Inequalityproblems

In a remarkably short time, the field of inequality problems has seen considerable development in mathematics and theoretical mechanics. Applied mechanics and the engineering sciences have also benefitted from these developments in that open problems have been treated and entirely new classes of problems have been formulated and solved. This book is an outgrowth of seven years of seminars and courses on inequality problems in mechanics for a variety of audiences in the Technical University of Aachen, the Aristotle University of Thessaloniki, the University of Hamburg and the Technical University of Milan. The book is intended for a variety of readers, mathematicians and engineers alike, as is detailed in the Guidelines for the Reader. It goes without saying that the work of G. Fichera, J. L. Lions, G. Maier, J. J. Moreau in originating and developing the theory of inequality problems has considerably influenced the present book. I also wish to acknowledge the helpful comments received from C. Bisbos, J. Haslinger, B. Kawohl, H. Matthies, H. O. May, D. Talaslidis and B. Werner. Credit is also due to G. Kyriakopoulos and T. Mandopoulou for their exceptionally diligent work in the preparation of the fmal figures. Many thanks are also due to T. Finnegan and J. Gateley for their friendly assistance from the linguistic standpoint. I would also like to thank my editors in Birkhiiuser Verlag for their cooperation, and all those who helped in the preparation of the manuscript.

1. Introductory Topics.- 1. Essential Notions and Propositions of Functional Analysis.- 1.1 Topological Vector Spaces and Related Subjects.- 1.1.1 Topological Spaces and Continuous Mappings.- 1.1.2 Locally Convex Topological Vector Spaces, Normed Spaces and Linear Mappings.- 1.2 Duality in Topological Vector Spaces.- 1.2.1 Duality. Weak and Strong Topologies.- 1.2.2 Topologically Dual Pairs of Vector Spaces.- 1.2.3 Duality in Normed and Hilbert Spaces.- 1.2.4 Transpose of a Continuous Linear Mapping Scales of Hilbert Spaces. The Lax-Milgram Theorem.- 1.3 Certain Function Spaces and Their Properties.- 1.3.1 The Spaces $$ {C^m}\left( \Omega \right),{C^m}\left( {\overline \Omega } \right),D\left( \Omega \right),D\left( {\overline \Omega } \right)and{L^p}\left( \Omega \right) $$.- 1.3.2 Spaces of Distributions.- 1.3.3 Sobolev Spaces.- 1.3.4 Trace Theorem. Imbedding Properties of Sobolev Spaces.- 1.3.5 The Space of Functions of Bounded Deformation.- 1.4 Additional Topics.- 1.4.1 Elements of the Theory of Vector-valued Functions and Distributions.- 1.4.2 Elements of Differential Calculus.- 1.4.3 Supplementary Notions and Propositions.- 2. Elements of Convex Analysis.- 2.1 Convex Sets and Functionals.- 2.1.1 Definitions.- 2.1.2 Lower Semicontinuous Convex Functionals.- 2.2 Minimization of Convex Functionals.- 2.2.1 Existence of a Minimum.- 2.2.2 Variational Inequalities.- 2.3 Subdifferentiability.- 2.3.1 Definitions and Related Propositions.- 2.3.2 One-Sided Directional Gâteaux-Differential.- 2.4 Subdifferential Calculus.- 2.4.1 The Subdifferential of a Sum of Functionals and of a Composite Functional.- 2.4.2 The Relative Interior of R(?f).- 2.5 Conjugates of Convex Functionals.- 2.5.1 The Classes ?(X) and ?0(X).- 2.5.2 The Conjugacy Operation.- 2.6 Maximal Monotone Operators.- 2.6.1 Definitions and Fundamental Results.- 2.6.2 Maximal Monotone Graphs in ?2.- 2. Inequality Problems.- 3. Variational Inequalities and Superpotentials.- 3.1 Mechanical Laws and Constraints.- 3.1.1 Generalized Forces and the Principle of Virtual Power.- 3.1.2 Multivalued Laws and Constraints in Mechanics.- 3.1.3 Minimization Problems and Variational Inequalities Characterizing the Equilibrium Configurations.- 3.1.4 Dissipative Laws. A Note on the Eigenvalue Problem for Superpotential Laws.- 3.2 Superpotentials and Duality.- 3.2.1 The Hypothesis of Normal Dissipation.- 3.2.2 Duality of Variational Principles.- 3.3 Subdifferential Boundary Conditions and Constitutive Laws.- 3.3.1 Subdifferential Boundary Conditions.- 3.3.2 Subdifferential Constitutive Laws I.- 3.3.3 Subdifferential Constitutive Laws II.- 3.3.4 Extension of Subdifferential Relations to Function Spaces.- 4. Variational Inequalities and Multivalued Convex and Nonconvex Problems in Mechanics.- 4.1 Two General Variational Inequalities and the Derivation of Variational Inequality “Principles” in Mechanics.- 4.1.1 Variational Inequalities of the Fichera Type.- 4.1.2 Variational Inequalities of Other Types.- 4.1.3 The Derivation of Variational Inequality “Principles” in Mechanics.- 4.2 Coexistent Phases. The Morphology of Material Phases.- 4.2.1 Neoclassical Processes and Gibbsian States. Rules for Coexistent Phases.- 4.2.2 Minimum Problems for Gibbsian States.- 4.2.3 Comparison of Gibbsian States. Some Results of the Dynamic Problem.- 4.3 Nonconvex Superpotentials.- 4.3.1 Introduction and Brief Survey of the Basic Mathematical Properties.- 4.3.2 Nonconvex Superpotentials. Hemivariational Inequalities and Substationarity Principles.- 4.3.3 Generalizations of the Hypothesis of Normal Dissipation.- 4.4 The Integral Inclusion Approach to Inequality Problems.- 5. Friction Problems in the Theory of Elasticity.- 5.1 The Static B.V.P.- 5.1.1 The Classical Formulation.- 5.1.2 The Variational Formulation.- 5.2 Existence and Uniqueness Propositions.- 5.2.1 Equivalent Minimum Problem. The case mes ?U>0.- 5.2.2 Study of the Case ?U=Ø.- 5.2.3 Further Properties of the Solution.- 5.3 Dual Formulation. Complementary Energy.- 5.3.1 Minimization of the Complementary Energy.- 5.3.2 Duality.- 5.4 The Dynamic B.V.P.- 5.4.1 Classical and Variational Formulations of the Problem.- 5.4.2 Existence of Solution.- 5.4.3 The Regularized Problem.- 5.4.4 The Uniqueness of the Solution.- 5.5 A Note on Other Types of Friction Problems.- 6. Subdifferential Constitutive Laws and Boundary Conditions.- 6.1 Subdifferential Material Laws and Classical Boundary Conditions.- 6.1.1 Formulation of the Problem.- 6.1.2 The Existence and Uniqueness of the Solution.- 6.1.3 Duality.- 6.2 Linear Elastic Material Law and Subdifferential Boundary Conditions.- 6.2.1 Formulation of the Boundary Conditions.- 6.2.2 Existence and Uniqueness Propositions.- 6.2.3 Duality.- 6.3 Subdifferential Material Laws and Subdifferential Boundary Conditions. Minimum Propositions for Nonmonotone Laws.- 6.3.1 Formulation and Study of the Problem.- 6.3.2 Nonmonotone Laws.- 6.4 The Corresponding Dynamic and Incremental Problems.- 7. Inequality Problems in the Theory of Thin Elastic Plates.- 7.1 Static Unilateral Problems of von Kármán Plates.- 7.1.1 Generalities.- 7.1.2 Boundary Conditions and Corresponding Variational Formulations.- 7.1.3 The Existence of the Solution.- 7.1.4 In-Plane Unilateral Boundary Conditions.- 7.2 The Unilateral Buckling Problem. Eigenvalue Problems for Variational Inequalities.- 7.2.1 Formulation of the Problem.- 7.2.2 A General Proposition on the Existence of the Solution.- 7.2.3 Application to the Buckling Problem.- 7.2.4 Extension of the Rayleigh-Quotient Rule to Unilateral Problems.- 7.3 Dynamic Unilateral Problems of von Kármán Plates.- 7.3.1 Boundary Conditions and Variational Inequalities.- 7.3.2 Existence Proposition.- 7.3.3 Uniqueness Proposition.- 8. Variational and Hemivariational Inequalities in Linear Thermoelasticity.- 8.1 B.V.P.s and their Variational Formulations.- 8.1.1 Classical Formulations.- 8.1.2 Variational Formulations.- 8.2 Existence and Uniqueness Propositions.- 8.2.1 Study of Problem 1.- 8.2.2 Study of Problem 2. Some Remarks on Related Problems.- 8.3 Generalizations and Related Variational Inequalities.- 8.4 Hemivariational Inequalities in Linear Thermoelasticity.- 8.4.1 Formulation of Certain General Problems.- 8.4.2 An Existence Result for a Hemivariational Inequality.—A Model Problem.- 9. Variational Inequalities in the Theory of Plasticity and Viscoplasticity.- 9.1 Elastic Viscoplastic Materials.- 9.1.1 Formulation of the Dynamic Problem, Existence and Uniqueness of the Solution.- 9.1.2 The Quasi-static Problem.- 9.2 Elastic Perfectly Plastic Materials.- 9.2.1 Formulation of the Quasi-static Problem.- 9.2.2 Existence and Uniqueness Propositions.- 9.3 Rigid Viscoplastic Flow Problems.- 9.3.1 Classical Formulation of the General Dynamic Problem.- 9.3.2 The Functional Framework and Existence Propositions.- 9.3.3 The Relation Between Velocity and Stress Fields.- 9.4 Other Problems on Bingham Fluids.- 9.4.1 Laminar Flow in a Cylindrical Pipe.- 9.4.2 Heat Transfer in Rigid Viscoplastic Flows.- 9.4.3 Variational Inequalities in the Case of the General Law ? ? ?w(D).- 3. Numerical Applications.- 10. The Numerical Treatment of Static Inequality Problems.- 10.1 Unilateral Contact and Friction Problems.- 10.1.1 Discrete Forms of the Problems of Minimum Potential and Complementary Energy.- 10.1.2 Applications.- 10.2 Torsion of Cylindrical or Prismatic Bars With Convex Strain-Energy Density.- 10.2.1 Formulation of the Problem.- 10.2.2 Discretization and Numerical Application.- 10.3 A Linear Analysis Approach to Certain Classes of Inequality Problems.- 10.3.1 Description of the Method.- 10.3.2 Applications.- 11. Incremental and Dynamic Inequality Problems.- 11.1 The Elastoplastic Calculation of Cable Structures.- 11.1.1 Formulation of the Problem as a Linear Complementarity Problem and Related Expressions.- 11.1.2 Multilevel Decomposition Techniques.- 11.1.3 Application.- 11.2 Incremental Elastoplastic Analysis.L.C.P.s, Variational Inequalities and Minimum Propositions.- 11.3 Dynamic Unilateral Contact Problems.- Epilogue.- Appendices.- Appendix I. Some Basic Notions [20] [112] [321] [322].- Appendix II. Rigidifying Velocity Fields. Objectivity [112] [197] [322].- Appendix III. Dissipation [112]..- Appendix IV. Plasticity and Thermodynamics [75] [196].- List of Notations.- References.