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)
MATHE	`MID`	`Model`		`NU`	`RHO`	`TEXP`	`TREF`
	`C10`	`C01`	`D`	`TAB1`	`TAB2`		`TAB4`	`TABD`
	`C20`	`C11`	`C02`	`D`	`NA`	`ND`
	`C30`	`C21`	`C12`	`C03`	`D`
	`C40`	`C31`	`C22`	`C13`	`C04`	`D`
	`C50`	`C41`	`C32`	`C23`	`C14`	`C05`	`D`

Format B

Arruda-Boyce Model (Model=ABOYCE):

(1)	(2)	(3)	(5)	(6)	(7)	(8)
MATHE	`MID`	`Model`	`NU`	`RHO`	`TEXP`	`TREF`
	`C`	$λ m$ $λ m$	`TAB1`	`TAB2`		`TAB4`
	`D`

Format C

Ogden Material Model (Model=OGDEN):

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
MATHE	`MID`	`Model`	`NA`	`NU`	`RHO`	`TEXP`	`TREF`
	`MU1`	`ALPHA1`	`D`	`TAB1`	`TAB2`		`TAB4`
	`MU2`	`ALPHA2`		`MU3`	`ALPHA3`
	`MU4`	`ALPHA4`		`MU5`	`ALPHA5`

Format D

Hill Foam Material Model (Model=FOAM):

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
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`

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. MOONEY (Default) Selects the generalized Mooney-Rivlin hyperelastic model MOOR Physical Mooney-Rivlin model RPOLY Reduced Polynomial model NEOH Neo-Hookean model YEOH Yeoh model ABOYCE Selects the Arruda-Boyce material model OGDEN Ogden material model FOAM Hill foam model blank (Character)
`NU`	Poisson's ratio. Default = 0.495 (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, `Cpq`, 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, `Cpq`, 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, `Cpq`, related to distortional deformation. The x-values in the TABLES1 entry should be the stretch ratios and y-values should be values of the nominal 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 pressure and y-values should be values of the volumetric change. `TABD` can only be used to fit Volumetric data for Format A, additionally, only first-order fitting is currently supported (only D1 value is sourced from the `TABD` data). (Integer > 0 or blank)
`C`	Initial shear modulus (`Model` = ABOYCE). 5 No default (Real)
$λ_{m}$ $λ_{m}$	Maximum locking stretch. Used to calculate the value of $β$ $β$ (`Model` = ABOYCE). 5 No default (Real)
`MUi`, `ALPHAi`	Material Constants for the Ogden Material Model (`Model` = OGDEN) 6; or Hill Foam Material Model (`Model` = FOAM. 7
`BETAi`	Material Constants for Hill Foam Material Model (`Model`=FOAM). 7

Comments

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.
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 ( $W$ $W$ ) is written in polynomial form, as:
Generalized polynomial form (MOONEY): (1)
$W = \sum_{p + q = 1}^{N_{1}} C_{p q} {({\bar{I}}_{1} - 3)}^{p} {({\bar{I}}_{2} - 3)}^{q} + \sum_{p = 1}^{N_{2}} \frac{1}{D_{p}} {(J_{e l a s} - 1)}^{2 p}$ $W = \sum_{p + q = 1}^{N_{1}} C_{p q} {({\bar{I}}_{1} - 3)}^{p} {({\bar{I}}_{2} - 3)}^{q} + \sum_{p = 1}^{N_{2}} \frac{1}{D_{p}} {(J_{e l a s} - 1)}^{2 p}$

Where,

$N_{1}$ $N_{1}$

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

$N_{2}$ $N_{2}$

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

$C_{p q}$ $C_{p q}$

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

${\bar{I}}_{1}$ ${\bar{I}}_{1}$ , ${\bar{I}}_{2}$ ${\bar{I}}_{2}$

Strain invariants, calculated internally by OptiStruct.

$D_{p}$ $D_{p}$

Material constants related to volumetric deformation ( $D_{p}$ $D_{p}$ ). These values define the compressibility of the material.

$J_{elas}$ $J_{elas}$

Elastic volume strain, calculated internally by OptiStruct.
The polynomial form can be used to model the following material types by specifying the corresponding coefficients ( $C_{p q}$ $C_{p q}$ , $D_{p}$ $D_{p}$ ) on the MATHE entry.
Physical Mooney-Rivlin Material (MOOR):

N₁ = N₂ =1 (2)
$W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$

Reduced Polynomial (RPOLY):

q=0, N₂ =1(3)
$W = \sum_{p = 1}^{N_{1}} C_{p 0} {({\bar{I}}_{1} - 3)}^{p} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = \sum_{p = 1}^{N_{1}} C_{p 0} {({\bar{I}}_{1} - 3)}^{p} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$

Neo-Hooken Material (NEOH):

N₁= N₂ =1, q=0(4)
$W = C_{10} ({\bar{I}}_{1} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = C_{10} ({\bar{I}}_{1} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$

Yeoh Material (YEOH):

N₁ =3 N₂ =1, q=0(5)
$W = C_{10} ({\bar{I}}_{1} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2} + C_{20} {({\bar{I}}_{1} - 3)}^{2} + C_{30} {({\bar{I}}_{1} - 3)}^{3}$ $W = C_{10} ({\bar{I}}_{1} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2} + C_{20} {({\bar{I}}_{1} - 3)}^{2} + C_{30} {({\bar{I}}_{1} - 3)}^{3}$

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

Three term Mooney-Rivlin Material: (6)
$W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$

Signiorini Material: (7)
$W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{20} {({\bar{I}}_{1} - 3)}^{2} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{20} {({\bar{I}}_{1} - 3)}^{2} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$

Third Order Invariant Material: (8)
$W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) + C_{20} {({\bar{I}}_{1} - 3)}^{2} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) + C_{20} {({\bar{I}}_{1} - 3)}^{2} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$

Third Order Deformation Material (James-Green-Simpson): (9)
$\begin{array}{l} W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) \\ + C_{20} {({\bar{I}}_{1} - 3)}^{2} + C_{30} {({\bar{I}}_{1} - 3)}^{3} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2} \end{array}$ $\begin{array}{l} W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) \\ + C_{20} {({\bar{I}}_{1} - 3)}^{2} + C_{30} {({\bar{I}}_{1} - 3)}^{3} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2} \end{array}$
The MATHE hyperelastic material supports CTETRA (4, 10), CPENTA (6, 15), and CHEXA (8, 20) element types.
The Arruda-Boyce model (ABOYCE) is defined as: (10)
$W = C_{1} \sum_{i = 1}^{5} α_{i} β^{i - 1} ({\bar{I}}_{1}^{i} - 3^{i}) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = C_{1} \sum_{i = 1}^{5} α_{i} β^{i - 1} ({\bar{I}}_{1}^{i} - 3^{i}) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$

Where,

$β = \frac{1}{N} = \frac{1}{λ_{m}^{2}}$ $β = \frac{1}{N} = \frac{1}{λ_{m}^{2}}$

$N$ $N$

Measure of the limiting locking stretch.

$λ_{m}$ $λ_{m}$

Maximum locking stretch.

$D_{1}$ $D_{1}$

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

${\bar{I}}_{1}$ ${\bar{I}}_{1}$

First strain invariant, internally calculated by OptiStruct.

Wherein, ${\bar{I}}_{1} = I_{1} J^{- \frac{2}{3}}$ ${\bar{I}}_{1} = I_{1} J^{- \frac{2}{3}}$ .

$J_{e l a s}$ $J_{e l a s}$

Elastic volume strain, internally calculated by OptiStruct.

$C_{1}$ $C_{1}$

Initial shear modulus.

$α_{1} = \frac{1}{2}; α_{2} = \frac{1}{20}; α_{3} = \frac{11}{1050}; α_{4} = \frac{19}{7000}; α_{5} = \frac{519}{673750}$ $α_{1} = \frac{1}{2}; α_{2} = \frac{1}{20}; α_{3} = \frac{11}{1050}; α_{4} = \frac{19}{7000}; α_{5} = \frac{519}{673750}$
The Ogden Material model (OGDEN) is defined as: (11)
$W = \sum_{i = 1}^{N_{1}} \frac{2 μ_{i}}{α_{i}^{2}} ({\bar{λ}}_{1}^{α_{i}} + {\bar{λ}}_{2}^{α_{i}} + {\bar{λ}}_{3}^{α_{i}} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = \sum_{i = 1}^{N_{1}} \frac{2 μ_{i}}{α_{i}^{2}} ({\bar{λ}}_{1}^{α_{i}} + {\bar{λ}}_{2}^{α_{i}} + {\bar{λ}}_{3}^{α_{i}} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$

Where,

${\bar{λ}}_{1}, {\bar{λ}}_{2}, {\bar{λ}}_{3}$ ${\bar{λ}}_{1}, {\bar{λ}}_{2}, {\bar{λ}}_{3}$

The three deviatoric stretches (deviatoric stretches are related to principal stretches by ${\bar{λ}}_{i} = J^{\frac{1}{3}} λ_{i}$ ${\bar{λ}}_{i} = J^{\frac{1}{3}} λ_{i}$ )

$μ_{i}$ $μ_{i}$

Defined by the MUi fields

$α_{i}$ $α_{i}$

Defined by the ALPHAi fields

$N_{1}$ $N_{1}$

Order of the deviatoric part of the strain energy function defined on the NA field
The Hill Foam Material model (FOAM) is defined as:(12)
$W = \sum_{i = 1}^{N_{1}} \frac{2 μ_{i}}{α_{i}^{2}} (λ_{1}^{α_{i}} + λ_{2}^{α_{i}} + λ_{3}^{α_{i}} - 3 + \frac{1}{β_{i}} (J^{- α_{i} β_{i}} - 1))$ $W = \sum_{i = 1}^{N_{1}} \frac{2 μ_{i}}{α_{i}^{2}} (λ_{1}^{α_{i}} + λ_{2}^{α_{i}} + λ_{3}^{α_{i}} - 3 + \frac{1}{β_{i}} (J^{- α_{i} β_{i}} - 1))$

Where,

$λ_{1}, λ_{2}, λ_{3}$ $λ_{1}, λ_{2}, λ_{3}$

Principle stretches

$μ_{i}$ $μ_{i}$

Defined by the MUi fields

$α_{i}$ $α_{i}$

Defined by the ALPHAi fields

$β_{i}$ $β_{i}$

Defined by the BETAi fields

$N_{1}$ $N_{1}$

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

Currently, the Hill material model is only supported for explicit analysis.
This card is represented as a material in HyperMesh.