Variational modelling of fracture in high-contrast microstructured materials: mathematical analysis and computational mechanics

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After the seminal work of Francfort and Marigo, free-discontinuity functionals of Mumford-Shah type have been established as simplified and yet relevant mathematical models to study fracture in brittle materials. For finite-contrast constituents, the homogenisation of brittle energies is by-now well-understood and provides a rigorous micro-to-macro upscaling for brittle fracture. Only recently, explicit high-contrast brittle microstructures have been provided, which show that, already for simple free-discontinuity energies of Mumford-Shah type, the high-contrast nature of the constituents can induce a complex effective behaviour going beyond that of the single constituents. In particular, macroscopic cohesive-zone models and damage models can be obtained by homogenising purely brittle microscopic energies with high-contrast coefficients. In this framework, the simple-to-complex transition originates from a microscopic bulk-surface energy-coupling which is possible due to the degeneracy of the functionals. Motivated by the need to understand the mathematical foundations of mechanical material-failure and to develop computationally tractable numerical techniques, the main goal of this project is to characterise all possible materials which can be obtained by homogenising simple high-contrast brittle materials. In mathematical terms, this amounts to determine the variational-limit closure of the set of high-contrast free-discontinuity functionals. This problem has a long history in the setting of elasticity, whereas is far less understood if fracture is allowed. For the variational analysis it will be crucial to determine novel homogenisation formulas which “quantify” the microscopic bulk-surface energy-coupling. Moreover, the effect of high-contrast constituents on macroscopic anisotropy will be investigated by providing explicit microstructures realising limit models with preferred crack-directions. The relevant mathematical tools will come from the Calculus of Variations and Geometric Measure Theory. Along the way, new ad hoc extension and approximation results for SBV-functions will be established. The latter will be of mathematical interest in their own right, and appear to be widely applicable in the analysis of scale-dependent free-discontinuity problems. The computational mechanics results will build upon the mathematical theory, and will complement it with relevant insights when the analysis becomes impracticable. High performance fast Fourier transform and adaptive tree-based computational methods will be developed to evaluate the novel cell formulas. The identified damage and cohesive-zone models will be transferred to simulations on component scale. The findings are expected to significantly enhance the understanding of the sources and mechanisms of material-failure and to provide computational tools for identifying anisotropic material-models useful for estimating the strength of industrial components.