Isotropic Hyperelasticity

Overview of Strain Energy Potentials

Hyperelastic materials are described in terms of a "strain energy potential," $U_{ε}$ , that defines the strain energy stored in the material per unit of reference volume (volume in the initial configuration) as a function of the strain at that point in the material. There are several strain energy potentials (strain energy stored in the material per unit of reference volume as a function of the strain at that point in the material) that you can choose from to define the material behavior.

Generally, when data from multiple experimental tests are available (typically, this requires at least uniaxial and equibiaxial test data), the Ogden and Van der Waals forms are more accurate in fitting experimental results. If limited test data are available for calibration, the Arruda-Boyce, Van der Waals, Yeoh, or reduced polynomial forms provide reasonable behavior. When only one set of test data (uniaxial, equibiaxial, or planar test data) is available, the Marlow form is recommended. In this case a strain energy potential is constructed that will reproduce the test data exactly and that will have reasonable behavior in other deformation modes.


Strain Energy Potential	Description
Arruda-Boyce	Simulates hyperelasticity using a representative cube of material stressed along eight diagonals. Under conditions of small deformation, this model reduces to the Neo-Hooke model.
Neo-Hooke	Simulates hyperelasticity using an algorithm equivalent to the reduced polynomial model, but with N=1.
Ogden	Uses data derived from results of stretching the material in each of the three principal directions.
Polynomial	Uses a strain energy density function that is a polynomial equation. It is the most complex hyperelastic model available for use.
Reduced Polynomial	Simulates hyperelasticity using an algorithm equivalent to the polynomial model, but with $C_{ij} = 0$ , for $j \neq 0$ .
Mooney Rivlin	Simulates hyperelasticity using an algorithm equivalent to the polynomial model, but with N=1.
Van der Waals	The Van der Waals model is also known as the Kilian model.
Yeoh	Simulates hyperelasticity using an algorithm equivalent to the reduced polynomial model, but with N=3.
Marlow	Uses a general first-invariant constitutive model suitable for analyses when only one set of test data (uniaxial, equibiaxial, or planar test data) is available.
User	User defined strain energy potential.

Arruda-Boyce

U = μ {\frac{1}{2} ({\bar{I}}_{1} - 3) + \frac{1}{20 λ_{m}^{2}} ({\bar{I}}_{1}^{2}_{} - 9) + \frac{11}{1050 λ_{m}^{4}} ({\bar{I}}_{1}^{3}_{} - 27) + \frac{19}{7000 λ_{m}^{6}} ({\bar{I}}_{1}^{4}_{} - 81) + \frac{519}{673750 λ_{m}^{8}} ({\bar{I}}_{1}^{5}_{} - 243)} + \frac{1}{D} (\frac{J_{e l}^{2} - 1}{2} - \ln J_{e l}),

where $U$ is the strain energy per unit of reference volume, $μ, λ_{m,}$ and $D$ are material parameters, and ${\bar{I}}_{1}$ is the first deviatoric strain invariant, defined as

{\bar{I}}_{1} = {\bar{λ}}_{1}^{2} + {\bar{λ}}_{2}^{2} + {\bar{λ}}_{3}^{2},

where ${\bar{λ}}_{i} = J^{- 1 / 3} λ_{i}$ ; $J$ is the total volume ratio; $J^{e l}$ is the elastic volume ratio; and $λ_{i}$ are the principal stretches.


Input Data	Description
mu	$μ$
lambda m	$λ_{m}$
D	$D$
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table.
Moduli time scale	Select Instantaneous or Long Term for the application of viscoelastic effects.

Neo-Hooke

U = C_{10} ({\bar{I}}_{1} - 3) + \frac{1}{D_{1}} {(J^{e l} - 1)}^{2},

where $J^{e l}$ is the elastic volume ratio, $C_{10}$ and $D_{1}$ are material parameters, and ${\bar{I}}_{1}$ is the first deviatoric strain invariant.


Input Data	Description
C10	$C_{10}$
D1	$D_{1}$
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table.
Moduli time scale	Select Instantaneous or Long Term for the application of viscoelastic effects.

Ogden

The Ogden strain energy potential includes a Strain Energy Potential Order.

The Ogden strain energy potential varies according to the following equation:

U = \sum_{i = 1}^{N} \frac{2 μ_{i}}{α_{i}^{2}} ({\bar{λ}}_{1}^{α_{i}} + {\bar{λ}}_{2}^{α_{i}} + {\bar{λ}}_{3}^{α_{i}} - 3) + \sum_{i = 1}^{N} \frac{1}{D_{i}} {(J^{e l} - 1)}^{2 i},

where ${\bar{λ}}_{i} = J^{- 1 / 3} λ_{i}$ ; $λ_{i}$ are the principal stretches; $N$ is the strain energy potential order; $J$ is the total volume ratio; $μ, α,$ and $D$ are material parameters; and $J^{e l}$ is the elastic volume ratio.

The table below lists the parameters for a first order potential. Each of the material parameters is repeated according to the chosen numeric order.


Input Data	Description
Strain Energy Potential Order	Numeric order, $N$ .
mu1	$μ_{1}$
alpha1	$α_{1}$
D1	$D_{1}$
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table.
Moduli time scale	Select Instantaneous or Long Term for the application of viscoelastic effects.

Polynomial

The Polynomial strain energy potential includes a Strain Energy Potential Order.

The polynomial form of strain energy potential varies according to the following equation:

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

where $U$ is the strain energy per unit of reference volume, $N$ is the strain energy potential order, $J^{e l}$ is the elastic volume ratio, and ${\bar{I}}_{1}$ and ${\bar{I}}_{2}$ are the first and the second deviatoric strain invariants.

The table below lists the parameters for a first-order potential.


Input Data	Description
Strain Energy Potential Order	Numeric order, $N$ .
$C_{10}$	The constant on the first deviatoric strain invariant term of the polynomial strain energy potential function.
$C_{01}$	The constant on the second deviatoric strain invariant term of the polynomial strain energy potential function.
$D_{1}$	The compressibility constant in the polynomial strain energy potential function.
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table.
Moduli time scale	Select Instantaneous or Long Term for the application of viscoelastic effects.

Reduced Polynomial

The Reduced Polynomial strain energy potential includes a Strain Energy Potential Order.

The reduced polynomial form of strain energy potential varies according to the following equation:

U = \sum_{i = 1}^{N} C_{i 0} {({\bar{I}}_{1} - 3)}^{i} + \sum_{i = 1}^{N} \frac{1}{D_{i}} {(J^{e l} - 1)}^{2 i},

where $U$ is the strain energy per unit of reference volume, $N$ is the strain energy potential order, $J^{e l}$ is the elastic volume ratio, and ${\bar{I}}_{1}$ is the first deviatoric strain invariant.

The table below lists the parameters for a first-order potential. Each of the material parameters is repeated according to the chosen numeric order.


Input Data	Description
Strain energy potential order	Numeric order, $N$ .
$C_{10}$	The constant on the first deviatoric strain invariant term of the polynomial strain energy potential function.
$D_{1}$	The compressibility constant in the polynomial strain energy potential function.
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table.
Moduli time scale	Select Instantaneous or Long Term for the application of viscoelastic effects.

Mooney Rivlin

The Mooney-Rivlin strain energy potential takes the form

U = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + \frac{1}{D_{1}} {(J^{e l} - 1)}^{2},

where $U$ is the strain energy per unit of reference volume, $C_{10}$ , $C_{01}$ , and $D_{1}$ are temperature-dependent material parameters, $J^{e l}$ is the elastic volume ratio, and ${\bar{I}}_{1}$ and ${\bar{I}}_{2}$ are the first and second deviatoric strain invariants, respectively.

The table below lists the parameters for a first-order potential. Each of the material parameters is repeated according to the chosen numeric order.


Input Data	Description
C10	Temperature-dependent material parameter, $C_{10}$ .
C01	Temperature-dependent material parameter, $C_{01}$ .
D1	Temperature-dependent material parameter, $D_{1}$ .
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table.
Moduli time scale	Select Instantaneous or Long Term for the application of viscoelastic effects.

Van der Waals

The Van der Waals strain energy potential takes the form

U = μ {- (λ_{m}^{2} - 3) [\ln (1 - η) + η] + \frac{2}{3} a {(\frac{​ \tilde{I} - 3}{2})}^{\frac{3}{2}}} + \frac{1}{D} (\frac{J_{e l}^{2} - 1}{2} - \ln J^{e l}),

where

\tilde{I} = (1 - β) {\bar{I}}_{1} + β {\bar{I}}_{2} a n d η = \sqrt{\frac{\tilde{I} - 3}{λ_{m}^{2} - 3}} .

Here, $U$ is the strain energy per unit of reference volume, $μ$ is the initial shear modulus, $λ_{m}$ is the locking stretch, $a$ is the global interaction parameter, $β$ is an invariant mixture parameter, and $D$ governs the compressibility. ${\bar{I}}_{1}$ and ${\bar{I}}_{2}$ are the first and second deviatoric strain invariants defined as

{\bar{I}}_{1} = {\bar{λ}}_{1}^{2} + {\bar{λ}}_{2}^{2} + {\bar{λ}}_{3}^{2} a n d {\bar{I}}_{2} = {\bar{λ}}_{1}^{(- 2)} + {\bar{λ}}_{2}^{(- 2)} + {\bar{λ}}_{3}^{(- 2)},

where the deviatoric stretches $\bar{λ_{i}} = J^{- \frac{1}{3}} λ_{i}$ ; $J$ is the total volume ratio; $J^{e l}$ is the elastic volume ratio; and $λ_{i}$ are the principal stretches. The initial shear modulus and bulk modulus are given by

μ_{0} = μ, Κ_{0} = \frac{2}{D} .

The table below lists the parameters for a first-order potential. Each of the material parameters is repeated according to the chosen numeric order.


Input Data	Description
mu	Initial shear modulus, $μ$ .
lambda m	Locking stretch, $λ_{m}$ .
alpha	Global interaction parameter, $a$ .
beta	An invariant mixture parameter, $β$ .
D	Compressibility, $D$ .
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table.
Moduli time scale	Select Instantaneous or Long Term for the application of viscoelastic effects.

Yeoh

The Yeoh strain energy potential takes the form

U = C_{10} ({\bar{I}}_{1} - 3) + C_{20} {({\bar{I}}_{1} - 3)}^{2} + C_{30} {({\bar{I}}_{1} - 3)}^{3} + \frac{1}{D_{1}} {(J^{e l} - 1)}^{2} + \frac{1}{D_{2}} {(J^{e l} - 1)}^{4} + \frac{1}{D_{3}} {(J^{e l} - 1)}^{6},

where $U$ is the strain energy per unit of reference volume, $C_{i 0}$ and $D_{i}$ are temperature-dependent material parameters; ${\bar{I}}_{1}$ is the first deviatoric strain invariant defined by

{\bar{I}}_{1} = {\bar{λ}}_{1}^{2} + {\bar{λ}}_{2}^{2} + {\bar{λ}}_{3}^{2},

where the deviatoric stretches $\bar{λ_{i}} = J^{- \frac{1}{3}} λ_{i}$ ; $J$ is the total volume ratio; $J^{e l}$ is the elastic volume ratio; and $λ_{i}$ are the principal stretches. The initial shear modulus and bulk modulus are given by

μ_{0} = 2 C_{10}, Κ_{0} = \frac{2}{D_{1}} .

The table below lists the parameters for a first-order potential. Each of the material parameters is repeated according to the chosen numeric order.


Input Data	Description
C10	Temperature-dependent material parameter, $C_{10}$ .
C20	Temperature-dependent material parameter, $C_{20}$ .
C30	Temperature-dependent material parameter, $C_{30}$ .
D1	Temperature-dependent material parameter, $D_{1}$ .
D2	Temperature-dependent material parameter, $D_{2}$ .
D3	Temperature-dependent material parameter, $D_{3}$ .
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table.
Moduli time scale	Select Instantaneous or Long Term for the application of viscoelastic effects.

Marlow

The Marlow strain energy potential takes the form

U = U_{d e v} ({\bar{I}}_{1}) + U_{v o l} (J_{e ℓ}),

where $U$ is the strain energy per unit of reference volume, with $U_{d e v}$ as its deviatoric part and $U_{v o l}$ as its volumetric part; ${\bar{I}}_{1}$ is the first deviatoric strain invariant defined as

{\bar{I}}_{1} = {\bar{λ}}_{1}^{2} + {\bar{λ}}_{2}^{2} + {\bar{λ}}_{3}^{2},

where the deviatoric stretches ${\bar{λ}}_{i} = J^{- \frac{1}{3}} λ_{i}$ , $J$ is the total volume ratio, $J_{e ℓ}$ is the elastic volume ratio, and $λ_{i}$ are the principal stretches. The deviatoric part of the potential is defined by providing uniaxial test data; while the volumetric part is defined by providing the volumetric test data, defining the Poisson's ratio, or specifying the lateral strains together with the uniaxial test data.

The table below lists the basic parameters for the Marlow strain energy potential.


Input Data	Description
Deviatoric response	Deviatoric response is determined by the uniaxial test data, which you can specify in Uniaxial Test Data Options for Marlow Hyperelasticity. Uniaxial test data is required.
Volumetric response	Type of data used to specify the volumetric response. Ignore test data: Leaves test data out of the calculation of volumetric response. Volumetric test data: Calculates volumetric response using test data and specifies test data behavior in Volumetric Test Data Options for Marlow Hyperelasticity. Lateral nominal strain: Specifies the lateral strains together with the uniaxial test data. Poisson's ratio: Specifies the Poisson's ratio value directly.
Moduli time scale	Select Instantaneous or Long Term for the application of viscoelastic effects.

User

You can specify strain energy potential as a function of strain invariants through the user subroutine UHYPER for implicit simulations.

Table 1. Options to specify user defined form
Input Data	Description
Compressiblity	Specify purely incompressible (enforce volume preserving) or compressible (allow volume changing) behavior occuring from mechanical forces.
User defined material parameters	User defined material parameters to calculate strain energy.