From Nonlinear to Linear Elasticity in a Coupled Rate-Dependent/Independent System for Brittle Delamination
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We revisit the weak, energetic-type existence results obtained in (Rossi and Thomas, ESAIM Control Optim. Calc. Var. 21, 1–59, (2015)) for a system for rate-independent, brittle delamination between two visco-elastic, physically nonlinear bulk materials and explain how to rigorously extend such results to the case of visco-elastic, linearly elastic bulk materials. Our approximation result is essentially based on deducing the Mosco-convergence of the functionals involved in the energetic formulation of the system. We apply this approximation result in two different situations at small strains: Firstly, to pass from a nonlinearly elastic to a linearly elastic, brittle model on the time-continuous level, and secondly, to pass from a time-discrete to a time-continuous model using an adhesive contact approximation of the brittle model, in combination with a vanishing, super-quadratic regularization of the bulk energy. The latter approach is beneficial if the model also accounts for the evolution of temperature.
This work has been partially supported by the DFG within the project Finite element approximation of functions of bounded variation and application to models of damage, fracture, and plasticity in the Priority Programme SPP 1748 Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretisation Methods, Mechanical and Mathematical Analysis.1 During a research stay of M.T. at DICATAM, University of Brescia, and of R.R. at WIAS Berlin major steps were made for the progress of this work. The two authors acknowledge the kind hospitality at their host institutions.
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