On choosing effective elasticity tensors using a Monte-Carlo method

Abstract

A generally anisotropic elasticity tensor can be related to its closest counterparts in various symmetry classes. We refer to these counterparts as effective tensors in these classes. In finding effective tensors, we do not assume a priori orientations of their symmetry planes and axes. Knowledge of orientations of Hookean solids allows us to infer properties of materials represented by these solids. Obtaining orientations and parameter values of effective tensors is a highly nonlinear process involving finding absolute minima for orthogonal projections under all three-dimensional rotations. Given the standard deviations of the components of a generally anisotropic tensor, we examine the influence of measurement errors on the properties of effective tensors. We use a global optimization method to generate thousands of realizations of a generally anisotropic tensor, subject to errors. Using this optimization, we perform a Monte Carlo analysis of distances between that tensor and its counterparts in different symmetry classes, as well as of their orientations and elasticity parameters.