Synopses & Reviews
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.
Description
Includes bibliographical references (p. [260]-267) and index.
Table of Contents
Introduction
1 Weak solutions to boundary value problems in the deformation theory of perfect elastoplasticity
1.0. Preliminaries
1.1. The classical boundary value problem for the equilibrium state of a perfect elastoplastic body and its primary functional formulation
1.2. Relaxation of convex variational problems in non reflexive spaces.
General construction
1.3. Weak solutions to variational problems of perfect elastoplasticity
2 Differentiability properties of weak solutions to boundary value problems in the deformation theory of plasticity
2.0. Preliminaries
2.1. Formulation of the main results
2.2. Approximation and proof of Lemma 2.1.1
2.3. Proof of Theorem 2.1.1 and local estimate of Caccioppoli-type for the stress tensor
2.4. Estimates for solutions of certain systems of PDE's with constant coeffcients
2.5. The main lemma and its iteration
2.6. Proof of Theorem 2.1.2
2.7. Open Problems
2.8. Remarks on the regularity of minimizers of variational functionals from the deformation theory of plasticity with power hardening
Appendix A
A.1 Density of smooth functions in spaces of tensor-valued functions
A.2 Density of smooth functions in spaces of vector-valued functions
A.3 Some properties of the space BD
A.4 Jensen's inequality
3 Quasi-static fluids of generalized Newtonian type
3.0. Preliminaries
3.1. Partial C1 regularity in the variational setting
3.2. Local boundedness of the strain velocity
3.3. The two-dimensional case
3.4. The Bingham variational inequality in dimensions two and three
3.5. Some open problems and comments concerning extensions
4 Fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening law
4.0. Preliminaries
4.1. Some functions spaces related to the Prandtl-Eyring fluid model
4.2. Existence of higher order weak derivatives and a Caccioppoli-type inequality
4.3. Blow-up: the proof of Theorem 4.1.1 for n=3
4.4. The two-dimensional case
4.5. Partial regularity for plastic materials with logarithmic hardening
4.6. A general class of constitutive relations
Appendix B
B.1 Density results
Notation and tools from functional analysis