Modeling phase transformations and damage processes under complex boundary conditions by means of extremal principles
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Phase transformations in solids can be observed for many materials among which shape memory alloys are a prominent example: additionally, to the usual temperature-driven phase transformation of the crystal lattice in metals, special alloys are able to undergo the very same transformations also induced by mechanical loads. This gives rise to the fascinating phenomena of pseudo-elasticity and pseudo-plasticity. The latter effect motivates the classification of a shape memory material since apparently permanent deformations are recovered after moderate heating and subsequent cooling. From a user’s perspective, the capability of transforming the crystal lattice and the related deformations, which are unexpectedly high for metals, allows for designing a variety of special products: medical devices such as stents or self-controlled actuators with inherent sensor functionality. Although providing very useful properties, the design process remains complicated due to the strongly coupled processes. Therefore, material modeling and numerical simulation provide a basis on which the understanding of the material and structural behavior can be improved. Damage processes are modeled by non-convex free energy densities to display the localization of cracks. The non-convexity renders damage models to be ill-posed such that regularization approaches are required. We present our gradient-enhanced model along with a tailored numerical treatment which results in minimal extra costs for the simulations. Having the models for phase transformations and damage processes at hand, we demonstrate how they can be used to simulate the complex and coupled behavior of an arteria into which a stent is deployed. This examples demonstrates both the modeling capacities of extremal principles and the robustness of our algorithm.
