Rate boundary value problems and variational inequalities in rate-independent finite elasto-plasticity

Abstract

The paper deals with variational inequalities related to the rate quasistatic boundary value problem and associated with a generic stage of the process in our approach to finite elasto-plastic materials. The Eulerian setting of the models can be associated with the various models, firstly postulated with respect to the plastically deformed configurations, by the pushing forward procedure, as well as the Lagragian formalism of the models can be adopted by pulled back to the reference configuration procedure. Only when the constitutive and the evolution functions which describe the behaviour of the elasto-plastic material are compatible with the restrictions imposed by the dissipation postulate, the variational inequalities are characterized by symmetric bilinear forms. In the present formulation of the variational inequalities we used an appropriate procedure that allows us to reformulate pointwise the consistency condition in the form of an inequality. We formulate the variational inequalities to be solved for the unknowns, namely the velocity field and the plastic factor, in the deformed configuration at time t, considered as a reference one. The coefficients of the variational inequalities are dependent on the current values of the stress, the elastic and plastic deformations, as well as the internal state variables. We propose an update algorithm which allows us to find the current state of the process at the time, by integrating the rate type constitutive equations, which characterizes the current state of the elasto-plastic materials through the objective derivative. We prove an existence and uniqueness result concerning the solution, namely the spatial velocity and plastic factor, for the symmetric variational inequality of the first kind and formulate an incrementally objective algorithm in order to update the state of the process. By solving the appropriate variational inequality at a fixed time t, we avoid the application of the Return-Mapping Algorithm proposed by Simo (1998).