Motivated by the work of K.R. Rajagopal, the objective of the talk is to discuss the construction and analysis of numerical approximations to a class of models that fall outside the realm of classical Cauchy elasticity. The models under consideration are implicit and nonlinear, and are referred to as strain-limiting, because the linearised strain remains bounded even when the stress is very large, a property that cannot be guaranteed within the framework of classical elastic or nonlinear elastic models. Strain-limiting models can be used to describe, for example, the behavior of brittle materials in the vicinity of fracture tips, or elastic materials in the neighborhood of concentrated loads where there is concentration of stress even though the magnitude of the strain tensor is limited.
We construct a finite element approximation of a strain-limiting elastic model and discuss the theoretical difficulties that arise in proving the convergence of the numerical method. The analytical results are illustrated by numerical experiments.
The talk is based on joint work with Andrea Bonito (Texas A&M University) and Vivette Girault (Sorbonne Université, Paris).
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- Numerical Analysis Group Internal Seminar