Quantitative stochastic homogenization: periodic representative volume element approximationsin non-linear elasticity

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Polycrystalline materials, reinforced rubber, foams, and biological tissues are examples for the huge class of random heterogeneous materials (RM). Those materials feature microstructural uncertainties: e.g. the distribution, geometry, and constitutive parameters of the individual phases of a composite might be known on a statistical level only. Typically RM display an effective behavior on large length-scales, which can be designed and optimized by changing the composition and geometry of the microstructure. Understanding the subtle interplay between microscopic and macroscopic scales and the question of how effective properties emerge in such systems is an exciting and fundamental problem --- both, for theory, as well as, for the development of new technologies. Science approaches this question in various ways, e.g.~experimentally, via modeling and simulation, and by mathematical analysis. In many cases, if the RM is statistically homogeneous, it displays on large length-scales an (almost) deterministic physical behavior. This allows for a tremendous reduction of complexity with regard to modeling and simulation: Instead of resolving microscopic and uncertain properties, one can consider a macroscopic and deterministic model describing a homogeneous material. This is at the bases of many methods in computational micromechanics, e.g. RVE-based methods that approximate an unknown macroscopic constitutive relation with help of a representative volume element (RVE) of the random microstructure. Although RVEs are widely used, only little is known about the convergence properties, and many questions are controversially discussed (e.g. size of the RVE, boundary conditions). The analytic study of RVEs in the random case (e.g. convergence rates, a priori error estimates) belongs to the field of quantitative stochastic homogenization (QSH) and is mainly based on methods from partial differential equations and Calculus of Variations. In the last ten years QSH developed into a very activearea in applied analysis. In a recent work (Gloria, Neukamm and Otto in Inventiones Mathematicae 2015), we obtained the first convergence results for a periodic RVE in the random case with optimal scaling in the size of the RVE and the number of Monte Carlo iterations. The result and method (in parts) are restricted to linear, scalar elliptic equations. In the present project, we develop a QSH theory for non-linearly elastic composites with random microstructures, which is a class of high relevance in mechanics. In particular, we study a priori estimates for periodic RVEs and investigate analytically the impact of statistical correlations on the convergence rates. The transition from linear QSH theory to the non-convex case is interesting and challenging: The latter already features a genuinely different classical homogenization theory that involves new objects, new phenomena, and difficulties that are absent in the linear case.