Complete damage in linear elastic materials - Modeling, weak formulation and existence results

TL;DR

This paper develops a mathematical model for complete damage in elastic materials using a degenerating PDE system, proving the existence of weak solutions and connecting classical and weak formulations.

Contribution

It introduces a new complete damage model with a degenerating PDE system, providing existence results and linking classical and weak formulations in an SBV framework.

Findings

Existence of weak solutions for the damage model.

Weak solutions can recover classical differential inclusions under regularity.

A novel SBV-framework formulation for complete damage modeling.

Abstract

In this work, we introduce a degenerating PDE system with a time-depending domain for complete damage processes under time-varying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a quasi-static balance equation for the displacement field which are strongly nonlinearly coupled. In our proposed model, the material may completely disintegrate which is indispensable for a realistic modeling of damage processes in elastic materials. Complete damage theories lead to several mathematical problems since for instance coercivity properties of the free energy are lost and, therefore, several difficulties arise. For the introduced complete damage model, we propose a classical formulation and a corresponding suitable weak formulation in an $SBV$-framework. The main aim is to prove existence of weak solutions for the introduced degenerating model. In addition, we show that the classical differential inclusion can be regained from the notion of weak solutions under certain regularity assumptions which is a novelty in the theory of complete damage models of this type.