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

zbMATH Open1487.74010arXiv2107.01082MaRDI QIDQ5026373

Adrian Muntean, Simon Grützner

Publication date: 7 February 2022

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.

Full work available at URL: https://arxiv.org/abs/2107.01082

zbMATH Keywords

existenceinverse problemdamage mechanicsBanach fixed point theoremNemytskii operatorforward operatorbalance momentum

Mathematics Subject Classification ID

Linear elasticity with initial stresses (74B10) Inverse problems in equilibrium solid mechanics (74G75) Theories of fracture and damage (74A45) PDEs in connection with mechanics of deformable solids (35Q74)

Related Items (2)

On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs ⋮ Quasistatic evolution of damage in an elastic body: numerical analysis and computational experiments

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