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 D TAB1 TAB2   TAB4 TABD  
  C20 C11 C02 D NA ND      
  C30 C21 C12 C03 D        
  C40 C31 C22 C13 C04 D      
  C50 C41 C32 C23 C14 C05 D    

Format B

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

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 D TAB1 TAB2   TAB4    
  MU2 ALPHA2   MU3 ALPHA3        
  MU4 ALPHA4   MU5 ALPHA5        

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      

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.
MOONEY (Default)
Selects the generalized Mooney-Rivlin hyperelastic model
MOOR
Physical Mooney-Rivlin model
RPOLY
Reduced Polynomial model
NEOH
Neo-Hookean model
YEOH
Yeoh model
ABOYCE
Selects the 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.

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, Cpq, 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, Cpq, 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, Cpq, related to distortional deformation. The x-values in the TABLES1 entry should be the stretch ratios and y-values should be values of the nominal 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 pressure and y-values should be values of the volumetric change. TABD can only be used to fit Volumetric data for Format A, additionally, only first-order fitting is currently supported (only D1 value is sourced from the TABD data).

(Integer > 0 or blank)

 
C Initial shear modulus (Model = ABOYCE). 5

No default (Real)

 
λ m Maximum locking stretch.

Used to calculate the value of β (Model = ABOYCE). 5

No default (Real)

 
MUi, ALPHAi Material Constants for the Ogden Material Model (Model = OGDEN) 6; or

Hill Foam Material Model (Model = FOAM. 7

 
BETAi Material Constants for Hill Foam Material Model (Model=FOAM). 7  

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 pq ( I ¯ 1 3 ) p ( I ¯ 2 3 ) q + p=1 N 2 1 D p ( J elas 1 ) 2p 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 MATHE hyperelastic material supports CTETRA (4, 10), CPENTA (6, 15), and CHEXA (8, 20) element types.
  5. The Arruda-Boyce model (ABOYCE) is defined as: (10)
    W= C 1 i=1 5 α i β i1 ( I ¯ 1 i 3 i )+ 1 D 1 ( J elas 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@

  6. 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 elas 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
  7. 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.

  8. This card is represented as a material in HyperMesh.