Global existence results for viscoplasticity at finite strain

TL;DR

This paper establishes the existence of global solutions for a complex rate-dependent gradient plasticity model at finite strain, addressing the mathematical challenges posed by nonconvex energies and geometric nonlinearities.

Contribution

It develops a novel existence theory for nonsmooth, nonconvex gradient systems in finite-strain viscoplasticity, introducing new solution concepts and applying variational methods.

Findings

Proved existence of energy-dissipation-balance solutions.

Extended the calculus of variations to complex nonlinear PDE systems.

Provided a framework for analyzing finite-strain viscoplastic models.

Abstract

We study a model for rate-dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of global-in-time solutions to the related PDE system. We reveal its underlying structure as a generalized gradient system, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite-strain elasticity as well as the multiplicative decomposition of finite-strain plasticity. Moreover, the dissipation potential depends on the left-invariant plastic rate and thus, depends on the plastic state variable. The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energy-dissipation-balance (EDB) and energy-dissipation-inequality (EDI) solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory.