b(v,µ)| ≤Mb∥v∥X∥µ∥Y,for allv∈X, µ∈Y,
there exists α>0such that inf
µ∈Y,µ̸=0Ysup
v∈X,v̸=0Xb(v,µ)
∥v∥X∥µ∥Y≥α.
Assumption 5.4. Λ⊂Yis a closed, convex set that contains 0Y.
Let us state the following abstract problem.
Problem 5.1. Givenf, g∈X,findu∈Xandλ∈Λsuch that
a(u,v) +b(v,λ) = (f, v)X, for allv∈X,
b(u,µ−λ) ≤b(g,µ−λ), for allµ∈Λ.
We emphasize that the bilinear form a(·,·) isnon-symmetric . Consequently, Problem 5.1
isnot a classical saddle point problem . Moreover, we are interested here in the case g̸= 0 X.
An analysis of the particular case g= 0 Xcan be found in [68]; see also Subsection 1.2.1 in the
present manuscript.
Refereing to Problem 5.1, the following theorem takes place.
Theorem 5.1. [Theorem 2 in [98]] Let f,g∈X.If Assumptions 5.1-5.4hold true, then there
exists a unique solution of Problem 5.1, (u,λ)∈X×Λ.Moreover, if (u1, λ1)and(u2, λ2)are
two solutions of Problem 5.1, corresponding to the data f1,g1∈Xandf2,g2∈X,then there
existsK=K(α,m a,Ma,Mb)>0such that
∥u1−u2∥X+∥λ1−λ2∥Y≤K(∥f1−f2∥X+∥g1−g2∥X).
We proceed with the analysis of a mechanical model. We consider an elasto-piezoelectric body
that occupies the bounded domain Ω ⊂R3,in contact with a rigid electrically nonconductive
foundation. We assume that the boundary Γ is partitioned into three disjoint measurable parts
Γ1, Γ2and Γ 3, such that meas(Γ 1)>0 and Γ3is a compact subset of Γ \Γ1. Let us denote by n3
the restriction of nto Γ 3.The body Ω is clamped on Γ 1, body forces of density f0act on Ω and
surface traction of density f2act on Γ 2.Moreover, we assume that Γ 3is the potential contact
73
zone and we denote by g: Γ3→Rthegap function . By gapin a given point of Γ 3we understand
the distance between the deformable body and the foundation measured along of the outward
normaln.Let us consider a second partition of the boundary Γ in two disjoint measurable parts
Γaand Γ bsuch that meas(Γ a)>0 and Γ b⊇Γ3. On Γ athe electrical potential vanishes and on
Γbwe assume electric charges of density q2.Since the foundation is electrically nonconductive,
and assuming that the gap zone is also electrically nonconductive, q2must vanish on Γ 3.Byq0
we will denote the density of the free electric charges on Ω .
Let us write the universal equilibrium equations
Divσ+f0=0in Ω, (5.1)
divD=q0in Ω, (5.2)
and the constitutive law,
σ=Cε(u) +E⊤∇φin Ω, (5.3)
D=Eε(u)−β∇φ in Ω, (5.4)
where C= (Cijls) is the elasticity tensor, E= (Eijl) is the piezoelectric tensor and βis the
permittivity tensor.
We prescribe the mechanical and the electrical boundary conditions, according to the physical
setting.
u=0on Γ 1, σn=f2on Γ 2, (5.5)
φ= 0 on Γ a, D·n=q2on Γ b. (5.6)
To model the contact process, we use the Signorini condition with non-zero gap. In addition,
we assume that the contact is frictionless. Consequently, we can express mathematically the
frictionless contact condition as follows,
στ=0, σ n≤0, u n≤g, σ n(un−g) = 0, on Γ 3. (5.7)
Knowing the displacement field uand the electric field φwe can compute the stress tensor σ
and the electric displacement Dusing (5.3) and (5.4), respectively. Therefore, the displacement
fielduand the electric field φare called the main unknowns.
To resume, we consider the following problem.
Problem 5.2. Find the displacement field u: Ω→R3and the electric potential field φ: Ω→R
such that (5.1)-(5.7)hold.
Assumption 5.5. C= (Cijls) : Ω×S3→S3,Cijls=Cijsl=Clsij∈L∞(Ω);
there exists mC>0such that Cijlsεijεls≥mC∥ε∥2,for allε∈S3,a.e. on Ω
74
Assumption 5.6. E= (Eijk) : Ω×S3→R3,Eijk=Eikj∈L∞(Ω).
Assumption 5.7. β= (βij) : Ω×R3→R3,βij=βji∈L∞(Ω);
there exists mβ>0such thatβijEiEj≥mβ∥E∥2,for allE∈R3,a.e. on Ω
Assumption 5.8. f0∈L2(Ω)3,f2∈L2(Γ2)3, q 0∈L2(Ω), q 2∈L2(Γb).
Assumption 5.9. There exists gext: Ω→Rsuch thatgext∈H1(Ω), gext= 0onΓ1, g ext≥0
onΓ\Γ1, g=gextonΓ3.
Assumption 5.10. The unit outward normal to Γ3denoted byn3is assumed to be constant.
Based on these assumptions, we present a mixed variational formulation of this mechanical
problem, using the Hilbert spaces,
V={v∈H1|v=0on Γ 1},
W={ψ∈H1(Ω)|ψ= 0 on Γ a},
˜V=V× W.
We consider the inner products ( ·,·)V:V×V→R,(·,·)W:W×W → Rand (·,·)˜V:˜VטV→R
defined as follows,
(u,v)V= (ε(u),ε(v))H,(φ,ψ)W= (∇φ,∇ψ)H
and
(˜u,˜v)˜V= (u,v)V+ (φ,ψ)W. (5.8)
Let us consider a:˜VטV→Rthe bilinear form,
a(˜u,˜v) =∫
ΩCε(u)·ε(v)dx+∫
ΩEε(v)· ∇φdx (5.9)
−∫
ΩEε(u)· ∇ψdx +∫
Ωβ∇φ· ∇ψdx
for all ˜u= (u,φ),˜v= (v,ψ)∈˜V.Also, we define ˜f∈˜Vsuch that for all ˜v= (v,ψ)∈˜V,
(˜f,˜v)˜V=∫
Ωf0·vdx+∫
Γ2f2·vda−∫
Γbq2ψda +∫
Ωq0ψdx. (5.10)
We define a dual Lagrange multiplier λ∈D=(
H1/2(Γ3)3)′
such that
⟨λ,v⟩Γ3=−∫
Γ3σnvnds, for allv∈H1/2(Γ3)3, (5.11)
75
whereH1/2(Γ3)3denotes the space of restrictions to Γ 3of the traces of all functions belonging
toVand⟨·,·⟩Γ3denotes the duality pairing between DandH1/2(Γ3)3.Moreover, we define a
bilinear form b:˜V×D→R,as follows
b(˜v,µ) =⟨µ,v⟩Γ3,for all ˜v= (v,ψ)∈˜V,µ∈D. (5.12)
Furthermore, we introduce a set as follows,
Λ ={
µ∈D:⟨µ,v⟩Γ3≤0 for allv∈K}
, (5.13)
where
K={v∈H1/2(Γ3)3:vn≤0 on Γ 3}.
We have the following weak formulation of Problem 5.2.
Problem 5.3. Find ˜u∈˜Vandλ∈Λ,such that
a(˜u,˜v) +b(˜v,λ) = ( ˜f,˜v)V, for all ˜v∈˜V,
b(˜u,µ−λ)≤b(˜gext,µ−λ), for allµ∈Λ.
Theorem 5.2. [Theorem 1 in [98]] If Assumptions 5.5-5.10hold true, then Problem 5.3 has a
unique solution (˜u,λ)∈˜V×Λ.Moreover, if (˜u,λ)and(˜u∗,λ∗)are two solutions of Problem
5.3 corresponding to the data (˜f,˜gext)∈˜VטVand(˜f∗,˜g∗
ext)∈˜VטV,respectively, then
∥˜u−˜u∗∥˜V+∥λ−λ∗∥D≤C(
∥˜f−˜f∗∥˜V+∥˜gext−˜g∗
ext∥˜V)
,
whereC=C(C,E,β,α,M b)>0.
The proof of Theorem 5.2 can be found in [98].
5.1.2 The case of conductive foundation
This subsection, based on the paper [70] is dedicated to the analysis of a new contact model
involving piezoelectric materials. We consider an elasto-piezoelectric body which occupies the
bounded domain Ω ⊂Rd, d∈ {2,3}with smooth enough boundary Γ .Also, we consider two
partitions of the boundary Γ .The first one is Γ 1, Γ 2and Γ 3such that meas(Γ 1)>0.The
second one is Γ a,Γband Γ 3such that meas(Γ a)>0.The partition Γ 1,Γ2and Γ 3applies to the
mechanical boundary conditions whereas the partition Γ a,Γband Γ 3to the electrical boundary
conditions. The body Ω is clamped on Γ 1, body forces of density f0act on Ω, and a surface
traction of density f2acts on Γ 2.
Moreover, we assume that on Γ 3the body can be in contact with a rigid electrically conductive
foundation. We denote the gap by g.On Γ athe electrical potential vanishes, and on Γ bwe assume
electric charges of density qb.Byq0we denote the density of the free electric charges on Ω .
76
The model under consideration is obtained from the equilibrium equations
Divσ+f0=0in Ω, (5.14)
divD=q0in Ω, (5.15)
the constitutive laws
σ=Cε(u) +E⊤∇φin Ω, (5.16)
D=Eε(u)−β∇φ in Ω, (5.17)
the mechanical and the electrical boundary conditions
u=0on Γ 1,σν=f2on Γ 2, (5.18)
φ= 0 on Γ a,D·ν=qbon Γ b, (5.19)
στ=0, σ ν≤0, u ν≤g, σ ν(uν−g) = 0 on Γ 3, (5.20)
D·ν=−k(σν)(φ−φ0), (5.21)
whereg: Γ3→R+is the gap function and −k(σν)≥0 is the conductivity.
The electric contact condition on Γ 3is described by a nonlinear Robin type condition for φ
which couples the mechanical stress with the electrical field.
The primary variables are the displacement field uand the electric field φ; the stress tensor
σand the electric displacement Dcan be computed from uandφby the constitutive relations
(5.16) and (5.17).
To resume, we consider the following problem.
Problem 5.4. Find the displacement field u:¯Ω→Rdand the electric potential field φ:¯Ω→R
such that (5.14)-(5.21)hold.
Assumption 5.11. (Elasticity tensor)
•C= (Cijls) : Ω×Sd→Sd,
•Cijls=Cijsl=Clsij∈L∞(Ω),
•There exists mC>0such that Cijlsεijεls≥mC∥ε∥2,ε∈Sd,a.e. in Ω.
Assumption 5.12. (Piezoelectric tensor)
•E= (Eijk) : Ω×Sd→Rd,
•Eijk=Eikj∈L∞(Ω).
Assumption 5.13. (Permittivity tensor)
77
•β= (βij) : Ω×Rd→Rd, β ij=βji∈L∞(Ω),
•There exists mβ>0such thatβijEiEj≥mβ∥E∥2, E∈Rd,a.e. in Ω.
Concerning the mechanical and the electrical data we assume
Assumption 5.14. f0∈L2(Ω)d,f2∈L2(Γ2)d, q 0∈L2(Ω), q b∈L2(Γb).
To simplify the presentation, we assume that φ0= 0 andg= 0.The general situation can be
transferred to this case by a transformation u−gext,→uandφ−φext,→φ, wheregext∈H1(Ω)
andφext∈H1(Ω) are extensions of the data gνandφ0.
We are interested in a variationally consistent formulation of this mechanical problem using
Lagrange multipliers.
The set of admissible functions for the displacement field is
X={v∈H1(Ω)d|γv=0a.e. on Γ 1}.
For the electric potential, we have the admissible set
Y={ψ∈H1(Ω)|γψ= 0 a.e. on Γ a}.
The restriction of νto Γ 3is denoted by ν3, and we restrict ourselves to geometries such that
the following assumption is satisfied.
Assumption 5.15. The vectorν3is constant on Γ3.
Let us introduce the space
S={w|w=vν=γv| 3·ν3,v∈X}.
The dual space of Sis denoted by Zand⟨·,·⟩stands for the duality paring between Zand
S.
Let us define the bilinear forms,
a:X×X→Ra(u,v) =∫
ΩCε(u)·ε(v)dx,u,v∈X, (5.22)
e:X×Y→Re(v,ψ) =∫
ΩEε(v)· ∇ψdx,v∈X,ψ∈Y, (5.23)
c:Y×Y→Rc(φ,ψ) =∫
Ωβ∇φ· ∇ψdx, φ,ψ ∈Y. (5.24)
Moreover, we define f∈Xandq∈Ysuch that
(f,v)X=∫
Ωf0·vdx+∫
Γ2f2·γvdΓ,v∈X, (q,ψ)Y=∫
Ωq0ψdx−∫
ΓbqbγψdΓ, ψ∈Y.
78
In order to obtain a well defined formulation, the conductivity operator has to have suitable
properties. Moreover, the proof of possible existence and uniqueness results depends crucially
on the properties of the conductivity operator.
We recall that f:EC→ECis called a weakly sequentially continuous map if, for all
sequences ( xn)n⊂EC, such that xn⇀x inEthenf(xn)⇀f(x) inE.LetXbe a real reflexive
Banach space, then an operator A:X→X′is called completely continuous if, for all sequences
(un)n⊂Xsuch thatun⇀u inXthenAun→AuinX′.We also recall the following embedding
results.
Lemma 5.1. LetΩ⊂Rdbe a bounded domain with Lipschitz continuous boundary ∂Ω, d≥1.
(i)If1< p < d then for 1≤q <(d−1)p
d−p, the operator γ:W1,p(Ω)→Lq(∂Ω)is completely
continuous.
(ii)Ifp≥dthen for any q∈[1,∞), the operator γ:W1,p(Ω)→Lq(∂Ω)is completely
continuous.
For a proof of this result, we refer the reader to [87].
We setK(µ) =−k(−µ) and require the following assumption to be true.
Assumption 5.16. (Conductivity operator I)
•K:Z→Ld−1+ϵ(Γ3), forϵ>0fixed;
•For eachµ∈Z, K (µ)≥0;
•If(µn)n⊂Zandµ∈Zsuch thatµn⇀ µ inZasn→ ∞,thenK(µn)⇀ K (µ)in
Ld−1+ϵ(Γ3)asn→ ∞.
Example 5.1. LetR:Z→Ld−1+ϵ(Γ3)be a linear continuous map and k∗>0. Then, we set
K(µ) =k∗|Rµ|. (5.25)
Example 5.2. Letφx(y) =φ(y−x)withφbeing a mollifier such that φ∈C∞
c(Rd), φ(x) =
φ(−x), φ≥0,∫
Rdφ(x)dx= 1,suppφis compact. Then, we define
K(µ)(x) =k∗|⟨µ,φx⟩|
1 +γ|⟨µ,φx⟩|, k∗>0,γ≥0. (5.26)
Now we define a functional j:Z×Y×Y→Rand a bilinear form b:X×Z→Rby
j(µ,φ,ψ ) =∫
Γ3K(µ)γφγψd Γµ∈Z,φ,ψ ∈Y, b (v, µ) =⟨µ,vν⟩,v∈X, µ∈Z
(5.27)
and note that both are well-defined under our assumptions. Introducing the dual cone Λ
Λ ={
µ∈Z:⟨µ, v⟩ ≥0v∈S,v≥0} (5.28)
and the Lagrange multiplier λ=−σν|Γ3, the weak formulation of Problem 5.4 is the following
one.
79
Problem 5.5. Find (u,φ,λ )∈X×Y×Λsuch that
a(u,v) +e(v,φ) +b(v,λ) = (f,v)X,v∈X,
c(φ,ψ)−e(u,ψ) +j(λ,φ,ψ ) = (q,ψ)Y,ψ∈Y,
b(u,µ−λ)≤0, µ∈Λ.
Notice that the spaces ( X,(·,·)X,∥·∥X),(Y,(·,·)Y,∥·∥Y) and (Z,(·,·)Z,∥·∥Z) are Hilbert
spaces and Λ is a closed, convex cone. The form a(·,·) :X×X→Ris a symmetric bilinear
form such that
•(a1) there exists Ma>0 :|a(u,v)| ≤Ma∥u∥X∥v∥X,u,v∈X;
•(a2)a(v,v)≥mC∥v∥2
X,v∈X.
We can take Ma=d∥C∥∞where ∥C∥∞= max i,j,k,l∥Cijkl∥L∞(Ω).
Also, it is worth to mention that c(·,·) :Y×Y→Ris a symmetric bilinear form such that
•(c1) there exists Mc>0 :|c(u,v)| ≤Mc∥u∥Y∥v∥Y,u,v∈Y;
•(c2)c(v,v)≥mβ∥v∥2
Y,v∈Y.
We can take Mc=d∥β∥∞where ∥β∥∞= max i,j∥βij∥L∞(Ω).
Moreover, the form b:X×Z→Ris a bilinear form such that
•(b1) there exists Mb>0 :|b(v, µ)| ≤Mb∥v∥X∥µ∥Zv∈X,µ∈Z;
•(b2) there exists α>0 : inf
µ∈Z,µ̸=0sup
v∈X,v̸=0b(v,µ)
∥v∥X∥µ∥Z≥α.
Next,e:X×Y→Ris a bilinear form and there exists Me>0 such that |e(v, φ)| ≤
Me∥v∥X∥φ∥Yfor allv∈X,φ∈Y.We can take Me=d∥E∥∞where ∥E∥∞= max i,j,k∥Eijk∥L∞(Ω).
The functional j(·,·,·) verifies the following properties:
•(j1) for each fixed ζ∈Z,j(ζ,·,·) is a continuous bilinear form on Y×Y;
•(j2)j(µ,ψ,ψ )≥0µ∈Z,ψ∈Y;
•(j3) if (ζn)n⊂Z,ζ n⇀ζ inZasn→ ∞ and (φn)n⊂Y, φ n⇀φ, inYasn→ ∞ then:
j(ζn,φn,ψ)→j(ζ,φ,ψ ) asn→ ∞ for allψ∈Y,andj(ζn,φn,φn)→j(ζ,φ,φ ) asn→
∞.
Theorem 5.3. (An existence result)[Theorem 3.1 in [70]] If Assumptions 5.11-5.16hold true,
then Problem 5.5has at least one solution, (u,φ,λ )∈X×Y×Λ.
The proof of Theorem 5.3 was given in [70]. The key of the proof is the following fixed point
result.
80
Lemma 5.2. LetEbe a metrizable locally convex topological vector space and let ECbe a weakly
compact convex subset of E.Then, any weakly sequentially continuous map f:EC→EChas a
fix point.
For the proof of Lemma 5.2, we refer to [5].
To obtain uniqueness of a solution we have to make one more assumption on the conductivity.
Assumption 5.17. (Conductivity operator II)
∥K(µ1)−K(µ2)∥Ld−1+ϵ(Γ3)≤LK∥µ1−µ2∥Z, µ 1,µ2∈ZandLK<∞fixed.
Taking into consideration Assumption 5.17, in addition to ( j1)-(j3),the functional jhas the
following property: for each pair ( φ,ψ)∈Y×Y,there exists L>0 such that
|j(ζ1,φ,ψ )−j(ζ2,φ,ψ )| ≤L∥ζ1−ζ2∥Z∥φ∥Y∥ψ∥Y, ζ 1,ζ2∈Z, (5.29)
whereL=c2
0LKandc0>0 is the continuity constant of the trace operator γassociated with
Lemma 5.1 and p= 2,q= 2(d−1 +ϵ)/(d−2 +ϵ).
Theorem 5.4. (A uniqueness result)[Theorem 3.6 in [70]] Let Assumptions 5.11-5.16and As-
sumption 5.17 be true. Additionally, we assume that
mC−c4
0L2
KBM2
a
mβα2>0andmβ
2−c4
0L2
KBM2
e
mβα2>0. (5.30)
Then Problem 5.5 has a unique solution.
The third result is the following one.
Theorem 5.5. (A stability result)[Theorem 3.7 in [70]] If Assumptions 5.11-5.17and the hy-
pothesis (5.30)hold true, then there exists S>0such that
∥u1−u2∥X+∥φ1−φ2∥Y+∥λ1−λ2∥Z≤ S(∥f1−f2∥X+∥q1−q2∥Y),
where (u1,φ1,λ1)and(u2,φ2,λ2)are the solutions of Problem 5.5 corresponding to the data
(f1,q1)∈X×Yand(f2,q2)∈X×Y,respectively.
The proofs of Theorems 5.4 and 5.5 can be found in [70]. See also [70] for a numerical
example.
81
5.2 The case of viscoplastic materials
This section is based on the paper [11]. Here, we consider a frictionless contact problem with
normal compliance and unilateral constraint and we investigate the behavior of the weak so-
lution with respect to the normal compliance function and the penetration bound. After the
description of the contact problem, we derive a new variational formulation which involves a
dual Lagrange multiplier. Then we provide the unique weak solvability of the problem, which
represents the first trait of novelty. The second trait of novelty consists in the fact that we prove
the continuous dependence of the weak solution with respect to the normal compliance function
and the penetration bound.
5.2.1 The model and its weak solvability
We consider a viscoplastic body that occupies the bounded domain Ω ⊂Rd(d= 1,2,3), with
the boundary ∂Ω = Γ partitioned into three disjoint measurable parts Γ 1, Γ2and Γ 3, such that
meas (Γ1)>0. We assume that the boundary Γ is Lipschitz continuous and we denote by νits
unit outward normal, defined almost everywhere. Let T > 0 and let [0 ,T] be the time interval.
The body is clamped on Γ 1×(0,T) and therefore the displacement field vanishes there. A
volume force of density f0acts in Ω ×(0,T), surface tractions of density f2act on Γ 2×(0,T)
and, finally, we assume that the body is in contact with a deformable foundation on Γ 3×(0,T).
The contact is frictionless and we model it with a normal compliance and unilateral constraint
condition.
Then, the classical formulation of the contact problem is the following.
Problem 5.6. Find a displacement field u: Ω×[0,T]→Rdand a stress field σ: Ω×[0,T]→Sd
such that
˙σ=Eε(˙u) +G(σ,ε(u)) in Ω ×(0,T), (5.31)
Divσ+f0=0 in Ω ×(0,T), (5.32)
u=0 on Γ 1×(0,T), (5.33)
σν=f2 on Γ 2×(0,T), (5.34)
uν≤g, σ ν+p(uν)≤0,
(uν−g)(σν+p(uν)) = 0
on Γ 3×(0,T), (5.35)
στ=0 on Γ 3×(0,T), (5.36)
u(0) =u0,σ(0) =σ0 in Ω. (5.37)
82
Equation (5.31) represents the viscoplastic constitutive law of the material. Equation (5.32)
is the equilibrium equation and we use it here since the process is assumed to be quasistatic.
Conditions (5.33) and (5.34) are the displacement and traction boundary conditions, respectively,
and condition (5.35) represents the normal compliance condition with unilateral constraint,
introduced in [78]. Recall that here g≥0 is a given bound for the penetration and prepresents
a given normal compliance function. Condition (5.36) shows that the tangential stress on the
contact surface, denoted στ, vanishes. We use it here since we assume that the contact process
is frictionless. Finally, (5.37) represents the initial conditions in which u0andσ0denote the
initial displacement and the initial stress field, respectively.
In the study of the mechanical problem (5.31)–(5.37) we made the following assumptions.
Assumption 5.18. E= (Eijkl) : Ω×Sd→Sd;
Eijkl=Eklij=Ejikl∈L∞(Ω),1≤i,j,k,l ≤d;
There exists mE>0 such that Eτ·τ≥mE∥τ∥2for allτ∈Sd,a.e.in Ω.
Assumption 5.19. G: Ω×Sd×Sd→Sd;
There exists LG>0 such that ∥G(x,σ1,ε1)− G(x,σ2,ε2)∥ ≤LG(∥σ1−σ2∥+∥ε1−ε2∥)
for allσ1,σ2,ε1,ε2∈Sd,a.e.x∈Ω.
The mapping x7→ G(x,σ,ε) is measurable on Ω ,for allσ,ε∈Sd.
The mapping x7→ G(x,0,0) belongs to Q.
Assumption 5.20. p:R→R+such that:
there exists Lp>0|p(r1)−p(r2)| ≤Lp|r1−r2|for allr1, r2∈R;
(p(r1)−p(r2))(r1−r2)≥0 for allr1, r2∈R;
p(r) = 0 for allr<0.
Assumption 5.21. f0∈C([0,T];L2(Ω)d),f2∈C([0,T];L2(Γ2)d).
Assumption 5.22. u0∈V,σ0∈Q.
Assumption 5.23. There existseθ∈Vsuch thateθ·ν= 1almost everywhere on Γ3.
We consider the space
V={v∈H1(Ω)d:v=0on Γ 1}.
We also consider the Hilbert space
S={w=v|Γ3v∈V},
wherev|Γ3denotes the restriction of the trace of the element v∈Vto Γ 3. Thus,S⊂H1/2(Γ3;Rd)
whereH1/2(Γ3;Rd) denotes the space of the restrictions on Γ 3of traces on Γ of functions of
H1(Ω)d. The dual of the space Swill be denoted by Dand the duality paring between Dand
83
Swill be denoted by ⟨·,·⟩Γ3. For more details on trace operators and trace spaces we refer to
[1, 92], for instance.
We define the operators L:V→V,P:V→Vand the function f: [0,T]→Vby equalities
(Lu,v)V=∫
ΩEε(u)·ε(v)dx, (5.38)
(Pu,v)V=∫
Γ3p(uν)vνda, (5.39)
(f(t),v)V=∫
Ωf0(t)·vdx+∫
Γ2f2(t)·vda (5.40)
for allu,v∈Vandt∈[0,T]. Also, let b:V×D→Rdenote the bilinear form defined by
b(v,µ) =⟨µ,v⟩Γ3 (5.41)
for allv∈Vandµ∈Dand consider the sets
K={v∈V:vν≤0 a.e. on Γ 3}, (5.42)
Λ ={µ∈D:⟨µ,v⟩Γ3≤0 for allv∈K}. (5.43)
Notice that
f∈C([0,T];V). (5.44)
Also, it is worth to mention that the bilinear form b(·,·) is continuous and satisfies the “inf-sup”
condition, i.e. there exists α>0 such that
inf
µ∈D,µ̸=0Dsup
v∈V,v̸=0Vb(v,µ)
∥v∥V∥µ∥D≥α. (5.45)
Denote byβ(t) andλ(t) the viscoplastic stress and the Lagrange multiplier given by
β(t) =σ(t)− Eε(u(t)), (5.46)
⟨λ(t),v⟩Γ3=−∫
Γ3(σν(t) +p(uν(t)))vνda for allv∈V. (5.47)
The weak formulation of the model is the following one.
Problem 5.7. Find a displacement field u: [0,T]→V, a viscoplastic stress field β: [0,T]→Q
and a Lagrange multiplier λ: [0,T]→Λsuch that, for all t∈[0,T],
(Lu(t),v)V+ (β(t),ε(v))Q+ (Pu(t),v)V (5.48)
+b(v,λ(t)) = (f(t),v)Vfor allv∈V,
b(u(t),µ−λ(t))≤b(geθ,µ−λ(t)) for allµ∈Λ, (5.49)
β(t) =∫t
0G(Eε(u(s)) +β(s),ε(u(s)))ds+σ0− Eε(u0). (5.50)
84
Letηbe an arbitrary element of the space C([0,T];V) and consider the following auxiliary
problem.
Problem P1
η.Find a displacement field uη: [0,T]→Vand a Lagrange multiplier λη: [0,T]→
Λsuch that, for all t∈[0,T],
(Luη(t),v)V+ (Puη(t),v)V+b(v,λη(t)) (5.51)
= (f(t)−η(t),v)Vfor allv∈V,
b(uη(t),µ−λη(t))≤b(geθ,µ−λη(t)) for allµ∈Λ. (5.52)
In the study of Problem P1
ηwe have the following result.
Lemma 5.3. [Lemma 4.1 in [11]] There exists a unique solution (uη,λη)of Problem P1
ηwhich
satisfies
uη∈C([0,T];V),λη∈C([0,T]; Λ). (5.53)
Moreover, if (ui,λi)represents the solution of Problem P1
ηforη=ηi∈C([0,T];V),i= 1,2,
then there exists c>0such that
∥u1(t)−u2(t)∥V+∥λ1(t)−λ2(t)∥D≤c∥η1(t)−η2(t)∥Vfor allt∈[0,T]. (5.54)
In the next step we construct the following auxiliary problem for the viscoplastic stress field.
Problem P2
η.Find a viscoplastic stress field βη: [0,T]→Qsuch that
βη(t) =∫t
0G(Eε(uη(s)) +βη(s),ε(uη(s)))ds+σ0− Eε(u0) (5.55)
for allt∈[0,T].
In the study of this problem we have the following result.
Lemma 5.4. [Lemma 4.2 in [11]] There exists a unique solution of Problem P2
ηwhich satisfies
βη∈C([0,T];Q). (5.56)
Moreover, ifβirepresents the solution of Problem P2
ηiforη=ηi∈C([0,T];V),i= 1,2, then
there exists c>0such that
∥β1(t)−β2(t)∥Q≤c∫t
0∥η1(s)−η2(s)∥Vds for allt∈[0,T]. (5.57)
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We now introduce the operator Θ : C([0,T];V)→C([0,T];V) which maps every element
η∈C([0,T];V) to the element Θ η∈C([0,T];V) defined as follows: for each η∈C([0,T];V)
and for each moment t∈[0,T], Θη(t) is the unique element in Vwhich satisfies the equality
(Θη(t),v)V= (βη(t),ε(v))Q for allv∈V; (5.58)
hereβηrepresents the viscoplastic stress obtained in Lemma 5.4.
We proceed with the following property of the operator Θ.
Lemma 5.5. The operator Θhas a unique fixed point η∗∈C([0,T];V).
The unique solvability of Problem 5.7 is given by the following result.
Theorem 5.6. [Theorem 3.1 in [11]] If Assumptions 5.18-5.23hold true, then Problem 5.7has
a unique solution (u,β,λ)which satisfies
u∈C([0,T];V),β∈C([0,T];Q),λ∈C([0,T]; Λ). (5.59)
The proof of Theorem 5.6 was given in [11].
A triple of functions ( u,β,λ) which satisfies (5.48)–(5.50) is called a weak solution of Problem
5.6. We conclude that, under Assumptions 5 .18-5.23, Problem 5.6 has a unique weak solution
with regularity (5.59). Moreover, we note that, once the weak solution is know, then the stress
fieldσcan be easily computed by using equality (5.46). And, using standard arguments, it can
be shown that σ∈C([0,T];Q1).
5.2.2 A convergence result
In this subsection we discuss the behavior of the solution with respect to a perturbation of the
normal compliance function pand the bound g. To this end, we assume in what follows that
Assumptions 5 .18–5.23 hold and we denote by ( u,β,λ) the solution of Problem 5.7. Also, for
eachρ>0 letgρ≥0 and consider a function pρwhich satisfies
Assumption 5.24. pρ:R→R+such that
There exists Lρ
p>0 :|pρ(r1)−pρ(r2)| ≤Lρ
p|r1−r2|for allr1, r2∈R;
(pρ(r1)−pρ(r2))(r1−r2)≥0 for allr1, r2∈R.
pρ(r) = 0 for allr<0.
We define the operator Pρ:V→Vby equality
(Pρu,v)V=∫
Γ3pρ(uν)vνdΓ for allu,v∈V. (5.60)
Then, we consider the following perturbation of the variational problem PV.
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Problem 5.8. Find a displacement field uρ: [0,T]→V, a viscoplastic stress field βρ: [0,T]→
Qand a Lagrange multiplier λρ: [0,T]→Λsuch that, for all t∈[0,T],
(Luρ(t),v)V+ (βρ(t),ε(v))Q+ (Pρuρ(t),v)V (5.61)
+b(v,λρ(t)) = (f(t),v)Vfor allv∈V,
b(uρ(t),µ−λρ(t))≤b(gρeθ,µ−λρ(t)) for allµ∈Λ, (5.62)
βρ(t) =∫t
0G(Eε(uρ(s)) +βρ(s),ε(uρ(s)))ds+σ0− Eε(u0). (5.63)
It follows from Theorem 5.6 that Problem 5.8 has a unique solution ( uρ,βρ,λρ) with the
regularity expressed in (5.59). Consider now the following assumption on the normal compliance
functionspρandp.
Assumption 5.25. There exists G:R+→R+such that |pρ(r)−p(r)| ≤G(ρ)(|r|+ 1) for all
r∈Randρ>0.
Then, we have the following estimate, which represents the main result in this subsection.
Theorem 5.7. [Theorem 5.1 in [11]] If Assumptions 5.18–5.25hold true, then there exists c>0
which depends on Ω,Γ1,Γ3,E,G,f0,f2,g,p,u0,σ0andT, but does not depend on ρ, such
that
∥uρ−u∥C([0,T];V)+∥βρ−β∥C([0,T];Q)+∥λρ−λ∥C([0,T];D) (5.64)
≤c(G(ρ) + 1)[
(G(ρ) + 1)|gρ−g|+G(ρ)]
.
Corollary 5.1. [Corollary 5.2 in [11]] If Assumptions 5.18–5.25hold true, and moreover, assume
that
gρ→g, G (ρ)→0 asρ→0, (5.65)
then the solution (uρ,λρ,βρ)of Problem 5.8 converges to the solution (u,λ,β)of Problem 5.7,
i.e.
uρ→uinC([0,T];V),βρ→βinC([0,T];Q),λρ→λinC([0,T];D),
asρ→0.
The proofs of Theorem 5.7 and Corollary 5.1 were given in [11]. In addition to the mathe-
matical interest, the convergence result in Corollary 5.1 is important from the mechanical point
of view, since it shows that the weak solution of the viscoplastic contact problem Pdepends
continuously on both the normal compliance function and the penetration bound.
Remark 5.1. In [11] it was provided a numerical validation of this convergence result.
Chapter 6
Contact problems involving
multi-contact zones
This chapter is based on the papers [110, 113]. We are interested on the weak solvability of a class
of contact models for elastic materials. Every model we propose is mathematically described by
a boundary value problem which consists of a system of partial differential equations associated
with four boundary conditions (the boundary being partitioned in four parts): a displacement
condition, a traction condition and two contact conditions. The weak solvability of the boundary
value problems we propose herein relies on new abstract results in the study of some generalized
saddle point problems.
6.1 The case of linear elastic operators
This section presents the results we have got in the paper [110]. In this section we firstly prove
abstract existence, uniqueness and boundedness results as well as abstract convergence results.
Next, we discuss the existence, the uniqueness, the boundedness and the approximation of the
weak solutions based on the abstract results.
6.1.1 Abstract results
Let (X,(·,·)X,∥ · ∥ X) and (Y,(·,·)Y,∥ · ∥ Y) be two real Hilbert spaces and Λ ⊂Y. We consider
the following problem.
Problem 6.1. Givenf,h∈X,find(u,λ)∈X×Ysuch thatλ∈Λand
a(u,v−u) +j(v)−j(u) +b(v−u,λ)≥(f, v−u)X for allv∈X, (6.1)
b(u,µ−λ) ≤b(h,µ−λ) for allµ∈Λ. (6.2)
Assumption 6.1. a(·,·) :X×X→Ris a symmetric bilinear form such that
87
88
•(i1)there exists Ma>0 :|a(u,v)| ≤Ma∥u∥X∥v∥Xfor allu,v∈X;
•(i2)there exists ma>0 :a(v,v)≥ma∥v∥2
X for allv∈X.
Assumption 6.2. The functional j:X→Ris convex. In addition, there exists Lj>0such
that
|j(v)−j(u)| ≤Lj∥v−u∥X for allu,v∈X.
Assumption 6.3. b(·,·) :X×Y→Ris a bilinear form such that
•(j1)there exists Mb>0 :|b(v,µ)| ≤Mb∥v∥X∥µ∥Yfor allv∈X, µ∈Y;
•(j2)there exists α>0 : inf
µ∈Y,µ̸=0Ysup
v∈X,v̸=0Xb(v,µ)
∥v∥X∥µ∥Y≥α.
Assumption 6.4. Λis a closed convex subset of Ysuch that 0Y∈Λ.
We can associate to Problem 6.1 the following functional:
L:X×Λ→R,L(v,µ) =1
2a(v,v) +j(v) +b(v−h,µ)−(f,v)X. (6.3)
According to the saddle point theory in [50], this functional Ladmits at least one saddle point
(u,λ)∈X×Λ.
Theorem 6.1 (An existence and uniqueness result) .[Theorem 2 in [110]] If Assumptions 6.1-6.4
hold true, then Problem 6.1has at least one solution, unique in the first argument.
Proposition 6.1. [A boundedness result][Proposition 2 in [110]] Assumptions 6.1-6.4hold true.
If(u,λ)∈X×Λis a solution of Problem 6.1, then there exist K1, K 2>0such that
∥u∥X≤K1;∥λ∥Y≤K2. (6.4)
Settingh= 0X,then Problem 6.1 leads us to the following semi-homogeneous problem.
Problem 6.2. Givenf∈X,find(u,λ)∈X×Ysuch thatλ∈Λand
a(u,v−u) +j(v)−j(u) +b(v−u,λ)≥(f, v−u)X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ.
Corollary 6.1. [Corollary 1 in [110]] If Assumptions 6.1-6.4hold true, then Problem 6.2has
at least one solution, unique in the first argument. In addition,
∥u∥X≤1
ma(∥f∥X+Lj);
∥λ∥Y≤1
α(
1 +Ma
ma)
(∥f∥X+Lj).
89
The proofs of Theorem 6.1, Proposition 6.1 and Corollary 6.1 were delivered in [110].
Letρbe a real positive number and jρ:X→Rbe a functional which fulfills the following
assumption.
Assumption 6.5. The functional jρ:X→Ris convex. In addition,
•there exists a positive real number L,which is independent of ρ,such that
|jρ(v)−jρ(u)| ≤L∥v−u∥Xfor allu,v∈X.
•jρis a Gˆ ateaux differentiable functional.
We denote by ∇jρthe Gˆ ateaux differential of jρ.
Assumption 6.6.
•There exists L∇jρ>0such that
∥∇jρ(v)− ∇jρ(w)∥X≤L∇jρ∥v−w∥Xfor allv,w∈X.
•There exists m∇jρ>0such that
(∇jρ(v)− ∇jρ(w),v−w)X≥m∇jρ∥v−w∥2
Xfor allv,w∈X.
Let us state the following regularized problem.
Problem 6.3. Givenρ>0andf,h∈X,find(uρ,λρ)∈X×Ysuch thatλρ∈Λ⊂Yand, for
allv∈X, µ∈Λ,
a(uρ,v−uρ) +jρ(v)−jρ(uρ) +b(v−uρ,λρ)≥(f, v−uρ)X (6.5)
b(uρ,µ−λρ) ≤b(h,µ−λρ). (6.6)
Lemma 6.1. [Lemma 2 in [110]] A pair (uρ,λρ)∈X×Λverifies (6.5)if and only if it verifies
a(uρ,v) + (∇jρ(uρ),v)X+b(v,λρ) = (f,v)X for allv∈X. (6.7)
Let us introduce the following notation:
M=∥f∥2
X+L2+maMb∥h∥X(∥f∥X+L)
α+M2
b∥h∥2
XM2
a
α2.
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Proposition 6.2. [Proposition 3 in [110]] Let ρ >0.If Assumption 6.1,Assumptions 6.3-6.6
hold true, then Problem 6.3has a unique solution (uρ,λρ).Moreover,
∥uρ∥X≤2M1/2
ma;
∥λρ∥Y≤∥f∥X+L+2MaM1/2
ma
α.(6.8)
Corollary 6.2. [Corollary 2 in [110]] Let (uρ,λρ)ρbe a sequence of solutions corresponding to a
sequence of regularized problems. There exists (uρ′,λρ′)ρ′a subsequence of the sequence (uρ,λρ)ρ,
and˜u∈X,˜λ∈Λsuch that,
uρ′⇀˜uinXasρ′→0
and
λρ′⇀˜λinYasρ′→0.
Assumption 6.7. There exists F:R+→R+such that:
•F(ρ)→0asρ→0;
•for eachρ>0,|jρ(v)−j(v)| ≤ F (ρ)for allv∈X.
Lemma 6.2. [Lemma 3 in [110]] Let uρbe the first component of the unique pair solution of
Problem 6.3. Then
uρ→uinXasρ→0,
whereuis the unique first component of a pair solution of Problem 6.1.
Corollary 6.3 (Corollary 3 in [110]) .The whole sequence (uρ)ρconverges strongly to ˜u=u.
Lemma 6.3. [Lemma 4 in [110]] Let (uρ,λρ)ρbe a sequence of solutions of a sequence of regu-
larized problems. Then
j(uρ)→j(u)asρ→0; (6.9)
jρ(uρ)→j(u)asρ→0. (6.10)
Proposition 6.3. [Proposition 4 in [110]] The pair (˜u,˜λ)is a solution of Problem 6.1.
For the proofs we send to [110].
Remark 6.1. We can compute the unique first component of a pair solution of Problem 6.1 by
computing the strong limit of the sequence (uρ)ρ.
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6.1.2 3D contact models
In this subsection we discuss two contact models. The first model is mathematically described
as follows.
Problem 6.4. Findu:¯Ω→R3andσ:¯Ω→S3such that
Divσ+f0=0 inΩ, (6.11)
σ=Eε(u) inΩ, (6.12)
u=0 onΓ1, (6.13)
σν=f2 onΓ2, (6.14)
−σν=F,∥στ∥ ≤k|σν|,στ=−k|σν|uτ
∥uτ∥ifuτ̸=0onΓ3, (6.15)
στ=0, σν≤0, uν−g≤0, σν(uν−g) = 0 onΓ4, (6.16)
where Eis the elastic tensor, F: Γ3→R+denotes the prescribed normal stress, k: Γ3→R+
denotes the coefficient of friction and g: Γ4→R+denotes the gap.
Let us make the following assumptions.
Assumption 6.8. E: Ω×S3→S3is a fourth order tensor such that:
•(i1)Eijkl=Eklij=Ejikl∈L∞(Ω),1≤i,j,k,l ≤d;
•(i2)there exists mE>0 :Eτ·τ≥mE∥τ∥2for allτ∈S3,a.e. in Ω.
Assumption 6.9. The density of the volume forces verifies f0∈L2(Ω)3and the density of
traction verifies f2∈L2(Γ2)3.
Assumption 6.10. There exists gext: Ω→Rsuch thatgext∈H1(Ω), γg ext= 0 almost
everywhere on Γ1, γg ext≥0almost everywhere on Γ\Γ1g=γgextalmost everywhere on Γ4.
Assumption 6.11. The prescribed normal stress verifies F∈L∞(Γ3)andF(x)≥0a.e.x∈
Γ3.
Assumption 6.12. The coefficient of friction verifies k∈L∞(Γ3)andk(x)≥0a.e.x∈Γ3.
Assumption 6.13. The unit outward normal to Γ4is a constant vector.
Let us introduce the space
V1={v∈H1(Ω)3:γv=0a.e. on Γ 1}. (6.17)
We define a bilinear form a1:V1×V1→Rsuch that
a1(u,v) =∫
ΩEε(u(x))·ε(v(x))dx for allu,v∈V1. (6.18)
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Next, we define f1∈V1such that, for all v∈V1,
(f1,v)V1=∫
Ωf0(x)·v(x)dx+∫
Γ2f2(x)·γv(x)dΓ−∫
Γ3F(x)vν(x)dΓ.
Besides, we introduce a functional j1as follows:
j1:V1→R+j1(v) =∫
Γ3F(x)k(x)∥vτ(x)∥dΓ. (6.19)
LetD1be the dual of the space
M1={ev=vν| 4v∈V1}.
We defineλ∈D1such that
⟨λ,w⟩=−∫
Γ4σν(x)w(x)dΓ for allw∈M1,
where ⟨·,·⟩denotes the duality pairing between D1andM1.Furthermore, we define a bilinear
form as follows,
b1:V1×D1→R, b 1(v, µ) =⟨µ,vν| 4⟩for allv∈V1, µ∈D1. (6.20)
Let us introduce the following subset of D1,
Λ1={
µ∈D1:⟨µ, v ν| 4⟩ ≤0 for allv∈ K 1}
, (6.21)
where
K1={v∈V1:vν≤0 almost everywhere on Γ 4}.
We are led to the following weak formulation of Problem 6.4.
Problem 6.5. Findu∈V1andλ∈Λ1such that, for all v∈V1, µ∈Λ1,
a1(u,v−u) +j1(v)−j1(u) +b1(v−u,λ)≥(f1,v−u)V1,
b1(u,µ−λ)≤b1(gextν4,µ−λ).
A solution of Problem 6.5 is called a weak solution to Problem 6.4. The well-posedness of
Problem 6.5 is given by the following theorem.
Theorem 6.2 (Theorem 3 in [110]) .If Assumptions 6.8-6.13hold true, then Problem 6.5has a
bounded solution (u,λ)∈V1×Λ1,unique in its first argument.
Letρ>0.We consider the following regularized problem.
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Problem 6.6. Finduρ∈V1andλρ∈Λ1such that, for all v∈V1, µ∈Λ1,
a1(uρ,v−uρ) +j1ρ(v)−j1ρ(uρ) +b1(v−uρ,λρ)≥(f1,v−uρ)V1,
b1(uρ,µ−λρ)≤b1(gextν4,µ−λρ)
where
j1ρ:V1→Rj1ρ(v) =∫
Γ3F(x)k(x)(√
∥vτ(x)∥2+ρ2−ρ)dΓ.
Notice that, according to [147], the functional j1ρis Gˆ ateaux differentiable and denoting by
∇j1ρits Gˆ ateaux differential we have:
•∇j1ρ:V1→V1(∇j1ρ(w),v)V1=∫
Γ3F(x)k(x)wτ(x)·vτ(x)√
∥wτ(x)∥2+ρ2dΓ;
•∥∇j1ρ(v)− ∇j1ρ(w)∥V1= sup
z∈V1,z̸=0V1(∇j1ρ(v)− ∇j1ρ(w),z)V1
∥z∥V1
≤2c2
tr∥kF∥L∞(Γ3)
ρ∥v−w∥V1for allw,v∈V1,
wherectris a positive constant which fulfills the following inequality
∥zτ∥L2(Γ3)≤ctr∥z∥V1for allz∈V1. (6.22)
In addition, for all v∈V1,
|j1ρ(v)−j1(v)| ≤ F (ρ),where F(ρ) =ρ∫
Γ3F(x)k(x)dΓ.
On the other hand, for all v,w∈V1we have
|j1ρ(v)−j1ρ(w)| ≤ctr∥Fk∥L2(Γ3)∥v−w∥V1. (6.23)
It is worth to emphasize that ( uρ,λρ)∈V1×Λ1is a solution of Problem 6.6 if and only if it
verifies
(A1ρuρ,v)V1+b1(v,λρ) = (f1,v)V1 for allv∈V1, (6.24)
b1(uρ,µ−λρ) ≤b1(gextν4,µ−λρ) for all µ∈Λ1, (6.25)
whereA1ρ:V1→V1,(A1ρv,w)V1=a1(v,w) + (∇j1ρ(v),w)V1.
Remark 6.2. The first component of a solution of Problem 6.5 (which is unique in the first
argument), is the strong limit of the sequence (uρ)ρ,uρbeing the first component of the solution
of the problem (6.24)-(6.25).
94
For details see [110].
Let us proceed with the second model we are interested on.
Problem 6.7. Findu:¯Ω→R3andσ:¯Ω→S3such that
Divσ+f0=0 inΩ, (6.26)
σ=Eε(u) inΩ, (6.27)
u=0 onΓ1, (6.28)
σν=f2 onΓ2, (6.29)
uν= 0,∥στ∥ ≤ζ,στ=−ζuτ
∥uτ∥ifuτ̸=0onΓ3, (6.30)
στ=0, σν≤0, uν−g≤0, σν(uν−g) = 0 onΓ4, (6.31)
where Eis the elastic tensor, ζ: Γ3→R+denotes the friction bound and g: Γ4→R+denotes
the gap. We keep Assumptions 6.8-6.10 and Assumption 6.13. In addition, we made the following
assumption.
Assumption 6.14. The friction bound verifies ζ∈L∞(Γ3)andζ(x)≥0a.e.x∈Γ3.
Let us introduce the space
V2={
v∈V1|vν= 0 a.e. on Γ 3}
which is a closed subspace of the space V1defined in (6.17).
We define a bilinear form a2:V2×V2→Rsuch that
a2(u,v) =∫
ΩEε(u(x))·ε(v(x))dx for allu,v∈V2. (6.32)
Next, we define f2∈V2such that,
(f2,v)V2=∫
Ωf0(x)·v(x)dx+∫
Γ2f2(x)·γv(x)dΓ for all v∈V2.
Besides, we introduce a functional j2as follows:
j2:V2→R+j2(v) =∫
Γ3ζ(x)∥vτ(x)∥dΓ. (6.33)
LetD2be the dual of the space
M2={ev=vν| 4v∈V2}.
We defineλ∈D2such that
⟨λ,w⟩=−∫
Γ4σν(x)w(x)dΓ for all w∈M2,
95
where ⟨·,·⟩denotes the duality pairing between D2andM2.Furthermore, we define a bilinear
form as follows,
b2:V2×D2→R, b 2(v,µ) =⟨µ,vν| 4⟩for allv∈V2, µ∈D2.
Let us introduce the following subset of D2,
Λ2={
µ∈D2:⟨µ, w⟩ ≤0 for allw∈ K}
,
where
K={w∈M2:w≤0 almost everywhere on Γ 4}.
Notice that λ∈Λ2.Moreover,
b2(u,λ) =b2(gextν4,λ)
b2(u,µ)≤b2(gextν4,µ) for allµ∈Λ2.
We have the following weak formulation of Problem 6.7.
Problem 6.8. Findu∈V2andλ∈Λ2such that, for all v∈V2, µ∈Λ2,
a2(u,v−u) +j2(v)−j2(u) +b2(v−u,λ)≥(f2,v−u)V2,
b2(u,µ−λ)≤b2(gextν4,µ−λ).
A solution of Problem 6.8 is called a weak solution of Problem 6.7.
The well-posedness of Problem 6.8 is given by the following theorem.
Theorem 6.3. [Theorem 4 in [110]] Assumptions 6.8-6.10, and Assumptions 6.13-6.14hold
true. Then, Problem 6.8has a bounded solution (u,λ)∈V2×Λ2,unique in its first argument.
Letρ>0.We consider the following regularized problem.
Problem 6.9. Letρ>0.Finduρ∈V2andλρ∈Λ2such that for all v∈V2, µ∈Λ2,
a2(uρ,v−uρ) +j2ρ(v)−j2ρ(uρ) +b2(v−uρ,λρ)≥(f2,v−uρ)V2,
b2(uρ,µ−λρ)≤b2(gextν4,µ−λρ)
wherej2ρ:V2→R, j2ρ(v) =∫
Γ3ζ(x)(√
∥vτ(x)∥2+ρ2−ρ)dΓ.
Notice that the functional j2ρis Gˆ ateaux differentiable and denoting by ∇j2ρits Gˆ ateaux
differential, we have
•∇j2ρ:V2→V2(∇j2ρ(w),v)V2=∫
Γ3ζ(x)wτ(x)·vτ(x)√
∥wτ(x)∥2+ρ2dΓ;
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•∥∇j2ρ(v)− ∇j2ρ(w)∥V2= sup
z∈V2,z̸=OV2(∇j2ρ(v)−∇j2ρ(w),z)V2
∥z∥V2
≤2c2
tr∥ζ∥L∞(Γ3)
ρ∥v−w∥V2for allw,v∈V2.
Furthermore, for all v∈V2,
|j2ρ(v)−j2(v)| ≤ F (ρ) where F(ρ) =ρ∫
Γ3ζ(x)dΓ
and, for allv,w∈V2,
j2ρ(v)−j2ρ(w) =∫
Γ3ζ(x)(∥vτ∥ − ∥wτ∥)(∥vτ∥+∥wτ∥)√
∥vτ∥2+ρ2+√
∥wτ∥2+ρ2dΓ.
Moreover,
|j2ρ(v)−j2ρ(w)| ≤Lj2ρ∥v−w∥V2;Lj2ρ=ctr∥ζ∥L2(Γ3).
It is worth to emphasize that ( uρ,λρ)∈V2×Λ2is a solution of Problem 6.8 if and only if it
verifies
(A2ρuρ,v)V2+b2(v,λρ) = (f2,v)V2 for allv∈V2, (6.34)
b2(uρ,µ−λρ) ≤b2(gextν4,µ−λρ) for all µ∈Λ2, (6.35)
whereA2ρ:V2→V2,(A2ρv,w)V2=a2(v,w) + (∇j2ρ(v),w)V2.
Remark 6.3. The unique first component of a pair solution of Problem 6.8 is the strong limit
of the sequence (uρ)ρ,uρbeing the first component of the solution of the problem (6.34)-(6.35).
For details see [110].
6.2 The case of nonlinear elastic operators
This section is based on Section 2 and on a part of Section 4 of the paper [113]. In this section
we firstly focus on an abstract problem governed by two convex functionals. Based on a saddle
point technique, we deliver existence and uniqueness results. To illustrate the applicability of
the abstract results we have got, two contact models are solved.
6.2.1 Abstract results
In this subsection we consider the following mixed variational problem.
Problem 6.10. Givenf∈X,find(u,λ)∈X×Ysuch thatλ∈Λ⊂Yand
J(v)−J(u) +b(v−u,λ) +φ(v)−φ(u)≥(f, v−u)X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ.
97
We made the following assumptions.
Assumption 6.15. (X,(·,·)X,∥ · ∥ X)and(Y,(·,·)Y,∥ · ∥ Y)are two Hilbert spaces.
Assumption 6.16. J:X→[0,∞)is a convex lower semicontinuous functional. In addition,
there existm1,m2>0such thatm1∥v∥2
X≥J(v)≥m2∥v∥2
Xfor allv∈X.
Assumption 6.17. b:X×Y→Ris a bilinear form such that
•there exists Mb>0 :|b(v,µ)| ≤Mb∥v∥X∥µ∥Yfor allv∈X, µ∈Y,
•there exists α>0 : inf
µ∈Y,µ̸=0Ysup
v∈X,v̸=0Xb(v,µ)
∥v∥X∥µ∥Y≥α.
Assumption 6.18. Λis a closed convex subset of Ythat contains 0Y.
Assumption 6.19. φ:X→[0,∞)is a convex lower semicontinuous functional. In addition,
there exists q1>0such that, for all v∈X, φ (v)≤q1∥v∥X.
Theorem 6.4. [An existence result][Theorem 3 in [113]] If Assumptions 6.15–6.19hold true,
then Problem 6.10has at least one solution.
The proof of Theorem 6.4 can be found in [113].
In order to establish the uniqueness of the solution, additional assumptions are necessary.
Assumption 6.20. J:X→[0,∞)is a Gˆ ateaux differentiable functional. In addition:
•there exists m> 0such that
(∇J(u)− ∇J(v),u−v)X≥m∥u−v∥2
X for allu,v∈X.
•there exists L>0such that
∥∇J(u)− ∇J(v)∥X≤L∥u−v∥X for allu,v∈X.
Assumption 6.21. φ:X→[0,∞)is a Gˆ ateaux differentiable functional.
Let us define
˜J:X→[0,∞)˜J=J+φ. (6.36)
We consider the following auxiliary problem.
Problem ˜1.Findu∈Xandλ∈Λsuch that
(∇˜J(u),v)X+b(v,λ) = (f,v)X for allv∈X,
b(u,µ−λ)≤0 for allµ∈Λ.
According to Lemma 2 in [113], the set of the solutions of Problem 6 .10 coincides with the
set of the solutions of Problem ˜1.
Theorem 6.5. [An uniqueness result][Theorem 4 in [113]] If Assumptions 6.15–6.21hold true,
then Problem 6.10has a unique solution.
For the proof of Theorem 6.5 we send the reader to [113].
98
6.2.2 3D contact models
To illustrate the applicability of the previous abstract results, two contact models are discussed
in this subsection. Each model involves a deformable body which occupies a bounded domain
Ω⊂R3with Lipschitz continuous boundary Γ partitioned in four parts. In order to describe the
behavior of the material, we use a nonlinear constitutive law expressed by the subdifferential of
a proper, convex, lower semicontinuous functional.
Problem 6.11. [The First Model] Find u:¯Ω→R3andσ:¯Ω→S3,such that
Divσ(x) +f0(x) =0 inΩ,(6.37)
σ(x)∈∂ω(ε(u(x))) inΩ,(6.38)
u(x) =0 onΓ1,(6.39)
σν(x) =f1(x) onΓ2,(6.40)
στ(x) =0, uν(x)≤0,σν(x)≤0,σν(x)uν(x) = 0 onΓ3,(6.41)
−σν(x) =F(x),
∥στ(x)∥ ≤K(x)|σν(x)|,στ(x) =−K(x)|σν(x)|uτ(x)
∥uτ(x)∥ifuτ(x)̸=0onΓ4,(6.42)
where Γ 1,Γ2,Γ3and Γ 4is a partition of Γ such that the Lebesgue measure of Γ 1is positive. Note
that (6.37) is the equilibrium equation, (6.38) is the constitutive law, (6.39) is the displacement
boundary condition, and (6.40) is the traction boundary condition. Finally, (6.41) is a frictionless
unilateral contact condition with zero gap and (6.42) is a frictional contact condition with
prescribed normal stress. The coefficient of friction Kas well as the prescribed normal stress F
are given functions. Details on the boundary contact conditions we use here can be found for
instance in [59, 147].
In order to give a weak formulation we make the following assumptions.
Assumption 6.22. f0∈L2(Ω)3;f1∈L2(Γ2)3.
Assumption 6.23. ω:S3→[0,∞)is a convex lower semicontinuous functional. In addition,
there existα1, α2>0such that
α1∥ε∥2≥ω(ε)≥α2∥ε∥2for allε∈S3.
Assumption 6.24. F∈L3(Γ4), F(x)≥0a.e. on Γ4;K∈L3(Γ4), K(x)≥0a.e. on Γ4.
The functional setting is as follows.
X={v∈H1(Ω)3:γv=0a.e. on Γ 1}. (6.43)
S={w=vν| 3v∈X}, (6.44)
Y=S′. (6.45)
99
Next, we define the functional
J:X→[0,∞), J (v) =∫
Ωω(ε(v(x)))dx. (6.46)
In addition, using Riesz’s representation theorem we define f∈Xas follows,
(f,v)X=∫
Ωf0(x)·v(x)dx+∫
Γ2f1(x)·γv(x)dΓ−∫
Γ3F(x)vν(x)dΓ for allv∈X.(6.47)
Furthermore, we can introduce the following convex and continuous functional.
φ:X→[0,∞)φ(v) =∫
Γ4F(x)K(x)∥vτ(x)∥dΓ. (6.48)
We defineλ∈Ysuch that
⟨λ,w⟩=−∫
Γ3σν(x)w(x)dΓ for allw∈S, (6.49)
where ⟨·,·⟩denotes the duality pairing between YandS.
Moreover, we define a bilinear form as follows,
b:X×Y→R, b(v, µ) =⟨µ,vν| 3⟩for allv∈X, µ∈Y. (6.50)
Let us introduce the following subset of Y,
Λ ={
µ∈Y:⟨µ, v ν| 3⟩ ≤0 for allv∈ K}
, (6.51)
where
K={v∈X:vν≤0 almost everywhere on Γ 3}.
This first contact model is related to Problem 6.10 for unbounded subset Λ.
According to Theorem 6.4, Problem 6.11 has at least one weak solution.
Problem 6.12. [The Second Model] Find u:¯Ω→R3andσ:¯Ω→S3,such that
Divσ(x) +f0(x) =0 inΩ,(6.52)
σ(x)∈∂ω(ε(u(x))) inΩ,(6.53)
u(x) =0 onΓ1,(6.54)
σν(x) =f1(x) onΓ2,(6.55)
uν(x) = 0,∥στ(x)∥ ≤ζ(x),στ(x) =−ζ(x)uτ(x)
∥uτ(x)∥ifuτ(x)̸=0 onΓ3,(6.56)
−σν(x) =F(x),
∥στ(x)∥ ≤K(x)|σν(x)|,στ(x) =−K(x)|σν(x)|uτ(x)
∥uτ(x)∥ifuτ(x)̸=0onΓ4,(6.57)
100
where Γ 1,Γ2,Γ3and Γ 4is a partition of Γ such that the Lebesgue measure of Γ 1is positive, as
in the previous example. Recall that (6.52) is the equilibrium equation, (6.53) is the constitutive
law, (6.54) is the displacement boundary condition and (6.55) is the traction boundary condition.
Herein (6.56) is a bilateral frictional contact condition with friction bound ζ.Finally, (6.57) is a
frictional contact condition with prescribed normal stress. The functions ζ, K andFare given
functions. For details on the boundary contact conditions written here see for instance [147] and
the references therein.
In order to analyze this second example we adopt Assumptions 6.22-6.24. In addition, we
make the following assumption.
Assumption 6.25. ζ∈L2(Γ3), ζ(x)≥0a.e. on Γ3.
Let us introduce the spaces
X={
v∈H1(Ω)3:γv=0a.e. on Γ 1, vν= 0 a.e. on Γ 3}
;
S={z=γw|Γ3|w∈X};
Y=S′.
We defineJ,fandφas in the previous subsection, see (6.46)-(6.48).
Next we introduce the following subset:
Λ ={
µ∈Y:⟨µ,z⟩ ≤∫
Γ3ζ(x)∥z(x)∥dΓ for allz∈S}
, (6.58)
⟨·,·⟩being the duality pairing between YandS.
Let us define λ∈Y,
⟨λ,z⟩=−∫
Γ3στ(x)·z(x)dΓ for allz∈S. (6.59)
We also define a bilinear form b(·,·),
b:X×Y→Rb(v,µ) =⟨µ,γv|Γ3⟩. (6.60)
This second model is related to Problem 6.10 for bounded subset Λ.
According to Theorem 6.4, Problem 6.12 has at least one weak solution. More details can
be found in [113].
Chapter 7
Unilateral frictional contact problems
This chapter, based on some results we have got in the paper [113], draws the attention to an
abstract mixed variational problem governed by a convex functional and a bifunctional which
depends on a Lagrange multiplier in the first argument and is convex in the second argument.
After we discuss the existence and the uniqueness of the solution of the abstract problem, we
illustrate the applicability of the abstract result to the weakly solvability of a unilateral frictional
contact problem.
7.1 Abstract results
In this section, based on Section 3 of the paper [113], we present the results in the study of the
following mixed variational problem.
Problem 7.1. Givenf∈X,find(u,λ)∈X×Ysuch thatλ∈Λ⊂Yand
J(v)−J(u) +b(v−u,λ) +j(λ,v)−j(λ,u)≥(f, v−u)X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ.
In order to study Problem 7.1 we adopt Assumptions 6.15-6.18 and 6.20 from the previous
chapter. In addition we made the following assumptions.
Assumption 7.1. j: Λ×X→[0,∞)is a bifunctional such that:
•j1)for allη∈Λ, j(η,·) :X→[0,∞)is a convex Gˆ ateaux differentiable functional;
•j2)for allη∈Λ,there exists q1>0(q1independent of η) such that
j(η,v)≤q1∥v∥X for allv∈X;
•j3)for allη∈Λ,(∇2j(η,u),u)X≥0for allu∈X;
101
102
•j4)for allη∈Λ,∇2j(η,0X) = 0 X;
•j5)for allη∈Λ,there exists Lj>0(Ljindependent of η) such that
∥∇ 2j(η,u)− ∇ 2j(η,v)∥X≤Lj∥u−v∥X for allu,v∈X.
•j6)if(un)n⊂Xand(ηn)n⊂Yare two sequences such that un⇀u inXasn→ ∞ and
ηn⇀η inYasn→ ∞,then lim supn→∞j(ηn,v)−j(ηn,un)≤j(η,v)−j(η,u).
Notice that for each ( u,η)∈X×Λ,∇2j(η,u) denotes the Gˆ ateaux differential of jinu.
Assumption 7.2. If(un)n⊂Xand(τn)n⊂Yare two sequences such that un⇀˜uinXas
n→ ∞ andτn⇀˜τinYasn→ ∞,then lim supn→∞b(un,τn) =b(˜u,˜τ).
Theorem 7.1. [An existence result][Theorem 5 in [113]] If Assumptions 6.15-6.18,6.20,7.1-7.2
hold true, then Problem 7.1has at least one solution.
The proof of Theorem 7.1 can be found in [113]. The key of the proof was the construction
of the following operator.
T: Λ→ΛT(η) =λη, (7.1)
which is a weakly sequentially continuous map. In addition, it is worth to mention that T|$has
a fixed point, where
£={µ∈Λ|∥µ∥Y≤∥f∥X
α+(L+Lj)∥f∥X
mα}.
Thus, there exists η∗∈£⊂Λ such that T(η∗) =λη∗=η∗.The pair (uη∗,λη∗)∈X×Λ is a
solution of Problem 7.1.
In order to investigate the uniqueness of the solution, we made a new assumption.
Assumption 7.3. For allµ1, µ2∈Λ, v1, v2∈Xthere exists G> 0such that
j(µ1,v2)−j(µ1,v1) +j(µ2,v1)−j(µ2,v2)≤G∥v1−v2∥X.
Theorem 7.2. [An uniqueness result][Theorem 6 in [113]] If Assumptions 6.15-6.18,6.20,7.1-
7.3hold true, then Problem 7.1has a solution, unique in its first argument.
The proof of Theorem 7.2 was given in [113].
7.2 A 3D contact model
To illustrate the applicability of the abstract results presented in Section 7.1, we consider the
following 3D model (the third example in Section 4 of the paper [113]).
103
Problem 7.2. Givenρ>0,findu:¯Ω→R3andσ:¯Ω→S3,such that
Divσ(x) +f0(x) =0 inΩ, (7.2)
σ(x)∈∂ω(ε(u(x))) inΩ, (7.3)
u(x) =0 onΓ1, (7.4)
σν(x) =f1(x) onΓ2, (7.5)
uν(x)≤0,σν(x)≤0,σν(x)uν(x) = 0
στ(x) =−k(σν)(x)uτ(x)√
∥uτ(x)∥2+ρ2onΓ3, (7.6)
where Γ 1,Γ2and Γ 3is a partition of Γ such that the Lebesgue measure of Γ 1is positive. Recall
that (7.2) is the equilibrium equation, (7.3) is the constitutive law, (7.4) is the displacement
boundary condition and (7.5) is the traction boundary condition. Herein (7.6) is a unilateral
contact condition with regularized Coulomb-type friction law. The friction law we use describes
a situation when slip appears even for small tangential shears, which is the case when the surfaces
are lubricated by a thin layer of non-Newtonian fluid, see [147] and the references therein.
In order to analyze the model, we adopt Assumptions 6.22 and 6.23 made in the previous
chapter. Moreover, herein we consider
ω:S3→[0,∞), ω (ε) =1
2Aε·ε+β
2∥ε−PKε∥2(7.7)
where Ais a fourth order symmetric tensor satisfying the ellipticity condition, βis a strictly
positive constant, K⊂S3denotes a closed, convex set containing the element 0 S3andPK:S3→
Kis the projection operator. Notice that the function ωfulfills the following property.
Property 7.1. ωis a Gˆ ateaux differentiable functional. In addition:
•ω1)there exists Lω>0such that
∥∇ω(ε)− ∇ω(τ)∥ ≤Lω∥ε−τ∥for allε,τ∈S3;
•ω2)there exists mω>0such that
(∇ω(ε)− ∇ω(τ))·(ε−τ)≥mω∥ε−τ∥2for allε,τ∈S3,
where
∇ω(ε)·τ=Aε·τ+β(ε−PKε)·τ.
We keep the definitions for Xin (6.43),Jin (6.46),Sin (6.44),Yin (6.45),λin (6.49),
b(·,·) in (6.50) and Λ in (6.51). Notice that λ=−σν| 3.
Herein we consider a coefficient of friction as follows:
k(σν) : Γ 3→[0,∞)k(σν)(x) =ϑ|(Rσν| 3)(x)|
1 +|(Rσν| 3)(x)|, (7.8)
104
whereϑ>0 and R:Y→L2(Γ3) is a linear and compact operator.
In addition, we define f∈Xas follows:
(f,v)X=∫
Ωf0(x)·v(x)dx+∫
Γ2f1(x)·γv(x)dΓ for allv∈X.
Finally, we define
j: Λ×X→R+j(η,v) =∫
Γ3K(η)(√
∥vτ(x)∥2+ρ2−ρ)dΓ, (7.9)
whereK: Λ→L∞(Γ3),
K(η)(x) =ϑ|(Rη)(x)|
1 +|(Rη)(x)|. (7.10)
Notice that K(µ) =−k(−µ) for allµ∈Λ.
According to Theorem 7.2, Problem 7.2 has a solution ( u,λ),unique in its first argument.
For details see [113].
Part II
A variational approach via bipotentials
105
Chapter 8
Unilateral frictionless contact problems
This chapter is based on the paper [106]. We consider a unilateral contact model for nonlinearly
elastic materials, under the small deformations hypothesis, for static processes. The contact is
modeled with Signorini’s condition with zero gap and the friction is neglected on the poten-
tial contact zone. The behavior of the material is modeled by a subdifferential inclusion, the
constitutive map being proper, convex, and lower semicontinuous. After describing the model,
we give a weak formulation using a bipotential which depends on the constitutive map and its
Fenchel conjugate. We look for the unknown into a Cartesian product of two nonempty, convex,
closed, unbounded subsets of two Hilbert spaces. We prove the existence and the uniqueness of
the weak solution based on minimization arguments for appropriate functionals associated with
the variational system. How the proposed variational approach is related to previous variational
approaches, is discussed too.
8.1 The model and its weak solvability via bipotentials
In this section we discuss the weak solvability via bipotentials theory for a unilateral frictionless
contact model in the following physical setting. A body occupies a bounded domain Ω ⊂R3
with Lipschitz continuous boundary, partitioned in three measurable parts, Γ 1,Γ2and Γ 3,such
that the Lebesgue measure of Γ 1is positive. The body Ω is clamped on Γ 1, body forces of
densityf0act on Ω and surface tractions of density f2act on Γ 2.On Γ 3the body can be in
contact with a rigid foundation. According to this physical setting we formulate the following
boundary value problem.
Problem 8.1. Findu:¯Ω→R3andσ:¯Ω→S3,such that
Divσ(x) +f0(x) =0inΩ, (8.1)
σ(x)∈∂ω(ε(u(x))) inΩ, (8.2)
106
107
u(x) =0 onΓ1, (8.3)
σν(x) =f2(x) onΓ2, (8.4)
στ(x) =0, uν(x)≤0, σν(x)≤0, σν(x)uν(x) = 0 onΓ3. (8.5)
Problem 8.1 has the following structure: (8.1) represents the equilibrium equation, (8.2)
represents the constitutive law, (8.3) represents the displacements boundary condition, (8.4)
represents the traction boundary condition and (8.5) represents the frictionless unilateral contact
condition.
Assumption 8.1. The constitutive function ω:S3→Ris convex and lower semicontinuous.
In addition, there exist α,β such that 1>β≥α>0andβ∥ε∥2≥ω(ε)≥α∥ε∥2for allε∈S3.
Assumption 8.2. The densities of the volume forces and traction verify
f0∈L2(Ω)3andf2∈L2(Γ2)3.
Let us introduce the space
V={v∈H1(Ω)3:γv=0a.e. on Γ 1}.
Letf∈Vbe such that
(f,v)V= (f0,v)L2(Ω)3+∫
Γ2f2(x)·γv(x)dΓ for allv∈V.
We introduce a subset of Vas follows,
U0={v∈V:vν≤0 a.e. on Γ 3}.
Lemma 8.1. [Lemma 1 in [106]] Let α, β be the constants in Assumption 8.1. Then
(1−β)∥τ∥2≤ω∗(τ)≤1
4α∥τ∥2for allτ∈S3. (8.6)
The proof of this lemma can be found in [106]. We associate with the constitutive map ω
the bipotential B:S3×S3→R,
B(τ,µ) =ω(τ) +ω∗(µ) for allτ,µ∈S3, (8.7)
whereω∗is Fenchel’s conjugate of the function ω.
Notice that there exists C=C(α,β)>0 such that
B(τ,µ)≥C(∥τ∥2+∥µ∥2) for allτ,µ∈S3. (8.8)
We introduce the Hilbert space
L2
s(Ω)3×3={µ= (µij) :µij∈L2(Ω), µij=µjifor alli,j∈ {1,2,3}}.
108
It is worth to note that
B(ε(v(·)),τ(·))∈L1(Ω) for allv∈V,τ∈L2
s(Ω)3×3.
We define now the form
b:V×L2
s(Ω)3×3→Rb(v,µ) =∫
ΩB(ε(v(x)),µ(x))dx. (8.9)
Consider the following subset of L2
s(Ω)3×3,
Λ ={µ∈L2
s(Ω)3×3: (µ,ε(v))L2(Ω)3×3≥(f,v)Vfor allv∈U0}. (8.10)
We have the following weak formulation of Problem 8.1.
Problem 8.2. Findu∈U0⊂Vandσ∈Λ⊂L2
s(Ω)3×3such that
b(v,σ)−b(u,σ)≥(f,v−u)Vfor allv∈U0
b(u,µ)−b(u,σ)≥0 for allµ∈Λ.
Each solution ( u,σ)∈U0×Λ of Problem 8 .2 is called a weak solution of Problem 8.1.
Theorem 8.1. (An existence result)[Theorem 3 in [106]] If Assumptions 8.1-8.2 hold true, then
Problem 8.2has at least one solution (u,σ)∈U0×Λ.
In order to get the uniqueness, additional assumptions were needed.
Assumption 8.3. The constitutive function ωand its Fenchel’s conjugate ω∗are strictly convex.
Theorem 8.2. (An uniqueness result)[Theorem 4 in [106]] If Assumptions 8.1-8.3 hold true,
then Problem 8.2 has a unique solution.
The proofs of Theorems 8.1 and 8.2 can be found in [106].
8.2 New approach versus previous approaches
In this section we discuss Problem 8.1 for a special class of nonlinear materials, such that the
following additional assumption holds true.
Assumption 8.4. The constitutive function ωis Gˆ ateaux differentiable and its gradient ∇ω
verifies:
there exists L>0such that ∥∇ω(ε1)− ∇ω(ε2)∥ ≤L∥ε1−ε2∥for allε1,ε2∈S3;
there exists m> 0such that (∇ω(ε1)− ∇ω(ε2))·(ε1−ε2)≥m∥ε1−ε2∥2for allε1,ε2∈S3.
109
In this special case, the constitutive law (8.2) becomes σ(x) =∇ω(ε(u(x))) in Ω,the
literature offering us two variational approaches: the primal variational formulation and the dual
variational formulation. More precisely, Problem 8.1 has the following variational formulation
in displacements.
Primal variational formulation. Findu∈U0such that
(Au,v−u)V≥(f,v−u)Vfor allv∈U0.
Herein the operator A:V→Vis defined as follows: for each u∈V, Auis the element of V
that satisfies
(Au,v)V=∫
Ω∇ω(ε(u(x)))·ε(v(x))dx
for allv∈V.The primal variational formulation has a unique solution u∈U0, see e.g. Theorem
5.10 in [147].
On the other hand, Problem 8.1 has the following weak formulation in terms of stress.
The dual variational formulation. Findσ∈Λsuch that
((∇ω)−1σ,τ−σ)L2(Ω)3×3≥0 for allτ∈Λ.
The dual variational formulation has a unique solution σ∈Λ, see e.g. Theorem 5.12 in [147].
Let us state the following auxiliary result.
Theorem 8.3. [Theorem 5 in [106]] Assumptions 8.1-8.2 and Assumption 8.4 hold true.
1) If ˜uis the solution of the primal variational formulation and ˜σis the function given by
˜σ=∇ω(ε(˜u))then ˜σis the unique solution of the dual variational formulation.
2) If ˜σis the solution of the dual variational formulation then ˜σ=∇ω(ε(˜u))where ˜uis the
solution of the primal variational formulation.
The proof of this theorem is a straightforward consequence of Theorem 5.13 in [147].
The main result of this section is the following theorem.
Theorem 8.4. [Theorem 6 in [106]] Assumptions 8.1-8.2 and Assumption 8.4 hold true.
i) If ˜uis the unique solution of the primal variational formulation and ˜σis the unique
solution of the dual variational formulation, then the pair (˜u,˜σ)is a solution of Problem 8.2.
ii) If (u,σ)is a solution of Problem 8.2,then the first component uis the unique solution
of the primal variational formulation.
iii) If, in addition, ωis strictly convex then the unique solution of Problem 8.2, (u,σ),coin-
cides with the pair (˜u,˜σ)consisting of the unique solution of the primal variational formulation
and the unique solution of the dual variational formulation.
We underline that, if the constitutive function ωfulfills Assumptions 8.1, Assumption 8.4
and, in addition is strictly convex, then the unique solution of Problem 8.2 coincides with the pair
110
consisting of the unique solution of the primal variational formulation and the unique solution
of the dual variational formulation. Let us give an example of such a constitutive function ω:
ω:S3→R, ω(τ) =1
2Eτ·τ+ζ
2∥τ−PKτ∥2, (8.11)
where E:S3→S3,E= (Eijkl),Eijkl=λδijδkl+µ(δikδjl+δilδjk),1≤i,j,k,l ≤3, λ, µ and
ζbeing positive coefficients of the material, small enough (e.g.3
2λ+µ+ζ
2<1),K ⊂S3is a
nonempty, closed and convex set and PK:S3→ K represents the projection operator on K.
In order to study the properties of the functional ωa very helpful reference was [139].
Chapter 9
Frictional contact problems
This chapter is based on the paper [108]. The frictional contact model we investigate in the
present paper is a 3D nonlinearly elastostatic model, under the small deformation hypothesis.
Mathematically, we describe it as a boundary value problem consisting of a system of a partial
differential vectorial equation (equilibrium equation) and a subdifferential inclusion (constitu-
tive law), associated with a homogeneous displacement boundary condition, a traction boundary
condition and a frictional contact condition. The constitutive law indicates us that the stress
belongs to the subdifferential of a proper, convex, lower semicontinuous functional. In order to
model the frictional contact we use a static version of Coulomb’s law of dry friction with pre-
scribed normal stress. Based on minimization arguments for appropriate functionals associated
with the variational system, the existence and the uniqueness of the weak solution of this model
it was proved. In addition to prove the existence and the uniqueness of the weak solution as
a ”global” solution, allowing to compute simultaneously the displacement field and the Cauchy
stress tensor, another relevant aspect of this approach it was discussed for a particular class of
constitutive functions: the weak solution in the new approach coincides with the pair consist-
ing of the unique solution of the primal variational formulation and the unique solution of the
dual variational formulation. Due to the particular feature of the mechanical model we treat in
this chapter, the weak formulation herein is more complex than those presented in the previous
chapter; it involves not only a bipotential function but also a potential which depends on the
prescribed normal stress and on the coefficient of friction.
9.1 The model and its weak solvability via bipotentials
We consider a body that occupies a bounded domain Ω ⊂R3,with smooth boundary, partitioned
in three measurable parts, Γ 1,Γ2and Γ 3,such that the Lebesgue measure of Γ 1is positive. The
body Ω is clamped on Γ 1, body forces of density f0act on Ω and surface tractions of density
f2act on Γ 2.On Γ 3the body is in frictional contact with a foundation, the normal stress
being prescribed. According to this physical setting we formulate the following boundary value
111
112
problem.
Problem 9.1. Findu:Ω→R3andσ:Ω→S3,such that
Divσ(x) +f0(x) =0inΩ, (9.1)
σ(x)∈∂ω(ε(u(x))) inΩ, (9.2)
u(x) =0 onΓ1, (9.3)
σν(x) =f2(x) onΓ2, (9.4)
−σν(x) =F(x) onΓ3 (9.5)
∥στ(x)∥ ≤k(x)|σν(x)|,
στ(x) =−k(x)|σν(x)|uτ(x)
∥uτ(x)∥ifuτ(x)̸=0on Γ 3. (9.6)
Problem 9.1 has the following structure: (9.1) represents the equilibrium equation, (9.2)
represents the constitutive law, (9.3) represents the homogeneous displacements boundary con-
dition, (9.4) represents the traction boundary condition and (9.5)-(9.6) model the frictional
contact with prescribed normal stress.
In order to study Problem 9.1 we keep Assumption 8.1 on the constitutive function ωand
Assumption 8.2 for the density of the volume forces f0and the density of the traction f2.
In addition we made the following assumptions.
Assumption 9.1. The prescribed normal stress verifies F∈L2(Γ3)andF(x)≥0a.e.x∈Γ3.
Assumption 9.2. The coefficient of friction verifies k∈L∞(Γ3)andk(x)≥0a.e.x∈Γ3.
On the other hand, we introduce the space
V={v∈H1(Ω)3:γv=0a.e. on Γ 1}
and definef∈V,
(f,v)V= (f0,v)L2(Ω)3+∫
Γ2f2(x)·γv(x)dΓ for allv∈V.
Notice that k(·)F(·)∥vτ(·)∥ ∈L1(Γ3).This allows us to consider the following functional
j:V→R+j(v) =∫
Γ3k(x)F(x)∥vτ(x)∥dΓ for allv∈V. (9.7)
As in the previous chapter, we associate with the constitutive map ωthe bipotential B:
S3×S3→R,
B(τ,µ) =ω(τ) +ω∗(µ) for allτ,µ∈S3, (9.8)
113
and we introduce a form b(·,·) as follows,
b:V×L2
s(Ω)3×3→Rb(v,µ) =∫
ΩB(ε(v(x)),µ(x))dx. (9.9)
Consider now the following subset of L2
s(Ω)3×3,
Λ ={µ∈L2
s(Ω)3×3: (µ,ε(v))L2(Ω)3×3+j(v)≥(f,v)Vfor allv∈V}. (9.10)
Lemma 9.1. [Lemma 2 in [108]] The subset Λis an unbounded, closed, convex subset of
L2
s(Ω)3×3.
The proof of Lemma 9.1 was given in [108].
Problem 9.1 has the following weak formulation.
Problem 9.2. Findu∈Vandσ∈Λ⊂L2
s(Ω)3×3such that
b(v,σ)−b(u,σ) +j(v)−j(u)≥(f,v−u)Vfor allv∈V
b(u,µ)−b(u,σ)≥0 for allµ∈Λ.
This is a new variational system governed by the functional j.
Each solution ( u,σ) of Problem 9 .2 is called a weak solution of Problem 9.1.
We define a functional Las follows,
L:V×Λ→RL(v,µ) =b(v,µ) +j(v)−(f,v)V.
Let us consider the following minimization problem.
L(u,σ) = min
(v,µ)∈V×ΛL(v,µ). (9.11)
Theorem 9.1. (An existence result)[Theorem 3 in [108]] If Assumptions 8.1,8.2,9.1and9.2
hold true, then Problem 9.2has at least one solution (u,σ)∈V×Λwhich is a solution of the
minimization problem (9.11).
The study of the uniqueness of the solution was made under the following additional assump-
tion.
Assumption 9.3. The constitutive function ωand its Fenchel’s conjugate ω∗are strictly convex.
Theorem 9.2. (An uniqueness result)[Theorem 4 in [106]] If Assumptions 8.1,8.2,9.1,9.2and
9.3hold true, then Problem 9.2has a unique solution.
The proofs of Theorems 9.1 and 9.2 were given in [108].
114
9.2 New approach versus previous approaches
In this section we discuss Problem 9.1 adopting Assumption 8.4 for the constitutive function ω.
In this special case, Problem 9.1 has the following variational formulation in displacements, see
[147].
Primal variational formulation. Findu∈Vsuch that
(Au,v−u)V+j(v)−j(u)≥(f,v−u)Vfor allv∈V.
The primal variational formulation has a unique solution u∈V, see for example Theorem 5.21
in [147].
Also, in the special case we treat in this section, Problem 9.1 has the following variational
formulation in terms of stress.
The dual variational formulation. Findσ∈Λsuch that
((∇ω)−1σ,τ−σ)L2(Ω)3×3≥0 for allτ∈Λ.
The dual variational formulation has a unique solution σ∈Λ, see Theorem 5.32 in [147].
A straightforward consequence of Theorem 5.34 in [147] is the following theorem.
Theorem 9.3. [Theorem 5 in [108]] Assumptions 8.1,8.2,9.1,9.2and Assumption 8.4hold
true.
1) Ifeuis the solution of the primal variational formulation and eσis the function given by
eσ=∇ω(ε(eu))theneσis the unique solution of the dual variational formulation.
2) Ifeσis the solution of the dual variational formulation then eσ=∇ω(ε(eu))whereeuis the
solution of the primal variational formulation.
The main result of this section is the following theorem.
Theorem 9.4. [Theorem 6 in [108]] Assumptions 8.1,8.2,9.1,9.2and Assumption 8.4hold
true.
i) Ifeuis the unique solution of the primal variational formulation and eσis the unique
solution of the dual variational formulation, then the pair (eu,eσ)is a solution of Problem 9.2.
ii) If (u,σ)is a solution of Problem 9.2,then the first component uis the unique solution
of the primal variational formulation.
iii) If, in addition, ωis strictly convex then the unique solution of Problem 9.2,(u,σ),coin-
cides with the pair (eu,eσ)consisting of the unique solution of the primal variational formulation
and the unique solution of the dual variational formulation.
The proof of Theorem 9.4 was given in [108].
Part III
A variational approach via
history-dependent quasivariational
inequalities on unbounded time interval
115
Chapter 10
Preliminaries
In this chapter we present some preliminaries. Firstly we recall an abstract fixed point result for
operators defined on the Fr´ echet space of continuous functions on R+= [0,∞) with values on a
real Banach space ( X,∥ · ∥ X), denotedC(R+,X). Then, we recall an existence and uniqueness
result of the solution for an abstract history-dependent quasivariational inequality formulated
on the unbounded time interval [0 ,∞).
10.1 A xed point result
Let (X,∥·∥X) be a real Banach space, N∗represents the set of positive integers and R+= [0,∞).
We consider the functional space of continuous functions defined on R+with values on X, that
is
C(R+;X) ={x:R+→X|xis continuous }.
Let us present some preliminaries on the space C(R+;X); details on the Fr´ echet space
C(R+,X) including some basic properties can be found in [39, 96].
For alln∈N∗, we denote by C([0,n];X) the space of continuous functions defined on [0 ,n]
with values on X, that is
C([0,n];X) ={x: [0,n]→X|xis continuous }.
The spaceC([0,n];X) is a real Banach space with the norm
∥x∥n= max
t∈[0,n]∥x(t)∥X (10.1)
and, moreover, for any λ>0 the norm (10.1) is equivalent with Bielecki’s norm,
∥x∥λ,n= max
t∈[0,n]{
e−λ t∥x(t)∥X}
. (10.2)
116
117
Consider now two sequences of real numbers ( λn)n∈N∗and (βn)n∈N∗such that
0<λ 1<λ 2<…<λ n<…., (10.3)
βn>0∀n∈N∗,∞∑
n=1βn<∞. (10.4)
For anyx,y∈C(R+;X) define
d(x,y) =∞∑
n=1βn|x−y|n
1 +|x−y|n, (10.5)
where, for all n∈N∗,
|x|n=∥x∥λn,n= max
t∈[0,n]{
e−λnt∥x(t)∥X}
. (10.6)
It is well known that dis a distance on C(R+;X) and the metric space ( C(R+;X),d) is complete,
i.e. is a Fr´ echet space.
We note that, for all n∈N∗,| · |nand∥ · ∥ nare equivalent norms on the space C([0,n];X).
Also, we recall that the convergence of a sequence ( xp)p∈N∗⊂C(R+;X) to the element x∈
C(R+;X), is characterized by the following equivalences:
d(xp,x)→0 asp→ ∞ ⇔ lim
p→∞|xp−x|n= 0∀n∈N∗(10.7)
⇔lim
p→∞∥xp−x∥n= 0∀n∈N∗.
According to (10.7), the convergence in the metric space ( C(R+;X),d) does not depend on the
choice of sequences ( λn)n∈N∗and (βn)n∈N∗which satisfy (10.3) and (10.4). For this reason, we
writeC(R+;X) instead of ( C(R+;X),d) and we refer to C(R+;X) as to a Fr´ echet space. Also,
note that:
(xp)p∈N∗⊂C(R+;X) is a Cauchy sequence if and only if (10.8)
∀ε>0,∀n∈N∗,∃N=N(ε,n) such that |xp−xq|n<ε∀p, q≥N.
Theorem 10.1. [Theorem 2.1 in [144]] Let Λ :C(R+;X)→C(R+;X)be a nonlinear operator.
Assume that there exists m∈N∗with the following property: for all n∈N∗there exist two
constantscn≥0andkn∈[0,1)such that
∥Λx(t)−Λy(t)∥m
X≤kn∥x(t)−y(t)∥m
X+cn∫t
0∥x(s)−y(s)∥m
Xds (10.9)
for allx, y∈C(R+;X)and for any t∈[0,n]. Then the operator Λhas a unique fixed point
η∗∈C(R+;X).
118
Corollary 10.1. [Corollary 2.5 in [144]] Let Λ :C(R+;X)→C(R+;X)be a nonlinear operator.
Assume that there exist m∈N∗,α∈[0,1)and a continuous function γ:R+→R+, such that
∥Λx(t)−Λy(t)∥m
X≤α∥x(t)−y(t)∥m
X+γ(t)∫t
0∥x(s)−y(s)∥m
Xds
for allx, y∈C(R+;X)and for any t∈R+. Then the operator Λhas a unique fixed point
η∗∈C(R+;X).
The proofs of Theorem 10.1 and Corollary 10.1 can be found in [144].
10.2 An abstract history-dependent quasivariational in-
equality
LetXbe a real Hilbert space with inner product ( ·,·)Xand associated norm ∥ · ∥ X. Let also
Ybe a normed space with the norm denoted ∥ · ∥ Yand let L(X,Y ) denote the space of linear
continuous operators from XtoYwith the usual norm ∥·∥L(X,Y ). Finally, for n∈N∗we denote
byC([0,n];L(X,Y )) the space of continuous functions defined on the bounded interval [0 ,n]
with values in L(X,Y ).
LetKbe a subset of Xand consider the operators A:K→X,S:C(R+;X)→C(R+;Y)
and the functionals φ:Y×X→R,j:X×X→R. Moreover, let f:R+→X. Then we
consider the problem of finding a function u∈C(R+;X), such that for all t∈R+, the inequality
below holds:
u(t)∈K, (Au(t),v−u(t))X+φ(Su(t),v)−φ(Su(t),u(t)) (10.10)
+j(u(t),v)−j(u(t),u(t))≥(f(t),v−u(t))X for allv∈K.
Note that (10.10) represents a time-dependent variational inequality governed by two func-
tionalsφandjwhich depend on the solution and, therefore, we refer to (10.10) as a quasi-
variational inequality. Also, to avoid any confusion, we note that here and below the notation
Au(t) and Su(t) are short hand notation for A(u(t)) and ( Su)(t), i.e.Au(t) =A(u(t)) and
Su(t) = (Su)(t), for allt∈R+.
In the study of (10.10) were used the following assumptions.
Assumption 10.1. Kis a closed,convex,nonempty subset of X.
Assumption 10.2.
There exists m> 0 such that ( Au1−Au2,u1−u2)X≥m∥u1−u2∥2
Xfor allu1,u2∈K.
There exists L>0such that ∥Au1−Au2∥X≤L∥u1−u2∥Xfor allu1,u2∈K.
119
Assumption 10.3. For ally∈Y, φ(y,·) is convex and lower semicontinuous on X.There exists
α>0 such that φ(y1,u2)−φ(y1,u1) +φ(y2,u1)−φ(y2,u2)≤α∥y1−y2∥Y∥u1−u2∥Xfor all
y1, y2∈Y, u 1, u2∈X.
Assumption 10.4. For allx∈X, j(u,·) is convex and lower semicontinuous on X.There exists
β > 0 such that j(u1,v2)−j(u1,v1) +j(u2,v1)−j(u2,v2)≤β∥u1−u2∥X∥v1−v2∥Xfor all
u1, u2, v1, v2∈X.
Assumption 10.5. β <m.
Assumption 10.6. S:C(R+;X)→C(R+;Y); for alln∈N∗there exists rn>0 such that
∥Su1(t)− Su2(t)∥Y≤rn∫t
0∥u1(s)−u2(s)∥Xdsfor allu1, u2∈C(R+;X), t∈[0,n].
Assumption 10.7. f∈C(R+;X).
Assumption 10.6 is satisfied for the operator S:C(R+;X)→C(R+;Y) given by
Sv(t) =R(∫t
0v(s)ds+v0)
for allv∈C(R+;X), t∈R+, (10.11)
whereR:X→Yis a Lipschitz continuous operator and v0∈X. It is also satisfied for the
Volterra operator S:C(R+;X)→C(R+;Y) given by
Sv(t) =∫t
0R(t−s)v(s)dsfor allv∈C(R+;X), t∈R+, (10.12)
where now R∈C(R+;L(X,Y )). In the case of the operator (10.11), inequality (10.6) holds
withcnbeing the Lipschitz constant of the operator R, for alln∈N∗, and in the case of the
operator (10.12) it holds with
rn=∥R∥C([0,n];L(X,Y ))= max
t∈[0,n]∥R(t)∥L(X,Y ) for alln∈N∗.
Clearly, in the case of the operators (10.11) and (10.12) the current value Sv(t) at the moment
tdepends on the history of the values of vat the moments 0 ≤s≤tand, therefore, we refer
the operators of the form (10.11) or (10.12) as history-dependent operators. We extend this
definition to all operators S:C(R+;X)→C(R+;Y) satisfying condition (10.6) and, for this
reason, we say that the quasivariational inequalities of the form (10.10) are history-dependent
quasivariational inequalities. Their main feature consists in the fact that, at any moment t∈R+
the functional φdepends on the history of the solution up to the moment t,Su(t). This feature
makes the difference with respect to the quasivariational inequalities studied in literature in
which, usually, φwas assumed to depend on the current value of the solution, u(t).
Based on arguments of monotonicity and convexity, combined with the fixed point result,
Corollary 10.1, we have the following result.
120
Theorem 10.2. [Theorem 2 in [146]] If Assumptions 10.1–10.7hold true, then the variational
inequality (10.10)has a unique solution u∈C(R+;X).
The proof of Theorem 10.2 can be found in [146].
Chapter 11
Viscoplastic problems
This chapter is based on the paper [10]. We consider two quasistatic problems which describe
the contact between a viscoplastic body and an obstacle, the so-called foundation. The contact
is frictionless and is modelled with normal compliance of such a type that the penetration is not
restricted in the first problem, but is restricted with unilateral constraint, in the second one. For
each problem we derive a variational formulation, then we prove its unique solvability. Next,
we prove the convergence of the weak solution of the first problem to the weak solution of the
second problem, as the stiffness coefficient of the foundation converges to infinity.
11.1 Mechanical models and their weak solvability
In this section we discuss the weak solvability of two viscoplastic contact models in the following
physical setting. A viscoplastic body occupies a bounded domain Ω ⊂Rd,(d∈ {2,3}) with
a Lipschitz continuous boundary Γ, divided into three measurable parts Γ 1, Γ2and Γ 3, such
that meas(Γ 1)>0. The body is subject to the action of body forces of density f0. We also
assume that it is fixed on Γ 1and surface tractions of density f2act on Γ 2. On Γ 3, the body is
in frictionless contact with a deformable obstacle, the so-called foundation. We assume that the
process is quasistatic, and we study the contact process in the interval of time R+= [0,∞).
In the first problem the contact is modeled with normal compliance in such a way that the
penetration is not limited. Under these conditions, the classical formulation of the problem is
the following.
Problem 11.1. Find a displacement field u: Ω×R+→Rdand a stress field σ: Ω×R+→Sd
such that
˙σ=Eε(˙u) +G(σ,ε(u)) inΩ×(0,∞), (11.1)
Divσ+f0=0 inΩ×(0,∞), (11.2)
(11.3)
121
122
u=0 on Γ1×(0,∞), (11.4)
σν=f2 on Γ2×(0,∞), (11.5)
−σν=p(uν) on Γ3×(0,∞), (11.6)
στ=0 on Γ3×(0,∞), (11.7)
u(0) =u0,σ(0) =σ0 in Ω. (11.8)
In order to simplify the notation, we do not indicate explicitly the dependence of various
functions on the variables xort. Equation (11.1) represents the viscoplastic constitutive law of
the material; equation (11.2) is the equilibrium equation; conditions (11.4) and (11.5) are the
displacement and traction boundary conditions, respectively, and condition (11.7) shows that
the tangential stress on the contact surface, denoted στ, vanishes. We use it here since we
assume that the contact process is frictionless. Finally, (11.8) represents the initial conditions
in whichu0andσ0denote the initial displacement and the initial stress field, respectively. The
functionpinvolved in the contact condition (11.6) verifies the following assumption.
Assumption 11.1. p:R→R+such that
There exists Lp>0such that |p(r1)−p(r2)| ≤Lp|r1−r2|for allr1, r2∈R.
(p(r1)−p(r2))(r1−r2)≥0for allr1, r2∈R.
p(r) = 0 for allr<0.
Condition (11.6) combined with Assumption 11.1 shows that when there is separation between
the body and the obstacle (i.e. when uν<0), then the reaction of the foundation vanishes
(sinceσν= 0); also, when there is penetration (i.e. when uν≥0), then the reaction of the
foundation is towards the body (since σν≤0) and it is increasing with the penetration (since p
is an increasing function). Finally, we note that in this condition the penetration is not restricted
and the normal stress is uniquely determined by the normal displacement.
Condition (11.6) was first introduced by Oden and Martin, see [95, 127], in the study of
dynamic contact problems with elastic and viscoelastic materials. The term normal compliance
for this condition was first used by Klarbring, Mikeliˇ c and Shillor, see [84, 85]. A first example
of normal compliance function pwhich satisfies condition (11.6) is
p(r) =cνr+ (11.9)
wherecνis a positive constant. In this case condition (11.6) shows that the reaction of the
foundation is proportional to the penetration. A second example of normal compliance function
pwhich satisfies condition (11.6) is given by
pν(r) =
cνr+ifr≤α,
cναifr>α,
123
whereαis a positive coefficient related to the wear and hardness of the surface and, again, cν>0.
In this case the contact condition (11.6) means that when the penetration is too large, i.e. when
it exceedsα, the obstacle backs off and offers no additional resistance to the penetration. We
conclude that in this case the foundation has an elastic-perfectly plastic behavior.
In the second problem the contact is again modeled with normal compliance but in such
a way that the penetration is limited and associated to a unilateral constraint. The classical
formulation of the problem is the following.
Problem 11.2. Find a displacement field u: Ω×R+→Rdand a stress field σ: Ω×R+→Sd
such that
˙σ=Eε(˙u) +G(σ,ε(u)) in Ω×(0,∞), (11.10)
Divσ+f0=0 in Ω×(0,∞), (11.11)
u=0 on Γ1×(0,∞), (11.12)
σν=f2 on Γ2×(0,∞), (11.13)
uν≤g, σ ν+p(uν)≤0,
(uν−g)(σν+p(uν)) = 0
on Γ3×(0,∞), (11.14)
στ=0 on Γ3×(0,∞), (11.15)
u(0) =u0,σ(0) =σ0 in Ω. (11.16)
Hereg≥0 is given and pis a function which satisfies the following assumption.
Assumption 11.2. p: ]− ∞,g]→R+is a given function such that:
There exists Lp>0 :|p(r1)−p(r2)| ≤Lp|r1−r2|for allr1, r2≤g.
(p(r1)−p(r2))(r1−r2)≥0for allr1, r2≤g.
p(r) = 0 for allr<0.
Recall that condition (11.14) was first introduced in [78]. Combined with Assumption 11.2
it shows that when there is separation between the body and the obstacle (i.e. when uν<0),
then the reaction of the foundation vanishes (since σν= 0); moreover, the penetration is limited
(sinceuν≤g) andgrepresents its maximum value. When 0 ≤uν< g then the reaction of
the foundation is uniquely determined by the normal displacement (since −σν=p(uν)) and,
whenuν=g, the normal stress is not uniquely determined but is submitted to the restriction
−σν≥p(g). Such a condition shows that the contact follows a normal compliance condition
124
of the form (11.6) but up to the limit gand then, when this limit is reached, the contact
follows a Signorini-type unilateral condition with the gap g. For this reason we refer to the
contact condition (11.14) as a normal compliance contact condition with finite penetration and
unilateral constraint or, for simplicity, a normal compliance condition with finite penetration.
We conclude from above that this case models an elastic-rigid behavior of the foundation. Also,
note that when g= 0 condition (11.14) becomes the classical Signorini contact condition in a
form with a zero gap function,
uν≤0, σ ν≤0, σ νuν= 0 on Γ 3×(0,∞).
Moreover, when g>0 andp= 0, condition (11.6) becomes the Signorini contact condition in a
form with a gap function,
uν≤g, σ ν≤0, σ ν(uν−g) = 0 on Γ 3×(0,∞).
The last two conditions model the contact with a perfectly rigid foundation.
We made the following assumptions.
Assumption 11.3. E= (Eijkl) : Ω×Sd→Sd;Eijkl=Eklij=Ejikl∈L∞(Ω),1≤i,j,k,l ≤d.
There exists mE>0 such that Eτ·τ≥mE∥τ∥2for allτ∈Sd,a.e.in Ω.
Assumption 11.4. G: Ω×Sd×Sd→Sd.
There exists LG>0 such that ∥G(x,σ1,ε1)− G(x,σ2,ε2)∥ ≤LG(∥σ1−σ2∥+∥ε1−ε2∥)
for allσ1,σ2,ε1,ε2∈Sd,a.e.x∈Ω.
The mapping x7→ G(x,σ,ε) is measurable on Ω ,for anyσ,ε∈Sd.
The mapping x7→ G(x,0,0) belongs to Q.
Assumption 11.5. f0∈C(R+;L2(Ω)d),f2∈C(R+;L2(Γ2)d).
Assumption 11.6. u0∈V,σ0∈Q.
Assumption 11.7. u0∈U,σ0∈Q,
In Assumption 11.7, Udenotes the set of admissible displacements defined by
U={v∈V:vν≤gon Γ 3}. (11.17)
We define the operator P:V→Vand the function f:R+→Vby equalities
(Pu,v)V=∫
Γ3p(uν)vνdΓ for allu,v∈V, (11.18)
(f(t),v)V=∫
Ωf0(t)·vdx+∫
Γ2f2(t)·vdΓ for allv∈V, t∈R+.(11.19)
We have the following variational formulation of Problem 11.1.
125
Problem 11.3. Find a displacement field u:R+→Vand a stress field σ:R+→Q, such that
σ(t) =Eε(u(t)) +∫t
0G(σ(s),ε(u(s)))ds+σ0− Eε(u0) (11.20)
and
(σ(t),ε(v))Q+ (Pu(t),v)V= (f(t),v)Vfor allv∈V (11.21)
for allt∈R+.
We have the following variational formulation for Problem 11.2.
Problem 11.4. Find a displacement field u:R+→Uand a stress field σ:R+→Q, such that
σ(t) =Eε(u(t)) +∫t
0G(σ(s),ε(u(s)))ds+σ0− Eε(u0) (11.22)
and
(σ(t),ε(v)−ε(u(t)))Q+ (Pu(t),v−u(t))V≥(f(t),v−u(t))Vfor allv∈U(11.23)
hold, for all t∈R+.
In the study of the Problem 11.3 we obtained the following results.
Lemma 11.1. [Lemma 4.3 in [10]] Assumptions 11.3,11.4and 11.6hold. Then, for each
functionu∈C(R+;V)there exists a unique function Su∈C(R+;Q)such that
Su(t) =∫t
0G(Su(s) +Eε(u(s)),ε(u(s)))ds+σ0− Eε(u0)∀t∈R+. (11.24)
Moreover, the operator S:C(R+;V)→C(R+;Q)satisfies the following property: for every
n∈Nthere exists rn>0such that
∥Su1(t)− Su2(t)∥Q≤rn∫t
0∥u1(s)−u2(s)∥Vds (11.25)
∀u1,u2∈C(R+;V),∀t∈[0,n].
Next, using the operator S:C(R+;V)→C(R+;Q) defined in Lemma 11.1 we obtained the
following equivalence result.
Lemma 11.2. [Lemma 4.4 in [10]] Assumptions 11.1and11.3–11.6hold and let (u,σ)be a
couple of functions such that u∈C(R+;V)andσ∈C(R+;Q). Then, (u,σ)is a solution of
Problem 11.3if and only if for all t∈R+, the following equalities hold:
σ(t) =Eε(u(t)) +Su(t), (11.26)
(Eε(u(t)),ε(v))Q+ (Su(t),ε(v))Q+ (Pu(t),v)V= (f(t),v)V∀v∈V. (11.27)
126
Let us define the operator A:V→Vand the functional φ:Q×V→Rby equalities
(Av,w)V= (Eε(v),ε(w))Q+ (Pv,w)V∀v,w∈V, (11.28)
φ(τ,v) = (τ,ε(v))Q∀τ∈Q,v∈V. (11.29)
With this notation we consider the problem of finding a function u:R+→Vsuch that, for all
t∈R+, the following inequality holds:
(Au(t),v−u(t))V+φ(Su(t),v)−φ(Su(t),u(t)) (11.30)
≥(f(t),v−u(t))V∀v∈V.
Applying Theorem 10.2, we have got the following result.
Theorem 11.1. [Theorem 4.1 in [10]] If Assumptions 11.1and11.3–11.6hold true, then Prob-
lem 11.3 has a unique solution, which satisfies
u∈C(R+;V),σ∈C(R+;Q). (11.31)
The proof of Theorem 11.1 can be found in [10].
In the study of the Problem 11.4 we obtained the following results.
Lemma 11.3. [Lemma 4.5 in [10]] Assumptions 11.2–11.5and11.7hold and let (u,σ)be a
couple of functions such that u∈C(R+;U)andσ∈C(R+;Q). Then, (u,σ)is a solution of
Problem 11.4if and only if for all t∈R+, the equality and the inequality below hold:
σ(t) =Eε(u(t)) +Su(t), (11.32)
(Eε(u(t)),ε(v)−ε(u(t)))Q+ (Su(t),ε(v)−ε(u(t)))Q (11.33)
+(Pu(t),v−u(t))V≥(f(t),v−u(t))V∀v∈U.
Using similar arguments with those used to prove the previous theorem we have got the
following result.
Theorem 11.2. [Theorem 4.2 in [10]] If Assumptions 11.2–11.5and11.7hold true, then Prob-
lem 11.4 has a unique solution, which satisfies
u∈C(R+;U),σ∈C(R+;Q). (11.34)
The proof of Theorem 11.2 was given in [10].
127
11.2 A convergence result
Everywhere in this section we assume that the function psatisfies Assumption 11.2 and let qbe
a function which satisfies the following assumption.
Assumption 11.8. q: [g,+∞)→R+is a given function such that:
There exists Lq>0 :|q(r1)−q(r2)| ≤Lq|r1−r2|for allr1, r2≥g.
(q(r1)−q(r2))(r1−r2)>0for allr1, r2≥g, r 1̸=r2.
q(g) = 0.
Letµ>0 and consider the function pµdefined by
pµ(r) =
p(r) ifr≤g,
1
µq(r) +p(g) ifr>g.(11.35)
Using Assumptions 11.2 and 11.8 it follows that the function pµsatisfies:
pµ:R→R+.
There exists Lpµ>0 such that
|pµ(r1)−pµ(r2)| ≤Lpµ|r1−r2|for allr1, r2∈R.
(pµ(r1)−pµ(r2))(r1−r2)≥0 for allr1, r2∈R.
pµ(r) = 0 for all r<0.
This allows us to consider the operator Pµ:V→Vdefined by
(Pµu,v)V=∫
Γ3pµ(uν)vνda for allu,v∈V. (11.36)
We note that Pµis a monotone, Lipschitz continuous operator.
We also consider the contact problem with normal compliance and infinite penetration when
the contact condition (11.6) is replaced with
−σν=pµ(uν) on Γ 3×(0,∞). (11.37)
In this condition µrepresents a penalization parameter which may be interpreted as a deforma-
bility of the foundation, and then1
µis the surface stiffness coefficient. Indeed, when µis smaller
the reaction force of the foundation to penetration is larger and so the same force will result in
a smaller penetration, which means that the foundation is less deformable. When µis larger the
reaction force of the foundation to penetration is smaller, and so the foundation is less stiff and
more deformable.
128
The variational formulation of the problem with normal compliance and finite penetration
associated to function pµis as follows.
Problem 11.5. Find a displacement field uµ:R+→Vand a stress field σµ:R+→Qsuch
that, for all t∈R+, the following equalities hold:
σµ(t) =Eε(uµ(t)) +∫t
0G(σµ(s),ε(uµ(s)))ds+σ0− Eε(u0),
(σµ(t),ε(v))Q+ (Pµuµ(t),v)V= (f(t),v)Vfor allv∈V.
It follows from Theorem 11.1 that Problem 11.5 has a unique solution ( uµ,σµ) which satisfies
(11.31). In addition, according to Theorem 11.2, Problem 11.4 has a unique solution ( u,σ) which
satisfies (11.34). The behavior of the solution ( uµ,σµ) asµ→0 is given in the following result.
Theorem 11.3. [Theorem 5.1 in [10]] If Assumptions 11.1–11.8hold, then the solution (uµ,σµ)
of Problem 11.5converges to the solution (u,σ)of Problem 11.4, that is
∥uµ(t)−u(t)∥V+∥σµ(t)−σ(t)∥Q→0 (11.38)
asµ→0, for allt∈R+.
For the proof of Theorem 11.3 we refer to [10].
In addition to the mathematical interest in the result above, this result is important from
the mechanical point of view, since it shows that the weak solution of the viscoplastic contact
problem with normal compliance and finite penetration may be approached as closely as one
wishes by the solution of the viscoplastic contact problem with normal compliance and infinite
penetration, with a sufficiently small deformability coefficient.
Remark 11.1. A numerical validation of this convergence result can be found in Section 6 of the
paper [10]. Fully discrete schemes for the numerical approximation of the contact problems were
introduced and implemented. Finally, numerical simulations in the study of a two-dimensional
example were presented.
Chapter 12
Electro-elasto-viscoplastic contact
problems
This chapter is based on the paper [20]. We consider a mathematical model which describes
the quasistatic frictionless contact between a piezoelectric body and a foundation. The novelty
of the model consists in the fact that the foundation is assumed to be electrically conductive,
the material’s behavior is described with an electro-elastic-visco-plastic constitutive law, the
contact is modelled with normal compliance and finite penetration and the problem is studied
on unbounded time interval. We derive a variational formulation of the problem and prove
existence, uniqueness and regularity results.
12.1 The mechanical model
In this section we describe an electro-elastic-visco-plastic model in the following physical setting.
An electro-elasto-viscoplastic body occupies a bounded domain Ω ⊂Rd(d= 1,2,3) with a
Lipschitz continuous boundary Γ. The body is subject to the action of body forces of density f0
and volume electric charges of density q0. The boundary of the body is subjected to mechanical
and electrical constraints. To describe the mechanical constraints we consider a partition of Γ
into three measurable parts Γ 1, Γ2and Γ 3such that meas(Γ 1)>0. We assume that the body is
fixed on Γ 1and surface tractions of density f2act on Γ 2. On Γ 3, the body is in contact with an
electrically conductive obstacle. The contact is frictionless and is modelled with a version of the
normal compliance condition with finite penetration, which takes into account the conductivity
of the foundation. To describe the electrical constraints we consider a partition of Γ 1∪Γ2into
two measurable sets Γ aand Γ bsuch that meas(Γ a)>0. We assume that the electrical potential
vanishes on Γ aand the surface electric charges of density qbare prescribed on Γ b. Also, during
the process, there may be electrical charges on the part of the body which is in contact with the
foundation and which vanish when contact is lost. We assume that the problem is quasistatic,
and we study the problem in the interval of time R+= [0,∞).
129
130
The classical formulation of the contact problem defined above, is as follows.
Problem 12.1. Find a displacement field u: Ω×R+→Rd, a stress field σ: Ω×R+→Sd,
an electric potential field φ: Ω×R+→R, and an electric displacement field D: Ω×R+→Rd
such that
˙σ=Aε(˙u)− E∗E( ˙φ) +G(σ,ε(u),D,E(φ)) inΩ×(0,∞), (12.1)
˙D=βE( ˙φ) +Eε(˙u) +G(D,E(φ),σ,ε(u)) inΩ×(0,∞), (12.2)
Divσ+f0=0 inΩ×(0,∞), (12.3)
divD=q0 inΩ×(0,∞), (12.4)
u=0 on Γ1×(0,∞), (12.5)
σν=f2 on Γ2×(0,∞), (12.6)
φ= 0 on Γa×(0,∞), (12.7)
D·ν=qb on Γb×(0,∞), (12.8)
uν≤g, σ ν+hν(φ−φF)pν(uν)≤0,
(uν−g)(σν+hν(φ−φF)pν(uν)) = 0,
on Γ3×(0,∞), (12.9)
στ=0 on Γ3×(0,∞), (12.10)
D·ν=pe(uν)he(φ−φF) on Γ3×(0,∞), (12.11)
u(0) =u0,σ(0) =σ0, φ(0) =φ0,D(0) =D0 in Ω. (12.12)
In order to simplify the notation, we do not indicate explicitly the dependence of various
functions on the variables x∈Ω∪Γ andt∈R+. Equations (12.1) and (12.2) represent the
electro-elasto-viscoplastic constitutive law of the material. Equations (12.3) and (12.4) are the
equilibrium equations for the stress and the electric displacement fields, respectively. Conditions
(12.5) and (12.6) are the displacement and traction boundary conditions, and conditions (12.7)–
(12.8) represent the electric boundary conditions. Also, (12.12) represents the initial conditions
in whichu0,σ0,φ0, andD0denote the initial displacement, the initial stress, the initial electric
potential field and the initial electric displacement field, respectively.
We turn to the boundary conditions (12.9)–(12.11) which describe the mechanical and elec-
trical conditions on the potential contact surface Γ 3; there,g>0 is a given bound for the normal
displacement and φFdenotes the electric potential of the foundation.
First, (12.9) represents the normal compliance contact condition with finite penetration in
whichpνis a prescribed nonnegative function which vanishes when its argument is negative and
131
hνis a positive function, the stiffness coefficient. Recall that this condition, first introduced in
[78] in the study of a purely mechanic contact problem, contains as particular cases both the
Signorini contact condition and the classical normal compliance contact condition described,
see for instance [59, 138]. We note that (12.9) shows that when there is no contact (i.e. when
uν<0) thenσν= 0 and, therefore, the normal pressure vanishes; when there is contact (i.e.
whenuν≥0) thenσν≤0 and, therefore, the reaction of the foundation is towards the body. The
functiongrepresents the maximum interpenetration of body’s and foundations’s asperities. Note
also that the stiffness coefficient is assumed to depend on the difference between the potential
of the foundation and the body’s surface which is one of the novelties of the model.
Next, condition (12.10) shows that the tangential stress on the contact surface vanishes. We
use it here since we assume that the contact process is frictionless. An important extension
of the results in this paper would take into consideration frictional conditions on the contact
surface Γ 3.
Finally, (12.11) is a regularized electrical contact condition on Γ 3, similar to that used in
[8, 9, 90]. Here perepresents the electrical conductivity coefficient, which vanishes when its
argument is negative, and heis a given function. Condition (12.11) shows that when there is no
contact at a point on the surface (i.e. when uν<0), then the normal component of the electric
displacement field vanishes, and when there is contact (i.e. when uν≥0) then there may be
electrical charges which depend on the difference between the potential of the foundation and
the body’s surface. Note also that if the foundation is assumed to be insulated then there are
no charges on Γ 3during the process and, therefore, D·ν= 0 on Γ 3×(0,∞). This condition
can be recovered from (12.11) by taking pe≡0.
Note that in (12.1)–(12.12) the coupling between the mechanical unknowns ( u,σ) and the
electrical unknowns ( φ,D) arises both in the constitutive equations (12.1)–(12.2) and the contact
conditions (12.9)–(12.11). This feature of the problem (12.1)–(12.12) is a consequence of the
assumption that the foundation is conductive. It represents one of the differences with respect
to the model treated in [60] and leads to additional mathematical difficulties.
12.2 Weak formulation and main results
In this section we discuss the weak solvability of the electro-elastic-visco-plastic model (12.1)–
(12.12). We assume that the elasticity tensor, the piezoelectric tensor and the electric permit-
tivity tensor satisfy the following conditions.
Assumption 12.1. A= (Aijkl) : Ω×Sd→Sd.
Aijkl=Aklij=Ajikl∈L∞(Ω),1≤i,j,k,l ≤d.
There exists mA>0 such that Aτ·τ≥mA∥τ∥2for allτ∈Sd,a.e.in Ω.
Assumption 12.2. E= (eijk) : Ω×Sd→Rd.
eijk=eikj∈L∞(Ω),1≤i,j,k≤d.
132
Assumption 12.3. β= (βij) : Ω×Rd→Rd.
βij=βji∈L∞(Ω),1≤i,j≤d.
There exists mβ>0such thatβE·E≥mβ∥E∥2for allE∈Rd,a.e.x∈Ω.
Assumption 12.4. G: Ω×Sd×Sd×Rd×Rd→Sd.
There exists LG>0 such that
∥G(x,σ1,ε1,D1,E1)− G(x,σ2,ε2,D2,E2)∥ ≤LG(∥σ1−σ2∥+∥ε1−ε2∥+∥D1−D2∥+
∥E1−E2∥)for allσ1,σ2,ε1,ε2∈Sd,D1,D2,E1,E2∈Rd,a.e.x∈Ω.
The mapping x7→ G (x,σ,ε,D,E) is measurable on Ω ,for anyσ,ε∈SdandD,E∈
Rd.
The mapping x7→ G(x,0,0,0,0) belongs to Q.
Assumption 12.5. G: Ω×Rd×Rd×Sd×Sd→Rd.
There exists LG>0 such that
∥G(x,D1,E1,σ1,ε1)−G(x,D2,E2,σ2,ε2)∥ ≤LG(∥D1−D2∥+∥E1−E2∥+∥σ1−σ2∥+
∥ε1−ε2∥)
for allD1,D2,E1,E2∈Rd,σ1,σ2,ε1,ε2∈Sd,a.e.x∈Ω.
The mapping x7→G(x,D,E,σ,ε) is measurable on Ω ,for anyD,E∈Rdandσ,ε∈
Sd.
The mapping x7→G(x,0,0,0,0) belongs to L2(Ω)d.
These assumptions are reasonable from physical point of view, see for instance [45, 59, 73,
138, 150]. In some applications, GandGare linear functions.
The functions prandhr(forr=ν, e) are such that the following assumptions hold true.
Assumption 12.6. pr: Γ3×R→R.
There exists Lr>0such that
|pr(x,u1)−pr(x,u2)| ≤Lr|u1−u2|for allu1, u2∈R,a.e.x∈Γ3.
There exists pr>0such that 0≤pr(x,u)≤prfor allu∈R,a.e.x∈Γ3.
The mapping x7→pr(x,u)is measurable on Γ3,for anyu∈R.
pr(x,u) = 0 for allu≤0,a.e.x∈Γ3.
Assumption 12.7. hr: Γ3×R→R.
There exists lr>0such that
|hr(x,φ1)−hr(x,φ2)| ≤lr|φ1−φ2|for allφ1, φ2∈R,a.e.x∈Γ3.
There exists hν>0such that 0≤hν(x,φ)≤hνfor allφ∈R,a.e.x∈Γ3.
There exists he>0such that |he(x,φ)| ≤hefor allφ∈R,a.e.x∈Γ3.
The mapping x7→hr(x,u)is measurable on Γ3,for anyφ∈R.
We also assume that the bound of the normal displacement and the electrical potential of
the foundation are as follows.
133
Assumption 12.8. g∈L2(Γ3), g≥0 a.e.on Γ 3.
Assumption 12.9. φF∈L2(Γ3).
Moreover, the density of the body forces and tractions, the volume and surface electric charge
densities have the following regularity.
Assumption 12.10. f0∈C(R+;L2(Ω)d),f2∈C(R+;L2(Γ2)d),
Assumption 12.11. q0∈C(R+;L2(Ω)), q b∈C(R+;L2(Γb)).
Finally, the initial data satisfy the following assumptions.
Assumption 12.12. u0∈U,σ0∈Q1,
Assumption 12.13. φ0∈W,D0∈ W 1.
Notice that Udenotes the set of admissible displacements defined by
U={v∈V:vν≤gon Γ 3}. (12.13)
Alternatively, we assume that there exists p∈[1,∞] such that
Assumption 12.14. f0∈W1,p
loc(R+;L2(Ω)d),f2∈W1,p
loc(R+;L2(Γ2)d),
Assumption 12.15. q0∈W1,p
loc(R+;L2(Ω)), q b∈W1,p
loc(R+;L2(Γb)).
Besides Assumptions 12.12–12.13, the initial data satisfy the following compatibility condi-
tions:
Assumption 12.16. (σ0,ε(v)−ε(u0))Q+Jν(φ0,u0,v−u0)≥(f(0),v−u0)Vfor allv∈U,
(D0,∇ψ)L2(Ω)d+ (q(0),ψ)W=Je(u0,φ0,ψ)for allψ∈W.
Here and below Jν:W×V×V→RandJe:V×W×W→Rdenote the functionals given by
Jν(φ,u,v) =∫
Γ3hν(φ−φF)pν(uν)vνda (12.14)
Je(u,φ,ψ ) =∫
Γ3pe(uν)he(φ−φF)ψda, (12.15)
for allu,v∈V, φ, ψ ∈W.
We consider the functions f:R+→Vandq:R+→Wdefined by
(f(t),v)V=∫
Ωf0(t)·vdx+∫
Γ2f2(t)·vda for allv∈V, t∈R+, (12.16)
(q(t),ψ)W=∫
Ωq0(t)ψdx−∫
Γbqb(t)ψda for allψ∈W, t∈R+. (12.17)
We have the following variational formulation of Problem 12.1.
134
Problem 12.2. Find a displacement field u:R+→U, a stress field σ:R+→Q1, an electric
potential field φ:R+→Wand an electric displacement field D:R+→ W 1such that
σ(t) =Aε(u(t))− E∗E(φ(t)) +∫t
0G(σ(s),ε(u(s)),D(s),E(φ(s)))ds (12.18)
+σ0− Aε(u0) +E∗E(φ0),
D(t) =βE(φ(t)) +Eε(u(t)) +∫t
0G(D(s),E(φ(s)),σ(s),ε(u(s)))ds (12.19)
+D0−βE(φ0)− Eε(u0)
and
(σ(t),ε(v)−ε(u(t)))Q+Jν(φ(t),u(t),v)−Jν(φ(t),u(t),u(t)) (12.20)
≥(f(t),v−u(t))Vfor allv∈U,
(D(t),∇ψ)L2(Ω)d+ (q(t),ψ)W=Je(u(t),φ(t),ψ)for allψ∈W, (12.21)
for allt∈R+.
We consider the spaces X=V×W,Y=Q×L2(Ω)d, together with the canonical inner
products ( ·,·)X, (·,·)Yand the associated norms ∥ · ∥ X,∥ · ∥ Y, respectively. In addition, for the
convenience of the reader we shall use the short hand notation
eG(u,φ,σ,D) =G(Aε(u)− E∗E(φ) +σ,ε(u),E(φ),βE(φ) +Eε(u) +D),
eG(u,φ,σ,D) =G(βE(φ)− Eε(u) +D,E(φ),ε(u),Aε(u)− E∗E(φ) +σ).
Lemma 12.1. [Lemma 5.1 in [20]] For all (u,φ)∈C(R+;X)there exists a unique couple of
functions (σI(u,φ),DI(u,φ))∈C1(R+;Y)such that, for all t∈R+, the following equalities
hold:
σI(u,φ)(t) =∫t
0eG(u(s),φ(s),σI(u,φ)(s),DI(u,φ)(s))ds (12.22)
+σ0− Aε(u0) +E∗E(φ0),
DI(u,φ)(t) =∫t
0eG(u(s),φ(s),σI(u,φ)(s),DI(u,φ)(s))ds (12.23)
+D0−βE(φ0)− Eε(u0).
Lemma 12.1 allows us to consider the operator S:C(R+,X)→C1(R+,Y) defined by
S(x) =(
σI(u,φ),−DI(u,φ))
∀x= (u,φ)∈C(R+,X). (12.24)
Moreover, it leads to the following equivalence result.
135
Lemma 12.2. [Lemma 5.2 in [20]] A quadruple of functions (u,σ,D,φ)which satisfy u∈
C(R+;U),σ∈C(R+;Q1), φ∈C(R+;W),D∈C(R+;W1)is a solution of Problem 12.2if and
only if
σ(t) =Aε(u(t)) +E∗∇φ(t) +σI(u,φ)(t), (12.25)
D(t) =−β∇φ(t) +Eε(u(t)) +DI(u,φ)(t), (12.26)
(Aε(u(t)),ε(v)−ε(u(t)))Q+ (E∗∇φ(t),ε(v)−ε(u(t)))Q (12.27)
+(σI(u,φ)(t),ε(v)−ε(u(t)))Q+Jν(φ(t),u(t),v)−Jν(φ(t),u(t),u(t))
≥(f(t),v−u(t))V∀v∈U,
(β∇φ(t),∇ψ)L2(Ω)d−(Eε(u(t)),∇ψ)L2(Ω)d−(
DI(u,φ)(t),∇ψ)
L2(Ω)d (12.28)
+Je(u(t),φ(t),ψ) = (q(t),ψ)W∀ψ∈W,
for allt∈R+.
To proceed, we consider the set K=U×W,the operator A:X→X, the functionals
φ:Y×X→Randj:X×X→R, and the function f:R+→Xdefined by
(Ax,y )X= (Aε(u),ε(v))Q+ (E∗∇φ,ε(v))Q (12.29)
−(Eε(u),∇ψ)L2(Ω)d+ (β∇φ,∇ψ)L2(Ω)d,
φ(z,x) = (σ,ε(u))Q+ (D,∇φ)L2(Ω)d, (12.30)
j(x,y) =Jν(φ,u,v) +Je(u,φ,ψ ), (12.31)
f= (f,q), (12.32)
for allx= (u,φ), y= (v,ψ)∈Xandz= (σ,D)∈Y. Note that the definition of the operator
Afollows by using Riesz’s representation theorem.
The next step is provided by the following result.
Lemma 12.3. [Lemma 5.3 in [20]] Let t∈R+,u∈C(R+,U),φ∈C(R+,W)and denote
x= (u,φ)∈C(R+,K). Then (12.27) –(12.28) hold if and only if x(t)satisfies the inequality
(Ax(t),y−x(t))X+φ(Sx(t),y)−φ(Sx(t),x(t)) (12.33)
+j(x(t),y)−j(x(t),x(t))≥(f(t),x(t)−y)X∀y∈K.
We continue with the following existence and uniqueness result.
Lemma 12.4. [Lemma 5.4 in [20]] There exists L0>0which depends on Ω,Γ1,Γ3,Aandβ
such that there exists a unique function x∈C(R+,K)which satisfies the inequality (12.33) for
allt∈R+, ifhνLν+heLe+pνlν+pele<L 0.
Based on these preliminaries steps, we have got the following results in the study of Problem
12.2 .
136
Theorem 12.1. [Theorem 4.1 in [20]] If Assumptions 12.1–12.13hold true, then there exists
L0>0which depends on Ω,Γ1,Γ3,Aandβsuch that Problem 12.2 has a unique solution, if
hνLν+heLe+pνlν+pele<L 0. (12.34)
Moreover, the solution satisfies
u∈C(R+;U),σ∈C(R+;Q1), φ∈C(R+;W),D∈C(R+;W1). (12.35)
Theorem 12.2. [Theorem 4.2 in [20]] If the inequality (12.34)and Assumptions 12.1–12.13hold
true, denoting by (u,σ,φ,D)the solution of Problem 12.2 obtained in Theorem 12.1, then:
1)Under Assumptions 12.14–12.15, the solution has the regularity
u∈W1,p
loc(R+;U),σ∈W1,p
loc(R+;Q1), φ∈W1,p
loc(R+;W),D∈W1,p
loc(R+,W1) (12.36)
and the following equalities hold, for almost any t∈R+:
˙σ(t) =Aε(˙u(t))− E∗E( ˙φ(t)) +G(σ(t),ε(u(t)),D(t),E(φ(t))), (12.37)
˙D(t) =βE( ˙φ(t)) +Eε(˙u(t)) +G(D(t),E(φ(t)),σ(t),ε(u(t))). (12.38)
2)Under Assumption 12.16, the solution satisfies the initial conditions
u(0) =u0,σ(0) =σ0, φ(0) =φ0,D(0) =D0. (12.39)
A quadruple of functions ( u,σ,φ,D) which satisfies (12 .18)–(12.21) for allt∈R+is called
a weak solution to the piezoelectric contact Problem 12.1. We conclude that Theorem 12.1
provides the unique weak solvability of Problem 12.1 and Theorem 12.2 provides a regularity
result of its weak solution.
Note that condition (12.34) represents a smallness assumption on the functions involved in
the boundary conditions of Problem 12.1. It is satisfied if, for instance, either the quantities pν,
hν,pe,he, or the quantities Lν,Le,lν,leare small enough. And, this means that either the
range of the functions pν,pe,hν,he, or the range of their derivatives with respect the second
variable (which exists, a.e.), is small enough. We conclude that the result in Theorem 12.1 works
in the case when either the normal compliance function, the stiffness coefficient, the electrical
conductivity coefficient and the electric charge function are small enough, or their variation is
small enough.
The proofs of Theorems 12.1 and 12.2, given in [20], are based on the abstract result, Theorem
10.2.
137
CAREER EVOLUTION AND DEVELOPMENT PLANS
138
Chapter 13
Further research directions
The overall goal is to improve the understanding of real-world problems governed by Partial
Differential Equations. The mathematical modeling via partial differential equations is founda-
tional to the field of mechanical engineering; it provides necessary information for efficient design
of technical systems. In particular, the contact models are used to analyze and test complex
industrial systems. Since it is not possible to find strong solutions for complex problems, a good
alternative is the weak solvability which allows to built efficient numerical approximations for
the weak solutions. This is a motivation for the candidate to continue to do research on the
direction of calculus of variations with applications in contact mechanics.
The most relevant further research directions envisaged from the candidate are the following:
•Qualitative and numerical analysis in the study of mixed variational problems The candi-
date intend to improve and extend the results in the papers [11, 68, 69, 70, 98, 99, 100, 101,
104, 105, 107, 111, 113]. Delivering uniqueness/multiplicity results, is one of the targets;
the following references can be helpful, [33, 120]. How we can approximate the solutions is
also of great interest. Regularization or perturbation techniques are envisaged. Also the
candidate is interested to the solvability of a class of mixed variational problems via hemi-
variational inequalities theory. The notion of hemivariational inequality was introduced in
[128] based on the properties of generalized gradient introduced and studied in [37, 38].
During the last two decades, a large number of works were devoted to the hemivariational
inequalities theory related to contact models; for a contribution of the candidate in the
field see the papers [4, 40, 41]. One target of the candidate is to extend and improve the
results obtained in the paper [107], by replacing into the mixed variational system the
variational inequality with a hemivariational inequality. Such a study allows to investigate
contact models with non-convex potentials via calculus of variations with Lagrange mul-
tipliers. The analysis of contact problems with adhesion or damage, via mixed variational
formulations, is also under attention; the following references can be helpful [140, 141, 142].
•Qualitative and numerical analysis in the study of variational systems via bipotentials The
139
140
candidate intends to improve and extend the results in [102, 106, 108] to more com-
plex variational systems via bipotentials (possibly non-separated). In particular time-
dependent/evolutionary models are envisaged. Numerical algorithms are also of interest
for the candidate.
•Variational formulations/ weak solutions via weighted Sobolev spaces in contact mechanics
A first step was already done, see [19]; see also [56, 89] for some fundamental mathematical
tools. The next steps will be related to the weak solvability of complex contact models for
various kind of materials; e.g. piezoelectric problems with singularities and degeneracies
(the following references can be helpful: [6, 9, 14, 17, 58, 60, 90, 123]).
•Variational formulations/ weak solutions via Lebesgue spaces with variable exponent, in
contact mechanics A first step was made in [21] for regularized antiplane contact problems
governed by nonlinearly elastic materials of Henky type. The candidate intends to improve
and extends the previous study to the non-regularized case. Also, the study of a class of
non-newtonian fluids is of interest for the candidate. The references [47, 51, 52, 53, 54, 86]
can be useful.
•Variational formulations/ weak solutions in contact mechanics, for materials with dry
porosity . The study of the behavior of non-classical materials (like materials with voids,
porous materials with dry porosity) is a challenging topic. To start, the candidate intends
to consult the following works [34, 35, 43, 71, 77, 126]. In the future the candidate envis-
ages to investigate the behavior of poro-therm materials, such a study being motivated by
the large applicability of such kind of materials.
•Optimal control problems in contact mechanics For the optimal control of variational in-
equalities we can refer for instance to [12, 18, 55, 91, 118, 119, 149]. Moreover, the recent
book [125] is devoted to the optimal control of linear or nonlinear elliptic problems, includ-
ing variational inequalities. Despite their mechanical relevance, optimal control problems
for contact models are not so frequent in the literature, the contact problems being strongly
nonlinear problems. The main aim of the candidate is to study optimal control problems
which consists of leading the stress tensor as close as possible to a given target, by acting
with a control on the boundary of the body. In the paper [103] a first step was already
done. Moreover, the candidate intends to study the optimal control for abstract variational
problems related to contact models, such as variational problems with Lagrange multipliers
and variational systems via bipotentials.
•Mathematical study for dissipative dynamic contact problems At this item, the main interest
lies in existence and uniqueness results for dynamic contact problems in elasticity, which
are dissipative. To give an example, we can consider a rod which is connected to a dashpot
at its left end and, at its right end it can come in contact with an obstacle; the obstacle
141
can be deformable (in such a case we have to use a normal compliance contact condition)
or rigid ( in this case we have to use the unilateral Signorini’s contact condition. The
techniques in the paper [82] and the references therein can be helpful. A first step was
already done, see the conference paper [46].
•Asymptotic analysis in contact mechanics We can found in the literature some stability
results for displacement-traction problems, see for example [73]; but not for contact prob-
lems. Due to their nonlinearity, the contact problems are difficult to be investigated from
the point of view of the asymptotic analysis. However there are some premises. For in-
stance, the papers [10, 20, 144, 146] were devoted to the existence and the uniqueness of
the solution of a class of contact problems on the unbounded time interval [0 ,∞). Using
the weak solutions at all moments t∈[0,∞),one idea is to define a dynamical system, to
seek for it the equilibrium points and to use a technique via Lyapunov functionals in order
to study the asymptotic behavior.
•Regularity results There are very few regularity results for contact problems. The field
is wide open and progress is likely to be slow. Let us give an example of a regularity
result we focus on. It is known that in the mixed variational approach, the weak solutions
of contact problems are pairs ( u,λ), λ∈DwhereDis a dual space, see e.g. D=S′,
S={γv|Γ3|v∈H1(Ω), γv = 0 a.e. on Γ 1}in Section 2.2 of the present manuscript.
However, the numerical analysis requires L2(Γ3)-regularity for λand, currently, this is an
open problem. The techniques in the book [49] and the references therein can be helpful.
•Convergence results Using similar techniques with those used in [11], the candidate intends
to investigate the convergence of the solutions of some regularized or perturbed problem
to the solution of the originate problem.
•Viscoelastic problems via fractional differential operators Fractional order operators are
suitable to model memory effects of various materials and systems of technical interest. In
particular, they can help to model viscoelastic materials, see e.g. [7, 31]. We also refer to
[2] for an efficient numerical method to integrate the constitutive law of fractional order
viscoelasticity. The fractional order derivatives were used to conceive a new component
spring-pot that interpolates between pure elastic and viscous behaviors. In [44] the authors
modified a standard linear solid model replacing a dashpot with a spring-pot of order α;
the fractional model was tested in human arterial segments. The candidate intends to
explore the weak formulations/weak solvability for spring-pot models.
More general, the candidate will focus on fractional calculus of variations, including weak
solutions of fractional partial differential equations. This topic began to be developed start-
ing with 1996 in order to better describe non-conservative mechanical systems. Currently,
the list of applications includes material sciences and mechanics of fractal and complex
media, see e.g. [32, 93, 94], just to mention a few.
142
Let us pick up a few open problems .
•In Section 7.1 it was discussed the following abstract problem: given f∈X,find (u,λ)∈
X×Ysuch thatλ∈Λ⊂Yand
J(v)−J(u) +b(v−u,λ) +j(λ,v)−j(λ,u)≥(f, v−u)X for allv∈X,
b(u,µ−λ) ≤0 for allµ∈Λ.
According to Theorem 7.2, this problem has a solution, unique in its first argument.
Let us draw the attention on a few points of interest.
1.Under the assumptions made in Section 7.1 the uniqueness/the multiplicity of the
solution in the second argument is an open question.
2.Let us focus on Assumptions 6.20 and 7.1 in the present manuscript; herein J:X→
[0,∞) is a Gˆ ateaux differentiable functional and for all η∈Λ, j(η,·) :X→[0,∞) is
a convex Gˆ ateaux differentiable functional. The approach adopted in previous work
essentially relies on this two hypotheses. The proof of the existence of the solutions
(u,λ) for a non-differential functional Jand a non-differential, in the second argument,
bifunctional jis of great interest from the mathematical point of view as well as
from the applications point of view, such a mathematical problem being connected
to more complex models with a better physical significance. The uniqueness/the
multiplicity of the solution in the second argument in a ”non-differential framework”
is also interesting.
3.to write an efficient approximating algorithm is also an unsolved problem at this
moment.
•In Subsection 5.1.2 it was formulated the following mixed problem: find ( u,φ,λ )∈X×
Y×Λ such that
a(u,v) +e(v,φ) +b(v,λ) = (f,v)X,v∈X,
c(φ,ψ)−e(u,ψ) +j(λ,φ,ψ ) = (q,ψ)Y,ψ∈Y,
b(u,µ−λ)≤0, µ∈Λ.
This variational formulation correspond to a frictionless unilateral contact model for electro-
elastic material. Let us mention a few points of interest here:
1.a better regularity of λ(L2-regularity)
2.to consider the frictional case; in this case existence, uniqueness, stability results are
expected and a numerical approach is also envisaged.
143
•In Part II of the present manuscript we discussed the weak solvability for a class of contact
problems via bipotentials theory. The following two variational problems were formulated.
(1) Findu∈U0⊂Vandσ∈Λ⊂L2
s(Ω)3×3such that
b(v,σ)−b(u,σ)≥(f,v−u)Vfor allv∈U0
b(u,µ)−b(u,σ)≥0 for allµ∈Λ,
see Problem 8.2, and
(2) Findu∈Vandσ∈Λ⊂L2
s(Ω)3×3such that
b(v,σ)−b(u,σ) +j(v)−j(u)≥(f,v−u)Vfor allv∈V
b(u,µ)−b(u,σ)≥0 for allµ∈Λ,
see Problem 9.2 in the present manuscript.
To solve such kind of variational problems in an abstract framework for ”non-separated”
formsbis of great interest in the next period.
•In Section 12.1 it was discussed the following model: find a displacement field u: Ω×R+→
Rd, a stress field σ: Ω×R+→Sd, an electric potential field φ: Ω×R+→R, and an
electric displacement field D: Ω×R+→Rdsuch that
˙σ=Aε(˙u)− E∗E( ˙φ) +G(σ,ε(u),D,E(φ)) in Ω ×(0,∞),
˙D=βE( ˙φ) +Eε(˙u) +G(D,E(φ),σ,ε(u)) in Ω ×(0,∞),
Divσ+f0=0 in Ω ×(0,∞),
divD=q0 in Ω ×(0,∞),
u=0 on Γ 1×(0,∞),
σν=f2 on Γ 2×(0,∞),
φ= 0 on Γ a×(0,∞),
D·ν=qb on Γ b×(0,∞),
uν≤g, σ ν+hν(φ−φF)pν(uν)≤0,
(uν−g)(σν+hν(φ−φF)pν(uν)) = 0,
on Γ 3×(0,∞),
στ=0 on Γ 3×(0,∞),
D·ν=pe(uν)he(φ−φF) on Γ 3×(0,∞),
u(0) =u0,σ(0) =σ0, φ(0) =φ0,D(0) =D0 in Ω.
We emphasize that this is a frictionless problem. To study the frictional case is an inter-
esting continuation of the present work.
144
•In Chapter 1 of the present thesis (Subsections 1.1.2 and 1.2.3), a priori error estimates
were presented for a class of piezoelectric contact problems, see [69, 68] for details; also,
in [69, 68], efficient algorithms to approximate the weak solutions were described. Such
kind of results are welcome in order to continue the study of the generalized saddle point
problems described in Chapters 2-7 of the present manuscript, firstly for the stationary
problems and nextly for the time-dependent or evolutionary problems.
•In the paper [4] it was studied the following mathematical model: find u,φ:¯Ω→Rsuch
that
(P) :
div (µ(x)∇u(x) +e(x)∇φ(x)) +f0(x) = 0 in Ω,
div (e(x)∇u(x)−β(x)∇φ(x)) =q0(x) in Ω,
u(x) = 0 on Γ 1,
φ(x) = 0 on Γ A,
µ(x)∂νu(x) +e(x)∂νφ(x) =f2(x) on Γ 2,
e(x)∂νu(x)−β(x)∂νφ(x) =qB(x) on Γ B,
−µ(x)∂νu(x)−e(x)∂νφ(x)∈h(x,u(x))∂j(x,u(x)) on Γ 3,
e(x)∂νu(x)−β(x)∂νφ(x)∈¯∂φ(x,φ(x)−φF(x)) on Γ 3.
This model describes the antiplane shear deformation of a piezoelectric cylinder in fric-
tional contact with a conductive foundation. The study was made under the following
assumptions.
(H1):µ∈L∞(Ω), β∈L∞(Ω), e∈L∞(Ω).There exist β∗,µ∗∈Rsuch that
β(x)≥β∗>0 andµ(x)≥µ∗>0 almost everywhere in Ω .
(H2):f0∈L2(Ω), q 0∈L2(Ω), f 2∈L2(Γ2), q B∈L2(ΓB), φ F∈L∞(Γ3).
(H3):h: Γ3×R→Ris a Carath´ eodory function (i.e. h(·,t) : Γ 3→Ris measurable,
for allt∈R, andh(x,·) :R→Ris continuous, a.e. on Γ 3). There exists a positive
constanth0such that 0 ≤h(x,t)≤h0, for allt∈R, a.e. on Γ 3.
(H4):j: Γ3×R→Ris a function which is measurable with respect to the first variable,
and there exists k∈L2(Γ3) such that a.e. on Γ 3and for allt1,t2∈Rwe have
|j(x,t1)−j(x,t2)| ≤k(x)|t1−t2|.
(H5):φ: Γ3×R→Ris a functional such that φ(·,t) : Γ 3→Ris measurable for each
t∈Randφ(x,·) :R→Ris convex and lower semicontinuous a.e. on Γ 3.
145
Theorem 13.1 (Theorem 2 in [4]) .Assume conditions (H1)-(H5)are fulfilled. Then
there exists at least one weak solution for problem ( P).
An interesting continuation of the previous work is related to the case of piezoelectric
materials having some ”perfect” insulators or ”perfect” conductors points. Such anisotropic
media lead to degenerate and singular mathematical problems. Notice that the presence of
some ”perfect” insulators or ”perfect” conductors points imposes, from the mathematical
point of view, some changes in the hypothesis ( H1). In particular, we have to assume that
infΩβ= 0, supΩβ=∞. Solving such a problem is an open question.
•In [103] the optimal control for an antiplane model it was investigated. Let us sketch
below the framework and the results. Let Ω ⊂R2be an open, bounded, connected set,
with Lipschitz continuous boundary Γ partitioned in three measurable parts Γ 1,Γ2,Γ3
such that the Lebesgue measures of Γ iis strictly positive, for each i∈ {1,2,3}.
We consider the following mechanical problem: find a displacement field u:¯Ω→Rsuch
that
div (µ(x)∇u(x)) +f0(x) = 0 in Ω, (13.1)
u(x) = 0 on Γ 1, (13.2)
µ(x)∂νu(x) =f2(x) on Γ 2, (13.3)
|µ(x)∂νu(x)| ≤g(x,|u(x)|),
µ(x)∂νu(x) =−g(x,|u(x)|)u(x)
|u(x)|ifu(x)̸= 0on Γ 3. (13.4)
Let us assume that
µ∈L∞(Ω), µ(x)≥µ∗>0 a.e. in Ω,µ∗big enough, (13.5)
f0∈L2(Ω), f 2∈L2(Γ2), (13.6)
g: Γ3×R+→R+such that there exists Lg>0 : (13.7)
|g(x,r 1)−g(x,r 2)| ≤Lg|r1−r2| ∀r1, r2∈R+,a.e.x∈Γ3; (13.8)
the mapping x7→g(x, r) is Lebesgue measurable on Γ 3,∀r∈R+; (13.9)
the mapping x7→g(x,0) belongs to L2(Γ3). (13.10)
Furthermore, we consider the Hilbert space
V={v∈H1(Ω)|γv= 0a.e.on Γ1}.
146
We are led to the following weak formulation of the problem (13.1)-(13.4): Givenf0∈
L2(Ω)andf2∈L2(Γ2),findu∈Vsuch that
(Au,v−u)V+j(u,v)−j(u,u)≥(f,v−u)V∀v∈V, (13.11)
where
A:V→V; (Au,v )V=∫
Ωµ(x)∇u(x)· ∇v(x)dx∀u, v∈V, (13.12)
j:V×V→R;j(u, v) =∫
Γ3g(x,|γu(x)|)|γv(x)|ds ∀u, v∈V,(13.13)
(f,v)V=∫
Ωf0(x)v(x)dx+∫
Γ2f2(x)γv(x)dΓ∀v∈V. (13.14)
Theorem 13.2. [Theorem 3.6 in [103]] Assume (13.5),(13.6)and (13.7). Then, the
problem (13.11)has a unique solution u∈Vwhich depends Lipschitz continuously on f.
For a fixed function f0∈L2(Ω),we consider the following state problem .
(PS1) Letf2∈L2(Γ2)(called control) be given. Find u∈Vsuch that
(Au,v−u)V+j(u,v)−j(u,u)≥∫
Ωf0(x) (v(x)−u(x))dx
+∫
Γ2f2(x)(γv(x)−γu(x))dΓ∀v∈V.(13.15)
For every control f2∈L2(Γ2),thestate problem (PS1) has a unique solution u∈V,
u=u(f2).
Now, we would like to act a control on Γ 2such that the resulting stress σbe as close as
possible to a given target
σd=
0 0 µ∂ud
∂x1
0 0 µ∂ud
∂x2
µ∂ud
∂x1µ∂ud
∂x20
whereudis a given function. Note that, since
∥σ−σd∥L2(Ω)3×3=√
2∥µ∇(u−ud)∥L2(Ω)≤√
2∥µ∥L∞(Ω)∥u−ud∥V,
σandσdwill be close one from another if the difference between the functions uandud
is small in the sense of V−norm. To give an example of a target of interest, ud,we can
147
considerud= 0.In this situation, by acting a control f2on Γ 2,the tensionσis small in
the sense of L2−norm, even if f0don’t vanishes in Ω .
Letα,β > 0 be two positive constants and let us define the following functional
L:L2(Γ2)×V→R, L (f2,u) =α
2∥u−ud∥2
V+β
2∥f2∥2
L2(Γ2). (13.16)
Furthermore, we denote
Vad={[u,f 2]|[u,f 2]∈V×L2(Γ2),such that (13 .15) is verified }
and we introduce the following optimal control problem ,
(POC1) Find [u∗,f∗
2]∈ V adsuch thatL(f∗
2,u∗) = min
[u,f2]∈Vad{
L(f2,u)}
.
Theorem 13.3. [Theorem 3.7 in [103]] Assume (13.5),(13.6),(13.7). Then, (POC 1)has
at least one solution (u∗,f∗
2).
Letρ>0.We define a functional jρ:V×V→Ras follows,
jρ(u, v) =∫
Γ3g(x,√
(γu(x))2+ρ2−ρ)(√
(γv(x))2+ρ2−ρ)dΓ∀u, v∈V. (13.17)
Let us state the following problem: Givenρ > 0, f0∈L2(Ω)andf2∈L2(Γ2),find the
displacement field uρ∈Vsuch that
(Auρ,v−uρ)V+jρ(uρ,v)−jρ(uρ,uρ)≥∫
Ωf0(x) (v(x)−uρ(x))dx
+∫
Γ2f2(x)(γv(x)−γuρ(x))dΓ∀v∈V.(13.18)
Theorem 13.4. [Theorem in 4.11 [103]] Assume (13.5),(13.6),(13.7). Then, problem
(13.18)has a unique solution uρ∈Vwhich depends Lipschitz continuously on f.
Let us fixρ>0 andf0∈L2(Ω).We introduce the following regularized problem
(PS2) Letf2∈L2(Γ2)(called control). Find u∈Vsuch that
(Au,v−u)V+jρ(u,v)−jρ(u,u)≥(f0,v−u)L2(Ω)
+(f2,γv−γu)L2(Γ2)∀v∈V.(13.19)
Let us define the following admissible set,
Vρ
ad={[u,f 2]|[u,f 2]∈V×L2(Γ2),such that (13 .19) is verified }.
(POC2) Find [¯u,¯f2]∈ Vρ
adsuch thatL(¯f2,¯u) = min
[u,f2]∈Vρ
ad{
L(f2,u)}
.
148
Theorem 13.5 (Theorem 4.13 in [103]) .Assume (13.5),(13.6),(13.7). Then, (POC2)
has at least one solution (¯u,¯f2).
Let us replace the hypotheses (b) and (d) in (13.7), with the following stronger ones,
g(x,·)∈C1a.e.x∈Γ3,
there exists Lg>0 : ∂2g(x,r) ≤Lg∀r∈R+,a.ex∈Γ3,
there exists M > 0 :|g(x,r)| ≤M∀r∈R+,a.e.x∈Γ3.
(13.20)
Theorem 13.6. (Optimality condition)[Theorem 4.14 in [103]] Any optimal control ¯f2of
the state problem (PS2) verifies
¯f2=−1
βγ(p(¯f2)), (13.21)
wherep(¯f2)is the unique solution of the variational equation
α(¯u−ud,w)V= (p(¯f2),Aw +D2
2jρ(¯u,¯u)w)V∀w∈V, (13.22)
and, for all v∈V,
(D2
2jρ(¯u,¯u)w,v)V=∫
Γ3∂2g(x,√
(γ¯u(x))2+ρ2−ρ)(γ¯u(x))2
(γ¯u(x))2+ρ2γw(x)γv(x)dΓ
+∫
Γ3g(x,√
(γ¯u(x))2+ρ2−ρ)ρ2
[(γ¯u(x))2+ρ2]3/2γw(x)γv(x)dΓ,
¯u=u(¯f2)being the solution of (PS2) with f2=¯f2.
Under the hypotheses (13.5), (13.6), (a) and (c) of (13.7) and (13.20), we have two con-
vergence results.
Theorem 13.7. [Theorem 5.16 in [103]] Let ρ>0, f0∈L2(Ω)andf2∈L2(Γ2)be given.
Ifuρ, u∈Vare the solutions of problems (PS2)and(PS1),respectively, then,
uρ→uinVasρ→0. (13.23)
Theorem 13.8. [Theorem 5.17 in [103]] Let [¯uρ,¯f2ρ]be a solution of the problem (POC2).
Then, there exists a solution of the problem (POC1), [u∗,f∗
2], such that
¯uρ→u∗inVasρ→0,
¯f2ρ⇀f∗
2inL2(Γ2)asρ→0.(13.24)
Following such a technique, the following questions are under attention in the future:
1.an extension of this study to the general case 3D;
2.to study the optimal control for 3D models taking into account various contact con-
ditions; to start, a contact condition with normal compliance is envisaged.
Chapter 14
Further plans
14.1 On the scienti c and professional career
After obtaining the Ph.D. degree in Mathematics, the candidate published in internationally
recognized journals such as SIAM Journal on Scientific Computing, Zeitschrift f¨ ur Angewandte
Mathematik und Mechanik, Nonlinear Analysis -Theory, Methods and Applications, Nonlinear
Analysis: Real World Applications, Journal of Mathematical Analysis and Applications, Math-
ematics and Mechanics of Solids, Communications on Pure and Applied Analysis, Journal of
Global Optimization, Proceedings of The Royal Society of Edinburgh, Section: A Mathematics,
The Quarterly Journal of Mechanics and Applied Mathematics, European Journal of Applied
Mathematics, The Australian and New Zealand Industrial and Applied Mathematical Journal,
Acta Applicandae Mathematicae, Quarterly of Applied Mathematics, Advanced Nonlinear Stud-
ies.In the future the candidate intends to do a research activity allowing to continue to publish
in international journals of hight level.
The dissemination of the results is also under attention. During the years the candidate
attended several international conferences. In the future the candidate intend to participate to
prestigious international meetings in order to disseminate the best results. Also the candidate
intends to be involved in the organization of scientific meetings.
The research activity of the candidate was realized mainly at the Department of Mathematics
of the University of Craiova, where the author has a permanent position, but also at some De-
partments of Mathematics from other universities in Europe: Stuttgart University, Technische
University of Munchen, University of Perpignan, where the candidate has had research collabo-
rations concretized in the publication of some scientific papers with colleagues from abroad. In
the future the candidate wishes to continue the collaborations started in the past and to establish
new contacts.
In recent years the candidate was reviewer at several journals. In the future the candidate
intends to extend the editorial activities for scientific journals.
The candidate intend to apply for national/international/interdisciplinary research projects
149
150
as manager or member of teams. A few steps were already made: director of a GRANT PN-
II-RU-TE CNCS-UEFISCDI; responsible of the Romanian side for a French-Romanian research
project LEA Math Mode CNRS-IMAR; member of several teams for national, international or
interdisciplinary research projects.
14.2 On the academical career
The teaching activities of the candidate were realized at the University of Craiova where along
the years his activity has concretized in teaching seminars or courses on different topics: Theo-
retical Mechanics (seminar); Real Analysis (seminar); PDE’s (seminar); Applied Nonlinear Anal-
ysis (course and seminar for MASTER); Control Theory (course and seminar for MASTER),
Mathematical Modeling by Differential Equations (course and laboratory for MASTER), Math-
ematical Modeling in Contact Mechanics (course and seminar for MASTER); Singular Problems
in Mathematical Physics (course and seminar for MASTER), Special Chapters of PDE’s (course
and seminar), Evolution Equations (course and seminar for MASTER), Numerical Analysis for
PDE’s (laborator for MASTER), etc. The candidate was co-author of two monographs pub-
lished at Springer and Cambridge University Press. These monographs can be found in several
libraries such as: Cornell University Library, McGill University Library, Stanford University,
Mathematics and Statistics Library, The University of Arizona, Denver University Libraries,
UCLA Library (University of California, Los Angeles Library), The University of Manchester,
University of Colorado, Eastern Michigan University Library, to give a few examples. In the
future , the candidate plans to publish Lecture Notes and new monographs addressed to students
or researchers.
During the last 10 years the candidate advised several bachelor’s degree or dissertation theses;
also, in recent years the candidate has collaborated with PH.D. students who became co-authors
and collaborators of the candidate (Ionicˇ a Andrei, Maria-Magdalena Boureanu, Raluca Ciurcea,
Nicu¸ sor Costea). By obtaining this habilitation the candidate plans to extend her advising
activity to Ph.D. theses.
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