Evolutionary-variational method in mathematical plasticity

Abstract

The elastic–plastic infinitesimal deformation of a solid is considered within the framework of the incremental flow theory using the constitutive relation in the general rate form. The appropriate initial boundary value problem is formulated for the displacement in the form of the evolutionary-variational problem (EVP), i.e., as the abstract Cauchy problem in the Hilbert space which coincides with a weak form of the equilibrium equation, known as the principle of possible displacements in mechanics. The general existence and uniqueness theorem for the EVP is discussed. The main sufficient condition has a simple algebraic form and does not coincide with the classical Drucker and similar thermodynamical postulates; therefore, it must be independently verified. Its independence is illustrated for the non-associated plastic model of linear isotropic-kinematic hardening with dilatation and internal friction. The classical and endochronic models are analyzed too. The initial EVP is reduced by a spatial finite element approximation to the Cauchy problem for an implicit system of essentially nonlinear ordinary differential equations which can be stiff. Therefore, for the numerical solution the implicit Euler scheme is proposed. All theoretical results are illustrated by means of original numerical experiments.