Variational Hemivariational Inequalities with Applications 1st Edition by Mircea Sofonea, Stanislaw Migorski 1498761585 9781498761581

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ISBN 10: 1498761585

ISBN 13: 9781498761581

Author: Mircea Sofonea, Stanislaw Migorski

This research monograph represents an outcome of the cross-fertilization between nonlinear functional analysis and mathematical modelling, and demonstrates its application to solid and contact mechanics. Based on authors’ original results, it introduces a general fixed point principle and its application to various nonlinear problems in analysis and mechanics. The classes of history-dependent operators and almost history-dependent operators are exposed in a large generality. A systematic and unified presentation contains a carefully-selected collection of new results on variational-hemivariational inequalities with or without unilateral constraints. A wide spectrum of static, quasistatic, dynamic contact problems for elastic, viscoelastic and viscoplastic materials illustrates the applicability of these theoretical results. Written for mathematicians, applied mathematicians, engineers and scientists, it is also a valuable tool for graduate students and researchers in nonlinear analysis, mathematical modelling, mechanics of solids, and contact mechanics.

Table of contents:

I: A Fixed Point Principle

1. Abstract Setting and Preliminary Applications

1.1 Statement of the principle

1.2 Background on functional analysis

1.3 Classical fixed point theorems

1.4 Applications to elliptic variational inequalities

1.5 Conclusions

2. History-Dependent Operators

2.1 Spaces of continuous functions

2.2 Definitions and basic properties

2.3 Fixed point properties

2.4 History‐dependent equations in Hilbert spaces

2.5 Nonlinear implicit equations in Banach spaces

2.6 History‐dependent variational inequalities

2.7 Relevant particular cases

3. Displacement-Traction Problems in Solid Mechanics

3.1 Modeling of displacement-traction problems

3.2 A viscoplastic displacement-traction problem

3.3 A viscoelastic displacement-traction problem

3.4 History-dependent constitutive laws

3.5 Primal variational formulation

3.6 Dual variational formulation

II: Variational-Hemivariational Inequalities

4. lements of Nonsmooth Analysis

4.1 Monotone and pseudomonotone operators

4.2 Bochner-Lebesgue spaces

4.3 Subgradient of convex functions

4.4 Subgradient in the sense of Clarke

4.5 Miscellaneous results

5. Elliptic Variational-Hemivariational Inequalities

5.1 A class of subdifferential inclusions

5.2 Dual formulation

5.3 A first existence and uniqueness result

5.4 A general existence and uniqueness result

5.5 A continuous dependence result

5.6 A penalty method

5.7 Relevant particular cases

6. History-dependent variational-hemivariational inequalities

6.1 An existence and uniqueness result

6.2 A continuous dependence result

6.3 A penalty method

6.4 Relevant particular cases

7. Evolutionary Variational-Hemivariational Inequalities

7.1 A class of evolutionary inclusions

7.2 An existence and uniqueness result

7.3 A continuous dependence result

7.4 Relevant particular cases

III: Applications to Contact Mechanics

8. Static contact problems

8.1 Modeling of static contact problems

8.2 A contact problem with normal compliance

8.3 A contact problem with subdifferential friction law

8.4 A first contact problem with unilateral constraints

8.5 A second contact problem with unilateral constraints

9. Time-dependent and quasistatic contact problems

9.1 Physical setting and mathematical models

9.2 Two time-dependent elastic contact problems

9.3 A quasistatic viscoplastic contact problem

9.4 A time-dependent viscoelastic contact problem

9.5 A quasistatic viscoelastic contact problem

10. Dynamic Contact Problems

10.1 A viscoelastic contact problem with normal damped response

10.2 A viscoplastic contact problem with normal compliance

10.3 A viscoelastic contact problem with normal compliance

10.4 Conclusions

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Tags: Mircea Sofonea, Stanislaw Migorski, Variational, Hemivariational