A General, Automated Method for Building Structural Tensors of Arbitrary Order for Anisotropic Function Representations

TL;DR

This paper introduces a fully automated linear algebra-based method for constructing basis tensors of arbitrary order to characterize material anisotropy, improving upon classical, intuition-dependent techniques.

Contribution

The authors develop a general, numerical linear algebra approach to find basis tensors for arbitrary order, applicable to diverse symmetry groups, enhancing the modeling of anisotropic materials.

Findings

Enumerated elastic modulus tensors for common symmetries.

Identified lowest-order structure tensors for all point groups.

Applied in neural network calibration for material symmetry and anisotropy.

Abstract

We present a general, constructive procedure to find the basis for tensors of arbitrary order subject to linear constraints by transforming the problem to that of finding the nullspace of a linear operator. The proposed method utilizes standard numerical linear algebra techniques that are highly optimized and well-behaved. Our primary applications are in mechanics where modulus tensors and so-called structure tensors can be used to characterize anisotropy of functional dependencies on other inputs such as strain. Like modulus tensors, structure tensors are defined by their invariance to transformations by symmetry group generators but have more general applicability. The fully automated method is an alternative to classical, more intuition-reliant methods such as the Pipkin-Rivlin polynomial integrity basis construction. We demonstrate the utility of the procedure by: (a) enumerating elastic modulus tensors for common symmetries, and (b) finding the lowest-order structure tensors that can represent all common point groups/crystal classes. Furthermore, we employ these results in two calibration problems using neural network models following classical function representation theory: (a) learning the symmetry class and orientation of a hyperelastic material given stress-strain data, and (b) representing strain-dependent anisotropy of the stress response of a soft matrix-stiff fiber composite in a sequence of uniaxial loadings. These two examples demonstrate the utility of the method in model selection and calibration by: (a) determining structural tensors of a selected order across multiple symmetry groups, and (b) determining a basis for a given group that allows the characterization of all subgroups. Using a common order in both cases allows sparse regression to operate on a common function representation to select the best-fit symmetry group for the data.