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

多晶材料、增强橡胶、泡沫和生物组织是随机非均质材料（RM）的一大类例子。这些材料具有微观结构的不确定性：例如，复合材料的各个相的分布、几何形状和本构参数可能仅在统计水平上已知。通常，RM在大的长度尺度上显示出有效的行为，这可以通过改变微结构的组成和几何形状来设计和优化。理解微观尺度和宏观尺度之间的微妙相互作用以及如何在这样的系统中出现有效属性的问题是一个令人兴奋的和基本的问题-无论是对于理论，还是对于新技术的发展。科学以各种方式处理这个问题，例如~实验上，通过建模和仿真，并通过数学分析。在许多情况下，如果RM在统计上是均匀的，它在大的长度尺度上显示出（几乎）确定性的物理行为。这使得建模和仿真的复杂性大大降低：人们可以考虑描述均匀材料的宏观和确定性模型，而不是解决微观和不确定性特性。这是计算微观力学中许多方法的基础，例如基于RVE的方法，其借助于随机微结构的代表性体积元（RVE）来近似未知的宏观本构关系。虽然RVE被广泛使用，只有很少的是已知的收敛性，许多问题是有争议的讨论（如大小的RVE，边界条件）。随机情况下RVE的分析研究（例如收敛速度，先验误差估计）属于定量随机均匀化（QSH）领域，主要基于偏微分方程和变分法的方法。近十年来，QSH已发展成为应用分析中一个非常活跃的领域.在最近的工作中（格洛丽亚，Neukamm和Otto in Inventiones Mathematicae 2015），我们在随机情况下获得了周期性RVE的第一个收敛结果，其中RVE的大小和Monte Carlo迭代次数具有最佳缩放。部分结果和方法仅限于线性标量椭圆型方程。在本项目中，我们发展了一个具有随机微结构的非线性弹性复合材料的QSH理论，这是一类与力学高度相关的理论。特别是，我们研究了先验估计周期RVES和调查分析的影响，统计相关性的收敛速度。 从线性QSH理论到非凸情形的过渡是有趣和具有挑战性的：后者已经具有真正不同的经典均匀化理论，涉及新对象，新现象和线性情形中不存在的困难。

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.