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This article is a description of elasticity theory for readers withmathematical background. The first sections are an abridgment of parts of thebook by Marsden and Hughes, including a compact identification of the equationsof motion as the Euler-Lagrange equations for the lagrangian density. The othersections describe the basic first-order classification of materials, from thepoint of view of representation theory as opposed to index calculus. Itincludes a computation of the axes of symmetry, when they exist, for most ofthe irreducible components of the elasticity tensor. When the two components ofthe 5-dimensional type $V_5$ have axes of symmetry, some invariants appear: 2angles 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 theclassification appearing for example in (Chadwick, Vianello, and Cowin) and(Bona, Bucataru, and Slawinski) by symmetry group in $SO(3)$. A somewhat morerepresentation-theoretic approach can be found in (Itin and Hehl), and acomplete list of polynomial invariants for generic elasticity tensors can befound in (Boehler, Kirillov Jr, and Onat).

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