Homogenisation and elliptic approximation of random free-discontinuity functionals

天然和工程复合材料通常具有令人难以置信的复杂微结构。为了降低这种复杂性，在材料建模中必须考虑合理的理想化。随机复合材料代表了一类相关的理想化。受(静态)断裂变分理论中的主要问题的启发，本研究项目的主要目的是研究可发生断裂的随机弹性复合材料的大尺度行为。从数学的观点来看，这将相当于发展了自由不连续类型的能量泛函的随机齐化理论。事实上，到目前为止，在SBV中定义的随机自由不连续泛函的第一个一般齐次化结果直到2017年才在[CDMSZ17-2]中建立。这一建议从这一最新结果出发，发展了自由间断泛函的随机齐次化的全面定性理论。这将通过结合两种相辅相成的方法来实现：“直接”方法和“间接”近似方法。直接的方法是将[CDMSZ17-2]中的SBV理论扩展到BV设置和具有退化系数的泛函的设置，后者是相关的，例如，在穿孔材料和高对比度脆性复合材料的研究中。相反，近似方法将在于提出适当的随机自由不连续泛函的椭圆相场近似，它可以提供均匀系数的规则近似，从而也为定量均匀化理论的发展奠定了基础。

Natural and engineered composite materials usually posses an incredibly complex microstructure. To reduce this complexity, in materials modelling reasonable idealizations have to be considered. Random composite materials represent a relevant class of such idealizations. Motivated by primary questions arising in the variational theory of (static) fracture, the main goal of this research project is to study the large-scale behavior of random elastic composites which can undergo fracture.From a mathematical standpoint this will amount to the development of a stochastic homogenization theory for energy-functionals of free-discontinuity type.The study of the limit behavior of random free-discontinuity functionals is very much at its infancy. Indeed, to date the first general homogenization result for random free-discontinuity functionals defined in SBV was established only in 2017 in [CDMSZ17-2]. This proposal starts from this very recent result and proposes to develop a comprehensive qualitative theory of stochastic homogenization for free-discontinuity functionals. This will be done by combining two complementary approaches: a "direct" approach and an "indirect" approximation-approach. The direct approach will consist in extending the SBV-theory in [CDMSZ17-2] both to the BV-setting and to the setting of functionals with degenerate coefficients, the latter being relevant, e.g., in the study of fracture in perforated materials and in high-contrast brittle composites. The approximation-approach, instead, will consist in proposing suitable elliptic phase-field approximations of random free-discontinuity functionals which can provide regular-approximations of the homogenized coefficients, thus also setting the stage for the development of a quantitative homogenization theory.