Euler-Rodrigues and Cayley formulas for rotation of elasticity tensors

TL;DR

This paper derives generalized Euler-Rodrigues and Cayley formulas for rotating higher-order tensors, especially elasticity tensors, simplifying their transformation representations and connecting them to symmetry and spectral properties.

Contribution

It introduces a generalized Euler-Rodrigues polynomial for tensor rotations and a new formula for elastic moduli transformation as a degree-8 polynomial, advancing tensor rotation theory.

Findings

Derived a degree-2n Euler-Rodrigues polynomial for tensor rotation matrices.

Presented a 21-vector rotation formula for elastic moduli with degree-8 polynomial.

Connected tensor rotation formulas to Cartan decomposition and symmetry projections.

Abstract

It is fairly well known that rotation in three dimensions can be expressed as a quadratic in a skew symmetric matrix via the Euler-Rodrigues formula. A generalized Euler-Rodrigues polynomial of degree 2n in a skew symmetric generating matrix is derived for the rotation matrix of tensors of order $n$. The Euler-Rodrigues formula for rigid body rotation is recovered by n=1. A Cayley form of the n-th order rotation tensor is also derived. The representations simplify if there exists some underlying symmetry, as is the case for elasticity tensors such as strain and the fourth order tensor of elastic moduli. A new formula is presented for the transformation of elastic moduli under rotation: as a 21-vector with a rotation matrix given by a polynomial of degree 8. Explicit spectral representations are constructed from three vectors: the axis of rotation and two orthogonal bivectors. The tensor rotation formulae are related to Cartan decomposition of elastic moduli and projection onto hexagonal symmetry.