2006/11/30 by Andrew N. Norris · 1 citation
Computer Science · Engineering · Physics and Astronomy · #Elasticity and Material Modeling #Geophysics and Sensor Technology #Matrix Theory and Algorithms #cond-mat.mtrl-sci
paper · pdf · doi:10.1177/1081286507077982
published as Mathematics and Mechanics of Solids, Vol. 13, No. 6, 465-498 (2008) · 26 pages
arxiv created 2007/01/14 · openalex publication_date 2008/04/03 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
It is 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 2 n 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 nth-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.