Theoretical and Computational Foundations of Multi-Scale Analysis of Inelastic Solid Materials

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The key aspect of the project is the development of new perspectives towards the formulation of micro-to-macro transitions and micro-structure developments in solid materials on multiple scales. The micro-structures may a priori be defined as representative volume elements of heterogeneous materials or may in homogeneous materials be generated at a certain stage of the deformation process as a result of a material instability phenomenon. A fundamentally new viewpoint to the homogenization analysis of inelastic materials is provided by recently developed incremental variational formulations of homogenization where fine-scale fluctuation fields are defined as energy minimizers of suitably defined homogenization functionals. Methodical basis is the application of mathematical concepts for the treatment of non-convex energy minimization problems and their convexifications. Specification of these concpts to engineering problems needs the development of a variational-based generic theory of stability for inelastic solids and the construction of new computational tools for an effective handling of hierarchical multi-scale homogenization procedures in nonlinear solid mechanics. We focus on constitutive models of finite elastoplasticity and damage mechanics with regard to applications to metals and geomaterials. The overall result of the project will be a better understanding of the mathematical basis, the physical mechanisms and the numerical of material instability phenomena in solids.