Homogenisation and elliptic approximation of random free-discontinuity functionals

点击翻译按钮获取中文摘要

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.