  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).
