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

by Fuchs, Martin; Seregin, Gregory

• ISBN13: 9783540413974
• ISBN10: 3540413979
• Format: Paperback
• Copyright: 2001-03-01
• Publisher: Springer Verlag

Summary

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.

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