Euler-Rodrigues and Cayley formulas for rotation of elasticity tensors

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.