Abstract nonlinear evolution inclusions of second order with applications in visco-elasto-plasticity

TL;DR

This paper proves the existence of solutions for a class of second-order nonlinear evolution inclusions using semi-implicit discretization, with applications to visco-elasto-plasticity.

Contribution

It introduces a novel semi-implicit discretization scheme for doubly nonlinear second-order inclusions and proves convergence and energy-dissipation properties.

Findings

Existence of strong solutions established.

Convergence of variational approximation scheme proven.

Application to visco-elasto-plasticity models.

Abstract

Existence of strong solutions of an abstract Cauchy problem for a class of doubly nonlinear evolution inclusion of second order is established via a semi-implicit time discretization method. The principal parts of the operators acting on $u$ and $u'$ are multi-valued subdifferential operators and are discretized implicitly. A non-variational and non-monotone perturbation acting nonlineary on $u$ and $u'$ is allowed and discretized explicitly in time. The convergence of a variational approximation scheme is established using methods from convex analysis. In addition, it is proven that the solution satisfies an energy-dissipation equality. Applications of the abstract theory to various examples, e.g., a model in visco-elastic-plasticity, are provided.