A lagrangian description of elastic motion in riemannian manifolds and an angular invariant of axially-symmetric elasticity tensors

Abstract

This article is a description of elasticity theory for readers with mathematical background. The first sections are an abridgment of parts of the book by Marsden and Hughes, including a compact identification of the equations of motion as the Euler-Lagrange equations for the lagrangian density. The other sections describe the basic first-order classification of materials, from the point of view of representation theory as opposed to index calculus. It includes a computation of the axes of symmetry, when they exist, for most of the irreducible components of the elasticity tensor. When the two components of the 5-dimensional type $V_5$ have axes of symmetry, some invariants appear: 2 angles in $S^{1}$ that measure the deviation of an associated decomposition $V_5\otimes \mathbb{R}^2=V_5\oplus V_5$ from the standard one. See also the classification appearing for example in (Chadwick, Vianello, and Cowin) and (Bona, Bucataru, and Slawinski) by symmetry group in $SO(3)$. A somewhat more representation-theoretic approach can be found in (Itin and Hehl), and a complete list of polynomial invariants for generic elasticity tensors can be found in (Boehler, Kirillov Jr, and Onat).