A model and a general problem on plastic flow under large deformations

Abstract

AbstractWe introduce a constitutive equation of the plastic flow such that a medium is considered as a fluid, whose viscosity depends on the second invariant of the tensor of strains accumulated under a plastic flow and on the second invariant of the deviator of the stress tensor. For this constitutive equation a general problem on plastic flow under large deformations is formulated and studied. It is supposed that the surface forces acting on a body are defined in the form of a feedback such that the forces at a moment t depend on the shape of the body at this moment. The problem consists of finding the configuration of the deformed body at each moment of time, velocities, strains, and stresses. The general problem is approximated by problems with delay. We prove the existence and the uniqueness of a solution for the problems with delay. The solution of the original problem is obtained as a limit of the solutions of the problems with delay as the delay parameter tends to zero.