## *HYPERELASTIC

Keyword type: model definition, material

This option is used to define the hyperelastic properties of a material. There are two optional parameters. The first one defines the model and can take one of the following strings: ARRUDA-BOYCE, MOONEY-RIVLIN, NEO HOOKE, OGDEN, POLYNOMIAL, REDUCED POLYNOMIAL or YEOH. The second parameter N makes sense for the OGDEN, POLYNOMIAL and REDUCED POLYMIAL model only, and determines the order of the strain energy potential. Default is the POLYNOMIAL model with N=1. All constants may be temperature dependent.

Let , and be defined by:

 (631) (632) (633)

where , and are the invariants of the right Cauchy-Green deformation tensor . The tensor is linked to the Lagrange strain tensor by:

 (634)

where is the Kronecker symbol.

The Arruda-Boyce strain energy potential takes the form:

 (635)

The Mooney-Rivlin strain energy potential takes the form:

 (636)

The Mooney-Rivlin strain energy potential is identical to the polynomial strain energy potential for .

The Neo-Hooke strain energy potential takes the form:

 (637)

The Neo-Hooke strain energy potential is identical to the reduced polynomial strain energy potential for .

The polynomial strain energy potential takes the form:

 (638)

In CalculiX .

The reduced polynomial strain energy potential takes the form:

 (639)

In CalculiX . The reduced polynomial strain energy potential can be viewed as a special case of the polynomial strain energy potential

The Yeoh strain energy potential is nothing else but the reduced polynomial strain energy potential for .

Denoting the principal stretches by , and ( , and are the eigenvalues of the right Cauchy-Green deformation tensor) and the deviatoric stretches by , and , where , the Ogden strain energy potential takes the form:

 (640)

The input deck for a hyperelastic material looks as follows:

First line:

• *HYPERELASTIC
• Enter parameters and their values, if needed

Following line for the ARRUDA-BOYCE model:

• .
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• Temperature
Repeat this line if needed to define complete temperature dependence.

Following line for the MOONEY-RIVLIN model:

• .
• .
• .
• Temperature
Repeat this line if needed to define complete temperature dependence.

Following line for the NEO HOOKE model:

• .
• .
• Temperature.
Repeat this line if needed to define complete temperature dependence.

Following line for the OGDEN model with N=1:

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• Temperature.
Repeat this line if needed to define complete temperature dependence.

Following line for the OGDEN model with N=2:

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• Temperature.
Repeat this line if needed to define complete temperature dependence.

Following lines, in a pair, for the OGDEN model with N=3: First line of pair:

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• .
Second line of pair:
• .
• Temperature.
Repeat this pair if needed to define complete temperature dependence.

Following line for the POLYNOMIAL model with N=1:

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• Temperature.
Repeat this line if needed to define complete temperature dependence.

Following line for the POLYNOMIAL model with N=2:

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• Temperature.
Repeat this line if needed to define complete temperature dependence.

Following lines, in a pair, for the POLYNOMIAL model with N=3: First line of pair:

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Second line of pair:
• .
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• Temperature.
Repeat this pair if needed to define complete temperature dependence.

Following line for the REDUCED POLYNOMIAL model with N=1:

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• Temperature.
Repeat this line if needed to define complete temperature dependence.

Following line for the REDUCED POLYNOMIAL model with N=2:

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• Temperature.
Repeat this line if needed to define complete temperature dependence.

Following line for the REDUCED POLYNOMIAL model with N=3:

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• Temperature.
Repeat this line if needed to define complete temperature dependence.

Following line for the YEOH model:

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• Temperature.
Repeat this line if needed to define complete temperature dependence.

Example:

*HYPERELASTIC,OGDEN,N=1
3.488,2.163,0.


defines an ogden material with one term: = 3.488, = 2.163, =0. Since the compressibility coefficient was chosen to be zero, it will be replaced by CalculiX by a small value to ensure some compressibility to guarantee convergence (cfr. page ).

Example files: beamnh, beamog.