  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.
