A random field formulation of Hooke's law in all elasticity classes

TL;DR

This paper develops a probabilistic framework for modeling elastic materials across all isotropy classes using homogeneous random fields, deriving correlation structures and spectral expansions.

Contribution

It introduces a novel random field formulation for elasticity classes, providing explicit correlation tensors and spectral expansions for each class.

Findings

Derived general forms of correlation tensors for elastic random fields.

Established spectral expansion methods for these fields.

Unified probabilistic approach applicable to all isotropy classes.

Abstract

For each of the $8$ isotropy classes of elastic materials, we consider a homogeneous random field taking values in the fixed point set $\mathsf{V}$ of the corresponding class, that is isotropic with respect to the natural orthogonal representation of a group lying between the isotropy group of the class and its normaliser. We find the general form of the correlation tensors of orders $1$ and $2$ of such a field, and the field's spectral expansion.