The equations of Euler-Lagrange elasticity describe elastic deformations
without reference to stress or strain. These equations as previously published
are applicable only to quasi-static deformations. This paper extends these
equations to include time dependent deformations. To accomplish this, an
appropriate Lagrangian is defined and an extrema of the integral of this
Lagrangian over the original material volume and time is found. The result is a
set of Euler equations for the dynamics of elastic materials without stress or
strain, which are appropriate for both finite and infinitesimal deformations of
both isotropic and anisotropic materials. Finally, the resulting equations are
shown to be no more than Newton's Laws applied to each infinitesimal volume of
the material.

Hardy, H.H. (2013) Euler-Lagrange Elasticity: Differential Equations for Elasticity without Stress or Strain. Journal of Applied Mathematics and Physics, 1, 26-30.

Todhunter, I. (1886) A History of the Theory of Elasticity and of the Strength of Materials from Galileo to the Present Time. Vol. 1, Cambridge University Press, New York.

Hardy, H.H. and Shmidheiser, H. (2011) A Discrete Region Model of Isotropic Elasticity. Mathematics and Mechanics of Solids, 16, 317-333. http://dx.doi.org/10.1177/1081286510391666