Formulations of the problems of the theory of an ideally elastic-plastic body

Abstract

Various functional formulations are given for the problems of the theory of quasistaic equilibrium of ideally elastic-plastic media. The first formulation (problem A) follows naturally form the classical formulation. The set of kinematically admissible fields corresponding to it has maximum permissible thickness under the assumption that the deformation rate tensor is summable It is shown that problem A is equivalent to two partial problems (problem B and C). Problem B represents an evolutionary variational inequality for the stress tensor, which has a unique solution. In problem C the known stress field is used to determine the velocity field as a solution of some variational problem depending on the load parameter. It is shown that problem C, and hence problem A, may have no solutions. A variational extension of problem C (problem C +) is constructed. Problems B and C + lead to an enlarged formulation of the classical problem (problem A +). It is shown that A + always has a solution. An example is given, in which A has no solution and A + has a unique solution. Problems concerning the mathematical correctness of the problem of ideal plasticity have been studied by many authors (see e.g. /1–12/. The approach proposed below removes a number of the restrictions in /4–11/. Problem C with fixed load parameter resembles, in the mathematical sense, the variational problem of deformation plasticity which has been intensely studied in the last few years (see e.g. /7–9/. In/7, 9/the problem was extended to the space of displacements for which the deformation tensor is a Radon measure. The necessary extremal conditions are expressed here in terms of functions of measures (7, 9/. In the present paper the extension is produced by another method which makes it possible to obtain the relations connecting the velocity and stress fields sought (problem A +) in terms of point functions only, and this simplifies the solution of specific problems. On the other hand, the definition of admissible sets of problem A + implies that the deformation rate tensor is a Radon measure which depends on the load parameter.