Models and analysis of quasistatic contact: variational methods

272 Pages·2004·1.045 MB·English

by  Meir Shillor, Mircea Sofonea, Józef Joachim Telega (auth.)

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M. Shillor M. Sofonea J.J. Telega
Models and Analysis
of Quasistatic Contact
Variational Methods
123
Authors
MeirShillor Jo´zefJoachimTelega
OaklandUniversity PolishAcademyofSciences
Dept.MathematicsandStatistics Inst.Fundamental
Rochester,MI48309,USA TechnologicalResearch
Swietokrzyska21
MirceaSofonea 00-049Warsaw,Poland
Universite´dePerpignan
LaboratoiredeThe´oriedesSyste`mes
52AvenuedeVilleneuve
66860PerpignanCedex,France
M.Shillor M.Sofonea J.J.Telega,ModelsandAnalysisofQuasistaticContact,Lect.Notes
Phys.655(Springer,BerlinHeidelberg2004),DOI10.1007/b99799
LibraryofCongressControlNumber:004095625
ISSN0075-8450
ISBN3-540-22915-9SpringerBerlinHeidelbergNewYork
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Preface
Currently the Mathematical Theory of Contact Mechanics is emerging from
its infancy, and a point has been reached where a unified presentation of the
results, scattered throughout a variety of publications, is needed.
The aim of this monograph is to partially address this need by providing
state-of-the-artmathematicalmodellingandanalysisofsomeofthephenom-
ena that take place when a deformable body comes into quasistatic contact.
We present models for the processes, describe the mathematical results, and
provide representative proofs. A comprehensive list of recent references sup-
plements this work. Between the time we started writing this monograph
and the present, W. Han and M. Sofonea published the book “Quasistatic
Contact Problems in Viscoelasticity and Viscoplasticity,” which focuses on
mathematicalandnumericalanalysisofcontactproblemsforviscoelasticand
viscoplastic materials.
Our book, divided into three parts, with 14 chapters, is intended as a
unifiedandreadilyaccessiblesourceformathematicians,appliedmathemati-
cians, mechanicians, engineers and scientists, as well as advanced students.
It is organized in three different levels, so that readers who are not fluent in
the Theory of Variational Inequalities can easily access the modelling part
and the main mathematical results.
Representative proofs, which may be skipped upon first reading, are pro-
vided for those who are interested in the mathematical methods. Part I con-
tainsmodelsoftheprocessesinvolvedincontact.Itiswrittenatthefirstlevel
for those who have an interest in Contact Mechanics or Tribology, and mini-
malbackgroundindifferentialequationsandinitial-boundaryvalueproblems.
The processes for which we provide various models are friction, heat gener-
ation and thermal effects, wear, adhesion and damage. Several sections are
devoted to each one of these topics and the relationships among them.
The second level of the book, which focuses on the settings of the models
as initial-boundary value problems and their variational formulations, can
be found in Part II. It requires some background in modern mathematics,
although preliminary material is provided in the first chapter. Each chapter
describes a few problems with a common underlying theme. The third level
deals with the proofs of the theorems. In each chapter in Part II, the proofs
of one or two theorems can be found as examples of the mathematical tools
VIII Preface
used.ThisisalsothelevelforthosemathematiciansinterestedintheTheory
of Variational Inequalities and its applications.
We observe that as a result of the specific problems posed by contact
models, the theory had to be extended and some of these generalizations are
also provided. Part III presents a short review and many references of re-
cent results for dynamic contact, one-dimensional contact and miscellaneous
problemsnotcoveredinthebook.Theconcludingchapterisasummaryand
a discussion of open problems and future directions. The topics of static and
evolutiongeometricallynonlinearcontactproblems,includingstructures,are
currently in preparation by the authors.
We would like to acknowledge and thank all of our collaborators for their
contributions that led to this book, especially to Professors Kevin T. An-
drews, Weimin Han and Kenneth L. Kuttler. We would also like to thank
Prof. Dr. Wolf Beiglbo¨ck, Senior Physics Editor, and his staff for their help
in bringing this monograph to your hand.
ThethirdauthorgratefullyacknowledgespartialsupportbytheMinistry
of Research and Information Technology (Poland) through the grant No.
T11F00325.
Auburn Hills, Michigan, USA Meir Shillor
Perpignan, France Mircea Sofonea
Warsaw, Poland J´ozef Joachim Telega
July 2004
Contents
1 Introduction.............................................. 1
Part I Modelling
2 Evolution Equations, Contact and Friction................ 9
2.1 The Modelling of Contact Processes ...................... 10
2.2 Physical Setting and Equations of Evolution ............... 11
2.3 Constitutive Relations .................................. 12
2.4 Boundary Conditions ................................... 15
2.5 Dimensionless Variables ................................. 16
2.6 Contact Conditions ..................................... 18
2.7 Friction Coefficient ..................................... 23
2.8 On Coulomb and Tresca Conditions....................... 28
3 Additional Effects Involved in Contact.................... 31
3.1 Thermal Effects ........................................ 32
3.2 Wear ................................................. 36
3.3 Adhesion .............................................. 39
3.4 Damage ............................................... 44
4 Thermodynamic Derivation............................... 49
4.1 The Formalism......................................... 50
4.2 Isothermal Unilateral Contact with Friction and Adhesion ... 57
4.3 Isothermal Contact with Normal Compliance, Friction
and Adhesion .......................................... 60
4.4 Thermoviscoelastic Material with Damage ................. 61
4.5 Short Summary ........................................ 63
5 A Detailed Representative Problem ...................... 65
5.1 Problem Statement ..................................... 66
5.2 Variational Formulation ................................. 69
5.3 An Existence and Uniqueness Result...................... 81
X Contents
Part II Models and Their Variational Analysis
6 Mathematical Preliminaries............................... 85
6.1 Notation .............................................. 85
6.2 Function Spaces ....................................... 88
6.3 Auxiliary Material...................................... 90
6.4 Constitutive Operators.................................. 96
7 Elastic Contact ........................................... 101
7.1 Frictional Contact with Normal Compliance ............... 101
7.2 Frictional Contact with Signorini’s Condition .............. 104
7.3 Bilateral Frictional Contact.............................. 106
7.4 Contact with Dissipative Friction Potential ................ 108
7.5 Proof of Theorems 7.3.1 and 7.4.1 ........................ 113
8 Viscoelastic Contact ...................................... 117
8.1 Frictionless Contact with Signorini’s Condition............. 118
8.2 Proof of Theorem 8.1.1.................................. 120
8.3 Frictional Contact with Normal Compliance ............... 122
8.4 Proof of Theorem 8.3.1.................................. 125
8.5 Bilateral Frictional Contact.............................. 126
8.6 Frictional Contact with Normal Damped Response ......... 131
9 Viscoplastic Contact...................................... 135
9.1 Frictionless Contact with Signorini’s Condition............. 136
9.2 Proof of Theorem 9.1.3.................................. 141
9.3 Frictional Contact with Normal Compliance ............... 142
9.4 Proof of Theorem 9.3.1.................................. 144
9.5 Bilateral Frictional Contact.............................. 156
9.6 Contact with Dissipative Friction Potential ................ 160
10 Slip or Temperature Dependent Frictional Contact ....... 163
10.1 Elastic Contact with Slip Dependent Friction .............. 164
10.2 Proof of Theorem 10.1.1................................. 167
10.3 Viscoelastic Contact
with Total Slip Rate Dependent Friction .................. 171
10.4 Thermoelastic Contact with Signorini’s Condition .......... 174
10.5 Thermoviscoelastic Bilateral Contact...................... 177
11 Contact with Wear or Adhesion .......................... 183
11.1 Bilateral Frictional Contact with Wear .................... 184
11.2 Frictional Contact with Normal Compliance and Wear ...... 186
11.3 Frictional Contact with Normal Compliance
and Wear Diffusion ..................................... 188
Contents XI
11.4 Adhesive Viscoelastic Bilateral Contact ................... 193
11.5 Proof of Theorem 11.4.1................................. 197
11.6 Membrane in Adhesive Contact .......................... 203
12 Contact with Damage .................................... 207
12.1 Viscoelastic Contact with Normal Compliance and Damage .. 208
12.2 Proof of Theorem 12.1.1................................. 211
12.3 Viscoelastic Contact
with Normal Damped Response and Damage .............. 216
12.4 Viscoplastic Contact with Dissipative Friction Potential
and Damage ........................................... 218
Part III Miscellaneous Problems and Conclusions
13 Dynamic, One-Dimensional and Miscellaneous Problems.. 225
13.1 Dynamic Contact Problems.............................. 226
13.2 One-Dimensional Dynamic or Quasistatic Contact .......... 230
13.3 Miscellaneous Results ................................... 232
14 Conclusions, Remarks and Future Directions ............. 235
References.................................................... 241
Index......................................................... 257