MATTHE

Bulk Data Entry Defines material properties for temperature-dependent nonlinear hyperelastic materials.

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`

Format E

Marlow Material Model (Model=MARLOW):


(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
MATTHE	`MID`	`Model`	2	`NU`	`RHO`	`TEXP`	`TREF`
	`MTIME`	`ND`	`ETYPE`	`D1`
	`SIGNOM`	`LAMBDA`	`D1/T`	`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 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. 2 No default (Integer ≥ 0)
`Cpq`	Material constants related to distortional deformation. No default (Real)
`Dp`	Material constants related to volumetric deformation. No default (Real > 0.0)
`C1`	Initial shear modulus (`Model` = ABOYCE). 4 No default (Real)
$λ m$	Maximum locking stretch ratio. 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
`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.
`ETYPE`	Experiment type for the Marlow model. UNIAX Uniaxial experiment. EQUIBI Equi-biaxial experiment. PLANAR Planar experiment. No default
`SIGNOM`	Nominal stress from the experiment for the Marlow model. No default (real).
`LAMBDA`	Stretch ratio from the experiment for the Marlow model. No default (Real > 0.0)
`D1/T`	`D1` is the material constant related to volumetric deformation. For the MARLOW material model, if `D1` is temperature dependent, it should be defined on line 3 in field 4, and temperature should be defined on line 3 in field 5. Otherwise, `D1` should be defined on line 2 in field 5, and temperature can be defined on line 3 in field 4. 7 No default (Real > 0.0)
`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):
$W = \sum_{p + q = 1}^{N_{1}} C_{p q} {({\bar{I}}_{1} - 3)}^{p} {({\bar{I}}_{2} - 3)}^{q} + \sum_{p = 1}^{N_{2}} \frac{1}{D_{p}} {(J_{e l a s} - 1)}^{2 p}$
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
$W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Reduced Polynomial (RPOLY):
q=0, N₂ =1
$W = \sum_{p = 1}^{N_{1}} C_{p 0} {({\bar{I}}_{1} - 3)}^{p} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Neo-Hooken Material (NEOH):
N₁= N₂ =1, q=0
$W = C_{10} ({\bar{I}}_{1} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Yeoh Material (YEOH):
N₁ =3 N₂ =1, q=0
$W = C_{10} ({\bar{I}}_{1} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2} + C_{20} {({\bar{I}}_{1} - 3)}^{2} + C_{30} {({\bar{I}}_{1} - 3)}^{3}$
Some other material models from the Generalized Mooney Rivlin model are:
Three term Mooney-Rivlin Material:
$W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Signiorini Material:
$W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{20} {({\bar{I}}_{1} - 3)}^{2} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
Third Order Invariant Material:
$W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) + C_{20} {({\bar{I}}_{1} - 3)}^{2} + \frac{1}{D_{1}} {(J_{e l a s} - 1)}^{2}$
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.

Currently, the Hill material model is only supported for explicit analysis.
The Marlow model is a Hyperelastic material model which directly defines the potential based on the experiment test data. Therefore, 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 the test data. 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.
The NA field should always be set to 2 for the Marlow material model.
For temperature-independent volumetric behavior, D1 should be specified on the 5^th field of the 2^nd line, ND should be set to 0, and the temperature is defined on the 4^th field of line 3. If D1 is changing with respect to the temperature, ND should be set to 1 with D1 specified on the 4^th field and the temperature specified on the 5^th field.
The temperature and the stretch ratio should both be specified in the ascending order.
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.
For all material models except Marlow, if Poisson’s ratio and D1 are both defined, 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.
For temperature-dependent material table lookup for an element with different temperatures at each node, the average temperature of all nodes of the element is considered as reference temperature for table lookup for that particular element.
This card is represented as a material in HyperMesh.