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 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 ratio.

   ${\lambda }_m$

   Maximum locking stretch ratio.

   $D_1$

   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.

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

5. 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 stretch ratios (deviatoric stretch ratios are related to principal stretch ratios 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.

6. 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 stretch ratios.

   ${\mu }_i$

   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.

   Additionally, MUi/ALPHAi can instead be fitted using TAB# table data, and BETAi are user specified values.

   If the TAB# fields are input, the MUi/ALPHAi values are overwritten by the fitted values. Any user specified values of MUi/ALPHAi will be overwritten.

   If both Poisson’s ratio NU (non-zero) and TAB# are specified, BETAi values will all be determined or overwritten by:(13)

   $${\beta }_i=\frac{\nu }{1-2\nu }$$

   If Poisson’s ratio NU is 0.0 or not specified, then it will be ignored. For parameter fitting, only the first value BETA1 will be used and BETA2-BETA5 are not used. It is recommended to use the same value of BETAi for parameter fitting.

   The Hill FOAM material is supported for both Implicit and Explicit Nonlinear analysis.

7. The Marlow model is a hyperelastic material model which directly defines the potential based on the experiment test data; there are no mathematical expressions based on the deformation tensors’ invariants or the deformation stretches for the potential. The isochoric deformation potential is determined by TAB1, TAB2 or TAB4. Only one test can be specified.

   A uniaxial tension test is equivalent to an equi-biaxial compression test; a uniaxial compression test is equivalent to an equi-biaxial tension test; a planar tension test is equivalent to a planar compression test. Either tension or compression test data can be specified but not at the same time.

   For Marlow, D1, TABD, or Poisson’s ratio can be defined to specify the volumetric behavior. Either D1 or TABD can be specified, but not both.

   1. If D1 or TABD is specified, the volumetric behavior is determined by D1 or TABD.
   2. If D1 and TABD are not specified and Poisson’s ratio is specified, Poisson’s ratio is used to determine volumetric behavior.
   3. If D1, TABD, or Poisson’s ratio are all not specified, the default Poisson’s ratio of 0.495 is used to determine volumetric behavior.
   4. If Poisson’s ratio and one of D1 or TABD are defined, D1 or TABD take precedence.
8. If Poisson’s ratio and D1 or TABD are both defined, Poisson’s ratio takes precedence.
9. The initial modulus used for linear analysis is:
   • Mooney, Neo-Hookean, Mooney-Rivlin, Yeoh, Reduced Polynomial

     $G=2(C_{10}+C_{01})$ and $K=\frac{2}{D_1}$

   • Ogden

     $G={\mu }_1+{\mu }_2+{\mu }_3+{\mu }_4+{\mu }_5={\displaystyle \sum _{t=1}^5{\mu }_t}$ and $K=\frac{2}{D_1}$

   • Arruda-Boyce

     $G=C_1\left(1+\frac{3}{5{\lambda }_m^2}+\frac{99}{175{\lambda }_m^4}+\frac{513}{875{\lambda }_m^6}+\frac{42039}{67375{\lambda }_m^8}\right)$ and $K=\frac{2}{D_1}$

   • Hill Foam

     $G={\mu }_1+{\mu }_2+{\mu }_3+{\mu }_4+{\mu }_5={\displaystyle \sum _{i=1}^5{\mu }_i}$ and $K={\displaystyle \sum _{i=1}^{5}2{\mu }_i}\left(\frac{1}{3}+{\beta }_i\right)$

   Additional treatments on bulk modulus $K$ are as:

   • If Poisson's ratio $\nu$ , is non-zero, bulk modulus $K$ is replaced with:(14)
     $$K=\frac{2G(1+\nu )}{3(1-2\nu )}$$
   • If $K=0,$ $K$ is set to be $30G$
   • If $K\ge 30G,$ $K$ is set to be $30G$
   The Young’s modulus and Poisson's ratio are given by:(15)

   $$E=\frac{9KG}{3K+G}$$
   and(16)

   $$\nu =\frac{3K-2G}{6K+2G}$$
   Where,

   $E$

   Young’s modulus.

   $G$

   Shear modulus.

   $K$

   Bulk modulus.

   $\nu$

   Poisson's ratio.

   $C_{10}$ , $C_{01}$ , $D_1$ , ${\mu }_i$ , and $C_1$

   Material coefficients.

   ${\lambda }_m$

   Stretch ratio at which the polymer chain network is locked.

10. MODULI continuation line is only applicable when used together with the MATVE entry. Refer to MATVE which provides additional information on how this material input is interpreted.
11. The support information for the available material models (in Model field) is:

    Analysis Type	Support Information
    Implicit Analysis Nonlinear Static Analysis Nonlinear Transient Analysis	All the material models are supported with: Plane strain elements Axisymmetric elements Solid elements
    Explicit Dynamic Analysis	All the material models are supported with: Solid elements

12. Temperature-dependent hyperelastic material data can be defined via the MATTHE entry.
13. This card is represented as a material in HyperMesh.