*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:

\begin{eqnarray}\bar{I}_1&=&III_C^{-1/3} I_C \\\bar{I}_2&=&III_C^{-1/3} II_C \\J&=&III_C^{1/2}\end{eqnarray} \displaystyle = \displaystyle III_C^{-1/3} I_C (797)

\displaystyle \bar{I}_2 \displaystyle = \displaystyle III_C^{-1/3} II_C (798)

\displaystyle J \displaystyle = \displaystyle III_C^{1/2} (799)

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:

2E_{KL}=C_{KL}-\delta_{KL} (800)

where \delta is the Kronecker symbol.

The Arruda-Boyce strain energy potential takes the form:

\begin{eqnarray}U&=&\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) \nonumber\\&+& \frac{19}{7000\lambda_m^6}(\bar{I}_1^4-81)+\frac{519}{673750\lambda_m^8}(\bar{I}_1^5-243) \Bigg\} \\&+& \frac{1}{D} \left( \frac{J^2-1}{2} - \ln J \right) \nonumber\end{eqnarray} \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\} (801)

\displaystyle + \displaystyle \frac{1}{D} \left( \frac{J^2-1}{2} - \ln J \right)

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-1)^2 (802)

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:

U=C_{10}(\bar{I}_1-3)+\frac{1}{D_1}(J-1)^2 (803)

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:

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} (804)

In CalculiX N\le 3.

The reduced polynomial strain energy potential takes the form:

U=\sum_{i=1}^{N} C_{i0}(\bar{I}_1-3)^i +\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i} (805)

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:

U=\sum_{i=1}^{N} \frac{2 \mu_i}{\alpha_i^2}(\bar{\lambda}_1^{\alpha_i}+\bar{\lambda}_2^{\alpha_i}+\bar{\lambda}_3^{\alpha_i}-3)+\sum_{i=1}^{N}\frac{1}{D_i}(J-1)^{2i}. (806)

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}.
• D.
• 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.