# Dynamic perfect plasticity and damage in viscoelastic solids

## TL;DR

This paper develops a rigorous mathematical framework for a complex viscoelastic model incorporating plasticity and damage, proving existence of weak solutions in a dynamic setting with advanced analytical techniques.

## Contribution

It introduces a novel analysis combining viscoelasticity, plasticity, and damage in a dynamic context, establishing existence of weak solutions using a new discretization scheme.

## Key findings

- Existence of weak solutions for the coupled viscoelastic, plastic, and damage model.
- Rigorous characterization of the interplay between rheology, inertia, and damage effects.
- Development of a constructive two-step time discretization method for the model.

## Abstract

In this paper we analyze an isothermal and isotropic model for viscoelastic media combining linearized perfect plasticity (allowing for concentration of plastic strain and development of shear bands) and damage effects in a dynamic setting. The interplay between the viscoelastic rheology with inertia, elasto-plasticity, and unidirectional rate-dependent incomplete damage affecting both the elastic and viscous response, as well as the plastic yield stress, is rigorously characterized by showing existence of weak solutions to the constitutive and balance equations of the model. The analysis relies on the notions of plastic-strain measures and bounded-deformation displacements, on sophisticated time-regularity estimates to establish a duality between acceleration and velocity of the elastic displacement, on the theory of rate-independent processes for the energy conservation in the dynamical-plastic part, and on the proof of the strong convergence of the elastic strains. Existence of a suitably defined weak solutions is proved rather constructively by using a staggered two-step time discretization scheme.