Quasi-Static behavior as a limit process of a dynamical one for an anisotropic hardening material

We formulate some problems modeling the local hardening behavior of a plastic material following a Prandtl-Reuss law. Directional linear hardening, which is similar to Bauschinger’s effect in metals, is characterized by an anisotropic factor. The flow process of the material is revealed by translation in the direction shown by deformation using an absolutely continuous guiding function. The possibility of choice of a plastic strain component for the mechanical system with hardening requires an algebraic factorization of the space of processes. If the quasi-static deformation is seen as a limit case of the dynamic deformation, corresponding to low speeds and low inertial forces, we obtain an existence and uniqueness result. In this context there exists an admissible plastic strain rate, as follows by choosing a permissible elastic stress as a solution for a sweeping process problem.