The Elastic Fluctuation Tensor: Quantifying Stochastic Fluctuations of the Apparent Stiffness of Non-representative Microstructure Volume Elements

Abstract

We introduce the elastic fluctuation tensor to quantify the stochastic fluctuation of the apparent stiffness of finite microstructural volume elements. Typically, in computational homogenization using volume elements of finite size, the apparent stiffness converges to the effective stiffness as the volume element size tends to infinity, such that the material can be approximated as homogeneous on the macroscale. For volume elements of finite size, the apparent stiffness fluctuates on the macroscale. In thermal conductivity homogenization, the fluctuations can be quantified using the fourth-order fluctuation tensor, which computes as the infinite-volume limit of the apparent conductivity covariance, rescaled with the volume. The fluctuation tensor for linear elasticity is of tensor order eight. We show that this fluctuation tensor inherits the symmetry of its ensemble. For instance, rotational statistical symmetry of the ensemble leads to isotropy of the elastic fluctuation tensor. Using results from group representation theory, we define efficient representations of the eighth-order fluctuation tensor for various microstructure symmetry classes and discuss the physical meaning of individual components for the statistically isotropic case. We furthermore leverage symmetry to mitigate numerical errors, thereby reducing the expense of computing the fluctuation tensor. As an example material, we consider polypropylene reinforced by fibers and spherical inclusions. Additionally, we examine polycrystalline copper microstructures. By numerically computing the elastic fluctuation tensor, we confirm theoretical asymptotic convergence rates and symmetry properties. For many of the considered statistically isotropic microstructures, the fluctuations of isotropic stiffness components, which are often the only fluctuations reported, are negligible compared to isotropic fluctuations of the anisotropic stiffness components. Therefore, the full fluctuation tensor must be considered when quantifying the uncertainty of stochastic homogenization.