Computation of minimal covariants bases for 2D coupled constitutive laws

TL;DR

This paper develops minimal covariant bases for 2D constitutive laws, including various physical tensors, providing explicit formulas and algorithms to facilitate applications in material science and continuum mechanics.

Contribution

It introduces a comprehensive method to compute minimal covariant bases for 2D constitutive tensors, extending invariant theory to covariants with explicit formulas and algorithms.

Findings

Minimal integrity bases for common 2D constitutive tensors derived.

Explicit complex and tensorial formulas provided for invariants and covariants.

A cleaning algorithm ensures minimality in isotropic cases.

Abstract

We produce minimal integrity bases for both isotropic and hemitropic invariant algebras (and more generally covariant algebras) of most common bidimensional constitutive tensors and -- possibly coupled -- laws, including piezoelectricity law, photoelasticity, Eshelby and elasticity tensors, complex viscoelasticity tensor, Hill elasto-plasticity, and (totally symmetric) fabric tensors up to twelfth-order. The concept of covariant, which extends that of invariant is explained and motivated. It appears to be much more useful for applications. All the tools required to obtain these results are explained in detail and a cleaning algorithm is formulated to achieve minimality in the isotropic case. The invariants and covariants are first expressed in complex forms and then in tensorial forms, thanks to explicit translation formulas which are provided. The proposed approach also applies to any $n$-uplet of bidimensional constitutive tensors.