Abstract
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.
Highlights
All modern theories of elasticity [1]-[4] build the equations to describe elasticity using stress and/or strain
Hardy [5] proposed to return to the approach of Euler, Lagrange, and Poisson [6] to build the equations of elasticity using point locations and forces instead of stress and strain
The equations of Euler-Lagrange elasticity are appropriate for quasi-static deformations, but do not include dynamics
Summary
All modern theories of elasticity [1]-[4] build the equations to describe elasticity using stress and/or strain. Hardy [5] proposed to return to the approach of Euler, Lagrange, and Poisson [6] to build the equations of elasticity using point locations and forces instead of stress and strain. The equations of Euler-Lagrange elasticity are appropriate for quasi-static deformations, but do not include dynamics. Hardy defined an elastic material as one which when deformed, stores energy; and when it is returned to its original state, the stored energy is returned to its surroundings. This is known as hyper-elasticity [7]. The advantage of Hardy’s approach is that Equation (3) is applicable to both infinitesimal and finite deformations as well as being appropriate for both anisotropic and isotropic materials. I will extend this approach to include dynamics
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