Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids

Abstract

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