8.15.6. Formulation and Parameters for a Hyper Elastic Material

A hyper elastic material has four available models: NEO-HOOKIAN, ARRUDA-BOYCE, OGDEN and MOONEY-RIVLIN. The parameters for a hyper elastic material are shown in the figure and table below.

Figure 8.84: Properties of a hyper elastic material

Figure 8.85: Properties of a hyper elastic material

Parameter	Symbol	Description	Dimension (Range)
1. C10		Use to set the elastic constant which corresponds to the Young's modulus for a linear material [NEO-HOOKIAN, MOONEY-RIVLIN].	Force/Length^2 (Real≥0)
2. D		Use to set the incompressibility parameter to determine the penalty of a volume constraint. As this value becomes smaller, the volume constraint is tighter. When a flexible body with hyper elastic material is contacted with the other body, the Motion solver sometimes has a small step size. In this case, if this value is large, the step size can be increased [All Models].	N/A (0<Real≤1)
3. Mu		Use to set the elastic constant which corresponds to the shear modulus in a linear material [ARRUDA-BOYCE, OGDEN]. Note: Be careful not to confuse the material constant with the initial shear modulus . The relationship between initial shear modulus , and is shown in the NEO-HOOKIAN and MOONEY-RIVLIN section of the Figure 8.86: Strain energy of a hyper elastic material.	Force/Length^2 (Real≥0)
4. Lambda		Use to set the locking stretch [ARRUDA-BOYCE].	N/A (Real)
5. List	N/A	Use to show the material parameters. The maximum number of the parameter set is six [OGDEN].	N/A
6. Operators	N/A	Use to add a parameter set, or modify or delete the selected set [OGDEN].	N/A
7. Alpha		Use to set the elastic constant [OGDEN].	N/A (Real)
8. C01		Use to set the 2^nd elastic constant which corresponds to the Young's modulus in a linear material [MOONEY-RIVLIN].	Force/Length^2 (Real≥0)

As with Equation 8–74, the strain energy of a hyper elastic material can be expressed as shown in the table below with the parameters as described in the table above.

Figure 8.86: Strain energy of a hyper elastic material

Model	Formulations
NEO-HOOKIAN	where is the volume strain and can be calculated with the invariant of the Cauchy-Green deformation tensor as follows. The relationship between Initial Shear Modulus and is as follows.
ARRUDA-BOYCE	where is defined in the formulation for NEO-HOOKIAN.
OGDEN	where can be calculated with the eigenvalue of the Cauchy-Green deformation tensor of as follows. In the equation for strain energy shown above, the terms for and can be expressed as follows using and . In the form of the above formula, the following relationship is established.
MOONEY-RIVLIN	where is defined in the formulation for NEO-HOOKIAN. The relationship between Initial Shear Modulus , . and is as follows.