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) (10)
MATHE MID Model   NU RHO TEXP TREF    
  C10 C01 D1 TAB1 TAB2   TAB4 TABD  
  C20 C11 C02 D2 NA ND      
  C30 C21 C12 C03 D3        
  C40 C31 C22 C13 C04 D4      
  C50 C41 C32 C23 C14 C05 D5    
  MODULI MTIME              

Format B

Arruda-Boyce Model (Model=ABOYCE):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATHE MID Model   NU RHO TEXP TREF    
  C1 λ m   TAB1 TAB2   TAB4    
  D1                
  MODULI MTIME              

Format C

Ogden Material Model (Model=OGDEN):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATHE MID Model NA NU RHO TEXP TREF    
  MU1 ALPHA1 D1 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) (9) (10)
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) (10)
MATHE MID Model   NU RHO TEXP TREF    
      D1 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 D1 value is sourced from the TABD data).

(Integer > 0 or blank)

 
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  
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

  1. 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.
  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 ratio.
    λ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaad2gaaeqaaaaa@38C9@
    Maximum locking stretch ratio.
    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 stretch ratios (deviatoric stretch ratios are related to principal stretch ratios 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 stretch ratios.
    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.

    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:(13)
    β i = ν 1 2 ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacqaH9oGBaeaacaaI XaGaeyOeI0IaaGOmaiabe27aUbaaaaa@3FA5@

    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.

  7. 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, 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.
  8. If Poisson’s ratio and D1 or TABD are both defined, Poisson’s ratio takes precedence.
  9. The initial modulus used for linear analysis is:
    • Mooney, Neo-Hookean, Mooney-Rivlin, Yeoh, Reduced Polynomial

      G = 2 ( C 10 + C 01 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9iaaikdacaGGOaGaam4qamaaBaaaleaacaaIXaGaaGimaaqabaGc cqGHRaWkcaWGdbWaaSbaaSqaaiaaicdacaaIXaaabeaakiaacMcaaa a@3FA5@ and K = 2 D 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9maalaaabaGaaGOmaaqaaiaadseadaWgaaWcbaGaaGymaaqabaaa aaaa@3A48@

    • Ogden

      G = μ 1 + μ 2 + μ 3 + μ 4 + μ 5 = t = 1 5 μ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9iabeY7aTnaaBaaaleaacaaIXaaabeaakiabgUcaRiabeY7aTnaa BaaaleaacaaIYaaabeaakiabgUcaRiabeY7aTnaaBaaaleaacaaIZa aabeaakiabgUcaRiabeY7aTnaaBaaaleaacaaI0aaabeaakiabgUca RiabeY7aTnaaBaaaleaacaaI1aaabeaakiabg2da9maaqahabaGaeq iVd02aaSbaaSqaaiaadshaaeqaaaqaaiaadshacqGH9aqpcaaIXaaa baGaaGynaaqdcqGHris5aaaa@522E@ and K = 2 D 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9maalaaabaGaaGOmaaqaaiaadseadaWgaaWcbaGaaGymaaqabaaa aaaa@3A48@

    • Arruda-Boyce

      G = C 1 ( 1 + 3 5 λ m 2 + 99 175 λ m 4 + 513 875 λ m 6 + 42039 67375 λ m 8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9iaadoeadaWgaaWcbaGaaGymaaqabaGcdaqadaqaaiaaigdacqGH RaWkdaWcaaqaaiaaiodaaeaacaaI1aGaeq4UdW2aa0baaSqaaiaad2 gaaeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaaGyoaiaaiMdaaeaa caaIXaGaaG4naiaaiwdacqaH7oaBdaqhaaWcbaGaamyBaaqaaiaais daaaaaaOGaey4kaSYaaSaaaeaacaaI1aGaaGymaiaaiodaaeaacaaI 4aGaaG4naiaaiwdacqaH7oaBdaqhaaWcbaGaamyBaaqaaiaaiAdaaa aaaOGaey4kaSYaaSaaaeaacaaI0aGaaGOmaiaaicdacaaIZaGaaGyo aaqaaiaaiAdacaaI3aGaaG4maiaaiEdacaaI1aGaeq4UdW2aa0baaS qaaiaad2gaaeaacaaI4aaaaaaaaOGaayjkaiaawMcaaaaa@5F25@ and K = 2 D 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9maalaaabaGaaGOmaaqaaiaadseadaWgaaWcbaGaaGymaaqabaaa aaaa@3A48@

    • Hill Foam

      G = μ 1 + μ 2 + μ 3 + μ 4 + μ 5 = i = 1 5 μ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9iabeY7aTnaaBaaaleaacaaIXaaabeaakiabgUcaRiabeY7aTnaa BaaaleaacaaIYaaabeaakiabgUcaRiabeY7aTnaaBaaaleaacaaIZa aabeaakiabgUcaRiabeY7aTnaaBaaaleaacaaI0aaabeaakiabgUca RiabeY7aTnaaBaaaleaacaaI1aaabeaakiabg2da9maaqahabaGaeq iVd02aaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaa baGaaGynaaqdcqGHris5aaaa@5218@ and K = i = 1 5 2 μ i ( 1 3 + β i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9maaqahabaGaaGOmaiabeY7aTnaaBaaaleaacaWGPbaabeaaaeaa caWGPbGaeyypa0JaaGymaaqaaiaaiwdaa0GaeyyeIuoakmaabmaaba WaaSaaaeaacaaIXaaabaGaaG4maaaacqGHRaWkcqaHYoGydaWgaaWc baGaamyAaaqabaaakiaawIcacaGLPaaaaaa@47BF@

    Additional treatments on bulk modulus K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C6@ are as:
    • If Poisson's ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4gaaa@37AE@ , is non-zero, bulk modulus K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C6@ is replaced with:(14)
      K = 2 G ( 1 + ν ) 3 ( 1 2 ν ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9maalaaabaGaaGOmaiaadEeacaGGOaGaaGymaiabgUcaRiabe27a UjaacMcaaeaacaaIZaGaaiikaiaaigdacqGHsislcaaIYaGaeqyVd4 Maaiykaaaaaaa@4444@
    • If K = 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9iaaicdacaGGSaaaaa@3936@ K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C6@ is set to be 30 G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic dacaWGhbaaaa@3839@
    • If K 30 G , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgw MiZkaaiodacaaIWaGaam4raiaacYcaaaa@3B7F@ K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C6@ is set to be 30 G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic dacaWGhbaaaa@3839@
    The Young’s modulus and Poisson's ratio are given by:(15)
    E = 9 K G 3 K + G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2 da9maalaaabaGaaGyoaiaadUeacaWGhbaabaGaaG4maiaadUeacqGH RaWkcaWGhbaaaaaa@3D70@
    and(16)
    ν = 3 K 2 G 6 K + 2 G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey ypa0ZaaSaaaeaacaaIZaGaam4saiabgkHiTiaaikdacaWGhbaabaGa aGOnaiaadUeacqGHRaWkcaaIYaGaam4raaaaaaa@40C0@
    Where,
    E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@
    Young’s modulus.
    G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@
    Shear modulus.
    K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@
    Bulk modulus.
    ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4gaaa@37AE@
    Poisson's ratio.
    C 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaaaaa@385F@ , C 01 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaaaaa@385F@ , D 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaaaaa@385F@ , μ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadMgaaeqaaaaa@38C6@ , and C 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaaaaa@385F@
    Material coefficients.
    λ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaad2gaaeqaaaaa@38C8@
    Stretch ratio at which the polymer chain network is locked.
  10. 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.
  11. The support information for the available material models (in Model field) is:
    Analysis Type Support Information
    Implicit Analysis All the material models are supported with:
    Explicit Dynamic Analysis All the material models are supported with:
  12. Temperature-dependent hyperelastic material data can be defined via the MATTHE entry.
  13. This card is represented as a material in HyperMesh.