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) (10)
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) (10)
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) (9) (10)
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) (9) (10)
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) (9) (10)
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) (9) (10)
MATTHE MID Model 2 NU RHO TEXP TREF
MTIME ND
C1 λ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaam yBaaaa@389C@ D1 T

Format C

Ogden Material Model (Model=OGDEN):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
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) (10)
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) (6) (7) (8) (9) (10)
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) (10)
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) (10)
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) (9) (10)
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)

 
λm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaam yBaaaa@389C@ 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

  1. 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.
  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 = p + q = 1 N 1 C p q ( I ¯ 1 3 ) p ( I ¯ 2 3 ) q + p = 1 N 2 1 D p ( J e l a s 1 ) 2 p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEfacqGH9a qpdaaeWbqaaiaadoeadaWgaaWcbaGaamiCaiaadghaaeqaaOWaaeWa aeaaceWGjbGbaebadaWgaaWcbaGaaGymaaqabaGccqGHsislcaaIZa aacaGLOaGaayzkaaWaaWbaaSqabeaacaWGWbaaaOWaaeWaaeaaceWG jbGbaebadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaaIZaaacaGLOa GaayzkaaWaaWbaaSqabeaacaWGXbaaaaqaaiaadchacqGHRaWkcaWG XbGaeyypa0JaaGymaaqaaiaad6eadaWgaaadbaGaaGymaaqabaaani abggHiLdGccqGHRaWkdaaeWbqaamaalaaabaGaaGymaaqaaiaadsea daWgaaWcbaGaamiCaaqabaaaaOWaaeWaaeaacaWGkbWaaSbaaSqaai aadwgacaWGSbGaamyyaiaadohaaeqaaOGaeyOeI0IaaGymaaGaayjk aiaawMcaamaaCaaaleqabaGaaGOmaiaadchaaaaabaGaamiCaiabg2 da9iaaigdaaeaacaWGobWaaSbaaWqaaiaaikdaaeqaaaqdcqGHris5 aaaa@63BC@
    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 ).
    I ¯ 1 , 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.
  3. 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 ( I ¯ 1 3 ) + C 01 ( I ¯ 2 3 ) + 1 D 1 ( J e l a s 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEfacqGH9a qpcaWGdbWaaSbaaSqaaiaaigdacaaIWaaabeaakmaabmaabaGabmys ayaaraWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaG4maaGaayjkai aawMcaaiabgUcaRiaadoeadaWgaaWcbaGaaGimaiaaigdaaeqaaOWa aeWaaeaaceWGjbGbaebadaWgaaWcbaGaaGOmaaqabaGccqGHsislca aIZaaacaGLOaGaayzkaaGaey4kaSYaaSaaaeaacaaIXaaabaGaamir amaaBaaaleaacaaIXaaabeaaaaGcdaqadaqaaiaadQeadaWgaaWcba GaamyzaiaadYgacaWGHbGaam4CaaqabaGccqGHsislcaaIXaaacaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@53EA@

    Reduced Polynomial (RPOLY):

    q=0, N2 =1(3)
    W = p = 1 N 1 C p 0 ( I ¯ 1 3 ) p + 1 D 1 ( J e l a s 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEfacqGH9a qpdaaeWbqaaiaadoeadaWgaaWcbaGaamiCaiaaicdaaeqaaOWaaeWa aeaaceWGjbGbaebadaWgaaWcbaGaaGymaaqabaGccqGHsislcaaIZa aacaGLOaGaayzkaaWaaWbaaSqabeaacaWGWbaaaaqaaiaadchacqGH 9aqpcaaIXaaabaGaamOtamaaBaaameaacaaIXaaabeaaa0GaeyyeIu oakiabgUcaRmaalaaabaGaaGymaaqaaiaadseadaWgaaWcbaGaaGym aaqabaaaaOWaaeWaaeaacaWGkbWaaSbaaSqaaiaadwgacaWGSbGaam yyaiaadohaaeqaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaa leqabaGaaGOmaaaaaaa@5398@

    Neo-Hooken Material (NEOH):

    N1= N2 =1, q=0(4)
    W = C 10 ( I ¯ 1 3 ) + 1 D 1 ( J e l a s 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEfacqGH9a qpcaWGdbWaaSbaaSqaaiaaigdacaaIWaaabeaakmaabmaabaGabmys ayaaraWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaG4maaGaayjkai aawMcaaiabgUcaRmaalaaabaGaaGymaaqaaiaadseadaWgaaWcbaGa aGymaaqabaaaaOWaaeWaaeaacaWGkbWaaSbaaSqaaiaadwgacaWGSb GaamyyaiaadohaaeqaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaaaaa@4B8A@

    Yeoh Material (YEOH):

    N1 =3 N2 =1, q=0(5)
    W = C 10 ( I ¯ 1 3 ) + 1 D 1 ( J e l a s 1 ) 2 + C 20 ( I ¯ 1 3 ) 2 + C 30 ( I ¯ 1 3 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEfacqGH9a qpcaWGdbWaaSbaaSqaaiaaigdacaaIWaaabeaakmaabmaabaGabmys ayaaraWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaG4maaGaayjkai aawMcaaiabgUcaRmaalaaabaGaaGymaaqaaiaadseadaWgaaWcbaGa aGymaaqabaaaaOWaaeWaaeaacaWGkbWaaSbaaSqaaiaadwgacaWGSb GaamyyaiaadohaaeqaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaakiabgUcaRiaadoeadaWgaaWcbaGaaGOmai aaicdaaeqaaOWaaeWaaeaaceWGjbGbaebadaWgaaWcbaGaaGymaaqa baGccqGHsislcaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYa aaaOGaey4kaSIaam4qamaaBaaaleaacaaIZaGaaGimaaqabaGcdaqa daqaaiqadMeagaqeamaaBaaaleaacaaIXaaabeaakiabgkHiTiaaio daaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaaaa@5E32@

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

    Three term Mooney-Rivlin Material: (6)
    W = C 10 ( I ¯ 1 3 ) + C 01 ( I ¯ 2 3 ) + C 11 ( I ¯ 1 3 ) ( I ¯ 2 3 ) + 1 D 1 ( J e l a s 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEfacqGH9a qpcaWGdbWaaSbaaSqaaiaaigdacaaIWaaabeaakmaabmaabaGabmys ayaaraWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaG4maaGaayjkai aawMcaaiabgUcaRiaadoeadaWgaaWcbaGaaGimaiaaigdaaeqaaOWa aeWaaeaaceWGjbGbaebadaWgaaWcbaGaaGOmaaqabaGccqGHsislca aIZaaacaGLOaGaayzkaaGaey4kaSIaam4qamaaBaaaleaacaaIXaGa aGymaaqabaGcdaqadaqaaiqadMeagaqeamaaBaaaleaacaaIXaaabe aakiabgkHiTiaaiodaaiaawIcacaGLPaaadaqadaqaaiqadMeagaqe amaaBaaaleaacaaIYaaabeaakiabgkHiTiaaiodaaiaawIcacaGLPa aacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGebWaaSbaaSqaaiaaigda aeqaaaaakmaabmaabaGaamOsamaaBaaaleaacaWGLbGaamiBaiaadg gacaWGZbaabeaakiabgkHiTiaaigdaaiaawIcacaGLPaaadaahaaWc beqaaiaaikdaaaaaaa@6155@
    Signiorini Material: (7)
    W = C 10 ( I ¯ 1 3 ) + C 01 ( I ¯ 2 3 ) + C 20 ( I ¯ 1 3 ) 2 + 1 D 1 ( J e l a s 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEfacqGH9a qpcaWGdbWaaSbaaSqaaiaaigdacaaIWaaabeaakmaabmaabaGabmys ayaaraWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaG4maaGaayjkai aawMcaaiabgUcaRiaadoeadaWgaaWcbaGaaGimaiaaigdaaeqaaOWa aeWaaeaaceWGjbGbaebadaWgaaWcbaGaaGOmaaqabaGccqGHsislca aIZaaacaGLOaGaayzkaaGaey4kaSIaam4qamaaBaaaleaacaaIYaGa aGimaaqabaGcdaqadaqaaiqadMeagaqeamaaBaaaleaacaaIXaaabe aakiabgkHiTiaaiodaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikda aaGccqGHRaWkdaWcaaqaaiaaigdaaeaacaWGebWaaSbaaSqaaiaaig daaeqaaaaakmaabmaabaGaamOsamaaBaaaleaacaWGLbGaamiBaiaa dggacaWGZbaabeaakiabgkHiTiaaigdaaiaawIcacaGLPaaadaahaa Wcbeqaaiaaikdaaaaaaa@5D3D@
    Third Order Invariant Material: (8)
    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 + 1 D 1 ( J e l a s 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEfacqGH9a qpcaWGdbWaaSbaaSqaaiaaigdacaaIWaaabeaakmaabmaabaGabmys ayaaraWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaG4maaGaayjkai aawMcaaiabgUcaRiaadoeadaWgaaWcbaGaaGimaiaaigdaaeqaaOWa aeWaaeaaceWGjbGbaebadaWgaaWcbaGaaGOmaaqabaGccqGHsislca aIZaaacaGLOaGaayzkaaGaey4kaSIaam4qamaaBaaaleaacaaIXaGa aGymaaqabaGcdaqadaqaaiqadMeagaqeamaaBaaaleaacaaIXaaabe aakiabgkHiTiaaiodaaiaawIcacaGLPaaadaqadaqaaiqadMeagaqe amaaBaaaleaacaaIYaaabeaakiabgkHiTiaaiodaaiaawIcacaGLPa aacqGHRaWkcaWGdbWaaSbaaSqaaiaaikdacaaIWaaabeaakmaabmaa baGabmysayaaraWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaG4maa GaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaalaaa baGaaGymaaqaaiaadseadaWgaaWcbaGaaGymaaqabaaaaOWaaeWaae aacaWGkbWaaSbaaSqaaiaadwgacaWGSbGaamyyaiaadohaaeqaaOGa eyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaa a@6AA8@
    Third Order Deformation Material (James-Green-Simpson): (9)
    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 + C 30 ( I ¯ 1 3 ) 3 + 1 D 1 ( J e l a s 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaam4vai abg2da9iaadoeadaWgaaWcbaGaaGymaiaaicdaaeqaaOWaaeWaaeaa ceWGjbGbaebadaWgaaWcbaGaaGymaaqabaGccqGHsislcaaIZaaaca GLOaGaayzkaaGaey4kaSIaam4qamaaBaaaleaacaaIWaGaaGymaaqa baGcdaqadaqaaiqadMeagaqeamaaBaaaleaacaaIYaaabeaakiabgk HiTiaaiodaaiaawIcacaGLPaaacqGHRaWkcaWGdbWaaSbaaSqaaiaa igdacaaIXaaabeaakmaabmaabaGabmysayaaraWaaSbaaSqaaiaaig daaeqaaOGaeyOeI0IaaG4maaGaayjkaiaawMcaamaabmaabaGabmys ayaaraWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaaG4maaGaayjkai aawMcaaaqaaiaaywW7cqGHRaWkcaWGdbWaaSbaaSqaaiaaikdacaaI WaaabeaakmaabmaabaGabmysayaaraWaaSbaaSqaaiaaigdaaeqaaO GaeyOeI0IaaG4maaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaa kiabgUcaRiaadoeadaWgaaWcbaGaaG4maiaaicdaaeqaaOWaaeWaae aaceWGjbGbaebadaWgaaWcbaGaaGymaaqabaGccqGHsislcaaIZaaa caGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaOGaey4kaSYaaSaaae aacaaIXaaabaGaamiramaaBaaaleaacaaIXaaabeaaaaGcdaqadaqa aiaadQeadaWgaaWcbaGaamyzaiaadYgacaWGHbGaam4CaaqabaGccq GHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa aa@7592@
  4. The Arruda-Boyce model (ABOYCE) is defined as: (10)
    W = C 1 i = 1 5 α i β i 1 ( I ¯ 1 i 3 i ) + 1 D 1 ( J e l a s 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9iaadoeadaWgaaWcbaGaaGymaaqabaGcdaaeWbqaaiabeg7aHnaa BaaaleaacaWGPbaabeaakiabek7aInaaCaaaleqabaGaamyAaiabgk HiTiaaigdaaaGcdaqadaqaaiqadMeagaqeamaaDaaaleaacaaIXaaa baGaamyAaaaakiabgkHiTiaaiodadaahaaWcbeqaaiaadMgaaaaaki aawIcacaGLPaaacqGHRaWkaSqaaiaadMgacqGH9aqpcaaIXaaabaGa aGynaaqdcqGHris5aOWaaSaaaeaacaaIXaaabaGaamiramaaBaaale aacaaIXaaabeaaaaGcdaqadaqaaiaadQeadaWgaaWcbaGaamyzaiaa dYgacaWGHbGaam4CaaqabaGccqGHsislcaaIXaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaIYaaaaaaa@59DA@

    Where,

    β = 1 N = 1 λ m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaey ypa0ZaaSaaaeaacaaIXaaabaGaamOtaaaacqGH9aqpdaWcaaqaaiaa igdaaeaacqaH7oaBdaqhaaWcbaGaamyBaaqaaiaaikdaaaaaaaaa@3F9C@
    N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@36CA@
    Measure of the limiting locking stretch.
    λ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaad2gaaeqaaaaa@38C9@
    Maximum locking stretch.
    D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadseadaWgaa WcbaGaaGymaaqabaaaaa@379C@
    Related to volumetric deformation. It defines the compressibility of the material.
    I ¯ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadMeagaqeam aaBaaaleaacaaIXaaabeaaaaa@37B9@
    First strain invariant, internally calculated by OptiStruct.
    Wherein, I ¯ 1 = I 1 J 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadMeagaqeam aaBaaaleaacaaIXaaabeaakiabg2da9iaadMeadaWgaaWcbaGaaGym aaqabaGccaWGkbWaaWbaaSqabeaacqGHsisldaWccaqaaiaaikdaae aacaaIZaaaaaaaaaa@3DFC@ .
    J e l a s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadQeadaWgaa WcbaGaamyzaiaadYgacaWGHbGaam4Caaqabaaaaa@3AA0@
    Elastic volume strain, internally calculated by OptiStruct.
    C 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadseadaWgaa WcbaGaaGymaaqabaaaaa@379C@
    Initial shear modulus.

    α 1 = 1 2 ; α 2 = 1 20 ; α 3 = 11 1050 ; α 4 = 19 7000 ; α 5 = 519 673750 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOm aaaacaGG7aGaeqySde2aaSbaaSqaaiaaikdaaeqaaOGaeyypa0ZaaS aaaeaacaaIXaaabaGaaGOmaiaaicdaaaGaai4oaiabeg7aHnaaBaaa leaacaaIZaaabeaakiabg2da9maalaaabaGaaGymaiaaigdaaeaaca aIXaGaaGimaiaaiwdacaaIWaaaaiaacUdacqaHXoqydaWgaaWcbaGa aGinaaqabaGccqGH9aqpdaWcaaqaaiaaigdacaaI5aaabaGaaG4nai aaicdacaaIWaGaaGimaaaacaGG7aGaeqySde2aaSbaaSqaaiaaiwda aeqaaOGaeyypa0ZaaSaaaeaacaaI1aGaaGymaiaaiMdaaeaacaaI2a GaaG4naiaaiodacaaI3aGaaGynaiaaicdaaaaaaa@5E69@

  5. The Ogden Material model (OGDEN) is defined as: (11)
    W = i = 1 N 1 2 μ i α i 2 ( λ ¯ 1 α i + λ ¯ 2 α i + λ ¯ 3 α i 3 ) + 1 D 1 ( J e l a s 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maaqahabaWaaSaaaeaacaaIYaGaeqiVd02aaSbaaSqaaiaadMga aeqaaaGcbaGaeqySde2aa0baaSqaaiaadMgaaeaacaaIYaaaaaaakm aabmaabaGafq4UdWMbaebadaqhaaWcbaGaaGymaaqaaiabeg7aHnaa BaaameaacaWGPbaabeaaaaGccqGHRaWkcuaH7oaBgaqeamaaDaaale aacaaIYaaabaGaeqySde2aaSbaaWqaaiaadMgaaeqaaaaakiabgUca RiqbeU7aSzaaraWaa0baaSqaaiaaiodaaeaacqaHXoqydaWgaaadba GaamyAaaqabaaaaOGaeyOeI0IaaG4maaGaayjkaiaawMcaaaWcbaGa amyAaiabg2da9iaaigdaaeaacaWGobWaaSbaaWqaaiaaigdaaeqaaa qdcqGHris5aOGaey4kaSYaaSaaaeaacaaIXaaabaGaamiramaaBaaa leaacaaIXaaabeaaaaGcdaqadaqaaiaadQeadaWgaaWcbaGaamyzai aadYgacaWGHbGaam4CaaqabaGccqGHsislcaaIXaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaIYaaaaaaa@6755@
    Where,
    λ ¯ 1 , λ ¯ 2 , λ ¯ 3
    The three deviatoric stretches (deviatoric stretches are related to principal stretches by λ ¯ i = J 1 3 λ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4UdWMbae badaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGkbWaaWbaaSqabeaa daWcbaadbaGaaGymaaqaaiaaiodaaaaaaOGaeq4UdW2aaSbaaSqaai aadMgaaeqaaaaa@3F55@ )
    μ i
    Defined by the MUi fields
    α i
    Defined by the ALPHAi fields
    N 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaSGaamOtamaaBa aameaacaaIXaaabeaaaaa@37B2@
    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 = i = 1 N 1 2 μ i α i 2 ( λ 1 α i + λ 2 α i + λ 3 α i 3 + 1 β i ( J α i β i 1 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maaqahabaWaaSaaaeaacaaIYaGaeqiVd02aaSbaaSqaaiaadMga aeqaaaGcbaGaeqySde2aa0baaSqaaiaadMgaaeaacaaIYaaaaaaakm aabmaabaGaeq4UdW2aa0baaSqaaiaaigdaaeaacqaHXoqydaWgaaad baGaamyAaaqabaaaaOGaey4kaSIaeq4UdW2aa0baaSqaaiaaikdaae aacqaHXoqydaWgaaadbaGaamyAaaqabaaaaOGaey4kaSIaeq4UdW2a a0baaSqaaiaaiodaaeaacqaHXoqydaWgaaadbaGaamyAaaqabaaaaO GaeyOeI0IaaG4maiabgUcaRmaalaaabaGaaGymaaqaaiabek7aInaa BaaaleaacaWGPbaabeaaaaGcdaqadaqaaiaadQeadaahaaWcbeqaai abgkHiTiabeg7aHnaaBaaameaacaWGPbaabeaaliabek7aInaaBaaa meaacaWGPbaabeaaaaGccqGHsislcaaIXaaacaGLOaGaayzkaaaaca GLOaGaayzkaaaaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6eadaWg aaadbaGaaGymaaqabaaaniabggHiLdaaaa@69DB@
    Where,
    λ 1 , λ 2 , λ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaaigdaaeqaaOGaaiilaiabeU7aSnaaBaaaleaacaaIYaaa beaakiaacYcacqaH7oaBdaWgaaWcbaGaaG4maaqabaaaaa@3F3E@
    Principle stretches
    μ i
    Defined by the MUi fields
    α i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadMgaaeqaaaaa@38AF@
    Defined by the ALPHAi fields
    β i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaadMgaaeqaaaaa@38B2@
    Defined by the BETAi fields
    N 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaaIXaaabeaaaaa@37B1@
    Order of the strain energy function defined on the NA field.

    Currently, the Hill material model is only supported for explicit analysis.

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