简介
Summary:
Publisher Summary 1
Variational methods are applied to prove the existence of weak solutions for boundary value problems from the deformation theory of plasticity as well as for the slow, steady state flow of generalized Newtonian fluids including the Bingham and Prandtl-Eyring model. For perfect plasticity the role of the stress tensor is emphasized by studying the dual variational problem in appropriate function spaces. The main results describe the analytic properties of weak solutions, e.g. differentiability of velocity fields and continuity of stresses. The monograph addresses researchers and graduate students interested in applications of variational and PDE methods in the mechanics of solids and fluids.
目录
Table Of Contents:
Introduction 1(4)
Weak solutions to boundary value problems in the deformation theory of perfect elastoplasticity 5(35)
Preliminaries 5(1)
The classical boundary value problem for the equilibrium state of a perfect elastoplastic body and its primary functional formulation 6(9)
Relaxation of convex variational problems in non reflexive spaces. General construction 15(12)
Weak solutions to variational problems of perfect elastoplasticity 27(13)
Differentiability properties of weak solutions to boundary value problems in the deformation theory of plasticity 40(67)
Preliminaries 40(2)
Formulation of the main results 42(10)
Approximation and proof of Lemma 2.1.1 52(5)
Proof of Theorem 2.1.1 and a local estimate of Caccioppoli-type for the stress tensor 57(14)
Estimates for solutions of certain systems of PDE's with constant coefficients 71(5)
The main lemma and its iteration 76(13)
Proof of Theorem 2.1.2 89(9)
Open Problems 98(2)
Remarks on the regularity of minimizers of variational functionals from the deformation theory of plasticity with power hardening 100(7)
Appendix A 107(144)
A.1 Density of smooth functions in spaces of tensor-valued functions 107(4)
A.2 Density of smooth functions in spaces of vector-valued functions 111(5)
A.3 Some properties of the space BD(Ω; Rn) 116(10)
A.4 Jensen's inequality 126(5)
Quasi-static fluids of generalized Newtonian type 131(76)
Preliminaries 131(12)
Partial C1 regularity in the variational setting 143(24)
Local boundedness of the strain velocity 167(13)
The two-dimensional case 180(13)
The Bingham variational inequality in dimensions two and three 193(11)
Some open problems and comments concerning extensions 204(3)
Fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening law 207(44)
Preliminaries 207(4)
Some function spaces related to the Prandtl-Eyring fluid model 211(5)
Existence of higher order weak derivatives and a Caccioppoli-type inequality 216(12)
Blow-up: the proof of Theorem 4.1.1 for n = 3 228(7)
The two-dimensional case 235(2)
Partial regularity for plastic materials with logarithmic hardening 237(11)
A general calss of constitutive relations 248(3)
Appendix B 251(3)
B.1 Density results 251(3)
Notation and tools from functional analysis 254(6)
Bibliography 260(8)
Index 268
Introduction 1(4)
Weak solutions to boundary value problems in the deformation theory of perfect elastoplasticity 5(35)
Preliminaries 5(1)
The classical boundary value problem for the equilibrium state of a perfect elastoplastic body and its primary functional formulation 6(9)
Relaxation of convex variational problems in non reflexive spaces. General construction 15(12)
Weak solutions to variational problems of perfect elastoplasticity 27(13)
Differentiability properties of weak solutions to boundary value problems in the deformation theory of plasticity 40(67)
Preliminaries 40(2)
Formulation of the main results 42(10)
Approximation and proof of Lemma 2.1.1 52(5)
Proof of Theorem 2.1.1 and a local estimate of Caccioppoli-type for the stress tensor 57(14)
Estimates for solutions of certain systems of PDE's with constant coefficients 71(5)
The main lemma and its iteration 76(13)
Proof of Theorem 2.1.2 89(9)
Open Problems 98(2)
Remarks on the regularity of minimizers of variational functionals from the deformation theory of plasticity with power hardening 100(7)
Appendix A 107(144)
A.1 Density of smooth functions in spaces of tensor-valued functions 107(4)
A.2 Density of smooth functions in spaces of vector-valued functions 111(5)
A.3 Some properties of the space BD(Ω; Rn) 116(10)
A.4 Jensen's inequality 126(5)
Quasi-static fluids of generalized Newtonian type 131(76)
Preliminaries 131(12)
Partial C1 regularity in the variational setting 143(24)
Local boundedness of the strain velocity 167(13)
The two-dimensional case 180(13)
The Bingham variational inequality in dimensions two and three 193(11)
Some open problems and comments concerning extensions 204(3)
Fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening law 207(44)
Preliminaries 207(4)
Some function spaces related to the Prandtl-Eyring fluid model 211(5)
Existence of higher order weak derivatives and a Caccioppoli-type inequality 216(12)
Blow-up: the proof of Theorem 4.1.1 for n = 3 228(7)
The two-dimensional case 235(2)
Partial regularity for plastic materials with logarithmic hardening 237(11)
A general calss of constitutive relations 248(3)
Appendix B 251(3)
B.1 Density results 251(3)
Notation and tools from functional analysis 254(6)
Bibliography 260(8)
Index 268
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