Variational methods for problems from plasticity theory and for generalized Newtonian fluids
type: Item
instance of: scholarly article
title: Variational methods for problems from plasticity theory and for generalized Newtonian fluids
zbMATH Open document ID: 0964.76003
author: G. Seregin
author: M. Fuchs
published in: Lecture Notes in Mathematics
publication date: 2001-01-14T00:00:00Z
review text: The monograph is devoted to a rigorous mathematical analysis of variational problems describing the equilibrium configuration of certain types of perfect elastoplastic solids, and also the stationary flows of some incompressible generalized Newtonian fluids. The authors concentrate on variational problems from the deformation theory of plasticity and on fluid models whose stress-strain relation can be formulated in terms of dissipative potential. The mathematical form of both problems is very close, and is reduced to the study of variational integrals with convex integrands depending only on the symmetric part of gradients of unknown vector-valued functions. The integrands have the linear growth, and the problem of finding suitable classes of admissible deformations is not trivial. Another difficulty of the considered problems consists in their formulations in non-reflexive spaces. The authors attack these problems by formulating a dual variational problem for the stress tensor, and by the relaxation method. Chapter 1 contains a general scheme for the relaxation of convex variational problems being coercive on some non-reflexive spaces like \(L^1\) or \(W^1_1\). The authors give a detailed description of the approach to the perfect plasticity. Besides plasticity with power hardening and perfect plasticity, the authors investigate here the model with logarithmic hardening. Hence the integrand is not of power growth, and it is necessary to introduce new function spaces for localization of solutions. Chapter 2 presents an analysis of regularity of weak solutions. The problems of regularity for integrands with linear growth and for vector-valued functions have been till now discussed only in a few papers. The approach of the authors is based on the study of dual variational problem for stresses. They obtain additional regularity for stress tensor, and with the help of duality relations establish the regularity of displacement field. The local regularity of solutions to …
reviewed by: Igor Bock
Mathematics Subject Classification ID: 76-02
Mathematics Subject Classification ID: 74-02
Mathematics Subject Classification ID: 74G65
Mathematics Subject Classification ID: 76A05
Mathematics Subject Classification ID: 76M30
Mathematics Subject Classification ID: 74C05
zbMATH DE Number: 1550540
zbMATH Keywords: incompressible generalized Newtonian fluids
zbMATH Keywords: weak solution
zbMATH Keywords: variational problems
zbMATH Keywords: minimax problem
zbMATH Keywords: dissipative potential
zbMATH Keywords: variational integrals
zbMATH Keywords: convex integrands
zbMATH Keywords: linear growth
zbMATH Keywords: non-reflexive spaces
zbMATH Keywords: dual variational problem
zbMATH Keywords: stress tensor
zbMATH Keywords: relaxation method
zbMATH Keywords: perfect plasticity
zbMATH Keywords: power hardening
zbMATH Keywords: logarithmic hardening
zbMATH Keywords: regularity
zbMATH Keywords: viscoplastic fluids
zbMATH Keywords: incompressible flows
zbMATH Keywords: minimizing deformation field
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0103751
OpenAlex ID: W1547458735
DOI: 10.1007/BFB0103751
description: scientific article; zbMATH DE number 1550540