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.

Attention: Valid for Implicit and Explicit Analysis

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`
	`MODULI`	`MTIME`

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`
	`MODULI`	`MTIME`

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`
	`MODULI`	`MTIME`

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`
	`MODULI`	`MTIME`

Format E

Marlow Material Model (Model=MARLOW):


(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
MATHE	`MID`	`Model`		`NU`	`RHO`	`TEXP`	`TREF`
			`D`	`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 `D` value is sourced from the `TABD` data). (Integer > 0 or blank)
`C`	Initial shear modulus (`Model` = ABOYCE). 4 No default (Real)
$λ_{m}$ $λ_{m}$	Maximum locking stretch ratio. Used to calculate the value of $β$ $β$ (`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.

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

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

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
$W = C_{10} ({¯ I}_{1} - 3) + C_{01} ({¯ I}_{2} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Reduced Polynomial (RPOLY):
q=0, N₂ =1
$W = N_{1} \sum p = 1 C_{p 0} {({¯ I}_{1} - 3)}^{p} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = \sum_{p = 1}^{N_{1}} C_{p 0} {({\bar{I}}_{1} - 3)}^{p} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Neo-Hooken Material (NEOH):
N₁= N₂ =1, q=0
$W = C_{10} ({¯ I}_{1} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = C_{10} ({\bar{I}}_{1} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Yeoh Material (YEOH):
N₁ =3 N₂ =1, q=0
$W = C_{10} ({¯ I}_{1} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2} + C_{20} {({¯ I}_{1} - 3)}^{2} + C_{30} {({¯ I}_{1} - 3)}^{3}$ $W = C_{10} ({\bar{I}}_{1} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2} + C_{20} {({\bar{I}}_{1} - 3)}^{2} + C_{30} {({\bar{I}}_{1} - 3)}^{3}$
Some other material models from the Generalized Mooney Rivlin model are:
Three term Mooney-Rivlin Material:
$W = C_{10} ({¯ I}_{1} - 3) + C_{01} ({¯ I}_{2} - 3) + C_{11} ({¯ I}_{1} - 3) ({¯ I}_{2} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Signiorini Material:
$W = C_{10} ({¯ I}_{1} - 3) + C_{01} ({¯ I}_{2} - 3) + C_{20} {({¯ I}_{1} - 3)}^{2} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{20} {({\bar{I}}_{1} - 3)}^{2} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Third Order Invariant Material:
$W = C_{10} ({¯ I}_{1} - 3) + C_{01} ({¯ I}_{2} - 3) + C_{11} ({¯ I}_{1} - 3) ({¯ I}_{2} - 3) + C_{20} {({¯ I}_{1} - 3)}^{2} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$ $W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) + C_{20} {({\bar{I}}_{1} - 3)}^{2} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Third Order Deformation Material (James-Green-Simpson):
$\begin{array}{l} W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) \\ + C_{20} {({\bar{I}}_{1} - 3)}^{2} + C_{30} {({\bar{I}}_{1} - 3)}^{3} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2} \end{array}$
The Arruda-Boyce model (ABOYCE) is defined as:
$W = C_{1} \sum_{i = 1}^{5} α_{i} β^{i - 1} ({\bar{I}}_{1}^{i} - 3^{i}) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Where,
$β = \frac{1}{N} = \frac{1}{λ_{m}^{2}}$

$N$

Measure of the limiting locking stretch ratio.

$λ_{m}$

Maximum locking stretch ratio.

$D_{1}$

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

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

First strain invariant, internally calculated by OptiStruct.

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

$J_{e l a s}$

Elastic volume strain, internally calculated by OptiStruct.

$C_{1}$

Initial shear modulus.

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

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

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

$μ_{i}$

Defined by the MUi fields.

$α_{i}$

Defined by the ALPHAi fields.

$N_{1}$

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

$λ_{1}, λ_{2}, λ_{3}$

Principle stretch ratios.

$μ_{i}$

Defined by the MUi fields.

$α_{i}$

Defined by the ALPHAi fields.

$β_{i}$

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:
$β_{i} = \frac{ν}{1 - 2 ν}$
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.
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, D, TABD, or Poisson’s ratio can be defined to specify the volumetric behavior. Either D or TABD can be specified, but not both.
1. If D or TABD is specified, the volumetric behavior is determined by D or TABD.
2. If D and TABD are not specified and Poisson’s ratio is specified, Poisson’s ratio is used to determine volumetric behavior.
3. If D, 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 D or TABD are defined, D or TABD take precedence.
If Poisson’s ratio and D or TABD are both defined, Poisson’s ratio takes precedence.
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
  $G = μ_{1} + μ_{2} + μ_{3} + μ_{4} + μ_{5} = \sum_{t = 1}^{5} μ_{t}$ and $K = \frac{2}{D_{1}}$
- Arruda-Boyce
  $G = C_{1} (1 + \frac{3}{5 λ_{m}^{2}} + \frac{99}{175 λ_{m}^{4}} + \frac{513}{875 λ_{m}^{6}} + \frac{42039}{67375 λ_{m}^{8}})$ and $K = \frac{2}{D_{1}}$
- Hill Foam
  $G = μ_{1} + μ_{2} + μ_{3} + μ_{4} + μ_{5} = \sum_{i = 1}^{5} μ_{i}$ and $K = \sum_{i = 1}^{5} 2 μ_{i} (\frac{1}{3} + β_{i})$
Additional treatments on bulk modulus $K$ are as:
- If Poisson's ratio $ν$ , is non-zero, bulk modulus $K$ is replaced with:
  $K = \frac{2 G (1 + ν)}{3 (1 - 2 ν)}$
- If $K = 0,$ $K$ is set to be $30 G$
- If $K \geq 30 G,$ $K$ is set to be $30 G$
The Young’s modulus and Poisson's ratio are given by:
$E = \frac{9 K G}{3 K + G}$
and
$ν = \frac{3 K - 2 G}{6 K + 2 G}$
Where,

$E$

Young’s modulus.

$G$

Shear modulus.

$K$

Bulk modulus.

$ν$

Poisson's ratio.

$C_{10}$ , $C_{01}$ , $D_{1}$ , $μ_{i}$ , and $C_{1}$

Material coefficients.

$λ_{m}$

Stretch ratio at which the polymer chain network is locked.
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.

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 Shell elements
Explicit Dynamic Analysis	All the material models are supported with: Solid elements Shell elements

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