The hyperelastic polynomial form can be fitted by ABAQUS up to order . Since the Mooney-Rivlin potential corresponds to the case , these remarks also apply by setting the higher-order coefficients to zero. The energy potential is as follows:

The deformation modes are characterized in terms of the principal stretches. The nominal stress-strain relations are now derived for the polynomial form with .

The hyperelastic reduced polynomial form can be fitted by ABAQUS up to order . For the reduced polynomial is identical to the Yeoh model, and for the neo-Hookean model is retained; hence, the following also applies to these forms. The reduced polynomial energy potential is as follows:

The hyperelastic Ogden form can be fitted up to order :

The hyperelastic Arruda-Boyce potential has the following form:

The hyperelastic Van der Waals potential, also known as the Kilian model, has the following form:

Following the same approach as for the polynomial form, we can derive the nominal stress-strain relations for the Van der Waals form.

The hyperfoam potential is a modified form of the Hill strain energy potential and can be fitted up to order :

The deformation modes are characterized in terms of the principal stretches and the volume ratio J. The elastomeric foams are not incompressible: . The transverse stretches and/or are independently specified in the test data either as individual values depending on the lateral deformations or through the definition of an effective Poisson's ratio.

Planar mode

The common nominal stress-strain relation for the three deformation modes above is

where is the nominal stress and is the stretch in the direction of loading.

Simple shear mode

The simple shear deformation is described in terms of the deformation gradient,

where is the shear strain. Note also that for this deformation, . The nominal shear stress is

where are the principal stretches in the plane of shearing, related to the shear strain as follows:

The stretch in the direction perpendicular to the plane of shearing is .

The transverse stress developed during simple shear deformation (as a result of the Poynting effect) is

Given experimental data, the material constants are determined through a least-squares-fit procedure, which minimizes the relative error in stress. For the n nominal stress–nominal strain data pairs, the relative error measure E is minimized,

The polynomial potential is linear in terms of the constants ; therefore, a linear least-squares procedure can be used. The Ogden, the Arruda-Boyce, and the Van der Waals potential are nonlinear in some of their coefficients, thus necessitating the use of a nonlinear least-squares procedure.

For the full polynomial model we can rewrite the expressions for the derived above as

To fit the volumetric coefficients, one needs to solve the system of N equations

For the reduced polynomial model we can rewrite the expressions for derived above as follows:

Following the same arguments as for the full polynomial, we arrive at the system of N equations

The Ogden, Arruda-Boyce, and Van der Waals potentials are nonlinear in some of their coefficients; hence, a nonlinear least-squares-fit procedure is required. We use the Marquard-Levenberg algorithm in the formulation by Twizell and Ogden (1986). Let , be the coefficients of these hyperelastic models, where m is the number of coefficients contributing to the deviatoric behavior. Specifically, for the Ogden model, for the Arruda-Boyce model, and for the Van der Waals model. The coefficients are found by iterating the equation

For , the Newton algorithm is obtained; for very large values of , the steepest descent method is obtained. Thus, the Marquard-Levenberg algorithm represents a compromise between these two approaches: the value of is increased if the error grows and is reduced otherwise.

After initializing the , the parameters are found with a linear least squares fit. In the iterative procedure outlined above, the following derivatives are used:

The Arruda-Boyce model is linear in the shear modulus but nonlinear in the locking stretch . The locking stretch is initialized as , where is the maximum stretch in the user-specified test data. Given this locking stretch, the initial shear modulus, , is obtained with a linear least squares fit.

In the iterative procedure outlined above, the following derivatives are used:

The Van der Waals model is linear in the shear modulus but nonlinear in the locking stretch , the global interaction parameter a, and the mixture parameter . The locking stretch is initialized as , where is the maximum stretch in the user-specified test data. Given this guess for the locking stretch, we make use of an expression proposed by Kilian et al. (1986) to initialize the global interaction parameter

In the iterative procedure outlined above, the following derivatives are used:

In the derivatives of

In the planar case ; hence, vanishes.

ABAQUS checks the Drucker stability of the material for the first three modes of deformation described above. The Drucker stability condition requires that the change in the Kirchhoff stress following from an infinitesimal change in the logarithmic strain satisfies the inequality

Using , the inequality becomes

For the isotropic elastic formulation considered here, the inequality can be represented in terms of the principal stresses and strains:

Polynomial form

With the incompressibility assumption for the two hyperelastic models, the Kirchhoff stress is equal to the Cauchy stress: and, thus,

In addition, we can choose any value for the hydrostatic pressure without affecting the strains. For the stability calculation a convenient choice is , which gives us

The infinitesimal strain changes are related to the changes in stretch ratios by the equations

The stresses follow from the strain energy, which in turn follow from the changes in the strain invariants or in the stretches.

The relation between changes in the stress and changes in strain are described by the matrix equation

where

For material stability must be positive definite; thus, it is necessary that

for all relevant values of , , and .

Ogden form

For the Ogden form we follow the same approach as the polynomial form. Using , we have

and the material stiffness that we check for positive definiteness is

Van der Waals form

When the Van der Waals model is employed in its admissible stretch range given by , its stability depends on the global interaction parameter, a, for positive values of the initial shear modulus, , and the locking stretch, . To verify the Drucker stability of the Van der Waals model, we can employ the equations derived for the polynomial models by making use of the fact that

To determine the admissible stretch range, we need to find the two positive real-valued roots neighboring of the equation

for each of the three stress states—uniaxial, biaxial, and planar—by using a simple bisection method.

Hyperfoam

The Kirchhoff stress-strain relation for the uniaxial, biaxial, planar, and volumetric deformation modes is

Taking the total differential of and using ,

Since we cannot use the incompressibility assumption, we have to use all three principal stress and strain components and a   matrix,

Specifically,

where ,

For the simple shear case the principal stretches and are computed from the shear strain (as given in an earlier expression). Thus, the same form of equations is used in checking material stability during simple shear deformation.

For material stability (i.e., for to be positive definite) the following conditions must be satisfied: