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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 $ J$ be defined by:

$\displaystyle \bar{I}_1$ $\displaystyle =$ $\displaystyle III_C^{-1/3} I_C$ (196)
$\displaystyle \bar{I}_2$ $\displaystyle =$ $\displaystyle III_C^{-1/3} II_C$ (197)
$\displaystyle J$ $\displaystyle =$ $\displaystyle III_C^{1/2}$ (198)

where $ I_C$, $ II_C$ and $ III_C$ 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:

$\displaystyle 2E_{KL}=C_{KL}-\delta_{KL}$ (199)

where $ \delta$ is the Kronecker symbol.

The Arruda-Boyce strain energy potential takes the form:

$\displaystyle U$ $\displaystyle =$ $\displaystyle \mu \Bigg\{ \frac{1}{2}(\bar{I}_1-3)+\frac{1}{20\lambda_m^2}(\bar{I}_1^2-9)+\frac{11}{1050\lambda_m^4}(\bar{I}_1^3-27)$  
  $\displaystyle +$ $\displaystyle \frac{19}{7000\lambda_m^6}(\bar{I}_1^4-81)+\frac{519}{673750\lambda_m^8}(\bar{I}_1^5-243) \Bigg\}$ (200)
  $\displaystyle +$ $\displaystyle \frac{1}{D} \left( \frac{J^2-1}{2} - \ln J \right)$  

The Mooney-Rivlin strain energy potential takes the form:

$\displaystyle U=C_{10}(\bar{I}_1-3)+C_{01}(\bar{I}_2-3)+\frac{1}{D_1}(J-1)^2$ (201)

The Mooney-Rivlin strain energy potential is identical to the polynomial strain energy potential for $ N=1$.

The Neo-Hooke strain energy potential takes the form:

$\displaystyle U=C_{10}(\bar{I}_1-3)+\frac{1}{D_1}(J-1)^2$ (202)

The Neo-Hooke strain energy potential is identical to the reduced polynomial strain energy potential for $ N=1$.

The polynomial strain energy potential takes the form:

$\displaystyle U=\sum_{i+j=1}^{N} C_{ij}(\bar{I}_1-3)^i(\bar{I}_2-3)^j +\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}$ (203)

In CalculiX $ N\le 3$.

The reduced polynomial strain energy potential takes the form:

$\displaystyle U=\sum_{i=1}^{N} C_{i0}(\bar{I}_1-3)^i +\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}$ (204)

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 $ N=3$.

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:

$\displaystyle U=\sum_{i=1}^{N} \frac{2 \mu_i}{\alpha_i^2}(\bar{\lambda}_1^{\alp...
...{\alpha_i}+\bar{\lambda}_3^{\alpha_i}-3)+\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}.$ (205)

The input deck for a hyperelastic material looks as follows:


First line:

Following line for the ARRUDA-BOYCE model:

Repeat this line if needed to define complete temperature dependence.

Following line for the MOONEY-RIVLIN model:

Repeat this line if needed to define complete temperature dependence.

Following line for the NEO HOOKE model:

Repeat this line if needed to define complete temperature dependence.

Following line for the OGDEN model with N=1:

Repeat this line if needed to define complete temperature dependence.

Following line for the OGDEN model with N=2:

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:

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

Following line for the POLYNOMIAL model with N=1:

Repeat this line if needed to define complete temperature dependence.

Following line for the POLYNOMIAL model with N=2:

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:

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

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

Repeat this line if needed to define complete temperature dependence.

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

Repeat this line if needed to define complete temperature dependence.

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

Repeat this line if needed to define complete temperature dependence.

Following line for the YEOH model:

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 up previous contents
Next: *HYPERFOAM Up: Input deck format Previous: *HEAT TRANSFER   Contents
guido dhondt 2012-10-06