/MAT/LAW100 (MNF)

Block Format Keyword The multi network framework or MNF is used to model polymers and elastomers with nonlinear viscous behavior.

It consists of having a specific number of networks with an elastic component and an optional flow component. This law is only compatible with solid elements.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW100/mat_ID/unit_ID or /MAT/MNF/mat_ID/unit_ID
mat_title
ρ i                
N_net Flag_HE Flag_Cr          
If Flag_HE = 1 (Polynomial form)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
C10 C01 C20 C11 C02
C30 C21 C12 C03  
D1 D2 D3    
If Flag_HE = 2 (Arruda Boyce model)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
μ D λ m    
Itype fct_IDAB ν FscaleAB    
If Flag_HE = 3 (Neo Hookean model)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
C10 D1      
If Flag_HE = 4 (Mooney-Rivlin model)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
C10 C01 D1    
If Flag_HE = 5 (Yeoh model)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
C10 C20 C30 D1  
If Flag_HE = 13 (Neo Hookean model with temperature dependency)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
fct_IDSM fct_IDBM FscaleSM FscaleBM    
If Flag_Cr =1 (Creep)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
A p l σ ^ 0 f f ε ^ n p l  
For Each Network
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
networkID Flag_visc stiffness      
If Flag_visc = 1 (Bergstrom-Boyce viscous model)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
A1 C M ξ Tau_ref
If Flag_visc = 2 (Hyperbolic sine viscous model)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
A2 B n2    
If Flag_visc = 3 (Power law viscous model)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
A3 n3 M3    

Definition

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 ]
N_net Total number of secondary networks.

(Integer)

 
Flag_HE Hyperelastic model flag.
=1
Polynomial form
=2
Arruda Boyce
=3
Neo Hookean
=4
Mooney-Rivlin
=5
Yeoh
=13
Neo Hookean with temperature dependency

(Integer)

 
Flag_Cr Creep in equilibrium network flag.
=0 (Default)
No creep (no additional line).
=1
Creep (read parameters in additional line).

(Integer)

 
Flag_visc Viscous model flag.
=1
Bergstrom Boyce.
=2
Hyperbolic sine.
=3
Power law.

(Integer)

 
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 ]
D1 Volumetric material parameter 1, used for bulk modulus computation.

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@
μ Shear modulus.

(Real)

[ Pa ]
D Material parameter for bulk modulus computation K = 2 D .

Default =1030 (Real)

[ 1 P a ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaaGymaaqaaiaaccfacaGGHbaaaaGaay5waiaaw2faaaaa @3AD5@
λ m The limit of stretch.

Default = 7.0 (Real)

 
Itype Test data type (stress strain curve).
=1 (Default)
Uniaxial data test
=2
Equibiaxial data test
=3
Planar data test

(Integer)

 
fct_IDAB Function identifier defining engineer stress versus engineer strain for the Arruda-Boyce material model.

(Integer)

 
ν Poisson ratio.

(Real)

 
FscaleAB Scale factor for fct_IDAB.

(Real)

[ Pa ]
fct_IDSM Function identifier for shear modulus versus temperature.

(Integer)

 
fct_IDBM Function identifier for bulk modulus versus temperature.

(Integer)

 
FscaleSM Shear modulus scale factor for fct_IDSM.

Default = 1.0 (Real)

[ Pa ]
FscaleBM Bulk modulus scale factor for fct_IDbM.

Default = 1.0 (Real)

[ Pa ]
Stiffness Stiffness weight factor for secondary networks, ( S i ).

Default = 0.0 (Real)

 
networkID Number of network (Must be left justified). 5
NETWORK1
For the first network.
NETWORK2
For the second network.
NETWORKi
For the ith network.

(Characters)

 
A1 Effective creep strain rate. 7

Default = 0.0 (Positive Real)

[ 1 s ]
A2 Effective creep strain rate.

Default = 0.0 (Positive Real)

[ 1 s ]
A3 Effective creep strain rate.

Default = 0.0 (Positive 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 ≥ 1 characterizing the effective stress dependence of the effective creep strain rate in secondary network.

Default = 1.0 (Real)

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

Default = 0.01 (Real)

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

Default = 1.0 (Real)

[ Pa ]
B Coefficient in hyperbolic sine viscous model multiplying the norm of the stress in the secondary network.

(Real)

 
n2 Exponent in hyperbolic sine viscous model in the secondary network.

(Real)

 
n3 Exponent in power law viscous model in the secondary network.

(Real)

 
M3 Exponent in power law viscous model in the secondary network.

(Real)

 
A p l Scaling factor for plastic flow rule.

(Real)

[ 1 s ]
σ ^ 0 Flow resistance for plastic flow rule.

Default = 1.0 (Real)

[ Pa ]
f f Weight factor for flow resistance in plastic flow rule.

Default = 1.0 (Real)

 
ε ^ Characteristic strain for plastic flow rule.

Default = 1.0 (Real)

 
n p l Exponent for plastic flow rule.

Default = 1 (Integer)

 

Example (Polynomial Model and One Network)

#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/LAW100/1/1
Hyperelastic mat with Polynomial form and one network
#              RHO_I        
 1.4200000000000E-06
#N_NETWORK   FLAG_HE   FLAG_Cr
         1         1          
#                C10                 C01                 C20                 C11                 C02
              0.2019                  0.             4.43E-5
#                C30                 C21                 C12                 C03 
            1.295E-4                  0.                  0.                  0. 
#                 D1                  D2                  D3     
           2.1839e-3
#   KEYNET FLAG_VISC          SCALESTIFF 
NETWORK1           1                 1.0
#                  A                EXPC                EXPM                 KSI             Tau_ref
               2000.                -1.0                  10                0.01
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Example (Polynomial Model and Three Networks)

#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/LAW100/1/1
Hyperelastic mat with Polynomial form and three networks
#              RHO_I        
 1.4200000000000E-06
#N_NETWORK   FLAG_HE   FLAG_Cr
         3         1          
#                C10                 C01                 C20                 C11                 C02
              0.2019                  0.             4.43E-5
#                C30                 C21                 C12                 C03 
            1.295E-4                  0.                  0.                  0. 
#                 D1                  D2                  D3     
           2.1839e-3
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#   KEYNET FLAG_VISC          SCALESTIFF 
NETWORK1           1                 0.6
#                 A1                EXPC                EXPM                 KSI             Tau_ref
               2000.                -1.0                  10                0.01
NETWORK3           2                 0.1
#                 A2                  B0                EXPN 
               1.000                1.0                   2. 
NETWORK2           3                 0.3
#                 A3                EXPN                EXPM       
                 1.0                 5.0                  2.             
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

  1. This material is only compatible with solid elements with Lagrange type total strain. The strain formulation flag is automatically set to Ismstr =10 in /PROP/SOLID.
  2. The response of the material can be represented using a set of parallel networks. Network 0 is the equilibrium network with a nonlinear hyperelastic component and optional creep component. In secondary networks, a nonlinear hyperelastic component is in series with a nonlinear viscoelastic flow element, and hence is the time-dependent network. All networks have the same hyperelastic behavior, scaled by a stiffness weight factor for the secondary networks.
    The sum of the stiffness weight factors must be equal to 1:(1)
    i = 0 N S i = 1


    Figure 1.
  3. Same polynomial strain energy potential is used for the hyperelastic components in all networks. In secondary networks, this potential is scaled by a factor S i .
  4. Flag_HE
    1. 1 = Polynomial form: The energy density is then written(2)
      W 0 = 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
    2. 2 = Arruda-Boyce: The energy density is then written:(3)
      W 0 =μ i=1 5 c i ( λ m ) 2i2 ( I ¯ 1 i 3 i )+ 1 D ( J 2 1 2 ln(J) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaaS baaSqaaiaaicdaaeqaaOGaeyypa0JaeqiVd02aaabCaeaadaWcaaqa aiaadogadaWgaaWcbaGaamyAaaqabaaakeaacaGGOaGaeq4UdW2aaS baaSqaaiaad2gaaeqaaOGaaiykamaaCaaaleqabaGaaGOmaiaadMga cqGHsislcaaIYaaaaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaaiw daa0GaeyyeIuoakmaabmaabaGabmysayaaraWaa0baaSqaaiaaigda aeaacaWGPbaaaOGaeyOeI0IaaG4mamaaCaaaleqabaGaamyAaaaaaO GaayjkaiaawMcaaiabgUcaRmaalaaabaGaaGymaaqaaiaadseaaaWa aeWaaeaadaWcaaqaaiaadQeadaahaaWcbeqaaiaaikdaaaGccqGHsi slcaaIXaaabaGaaGOmaaaacqGHsislciGGSbGaaiOBaiGacIcacaWG kbGaaiykaaGaayjkaiaawMcaaaaa@5EB5@
    3. 3 = Neo-Hook: The energy density is then written:(4)
      W 0 = C 10 ( I ¯ 1 3 ) + 1 D ( J 1 ) 2
    4. 4 = Mooney-Rivlin: The energy density is then written:(5)
      W 0 = C 10 ( I ¯ 1 3 ) + C 01 ( I ¯ 2 3 ) + 1 D ( J 1 ) 2
    5. 5 = Yeoh: The energy density is then written:(6)
      W 0 = C 10 ( I ¯ 1 3 ) + C 20 ( I ¯ 1 3 ) 2 + C 30 ( I ¯ 1 3 ) 3 + 1 D ( J 1 ) 2
    6. 13 = Neo-Hook with temperature: The energy density is then written:(7)
      W 0 = μ ( T ) 2 ( I ¯ 1 3 ) + K ( T ) 2 ( J 1 ) 2
    and the energy density for each secondary network: (8)
    W i = S i W 0
    Then, the total energy density for the secondary network is (9)
    W = i = 0 N W i
    Note: (10)
    i = 0 N S i = 1
    (11)
    I ¯ 1 = λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2
    (12)
    I ¯ 2 = λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2
    (13)
    λ ¯ i = J 1 3 λ i
    The Cauchy stress is computed as:(14)
    σ i = λ i J W λ i
  5. The networkID must be left justified, and the name must use the form "NETWORKi"

    Where, i is the networkID. Other names like "network1" or "NET1" are not allowed.

  6. Polynomial form:
    1. The initial shear modulus and the bulk modulus are computed as:(15)
      G = 2 ( S i + 1 ) ( C 10 + C 01 )
      and (16)
      K = 2 D 1 ( 1 + S i )
    2. If D1 = 0, an incompressible material is considered.
  7. The effective creep strain rate
    1. For Bergstrom Boyce viscous model, the expression is:(17)
      ε ˙ B v = A 1 ( 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 aaGccqGH9aqpcaWGbbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbm aabmaapaqaamaaxacabaWdbiabeU7aSbWcpaqabeaapeGaaii3caaa kiabgkHiTiaaigdacqGHRaWkcqaH+oaEaiaawIcacaGLPaaapaWaaW baaSqabeaapeGaam4qaaaak8aadaqadaqaamaalaaabaWdbiqbeo8a Z9aagaqeamaaBaaaleaapeGaamOqaaWdaeqaaaGcbaGaeqiXdq3aaS baaSqaaiaadkhacaWGLbGaamOzaaqabaaaaaGccaGLOaGaayzkaaWa aWbaaSqabeaapeGaamytaaaaaaa@5227@
      Where, (18)
      λ ˜ = I ¯ 1 3
    2. For Hyperbolic sine viscous model, the expression is:(19)
      ε ˙ B v = A 2 ( sinh B σ ¯ ) n 2
    3. For Power law viscous model, the expression is:
      (20)
      ε ˙ B v = A 3 { σ ¯ n 3 [ ( M 3 + 1 ) ε v ] M 3 } 1 M 3 + 1
      Flow rule for equilibrium network:(21)
      ε ˙ c r = A p l ( σ ¯ σ ^ ) n p l
      (22)
      σ ^ = σ ^ 0 [ f f + ( 1 f f ) exp ( ε c r ε ^ ) ]
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