MATHE

Bulk Data Entry Defines material properties for nonlinear hyperelastic materials. The Polynomial form is available and various material types 3 can be defined by specifying the corresponding coefficients.

Format A

Generalized Mooney-Rivlin Polynomial (MOONEY), Reduced Polynomial (RPOLY), Physical Mooney-Rivlin (MOOR), Neo-Hookean (NEOH), and Yeoh model (YEOH):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATHE MID Model   NU RHO TEXP TREF
C10 C01 D1 TAB1 TAB2   TAB4 TABD
C20 C11 C02 D2 NA ND
C30 C21 C12 C03 D3
C40 C31 C22 C13 C04 D4
C50 C41 C32 C23 C14 C05 D5
MODULI MTIME

Format B

Arruda-Boyce Model (Model=ABOYCE):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATHE MID Model   NU RHO TEXP TREF
C1 $\lambda m$   TAB1 TAB2   TAB4
D1
MODULI MTIME

Format C

Ogden Material Model (Model=OGDEN):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATHE MID Model NA NU RHO TEXP TREF
MU1 ALPHA1 D1 TAB1 TAB2   TAB4
MU2 ALPHA2   MU3 ALPHA3
MU4 ALPHA4   MU5 ALPHA5
MODULI MTIME

Format D

Hill Foam Material Model (Model=FOAM):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATHE MID Model NA NU RHO TEXP TREF
MU1 ALPHA1 BETA1 TAB1 TAB2   TAB4
MU2 ALPHA2 BETA2 MU3 ALPHA3 BETA3
MU4 ALPHA4 BETA4 MU5 ALPHA5 BETA5
MODULI MTIME

Format E

Marlow Material Model (Model=MARLOW):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATHE MID Model   NU RHO TEXP TREF
D1 TAB1 TAB2   TAB4 TABD
MODULI MTIME

Example

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATHE 2 MOONEY
80 20 0.001

Definitions

Field Contents SI Unit Example
MID Unique material identification number.

No default (Integer > 0)

Model Hyperelastic material model type. See Comments.
MOONEY (Default)
Generalized Mooney-Rivlin hyperelastic model
MOOR
Physical Mooney-Rivlin model
RPOLY
Reduced Polynomial model
NEOH
Neo-Hookean model
YEOH
Yeoh model
ABOYCE
Arruda-Boyce model
OGDEN
Ogden model
FOAM
Hill foam model
MARLOW
Marlow model
blank

(Character)

NU Poisson's ratio.

Default = 0.495 for all models (except FOAM)

Default = 0.0 for FOAM (Real)

RHO Material density.

No default (Real)

TEXP Coefficient of thermal expansion.

No default (Real)

TREF Reference temperature.

No default (Real)

NA Order of the distortional strain energy polynomial function if the type of the model is generalized polynomial (MOONEY) or Reduced Polynomial (RPOLY).

It is also the Order of the Deviatoric Part of the Strain Energy Function of the OGDEN material (Format C).

Default = 2 (0 < Integer ≤ 5)

ND Order of the volumetric strain energy polynomial function. 3

Default = 1 (Integer > 0)

Cpq Material constants related to distortional deformation.

No default (Real)

Dp Material constant related to volumetric deformation (MODEL=BOYCE).

No defaults (Real ≥ 0.0)

TAB1 Table identification number of a TABLES1 entry that contains simple tension-compression data to be used in the estimation of the material constants, related to distortional deformation. The x-values in the TABLES1 entry should be the stretch ratios and y-values should be values of the engineering stress.

(Integer > 0 or blank)

TAB2 Table identification number of a TABLES1 entry that contains equi-biaxial tension data to be used in the estimation of the material constants, related to distortional deformation. The x-values in the TABLES1 entry should be the stretch ratios and y-values should be values of the engineering stress.

(Integer > 0 or blank)

TAB4 Table identification number of a TABLES1 entry that contains pure shear data to be used in the estimation of the material constants, related to distortional deformation. The x-values in the TABLES1 entry should be the stretch ratios and y-values should be values of the engineering stress.

(Integer > 0 or blank)

TABD Table identification number of a TABLES1 entry that contains volumetric part (Dp) of the data to be used in the estimation of the material constants. The x-values in the TABLES1 entry should be the volumetric ratio and y-values should be values of the pressure.

TABD can only be used to fit volumetric data for formats A, B, C, and E. Additionally, only first-order fitting is currently supported (only D1 value is sourced from the TABD data).

(Integer > 0 or blank)

C1 Initial shear modulus (Model = ABOYCE). 4

No default (Real)

${\lambda }_{m}$ Maximum locking stretch ratio.

Used to calculate the value of $\beta$ (Model = ABOYCE). 4

No default (Real)

MUi, ALPHAi Material constants for the Ogden Material Model (Model = OGDEN) 5; or

Hill Foam Material Model (Model = FOAM. 6

BETAi Material constants for Hill Foam Material Model (Model=FOAM). 6
MODULI Continuation line flag for moduli temporal property. 10
MTIME Material temporal property. This field controls the interpretation of the input material property for viscoelasticity.
INSTANT
This material property is considered as the Instantaneous material input for viscoelasticity on the MATVE entry.
LONG (Default)
This material property is considered as the Long-term relaxed material input for viscoelasticity on the MATVE entry.

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)
$W=\sum _{p+q=1}^{{N}_{1}}{C}_{pq}{\left({\overline{I}}_{1}-3\right)}^{p}{\left({\overline{I}}_{2}-3\right)}^{q}+\sum _{p=1}^{{N}_{2}}\frac{1}{{D}_{p}}{\left({J}_{elas}-1\right)}^{2p}$
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)
$W=\sum _{p=1}^{{N}_{1}}{C}_{p0}{\left({\overline{I}}_{1}-3\right)}^{p}+\frac{1}{{D}_{1}}{\left({J}_{elas}-1\right)}^{2}$

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)
$\begin{array}{l}W={C}_{10}\left({\overline{I}}_{1}-3\right)+{C}_{01}\left({\overline{I}}_{2}-3\right)+{C}_{11}\left({\overline{I}}_{1}-3\right)\left({\overline{I}}_{2}-3\right)\\ \text{ }+{C}_{20}{\left({\overline{I}}_{1}-3\right)}^{2}+{C}_{30}{\left({\overline{I}}_{1}-3\right)}^{3}+\frac{1}{{D}_{1}}{\left({J}_{elas}-1\right)}^{2}\end{array}$
4. The Arruda-Boyce model (ABOYCE) is defined as: (10)
$W={C}_{1}\sum _{i=1}^{5}{\alpha }_{i}{\beta }^{i-1}\left({\overline{I}}_{1}^{i}-{3}^{i}\right)+\frac{1}{{D}_{1}}{\left({J}_{elas}-1\right)}^{2}$

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, ${\overline{I}}_{1}={I}_{1}{J}^{-2}{3}}$ .
${J}_{elas}$
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)
$W=\sum _{i=1}^{{N}_{1}}\frac{2{\mu }_{i}}{{\alpha }_{i}^{2}}\left({\overline{\lambda }}_{1}^{{\alpha }_{i}}+{\overline{\lambda }}_{2}^{{\alpha }_{i}}+{\overline{\lambda }}_{3}^{{\alpha }_{i}}-3\right)+\frac{1}{{D}_{1}}{\left({J}_{elas}-1\right)}^{2}$
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)
$W=\sum _{i=1}^{{N}_{1}}\frac{2{\mu }_{i}}{{\alpha }_{i}^{2}}\left({\lambda }_{1}^{{\alpha }_{i}}+{\lambda }_{2}^{{\alpha }_{i}}+{\lambda }_{3}^{{\alpha }_{i}}-3+\frac{1}{{\beta }_{i}}\left({J}^{-{\alpha }_{i}{\beta }_{i}}-1\right)\right)$
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

$G=2\left({C}_{10}+{C}_{01}\right)$ and $K=\frac{2}{{D}_{1}}$

• Ogden

$G={\mu }_{1}+{\mu }_{2}+{\mu }_{3}+{\mu }_{4}+{\mu }_{5}=\sum _{t=1}^{5}{\mu }_{t}$ 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

$G={\mu }_{1}+{\mu }_{2}+{\mu }_{3}+{\mu }_{4}+{\mu }_{5}=\sum _{i=1}^{5}{\mu }_{i}$ and $K=\sum _{i=1}^{5}2{\mu }_{i}\left(\frac{1}{3}+{\beta }_{i}\right)$

Additional treatments on bulk modulus $K$ are as:
• If Poisson's ratio $\nu$ , is non-zero, bulk modulus $K$ is replaced with:(14)
$K=\frac{2G\left(1+\nu \right)}{3\left(1-2\nu \right)}$
• 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 All the material models are supported with:
Explicit Dynamic Analysis All the material models are supported with:
12. Temperature-dependent hyperelastic material data can be defined via the MATTHE entry.
13. This card is represented as a material in HyperMesh.