Stochastic Modelling of Elasticity Tensor Fields

TL;DR

This paper introduces a probabilistic framework for modeling random elasticity tensor fields that are symmetric and positive definite, incorporating spatial symmetries and invariances, with applications in elasticity analysis.

Contribution

It develops a Lie algebra-based stochastic modeling approach for elasticity tensors that respects physical constraints and symmetries, enabling controlled ensemble generation and geometric analysis.

Findings

Generated ensembles with independent control over strength, eigenstrain, and orientation.

Ensured ensemble mean conforms to specified spatial invariances.

Introduced a new 'elasticity metric' for tensor comparison and visualization.

Abstract

We present a novel framework for the probabilistic modelling of random fourth order material tensor fields, with a focus on tensors that are physically symmetric and positive definite (SPD), of which the elasticity tensor is a prime example. Given the critical role that spatial symmetries and invariances play in determining material behaviour, it is essential to incorporate these aspects into the probabilistic description and modelling of material properties. In particular, we focus on spatial point symmetries or invariances under rotations, a classical subject in elasticity. Following this, we formulate a stochastic modelling framework using a Lie algebra representation via a memoryless transformation that respects the requirements of positive definiteness and invariance. With this, it is shown how to generate a random ensemble of elasticity tensors that allows an independent control of strength, eigenstrain, and orientation. The procedure also accommodates the requirement to prescribe specific spatial symmetries and invariances for each member of the whole ensemble, while ensuring that the mean or expected value of the ensemble conforms to a potentially 'higher' class of spatial invariance. Furthermore, it is important to highlight that the set of SPD tensors forms a differentiable manifold, which geometrically corresponds to an open cone within the ambient space of symmetric tensors. Thus, we explore the mathematical structure of the underlying sample space of such tensors, and introduce a new distance measure or metric, called the 'elasticity metric', between the tensors. Finally, we model and visualize a one-dimensional spatial field of orthotropic Kelvin matrices using interpolation based on the elasticity metric.