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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.

Virtually all modern theories of elasticity [

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 [

relative change in distances between points,

To obtain the Euler-Lagrange differential equations, Hardy minimized the total energy,

which resulted in three Euler equations,

The advantage of Hardy’s approach is that Equation (3) is applicable to both infinitesimal and finite defor- mations as well as being appropriate for both anisotropic and isotropic materials. The disadvantage of this approach is that it is only appropriate for quasi-static deformations, since time dependence is not included. In this paper, I will extend this approach to include dynamics.

To add dynamics to the Euler-Lagrange elasticity equations several changes are needed to the quasi-static approach. First define each

with

Define the potential energy per unit original volume as

Substitute Equation (1) into Equation (6) with

Now find the extrema of

Since

Substituting

or

Equation (11) are the equations of dynamics for deformation of elastic materials. All that is required is to define

tions. One method is to define

Note that no assumptions of infinitesimal deformation or isotropy have been made to derive Equation (11), so they are applicable for both infinitesimal and finite deformations of both isotropic and anisotropic materials. The most surprising thing about Equation (11) is that each term in Equation (11) can be given a simple physical interpretation.

In order to give a physical interpretation to the individual terms in Equation (11) consider a small cuboid defined

as

unit original volume of this cuboid with respect to time in the limit as

force per unit original volume applied to all the surfaces of the cuboid as the volume of the cuboid shrinks to

zero. In other words, Equation (11) is just an expression of Newton’s laws

volume of the material.

To see that

recall that Hardy [

Let

For our cuboid, defined as

For example,

Taking the limit as the dimensions of the cube go to zero gives the net force per unit original volume on region b in the

A similar argument using

and

Next consider

and in general

Combining these results, we have the total force in the

for i = 1, 2, 3, and summed over j = 1, 2, 3, which is the third term in Equation (11). Thus

is the net surface force per unit original volume in the

shown. There are forces on the rear surfaces that also contribute to each

The procedure outlined in the last section to calculate the force on a plane after a deformation seems a bit convoluted in that the location of the plane before any deformation must be found in order to find the force on the plane after deformation. However, Equation (12) are excellent for applying Neumann boundary conditions to Equation (11). As an example, consider the case of deforming a rectangular body as shown in

If the force is applied uniformly over the area,

Finite deformations may displace and distorted planes in the cuboid from their original positions, but as long as inversions are not allowed, the same bounding surfaces of the cuboid are found regardless of how the material

is deformed. The values of

vectors are unchanged by the deformation. Thus the forces shown in Figures 1-3 may be displaced due to the finite deformation, but the orientation of each component of each force from each surface is the same and the

form of the sum of the forces,

Lastly, it is tempting to consider the second order tensor quantity

stress for infinitesimal deformations. This is because

surface vector, not the current one to get the force at the current location.

The equations for dynamics in Euler-Lagrange elasticity have been derived. These equations are shown to be a

simple statement of Newton’s Law

equations, Equation (11), are applicable to infinitesimal and finite deformations for both isotropic and anisotropic materials.