Quadratic invariants of the elasticity tensor

TL;DR

This paper systematically constructs a complete set of seven quadratic invariants for the elasticity tensor using its irreducible decomposition, providing tools to measure how close materials are to idealized elastic models.

Contribution

It introduces a natural basis of quadratic invariants for the elasticity tensor based on its irreducible decomposition, ensuring their independence and completeness.

Findings

Seven quadratic invariants form a complete basis for the elasticity tensor.

Defined Cauchy and isotropic factors as measures of material closeness to ideal models.

Derived explicit formulas for cubic crystals and their averages.

Abstract

We study the quadratic invariants of the elasticity tensor in the framework of its unique irreducible decomposition. The key point is that this decomposition generates the direct sum reduction of the elasticity tensor space. The corresponding subspaces are completely independent and even orthogonal relative to the Euclidean (Frobenius) scalar product. We construct a basis set of seven quadratic invariants that emerge in a natural and systematic way. Moreover, the completeness of this basis and the independence of the basis tensors follow immediately from the direct sum representation of the elasticity tensor space. We define the Cauchy factor of an anisotropic material as a dimensionless measure of a closeness to a pure Cauchy material and a similar isotropic factor is as a measure for a closeness of an anisotropic material to its isotropic prototype. For cubic crystals, these factors are explicitly displayed and cubic crystal average of an arbitrary elastic material is derived.