MATTHE

The Polynomial form is available and various material types (3) can be defined by specifying the corresponding coefficients.

Format A1

Generalized Mooney-Rivlin Polynomial (Model=MOONEY):

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
MATTHE	`MID`	`Model`	`NA`	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	`ND`
	`C10`	`C01`	`C20`	`C11`	`C02`	`C30`	`C21`	`C12`
	`C03`	`C40`	`C31`	`C22`	`C13`	`C04`	`C50`	`C41`
	`C32`	`C23`	`C14`	`C05`	`D1`	`D2`	`D3`	`D4`
	`D5`	`T`

Format A2

Reduced Polynomial (Model=RPOLY):

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
MATTHE	`MID`	`Model`	`NA`	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	`ND`
	`C10`	`C20`	`C30`	`C40`	`C50`	`D1`	`D2`	`D3`
	`D4`	`D5`	`T`

Format A3

Physical Mooney-Rivlin (Model=MOOR):

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
MATTHE	`MID`	`Model`	2	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	`ND`
	`C10`	`C01`	`D1`	`T`

Format A4

Neo-Hookean (Model=NEOH):

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
MATTHE	`MID`	`Model`	1	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	`ND`
	`C10`	`D1`	`T`

Format A5

Yeoh Model (Model=YEOH):

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
MATTHE	`MID`	`Model`	3	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	`ND`
	`C10`	`C20`	`C30`	`D1`	`D2`	`D3`	`T`

Format B

Arruda-Boyce Model (Model=ABOYCE):

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
MATTHE	`MID`	`Model`	2	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	`ND`
	`C1`	$λ m$	`D1`	`T`

Format C

Ogden Material Model (Model=OGDEN):

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
MATTHE	`MID`	`Model`	`NA`	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	`ND`
	`MU1`	`ALPHA1`	`MU2`	`ALPHA2`	`MU3`	`ALPHA3`	`MU4`	`ALPHA4`
	`MU5`	`ALPHA5`	`D1`	`D2`	`D3`	`D4`	`D5`	`T`

Format D

Hill Foam Material Model (Model=FOAM):

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
MATTHE	`MID`	`Model`	`NA`	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	0
	`MU1`	`ALPHA1`	`BETA1`	`MU2`	`ALPHA2`	`BETA2`	`MU3`	`ALPHA3`
	`BETA3`	`MU4`	`ALPHA4`	`BETA4`	`MU5`	`ALPHA5`	`BETA5`	`T`

Example

(1)	(2)	(3)	(4)	(5)
MATTHE	2	NEOH	1	0.495
	LONG	0
	5.2 (C10)	10.0 (T)
	5.1 (C10)	20.0 (T)

The following examples show the field names instead of actual values to showcase the variation in each temperature-dependent data block depending on the NA and ND values.

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
MATTHE	2	MOONEY	5	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	1
	C10	C01	C20	C11	C02	C30	C21	C12
	C03	C40	C31	C22	C13	C04	C50	C41
	C32	C23	C14	C05	D1	`T`

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
MATTHE	2	MOONEY	3	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	1
	C10	C01	C20	C11	C02	C30	C21	C12
	C03	D1	`T`

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
MATTHE	2	FOAM	2	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	0
	`MU1`	`ALPHA1`	`BETA1`	`MU2`	`ALPHA2`	`BETA2`	`T`

Definition

Field	Contents	SI Unit Example
`MID`	Unique material identification number. No default (Integer > 0)
`Model`	Hyperelastic material model type. The input format can vary for each material model. See the format details in the table above. 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 material model OGDEN Ogden material model FOAM Hill foam model blank (Character)
`NU`	Poisson’s ratio. Default = 0.495 (Real)
`RHO`	Material density. Default = 0.495 (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. 2 No default (Integer ≥ 0)
`Cpq`	Material constants related to distortional deformation. No default (Real)
`Dp`	Material constant related to volumetric deformation (`Model`=BOYCE). No default (Real > 0.0)
`C1`	Initial shear modulus (`Model` = ABOYCE). 4 No default (Real)
$λ m$	Maximum locking stretch. Used to calculate the value of $β$ (`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. 9
`MTIME`	Material temporal property. This field controls the interpretation of the input material property for viscoelasticity. INSTANT The instantaneous material input for viscoelasticity on the MATVE entry. LONG (Default) The long-term relaxed material input for viscoelasticity on the MATVE entry.
`T`	Temperature at which the defined material properties are specified. The material data and temperature set can be repeated as required to define temperature-dependent material data for hyperelasticity. 1 No default (Real)

Comments

MATTHE Bulk Data is an extension of MATHE Bulk Data to allow definition of temperature-dependent hyperelastic materials. Currently direct table input is not supported for MATTHE and only fitted parameters are allowed. Table inputs should be calibrated by curve-fitting first for each temperature and then input on the MATTHE entry.
- The general rule to specify each data block is to define NA distortional strain energy parameters followed by ND volumetric strain energy parameters, subsequently followed by the temperature for these parameters.
- Each temperature material data block may extend into more than one line. Therefore, NA and ND should be specified accurately to indicate how many terms are expected.
- The order of parameters follow the same order as MATHE entry for distortional and bulk parameters, respectively.
- The different temperature values should be defined in ascending order.
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)
$W = \sum_{p + q = 1}^{N_{1}} C_{p q} {({\bar{I}}_{1} - 3)}^{p} {({\bar{I}}_{2} - 3)}^{q} + \sum_{p = 1}^{N_{2}} \frac{1}{D_{p}} {(J_{e l a s} - 1)}^{2 p}$

Where,

$N_{1}$

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).

$C_{p q}$

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

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

Strain invariants, calculated internally by OptiStruct.

$D_{p}$

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

$J_{elas}$

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

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

Reduced Polynomial (RPOLY):

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

Neo-Hooken Material (NEOH):

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

Yeoh Material (YEOH):

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

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

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

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

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

Third Order Deformation Material (James-Green-Simpson): (9)
$\begin{array}{l} W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) \\ + C_{20} {({\bar{I}}_{1} - 3)}^{2} + C_{30} {({\bar{I}}_{1} - 3)}^{3} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2} \end{array}$
The Arruda-Boyce model (ABOYCE) is defined as: (10)
$W = C_{1} \sum_{i = 1}^{5} α_{i} β^{i - 1} ({\bar{I}}_{1}^{i} - 3^{i}) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$

Where,

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

$N$

Measure of the limiting locking stretch.

$λ_{m}$

Maximum locking stretch.

$D_{1}$

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: (11)
$W = \sum_{i = 1}^{N_{1}} \frac{2 μ_{i}}{α_{i}^{2}} ({\bar{λ}}_{1}^{α_{i}} + {\bar{λ}}_{2}^{α_{i}} + {\bar{λ}}_{3}^{α_{i}} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$

Where,

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

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

$μ_{i}$

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:(12)
$W = \sum_{i = 1}^{N_{1}} \frac{2 μ_{i}}{α_{i}^{2}} (λ_{1}^{α_{i}} + λ_{2}^{α_{i}} + λ_{3}^{α_{i}} - 3 + \frac{1}{β_{i}} (J^{- α_{i} β_{i}} - 1))$

Where,

$λ_{1}, λ_{2}, λ_{3}$

Principle stretches

$μ_{i}$

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.

Currently, the Hill material model is only supported for explicit analysis.
If Poisson’s ratio and D1 are both defined, then Poisson’s ratio takes precedence.
MATTHE is currently only supported for implicit Large Displacement Nonlinear Analysis.
The MATTHE hyperelastic material supports CTETRA (4, 10), CPENTA (6, 15), and CHEXA (8, 20) element types.
This card is represented as a material in HyperMesh.