Following are several forms of strain energy potential (Ψ) provided for the simulation of nearly incompressible hyperelastic materials. The different models are generally applicable over different ranges of strain as illustrated in the table below, however these numbers are not definitive and users should verify the applicability of the model chosen prior to use.
Currently hyperelastic materials may only be used in solid elements for explicit dynamics simulations.
| Model | Applied Strain Range |
|---|---|
| Neo-Hookean | 30% |
| Mooney-Rivlin | 30%-200% depending on order |
| Polynomial | |
| Ogden | Up to 700% |
Neo-Hookean
The strain energy function for the Neo-Hookean hyperelastic model is,
where 
is the deviatoric first principal invariant, J is the Jacobian and the required input parameters are defined as:
| µ = initial shear modulus of the material |
| d= incompressibility parameter. |
and the initial bulk modulus is defined as:
K = 2/d
Mooney-Rivlin
The strain energy function of a hyperelastic material can be expanded as an infinite
series in terms of the first and second deviatoric principal invariants 
and 
, as follows,
The 2, 3, 5 and 9 parameter Mooney-Rivlin hyperelastic material models have been implemented and are described in turn below.
2–Parameter Mooney-Rivlin Model
The strain energy function for the 2–parameter model is,
where:
| C10, C01 = material constants |
| d = material incompressibility parameter. |
The initial shear modulus is defined as:
and the initial bulk modulus is defined as:
K = 2/d
3–Parameter Mooney-Rivlin Model
The strain energy function for the 3–parameter model is,
where the required input parameters are defined as:
| C 10 , C 01 ,C 11 = material constants |
| d = material incompressibility parameter |
The bulk and shear modulus are as defined for the 2–parameter Mooney-Rivlin model.
5–Parameter Mooney-Rivlin Model
The strain energy function for the 5–parameter model is,
 | (11–1) |
where the required input parameters are defined as:
| C 10 ,C 01 ,C 20 ,C 11 ,C 02 = material constants |
| d = material incompressibility parameter. |
The bulk and shear modulus are as defined for the 2–parameter Mooney-Rivlin model.
9–Parameter Mooney-Rivlin Model
The strain energy function for the 9–parameter hyperelastic model is,
 | (11–2) |
where the required input parameters are defined as:
| C 10 ,C 01 ,C 20 ,C 11 , C 02 , C 30 , C 21 , C 12 ,C 03 = material constants |
| d = material incompressibility parameter. |
The bulk and shear modulus are as defined for the 2–parameter Mooney-Rivlin model.
Polynomial
The strain energy function of a hyperelastic material can be expanded as an infinite series of the first and second deviatoric principal invariants l 1 and l 2. The polynomial form of strain energy function is given below:
1st, 2nd, and 3rd order polynomial hyperelastic material models have been implemented in the solver where N is 1, 2 or 3 respectively.
| Cmn = material constants |
| dk = material incompressibility parameters. |
The initial shear modulus is defined as:
and the initial bulk modulus is defined as:
K = 2/d1
Yeoh
The Yeoh hyperelastic strain energy function is similar to the Mooney-Rivlin models described above except that it is only based on the first deviatoric strain invariant. It has the general form,
Yeoh 1st order
The strain energy function for the first order Yeoh model is,
where:
| N = 1 |
| C10 = material constant |
| d1 = incompressibility parameter |
The initial shear modulus is defined as:
µ = 2c10
and the initial bulk modulus is defined as:
K = 2/d1
Yeoh 2nd order
The strain energy function for the second order Yeoh hyperelastic model is
where the required input parameters are defined as:
| N = 2. |
| C10, C20 = material constants |
| d1, d2 = incompressibility parameters |
See 1st order Yeoh model for definitions of the initial shear and bulk modulus.
Yeoh 3rd order
The strain energy function for the third order Yeoh hyperelastic model is,
where the required input parameters are defined as:
| N = 3. |
| C10, C20, C30 = material constants |
| d1, d2, d3 = incompressibility parameters |
See 1st order Yeoh model for definitions of the initial shear and bulk modulus.
Ogden
The Ogden form of the strain energy function is based on the deviatoric principal stretches of the left-Cauchy-Green tensor and has the form,
Ogden 1st Order
The strain energy function for the first order Ogden hyperelastic model is,
where:
| λ p = deviatoric principal stretches of the left-Cauchy-Green tensor |
| J = determinant of the elastic deformation gradient |
| µp, αp and dp = material constants |
The initial shear modulus is given as:
and the initial bulk modulus is:
Ogden 2nd order
The strain energy function for the first order Ogden hyperelastic model is,
where:
| λ p= deviatoric principal stretches of the left-Cauchy-Green tensor |
| J = determinant of the elastic deformation gradient |
| µp, α p and dp = material constants |
The initial shear modulus is given as:
and the initial bulk modulus is:
Ogden 3rd order
The strain energy function for the first order Ogden hyperelastic model is,
where:
| λ p= deviatoric principal stretches of the left-Cauchy-Green tensor |
| J = determinant of the elastic deformation gradient |
| µp, α p and dp = material constants |
The initial shear modulus is given as:
and the initial bulk modulus is:
