Quadratic invariants of elastic moduli

TL;DR

This paper identifies the number of quadratic invariants of elastic moduli tensors under specific symmetry groups, providing answers to previously open questions in tensor invariance.

Contribution

It proves the exact count of quadratic invariants of elastic moduli tensors under SO(3) and SO(2) groups, resolving earlier open problems.

Findings

7 quadratic invariants under SO(3)

35 quadratic invariants under SO(2)

Addresses open questions from Ting (1987) and Ahmad (2002)

Abstract

A quadratic invariant is defined as a quadratic form in the elements of a tensor that remains invariant under a group of coordinate transformations. It is proved that there are 7 quadratic invariants of the 21-element elastic modulus tensor under SO(3) and 35 under SO(2). This answers some open questions raised by Ting (1987) and Ahmad (2002).