Fracture models in SBD: Homogenization and quasistatic evolution

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The analysis of fracture models presents a variety of challenging mathematical questions, including existence results for crack growth, prediction of time evolution of the crack along its path, or understanding the effective mechanical behavior of brittle materials with heterogeneities. Formulations by variational methods, where solutions are determined from an energy minimization principle, provide efficient tools for modeling, analysis, and simulations.We propose a research project focusing on homogenization and quasistatic evolution for fracture models in linearized elasticity. More specifically, we aim at investigating the effective asymptotic behavior of brittle materials with fine microstructures by identifying variational models via effective bulk and surface densities. This static approach based on Gamma-convergence will be combined with the investigation of evolutionary problems to establish existence results for quasistatic crack growth for composite materials and to study qualitative properties of the corresponding solutions. The problems will be tackled with advanced tools from the calculus of variations and geometric measure theory, departing from new fundamental results for the space SBD of functions of bounded deformation which have been obtained in the last years. Besides its applications to Materials Science, the proposed project will further develop the mathematical foundations of this underlying function space by addressing general questions about Gamma-convergence, integral representation, and lower semicontinuity.