Strain energy potential

The polynomial form of the strain energy potential is one that is commonly used. Its form is

U = \overset{N}{\sum_{i + j = 1}} C_{i j} {({\bar{I}}_{1} - 3)}^{i} {({\bar{I}}_{2} - 3)}^{j} + \overset{N}{\sum_{i = 1}} \frac{1}{D_{i}} {(J_{e l} - 1)}^{2 i},

where U is the strain energy potential; $J_{e l}$ is the elastic volume ratio; ${\bar{I}}_{1}$ and ${\bar{I}}_{2}$ are measures of the distortion in the material; and N, $C_{i j}$ , and $D_{i}$ are material parameters, which may be functions of temperature. The $C_{i j}$ parameters describe the shear behavior of the material, and the $D_{i}$ parameters introduce compressibility. If the material is fully incompressible (a condition not allowed in Abaqus/Explicit), all the values of $D_{i}$ are set to zero and the second part of the equation shown above can be ignored. If the number of terms, N, is one, the initial shear modulus, $μ_{0}$ , and bulk modulus, $K_{0}$ , are given by

μ_{0} = 2 (C_{01} + C_{10}),

and

K_{0} = \frac{2}{D_{1}} .

If the material is also incompressible, the equation for the strain energy density is

U = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) .

This expression is commonly referred to as the Mooney-Rivlin material model. If $C_{01}$ is also zero, the material is called neo-Hookean.

The other hyperelastic models are similar in concept and are described in Hyperelasticity.

You must provide Abaqus with the relevant material parameters to use a hyperelastic material. For the polynomial form these are $C_{i j}$ and $D_{i}$ . It is possible that you will be supplied with these parameters when modeling hyperelastic materials; however, more likely you will be given test data for the materials that you must model. Fortunately, Abaqus can accept test data directly and calculate the material parameters for you (using a least squares fit).