Fracture models in SBD: Homogenization and quasistatic evolution

断裂模型的分析提出了各种具有挑战性的数学问题，包括裂纹扩展的存在性结果，裂纹沿着其路径的时间演化的预测，或理解具有非均匀性的脆性材料的有效力学行为。通过变分方法，其中的解决方案是从能量最小化原则确定的配方，提供了有效的工具，建模，分析和simulation.We提出了一个研究项目，专注于均匀化和准静态演化的断裂模型在线性弹性。更具体地说，我们的目标是调查有效的渐近行为的脆性材料与精细的微观结构通过识别变分模型，通过有效的体积和表面密度。这种静态方法的基础上伽玛收敛将结合进化问题的调查，建立复合材料的准静态裂纹扩展的存在性结果，并研究相应的解决方案的定性性质。这些问题将处理先进的工具，从变分法和几何测量理论，从新的基本结果的空间SBD的有界变形的功能，已在过去几年中获得。除了其在材料科学中的应用外，该项目还将通过解决有关伽玛收敛、积分表示和较低连续性的一般问题，进一步发展这一基础函数空间的数学基础。

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.