Identifying Processes Governing Damage Evolution in Quasi-Static Elasticity Part I -- Analysis

TL;DR

This paper develops a nonlinear elasticity model incorporating damage evolution, analyzes its well-posedness, and introduces an inverse problem framework to identify the damage-displacement relationship via operator shape identification.

Contribution

It presents a novel inverse problem approach to identify the shape of the damage-displacement operator in a nonlinear elasticity model.

Findings

The inverse problem is ill-posed.

Fréchet-derivative and adjoint are characterized by linear differential systems.

Convergence conditions for the Landweber method are established.

Abstract

We present a quasi-static elasticity model that accounts for damage evolution based on the ideas of Kachanov 1958 and and Rabotnov 1968. We analyze the resulting strongly nonlinear system of differential equations in view of well-posedness. The specific feature is that the displacements are connected to the damage evolution equation via a Nemytskii- or superposition- operator. The novelty in this work is that we present an inverse problem in a parameter identification setting, in which we are able to identify the shape of this Nemytskii-operator. From the material modelling point of view, the relation of material damage and displacements is modelled by the shape of this operator. We establish the Fr\'echet-derivative of the forward operator as well as the adjoint of the derivative and characterize both via systems of linear differential equations. We prove ill-posedness of the inverse problem and provide a sufficient condition for the classical nonlinear Landweber method to converge.