Homogenisation and elliptic approximation of random free-discontinuity functionals

天然和工程复合材料通常具有极其复杂的微观结构。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这类理想化的一个相关类别。受（静态）断裂变分理论中出现的主要问题的启发，本研究计划的主要目标是研究随机弹性复合材料的大尺度断裂行为。从数学观点来看，这将相当于发展一种自由间断型能量泛函的随机均匀化理论。随机自由间断型能量泛函的极限行为的研究，不连续泛函还处于起步阶段。事实上，到目前为止，SBV中定义的随机自由不连续泛函的第一个通用均匀化结果仅在2017年在[CDMSZ 17 -2]中建立。这个建议从这个最近的结果，并建议发展一个全面的定性理论的随机均匀化的自由不连续泛函。这将通过结合两种互补的方法来实现：“直接”方法和“间接”近似方法。直接的方法是将[CDMSZ 17 -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断裂研究中。近似的方法，而不是，将包括在提出合适的椭圆相场近似的随机自由不连续泛函，可以提供定期近似的均匀化系数，从而也设置了阶段的发展定量均匀化理论。

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.