/MAT/LAW95 (BERGSTROM_BOYCE)

Block Format Keyword This law is a constitutive model for predicting the nonlinear time dependency of elastomer like materials.

It uses a polynomial material model for the hyperelastic material response and the Bergstrom-Boyce material model 1 to represent the nonlinear viscoelastic time dependent material response. This law is only compatible with solid elements.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW95/mat_ID/unit_ID or /MAT/BERGSTROM_BOYCE/mat_ID/unit_ID
mat_title
ρ i                
C10 C01 C20 C11 C02
C30 C21 C12 C03 sb
D1 D2 D3    
A C M ξ Tau_ref

Definitions

Field Contents SI Unit Example
mat_ID Material identifier.

(Integer, maximum 10 digits)

 
unit_ID Unit Identifier.

(Integer, maximum 10 digits)

 
mat_title Material title.

(Character, maximum 100 characters)

 
ρ i Initial density.

(Real)

[ kg m 3 ]
C10 Material parameter for hyperelastic model.

Default = 0.0 (Real)

[ Pa ]
C01 Material parameter for hyperelastic model.

Default = 0.0 (Real)

[ Pa ]
C20 Material parameter for hyperelastic model.

Default = 0.0 (Real)

[ Pa ]
C11 Material parameter for hyperelastic model.

Default = 0.0 (Real)

[ Pa ]
C02 Material parameter for hyperelastic model.

Default = 0.0 (Real)

[ Pa ]
C30 Material parameter for hyperelastic model.

Default = 0.0 (Real)

[ Pa ]
C21 Material parameter for hyperelastic model.

Default = 0.0 (Real)

[ Pa ]
C12 Material parameter for hyperelastic model.

Default = 0.0 (Real)

[ Pa ]
C03 Material parameter for hyperelastic model.

Default = 0.0 (Real)

[ Pa ]
Sb Stress scaling factor for network B.

Default = 0.0 (Real)

 
D1 Volumetric material parameter 1, for bulk modulus computation.

K = 2 D 1

Default = 0.0 (Real)

[ 1 P a ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaaGymaaqaaiaaccfacaGGHbaaaaGaay5waiaaw2faaaaa @3AD5@
D2 Volumetric material parameter 2.

Default = 0.0 (Real)

[ 1 P a ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaaGymaaqaaiaaccfacaGGHbaaaaGaay5waiaaw2faaaaa @3AD5@
D3 Volumetric material parameter 3.

Default = 0.0 (Real)

[ 1 P a ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaaGymaaqaaiaaccfacaGGHbaaaaGaay5waiaaw2faaaaa @3AD5@
A Effective creep strain rate.

Default = 0.0 (Postive Real)

[ 1 s ]
C Exponent characterizing the creep strain dependence of the effective creep strain rate in network B (-1 < C < 0).

Default = -0.7 (Real)

 
M Positive exponent ( M 1.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaey yzImRaaGymaiaac6cacaaIWaaaaa@3C27@ ) characterizing the effective stress dependence of the effective creep strain rate in network B.

Default = 1.0 (Real)

 
ξ Constant for regularization of the creep strain rate near undeformed state.

Default = 0.01 (Real)

 
Tau_ref Reference stress for the Effective creep strain rate in secondary network.

Default = 1.0 (Real)

[ Pa ]

Example

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  kg                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW95/1/1
BERGSTROM 
#              RHO_I        
             1.42E-6 		
#                C10                 C01                 C20                 C11                 C22
              0.2019                  0.             4.43E-5
#                C30                 C21                 C12                 C03                  Sb
            1.295E-4                  0.                  0.                  0.                 2.0
#                 D1                  D2                  D3
           2.1839E-3             8.68E-5           -1.794E-5
#                  A                EXPC                EXPM                 KSI             Tau_ref
              1.0E-1                -0.7                   5                0.01
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

  1. The response of the material can be represented using two parallel networks A and B. Network A is the equilibrium network with a nonlinear hyperelastic component. In network B, a nonlinear hyperelastic component is in series with a nonlinear viscoelastic flow element, and hence the time-dependent network.


    Figure 1. Time-dependent Network
  2. The same polynomial strain energy potential is used for the hyperelastic components in both networks. In network B, this potential is scaled by a factor Sb. The strain energy density is then written for the hyperelastic component of the network:(1)
    W A = i + j = 1 3 C i j ( I ¯ 1 3 ) i ( I ¯ 2 3 ) j + i = 1 3 1 D i ( J 1 ) 2 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaaS baaSqaaiaadgeaaeqaaOGaeyypa0ZaaabCaeaacaWGdbWaaSbaaSqa aiaadMgacaWGQbaabeaakmaabmaabaGabmysayaaraWaaSbaaSqaai aaigdaaeqaaOGaeyOeI0IaaG4maaGaayjkaiaawMcaamaaCaaaleqa baGaamyAaaaakiabgwSixpaabmaabaGabmysayaaraWaaSbaaSqaai aaikdaaeqaaOGaeyOeI0IaaG4maaGaayjkaiaawMcaamaaCaaaleqa baGaamOAaaaaaeaacaWGPbGaey4kaSIaamOAaiabg2da9iaaigdaae aacaaIZaaaniabggHiLdGccqGHRaWkdaaeWbqaamaalaaabaGaaGym aaqaaiaadseadaWgaaWcbaGaamyAaaqabaaaaOWaaeWaaeaacaWGkb GaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaiaa dMgaaaaabaGaamyAaiabg2da9iaaigdaaeaacaaIZaaaniabggHiLd aaaa@614A@
    and(2)
    W B = S b W A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaaS baaSqaaiaadkeaaeqaaOGaeyypa0Jaam4uamaaBaaaleaacaWGIbaa beaakiabgwSixlaadEfadaWgaaWcbaGaamyqaaqabaaaaa@3F4B@
    Where,
    • I ¯ 1 = λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGjbGbae badaWgaaWcbaGaaGymaaqabaGccqGH9aqpcuaH7oaBgaqeamaaDaaa leaacaaIXaaabaGaaGOmaaaakiabgUcaRiqbeU7aSzaaraWaa0baaS qaaiaaikdaaeaacaaIYaaaaOGaey4kaSIafq4UdWMbaebadaqhaaWc baGaaG4maaqaaiaaikdaaaaaaa@4567@
    • I ¯ 2 = λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGjbGbae badaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcuaH7oaBgaqeamaaDaaa leaacaaIXaaabaGaeyOeI0IaaGOmaaaakiabgUcaRiqbeU7aSzaara Waa0baaSqaaiaaikdaaeaacqGHsislcaaIYaaaaOGaey4kaSIafq4U dWMbaebadaqhaaWcbaGaaG4maaqaaiabgkHiTiaaikdaaaaaaa@482F@
    • λ ¯ i = J 1 3 λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4UdWMbae badaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGkbWaaWbaaSqabeaa cqGHsisldaWcaaqaaiaaigdaaeaacaaIZaaaaaaakiabeU7aSnaaBa aaleaacaWGPbaabeaaaaa@4036@
  3. For special value of C i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@3930@ , the polynomial model can be reduced to the following material models:
    • Yeoh: j=0

      Where, C10, C20, C30 are not zero

    • Mooney-Rivlin: i+j =1

      Where, C10 and C01 are not zero, and D2 =D3=0

    • Neo-Hookean:

      Only C10 and D1 are not zero

  4. The initial shear modulus and the bulk modulus are computed as:(3)
    μ = 2 ( S b + 1 ) ( C 10 + C 01 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcq GH9aqpcaaIYaWaaeWaaeaacaWGtbWaaSbaaSqaaiaadkgaaeqaaOGa ey4kaSIaaGymaaGaayjkaiaawMcaamaabmaabaGaam4qamaaBaaale aacaaIXaGaaGimaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaaicda caaIXaaabeaaaOGaayjkaiaawMcaaaaa@4643@
    and(4)
    K = 2 D 1 ( 1 + S b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbGaey ypa0ZaaSaaaeaacaaIYaaabaGaamiramaaBaaaleaacaaIXaaabeaa aaGcdaqadaqaaiaaigdacqGHRaWkcaWGtbWaaSbaaSqaaiaadkgaae qaaaGccaGLOaGaayzkaaaaaa@3FD6@
  5. If D1= 0, an incompressible material is considered.
  6. If A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbaaaa@3725@ =0, then only the hyperelastic polynomial material model is used with no viscoelastic time dependent response.
  7. The effective creep strain rate in network B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbaaaa@3725@ is given by the expression:(5)
    ε ˙ B v = A ( λ ˜ 1 + ξ ) C ( σ ¯ B τ r e f ) M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacuaH1oqzpaGbaiaadaqhaaWcbaWdbiaadkeaa8aabaWdbiaadAha aaGccqGH9aqpcaWGbbWaaeWaa8aabaWaaCbiaeaapeGaeq4UdWgal8 aabeqaa8qacaGGClaaaOGaeyOeI0IaaGymaiabgUcaRiabe67a4bGa ayjkaiaawMcaa8aadaahaaWcbeqaa8qacaWGdbaaaOWdamaabmaaba WaaSaaaeaapeGafq4Wdm3dayaaraWaaSbaaSqaa8qacaWGcbaapaqa baaakeaacqaHepaDdaWgaaWcbaGaamOCaiaadwgacaWGMbaabeaaaa aakiaawIcacaGLPaaadaahaaWcbeqaa8qacaWGnbaaaaaa@50F8@
    Where,
    λ ˜ = I ¯ 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBga acaiabg2da9maakaaabaWaaSaaaeaaceWGjbGbaebadaWgaaWcbaGa aGymaaqabaaakeaacaaIZaaaaaWcbeaaaaa@3BE7@ and σ ¯ B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qeamaaBaaaleaacaWGcbaabeaaaaa@392D@
    Effective stress in Network B
    ξ , M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbaaaa@3725@ , C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbaaaa@3725@ and τ ref MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaadkhacaWGLbGaamOzaaqabaaaaa@3AB4@
    Input material parameters
1 Bergström, J. S., and M. C. Boyce. "Constitutive modeling of the large strain time-dependent behavior of elastomers." Journal of the Mechanics and Physics of Solids" 46, No. 5 (1998): 931-954