In this article, we extend the study of embedded corrector problems, that we have previously introduced in the context of the homogenization of scalar diffusive equations, to the context of homogenized elastic properties of materials. This extension is not trivial and requires mathematical arguments specific to the elasticity case. Starting from a linear elasticity model with highly-oscillatory coefficients, we introduce several effective approximations of the homogenized tensor. These approximations are based on the solution to an embedded corrector problem, where a finite-size domain made of the linear elastic heterogeneous material is embedded in a linear elastic homogeneous infinite medium, the constant elasticity tensor of which has to be appropriately determined. The approximations we provide are proven to converge to the homogenized elasticity tensor when the size of the embedded domain tends to infinity. Some particular attention is devoted to the case of isotropic materials.

翻译：本文将对之前针对标量扩散方程均匀化提出的嵌入式校正问题进行扩展研究，以应用于材料弹性性质的均匀化分析。这一扩展并非平凡，需要针对弹性力学情形提出特定的数学论证。从具有高度振荡系数的线性弹性模型出发，我们提出了均匀化张量的几种有效近似方法。这些近似方法基于嵌入式校正问题的解：将线性弹性非均质材料构成的有限尺寸域嵌入线性弹性均匀无限介质中，其中后者常弹性张量需适当确定。当嵌入域尺寸趋于无穷时，我们证明所提出的近似值收敛于均匀化弹性张量。特别关注了各向同性材料的情形。