MATHE

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Example

Definitions

Comments

1. If the Cpq and TAB# fields are input, the Cpq (≠ 0.0) values are overwritten with the curve fit values based on the corresponding TAB# tables. However, any Cpq values set to 0.0 are not overwritten.
2. The Generalized polynomial form (MOONEY) of the Hyperelastic material model is written as a combination of the deviatoric and volumetric strain energy of the material. The potential or strain energy density ( $W$ ) is written in polynomial form, as:
   Generalized polynomial form (MOONEY): (1)

   $$W={\displaystyle \sum _{p+q=1}^{N_1}{C}_{pq}{\left({\overline{I}}_1-3\right)}^p{\left({\overline{I}}_2-3\right)}^q}+{\displaystyle \sum _{p=1}^{N_2}\frac{1}{D_p}{\left(J_{elas}-1\right)}^{2p}}$$
   Where,

   $N_1$

   Order of the distortional strain energy polynomial function (NA).

   $N_2$

   Order of the volumetric strain energy polynomial function (ND). Currently only first order volumetric strain energy functions are supported (ND=1).

   $C_{pq}$

   The material constants related to distortional deformation ( $C_{pq}$ ).

   ${\overline{I}}_1$ , ${\overline{I}}_2$

   Strain invariants, calculated internally by OptiStruct.

   $D_p$

   Material constants related to volumetric deformation ( $D_p$ ). These values define the compressibility of the material.

   $J_{\mathit{elas}}$

   Elastic volume strain, calculated internally by OptiStruct.

3. The polynomial form can be used to model the following material types by specifying the corresponding coefficients ( $C_{pq}$ , $D_p$ ) on the MATHE entry.

   Physical Mooney-Rivlin Material (MOOR):

   N₁ = N₂ =1 (2)

   $$W=C_{10}\left({\overline{I}}_1-3\right)+C_{01}\left({\overline{I}}_2-3\right)+\frac{1}{D_1}{\left(J_{elas}-1\right)}^2$$

   Reduced Polynomial (RPOLY):

   q=0, N₂ =1(3)

   $$W={\displaystyle \sum _{p=1}^{N_1}{C}_{p0}{\left({\overline{I}}_1-3\right)}^p}+\frac{1}{D_1}{\left(J_{elas}-1\right)}^2$$

   Neo-Hooken Material (NEOH):

   N₁= N₂ =1, q=0(4)

   $$W=C_{10}\left({\overline{I}}_1-3\right)+\frac{1}{D_1}{\left(J_{elas}-1\right)}^2$$

   Yeoh Material (YEOH):

   N₁ =3 N₂ =1, q=0(5)

   $$W=C_{10}\left({\overline{I}}_1-3\right)+\frac{1}{D_1}{\left(J_{elas}-1\right)}^2+C_{20}{\left({\overline{I}}_1-3\right)}^2+C_{30}{\left({\overline{I}}_1-3\right)}^3$$

   Some other material models from the Generalized Mooney Rivlin model are:

   Three term Mooney-Rivlin Material: (6)

   $$W=C_{10}\left({\overline{I}}_1-3\right)+C_{01}\left({\overline{I}}_2-3\right)+C_{11}\left({\overline{I}}_1-3\right)\left({\overline{I}}_2-3\right)+\frac{1}{D_1}{\left(J_{elas}-1\right)}^2$$
   Signiorini Material: (7)

   $$W=C_{10}\left({\overline{I}}_1-3\right)+C_{01}\left({\overline{I}}_2-3\right)+C_{20}{\left({\overline{I}}_1-3\right)}^2+\frac{1}{D_1}{\left(J_{elas}-1\right)}^2$$
   Third Order Invariant Material: (8)

   $$W=C_{10}\left({\overline{I}}_1-3\right)+C_{01}\left({\overline{I}}_2-3\right)+C_{11}\left({\overline{I}}_1-3\right)\left({\overline{I}}_2-3\right)+C_{20}{\left({\overline{I}}_1-3\right)}^2+\frac{1}{D_1}{\left(J_{elas}-1\right)}^2$$
   Third Order Deformation Material (James-Green-Simpson): (9)

   $$\begin{array}{l}W=C_{10}\left({\overline{I}}_1-3\right)+C_{01}\left({\overline{I}}_2-3\right)+C_{11}\left({\overline{I}}_1-3\right)\left({\overline{I}}_2-3\right)\\ +C_{20}{\left({\overline{I}}_1-3\right)}^2+C_{30}{\left({\overline{I}}_1-3\right)}^3+\frac{1}{D_1}{\left(J_{elas}-1\right)}^2\end{array}$$
4. The MATHE hyperelastic material supports CTETRA (4, 10), CPENTA (6, 15), and CHEXA (8, 20) element types.
5. The Arruda-Boyce model (ABOYCE) is defined as: (10)
   $$W=C_1{\displaystyle \sum _{i=1}^5{\alpha }_i{\beta }^{i-1}\left({\overline{I}}_1^i-3^i\right)+}\frac{1}{D_1}{\left(J_{elas}-1\right)}^2$$

   Where,

   $\beta =\frac{1}{N}=\frac{1}{{\lambda }_m^2}$

   $N$

   Measure of the limiting locking stretch.

   ${\lambda }_m$

   Maximum locking stretch.

   $D_1$

   Related to volumetric deformation. It defines the compressibility of the material.

   ${\overline{I}}_1$

   First strain invariant, internally calculated by OptiStruct.

   Wherein, ${\overline{I}}_1=I_1{J}^{-\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ .

   $J_{elas}$

   Elastic volume strain, internally calculated by OptiStruct.

   $C_1$

   Initial shear modulus.

   ${\alpha }_1=\frac{1}{2};{\alpha }_2=\frac{1}{20};{\alpha }_3=\frac{11}{1050};{\alpha }_4=\frac{19}{7000};{\alpha }_5=\frac{519}{673750}$

6. The Ogden Material model (OGDEN) is defined as: (11)
   $$W={\displaystyle \sum _{i=1}^{N_1}\frac{2{\mu }_i}{{\alpha }_i^2}\left({\overline{\lambda }}_1^{{\alpha }_i}+{\overline{\lambda }}_2^{{\alpha }_i}+{\overline{\lambda }}_3^{{\alpha }_i}-3\right)}+\frac{1}{D_1}{\left(J_{elas}-1\right)}^2$$
   Where,

   ${\overline{\lambda }}_1,{\overline{\lambda }}_2,{\overline{\lambda }}_3$

   The three deviatoric stretches (deviatoric stretches are related to principal stretches by ${\overline{\lambda }}_i=J^{\frac{1}{3}}{\lambda }_i$ )

   ${\mu }_i$

   Defined by the MUi fields

   ${\alpha }_i$

   Defined by the ALPHAi fields

   $N_1$

   Order of the deviatoric part of the strain energy function defined on the NA field

7. The Hill Foam Material model (FOAM) is defined as:(12)
   $$W={\displaystyle \sum _{i=1}^{N_1}\frac{2{\mu }_i}{{\alpha }_i^2}\left({\lambda }_1^{{\alpha }_i}+{\lambda }_2^{{\alpha }_i}+{\lambda }_3^{{\alpha }_i}-3+\frac{1}{{\beta }_i}\left(J^{-{\alpha }_i{\beta }_i}-1\right)\right)}$$
   Where,

   ${\lambda }_1,{\lambda }_2,{\lambda }_3$

   Principle stretches

   ${\mu }_i$

   Defined by the MUi fields

   ${\alpha }_i$

   Defined by the ALPHAi fields

   ${\beta }_i$

   Defined by the BETAi fields

   $N_1$

   Order of the strain energy function defined on the NA field.

   Currently, the Hill material model is only supported for explicit analysis.

8. This card is represented as a material in HyperMesh.