_{1}

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Differential equations to describe elasticity are derived without the use of stress or strain. The points within the body are the independent parameters instead of strain and surface forces replace stress tensors. These differential equations are a continuous analytical model that can then be solved using any of the standard techniques of differential equations. Although the equations do not require the definition stress or strain, these quantities can be calculated as dependent parameters. This approach to elasticity is simple, which avoids the need for multiple definitions of stress and strain, and provides a simple experimental procedure to find scalar representations of material properties in terms of the energy of deformation. The derived differential equations describe both infinitesimal and finite deformations.

Euler, Lagrange, and Poisson all described the deformation of materials in terms of positions of points and forces within the body. Cauchy introduced the concepts of stress and strain, which are now the standard for elasticity equations (An accessible early history is found in Ref [

Using the notation of Spencer [_{i} = (X_{1}, X_{2}, X_{3}), with corresponding to the initial x, y, and z coordinates of each point, respectively. The corresponding points within the body after deformation are described as. The position of each point after deformation is a function of the original location of that point, i.e., x_{2} = f_{2}(X_{1}, X_{2}, X_{3}), and.

When an elastic material is deformed, work is done on the material and energy is stored in the material. As the material is returned to its original shape, the energy returns to its original value. Thus the energy depends only upon the final position of the points within the body. Experts in elasticity theory call this hyperelasticity. Hyperelasticity will be assumed for the remainder of this paper.

The differential equations are derived by assuming the energy per unit volume of the material is a function of the final point locations, , and the relative displacements of near-by points,. The total energy of the body is then

It is also assumed that when the body is moved or deformed, the internal points within the body move so as to minimize the total energy. That is

for i = 1, 2, 3 and j = 1, 2, 3 and where the integral is taken over the entire body.

Minimizing this energy function Equation (2) results in the following three Euler-Lagrange equations,

These are the differential equations of elasticity. All that remains is to appropriately describe the energy function E and the boundary conditions.

To describe the energy per unit volume E, divide the energy into two parts. which defines the energy associated with body forces and which defines the energy associated with the deformation of the body. is typically only a function of the positions of the points within the body. (e.g., the energy per unit volume associated with a gravitational force would be just, with the axis vertical.) is typically a function only of the relative positions of the body, i.e. a function only of. The total energy associated with the body is then the sum of the contributions from these different energies,

The energy of deformation, , must be invariant to coordinate translations and rotations. A common way of accomplishing this [

Hardy and Shmidheiser [

which can also be written directly in terms of the matrix elements of as

where, , and are the column vectors of.

The invariants for anisotropic bodies can be the six values produced by a Gram-Schmidt QRDecomposition of. This QRDecomposition results in an upper triangular matrix,

The elements of this upper triangular matrix can also be written in terms of the column vectors of as follows:

Any algebraic combination of these 6 values can also be used as invariants for anisotropic bodies.

Boundary conditions for material deformation problems usually consist of either specifying the final positions of boundary points, or the forces on the surfaces. The Dirichlet boundary conditions consist of simply defining at the desired surfaces. The Neumann boundary conditions require converting the surface forces into derivatives of. This can be accomplished by first noting that once are known for a given deformation, E and are known. Thus the change in energy due to the work done on the body is a function of only the initial and final positions of the points within the body. This implies that the deformation forces are conservative. As a result the internal forces within the body can be written as the negative gradient of the energy, , which is equivalent to

Care must be taken here, however, because the gradient of E is with respect to, whereas the integral is over the original. Using Equation (3),

Applying the n-dimensional divergence theorem gives

where is the component of the force at the surface defined by the vector with, ,.

These internal forces must be balanced by an equivalent surface force in the opposite direction. Since this must be true for all surfaces, the surface forces are

Once the energy per unit volume, E, is expressed in terms of the invariants of, which are in turn functions of, Equation (12) provide the constraint equations for the Neumann boundary conditions.

The differential equations of elasticity have now been completely described. No reference has been made to the magnitude of the deformation. As a result, Equation (3) apply to both infinitesimal and finite elastic deformations. Also they apply to both isotropic and anisotropic materials. Although the equations are general, for simplicity I will limit the rest of this paper to isotropic materials.

In order to make the connection between the differential equations of elasticity given here and the “standard” differential equations of infinitesimal elasticity, I will show that a Taylor expansion of for an isotropic body yields the infinitesimal free energy as described by Landau [

The Taylor expansion of yields

where the “0” subscript corresponds to no deformation (i.e. when and).

The first term in the expansion is a constant and is physically irrelevant. To evaluate the second term, I will choose to define the initial state, when no deformation has occurred, as corresponding to the state of the body when all internal forces are zero. Since this must be true for all points within the body, Equation (11) gives

which is the coefficient of the second term in the Taylor expansion, Equation (13). As a result, the third term, is the leading term in the Taylor expansion. Before evaluating this however, let’s see what constraint Equation (14) provides. To do this, express in terms of the three invariants for an isotropic body given in Equation (6) Then

Direct substitution of the invariants in terms of into this equation yields

It now remains to expand and evaluate the third term in the Taylor expansion. Again writing in terms of allows us to proceed. The third term in the Taylor expansion can be expanded quickly using algebraic computation software like Mathematica. It is a bit more tedious by hand, but the result of either is the free energy as defined by Landau [

with and, so that where

and

If this Taylor expansion of is substituted into Equation (3) and a gravitational body force is added, we get Landau’s equation [

where Young’s modulus and Poisson’s ratio.

The boundary conditions for classical infinitesimal elasticity consist of setting stress and strain on boundaries. To complete the comparison, stress and strain need to be defined. There are many different definitions of stress and strain in the literature. As an example, the Lagrangian strain tensor is, and stress as defined by Landau [

Note that this stress corresponds to force divided by original area. We can now conclude that all of the solutions to problems in classical infinitesimal elasticity described by Equation (20) are also solutions of Equation (3).

One other observation should be made here. As defined by Landau, defined in Equation (21) is the stress exerted by the surroundings on the material volume. Noting this and considering the case where the material is in the gravitational field of the earth, we see that Equation (2) are just the equations of equilibrium in terms of stress, i.e.

where.

The real power of Equation (3) is not that they can reproduce standard infinitesimal elasticity equations, but that they apply equally well to finite elasticity. Consider for example a finite deformation with no body forces.

Since E is a function only of, the derivative of E in Equation (3) results in equations with every term containing a second derivative of with respect to. As a result, any homogeneous deformation without body forces automatically solves Equation (3), depending only upon the boundary conditions.

Consider two such homogeneous deformations. The first homogeneous deformation simulates the experimental results of Rivlin [

Any of the many models of elasticity expressed in terms of invariants of (e.g. Ogden [

Note that the coefficient of has been chosen so that Equation (16) is satisfied (i.e. so that the forces with no deformation are zero). Considering solutions of Equation (3) of the form, the deformation gradient matrix is

and the singular value decomposition of gives.

, , , and were measured by Rivlin. No force was applied in the z direction, so.

where w = original width = 8 cm, h =

original thickness = 0.7 mm of the deformed rubber sample. This information allows the measured and to be reconstructed. Substituting these values into Equation (12) provides three constrain equations for finding a, b, , and. Unfortunately Rivlin assumed the rubber he deformed to be incompressible, so and b can not be evaluated using his data. However, if a, , are computed assuming incompressibility, an estimate of b can be obtained using compressibility from Wood [_{1} = 2.0902, c_{2} = −0.127399 all in Rivlin’s units of . Poisson’s ratio is 0.499888 and Young’s modulus is. These results are at least consistent with Rivlin’s data. The process followed here to derive is not ideal. What is really needed is experimental data for the entire energy cube as describe by Hardy and Shmidheiser [

A simple shear, corresponding to a 25˚ deformation in the y direction, is, , and. The forces required for this deformation on a cm cube can be found using Equation (12) and the energy function that fits Rivlin’s data, Equation (23). The result is, , and all in Newtons, with, , and being the force against what was the original yz, xz, and xy face, respectively.

Of course not all deformations are homogeneous, but Equation (3) are appropriate for finite element or any other technique for solving differential equations for more complicated problems when homogeneous deformations do not satisfy the boundary conditions of a particular problem.

Elasticity theory has been dominated by Cauchy’s stress and strain. Cauchy’s approach works well for infinitesimal elasticity as used in most engineering applications. For finite deformations, however, Cauchy’s approach becomes unduly complicated, spawning a number of new definitions of stress and strain. The Euler-Lagrange approach presented here avoids these complications. In addition, material properties in terms of the energy of deformation are easily input into the elasticity equations, Equation (3). It is my hope that engineers and physicists who require computer models of finite deformations will consider the Euler-Lagrange approach. I also hope that it will encourage experimentalists to “map out” the energy of deformation for more materials as Rivlin has done with rubber.

I would like to thank Jens Feder, Torstein Jossang, Richard Beier, Paul Meakin, and Jonathan Gaston for their support and encouragement of this work.