An application of polynomial optimization to the study of Hookean solids

Abstract

A Hookean solid is described by a so-called elasticity tensor, which is a fourth-order tensor with certain index symmetries that yield, in general, 21 independent components. However, a Hookean solid can also exhibit material symmetries. For example, an isotropic solid is described by an elasticity tensor that is invariant under all rotations of the 3D space and, as a consequence, has only two independent components. Material symmetries give insight into the mechanical properties of the solid and facilitate mathematical modelling. Therefore, it is desirable to approximate a given general elasticity tensor (for example, measured experimentally) by one that belongs to a chosen symmetry class. In this paper, we compare two di erent formulations of the problem of nding the closest element in the symmetry class as a polynomial optimization problem. We apply the polynomial optimization techniques due to Lasserre to speci c elasticity tensors, utilizing the implementation of Lasserre's method in MATLAB through the Gloptipoly 3 toolbox.