*HYPERELASTIC

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*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 $\bar{I}_1$ , $\bar{I}_2$ and be defined by:

$\displaystyle \bar{I}_1$	$\displaystyle =$	$\displaystyle III_C^{-1/3} I_C$	(173)
$\displaystyle \bar{I}_2$	$\displaystyle =$	$\displaystyle III_C^{-1/3} II_C$	(174)
$\displaystyle J$	$\displaystyle =$	$\displaystyle III_C^{1/2}$	(175)

where , and are the invariants of the right Cauchy-Green deformation tensor $C_{KL}=x_{k,K}x_{k,L}$ . The tensor $C_{KL}$ is linked to the Lagrange strain tensor $E_{KL}$ by:

$\displaystyle 2E_{KL}=C_{KL}-\delta_{KL}$

(176)

where $\delta$ is the Kronecker symbol.

The Arruda-Boyce strain energy potential takes the form:

$\displaystyle U$	$\displaystyle =$	$\displaystyle \mu \Bigg\{ \frac{1}{2}(\bar{I}_1-3)+\frac{1}{20\lambda_m^2}(\bar{I}_1^2-9)+\frac{11}{1050\lambda_m^4}(\bar{I}_1^3-27)$
	$\displaystyle +$	$\displaystyle \frac{19}{7000\lambda_m^6}(\bar{I}_1^4-81)+\frac{519}{673750\lambda_m^8}(\bar{I}_1^5-243) \Bigg\}$	(177)
	$\displaystyle +$	$\displaystyle \frac{1}{D} \left( \frac{J^2-1}{2} - \ln J \right)$

The Mooney-Rivlin strain energy potential takes the form:

$\displaystyle U=C_{10}(\bar{I}_1-3)+C_{01}(\bar{I}_2-3)+\frac{1}{D_1}(J-1)^2$

(178)

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

The Neo-Hooke strain energy potential takes the form:

$\displaystyle U=C_{10}(\bar{I}_1-3)+\frac{1}{D_1}(J-1)^2$

(179)

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

The polynomial strain energy potential takes the form:

$\displaystyle U=\sum_{i+j=1}^{N} C_{ij}(\bar{I}_1-3)^i(\bar{I}_2-3)^j +\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}$

(180)

In CalculiX $N\le 3$ .

The reduced polynomial strain energy potential takes the form:

$\displaystyle U=\sum_{i=1}^{N} C_{i0}(\bar{I}_1-3)^i +\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}$

(181)

In CalculiX $N\le 3$ . 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 $\lambda_1$ , $\lambda_2$ and $\lambda_3$ ( $\lambda_1^2$ , $\lambda_2^2$ and $\lambda_3^2$ are the eigenvalues of the right Cauchy-Green deformation tensor) and the deviatoric stretches by $\bar{\lambda}_1$ , $\bar{\lambda}_2$ and $\bar{\lambda}_3$ , where $\bar{\lambda}_i=III_C^{-1/6}\lambda_i$ , the Ogden strain energy potential takes the form:

$\displaystyle U=\sum_{i=1}^{N} \frac{2 \mu_i}{\alpha_i^2}(\bar{\lambda}_1^{\alp... ...{\alpha_i}+\bar{\lambda}_3^{\alpha_i}-3)+\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}.$

(182)

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:

$\mu$ .
$\lambda_{m}$ .
.
Temperature

Repeat this line if needed to define complete temperature dependence.

Following line for the MOONEY-RIVLIN model:

$C_{10}$ .
$C_{01}$ .
$D_{1}$ .
Temperature

Repeat this line if needed to define complete temperature dependence.

Following line for the NEO HOOKE model:

$C_{10}$ .
$D_{1}$ .
Temperature.

Repeat this line if needed to define complete temperature dependence.

Following line for the OGDEN model with N=1:

$\mu_{1}$ .
$\alpha_{1}$ .
$D_{1}$ .
Temperature.

Repeat this line if needed to define complete temperature dependence.

Following line for the OGDEN model with N=2:

$\mu_{1}$ .
$\alpha_{1}$ .
$\mu_{2}$ .
$\alpha_{2}$ .
$D_{1}$ .
$D_{2}$ .
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:

$\mu_{1}$ .
$\alpha_{1}$ .
$\mu_{2}$ .
$\alpha_{2}$ .
$\mu_{3}$ .
$\alpha_{3}$ .
$D_{1}$ .
$D_{2}$ .

Second line of pair:

$D_{3}$ .
Temperature.

Repeat this pair if needed to define complete temperature dependence.

Following line for the POLYNOMIAL model with N=1:

$C_{10}$ .
$C_{01}$ .
$D_{1}$ .
Temperature.

Repeat this line if needed to define complete temperature dependence.

Following line for the POLYNOMIAL model with N=2:

$C_{10}$ .
$C_{01}$ .
$C_{20}$ .
$C_{11}$ .
$C_{02}$ .
$D_{1}$ .
$D_{2}$ .
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:

$C_{10}$ .
$C_{01}$ .
$C_{20}$ .
$C_{11}$ .
$C_{02}$ .
$C_{30}$ .
$C_{21}$ .
$C_{12}$ .

Second line of pair:

$C_{03}$ .
$D_{1}$ .
$D_{2}$ .
$D_{3}$ .
Temperature.

Repeat this pair if needed to define complete temperature dependence.

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

$C_{10}$ .
$D_{1}$ .
Temperature.

Repeat this line if needed to define complete temperature dependence.

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

$C_{10}$ .
$C_{20}$ .
$D_{1}$ .
$D_{2}$ .
Temperature.

Repeat this line if needed to define complete temperature dependence.

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

$C_{10}$ .
$C_{20}$ .
$C_{30}$ .
$D_{1}$ .
$D_{2}$ .
$D_{3}$ .
Temperature.

Repeat this line if needed to define complete temperature dependence.

Following line for the YEOH model:

$C_{10}$ .
$C_{20}$ .
$C_{30}$ .
$D_{1}$ .
$D_{2}$ .
$D_{3}$ .
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: $\mu_{1}$ = 3.488, $\alpha_{1}$ = 2.163, $D_{1}$ =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.

Next: *HYPERFOAM Up: Input deck format Previous: *HEAT TRANSFER Contents

guido dhondt 2014-03-02