/MAT/LAW120 (TAPO)

Block Format Keyword This is a non-associated elasto-plastic model for polymer adhesives. The constitutive model is based on a I1-J2 criterion that can be reduced either to a von Mises or Drucker-Prager type in compression.

It can be used to represent the mechanical behavior of adhesives under complex loading paths with combined shear and tension. The material model includes a nonlinear damage model depending on plastic strain, triaxiality and strain rate. This material is applicable only to solid hexahedron elements (/BRICK).

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW120/mat_ID/unit_ID or /MAT/TAPO/mat_ID/unit_ID
mat_title
ρ i                
E ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBaa a@3816@ Iform Itrx Idam      
Table_ID Xscale Yscale          
τ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3damaaBaaaleaapeGaaGimaaWdaeqaaaaa@38E5@ Q β H    
A1F A2F A1H A2H AS
C ε ˙ ref MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqyTdu2dayaacaWaaSbaaSqaa8qacaWGYbGaamyzaiaadAgaa8aa beaaaaa@3AE2@ ε ˙ max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqyTdu2dayaacaWaaSbaaSqaa8qacaWGTbGaamyyaiaadIhaa8aa beaaaaa@3AEB@        
D1c D2c D1f D2f    
Dtrx DJC Exp_n        

Definition

Field Contents SI Unit Example
mat_ID Material identifier.

(Integer, maximum 10 digits)

unit_ID (Optional) Unit identifier.

(Integer, maximum 10 digits)

mat_title Material title.

(Character, maximum 100 characters)

ρ i Initial density.

(Real)

[ kg m 3 ]
E Young’s (stiffness) modulus.

(Real)

[ Pa ]
ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBaa a@3816@ Poisson's coefficient.

(Real)

 
Iform Yield criterion formulation flag.
= 1 (Default)
Drucker-Prager model in compression.
= 2
von Mises in compression.

(Integer)

Itrx Damage dependency on triaxiality in compression flag.
= 1
Damage depends on triaxiality in tension and compression.
= 2 (Default)
Damage depends on triaxiality in tension only.

(Integer)

Idam Strain rate definition in damage model flag.
= 1
Damage factor defined with undamaged plastic strain rate.
= 2 (Default)
Damage factor defined with damaged plastic strain rate.

(Integer)

Table_ID Table identifier to define yield stress as a function of plastic strain, strain rate and temperature.

(Integer)

 
Xscale Scale factor for strain rate variable in Table_ID.

(Real)

[Hz]
Yscale Scale factor for yield stress value defined by Table_ID.

(Real)

[ Pa ]
τ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3damaaBaaaleaapeGaaGimaaWdaeqaaaaa@38E5@ Initial shear yield stress.

(Real)

[ Pa ]
Q Voce hardening modulus.

(Real)

[ Pa ]
β Voce nonlinear hardening exponent.

Default = 1.0 (Real)

H Linear hardening exponent.

Default = 1.0 (Real)

[ Pa ]
A1F Yield function parameter.

(Real)

A2F Yield function parameter.

(Real)

A1H Yield function distortional hardening parameter.

(Real)

A2H Yield function distortional hardening parameter.

(Real)

AS Plastic flow function parameter for hydrostatic term.

(Real)

C Johnson-Cook strain rate coefficient for hardening.

(Real)

ε ˙ r e f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqyTdu2dayaacaWaaSbaaSqaa8qacaWGYbGaamyzaiaadAgaa8aa beaaaaa@3AE2@ Quasi-static threshold strain rate in Johnson-Cook term.

(Real)

[Hz]
ε ˙ m a x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqyTdu2dayaacaWaaSbaaSqaa8qacaWGTbGaamyyaiaadIhaa8aa beaaaaa@3AEB@ Maximum dynamic threshold strain rate in Johnson-Cook term.

(Real)

[Hz]
D1c Johnson-Cook parameter for damage initiation.

(Real)

D2c Johnson-Cook parameter for damage initiation.

(Real)

D1f Johnson-Cook parameter for failure strain.

(Real)

D2f Johnson-Cook parameter for failure strain.

(Real)

Dtrx Johnson-Cook damage parameter for triaxiality term.

(Real)

DJC Johnson-Cook strain rate parameter for damage.

(Real)

Exp_n Exponential coefficient for damage strain rate dependency.

(Real)

Example (Adhesive Polymer)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/20
Material model units
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/TAPO/1/20
Adhesive polymer
#              RHO_I
              1.2E-9                   0
#                  E                  Nu     Iform      Itrx      Idam
                1588                 .34         1         0         0          
#               TAU0                   Q                beta                   H
               19.66               2.746               24.98               13.35
#                A1F                 A2F                 A1H                 A2H                  AS
               0.446               0.218                0.24                 0.1               0.338
#                 CC            Epsp_ref            Epsp_max
                 0.1               0.002                1726
#                D1c                 D2c                 D1f                 D2f
               0.345               1.094               6.935                0.00 
#              D_trx                D_JC               Exp_n
               0.001               1.044                   0

Comments

  1. The yield function is described depending on the Iform flag:
    • Iform = 1: Drucker-Prager formulation:(1)
      f =   J 2 + a 1 3 τ 0 I 1 + a 2 3 I 1 2 τ y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzaiabg2da9iaacckacaWGkbWdamaaBaaaleaapeGaaGOmaaWd aeqaaOWdbiabgUcaRmaalaaapaqaa8qacaWGHbWdamaaBaaaleaape GaaGymaaWdaeqaaaGcbaWdbmaakaaapaqaa8qacaaIZaaaleqaaaaa kiabes8a09aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaWGjbWdam aaBaaaleaapeGaaGymaaWdaeqaaOWdbiabgUcaRmaalaaapaqaa8qa caWGHbWdamaaBaaaleaapeGaaGOmaaWdaeqaaaGcbaWdbiaaiodaaa Gaamysa8aadaWgaaWcbaWdbiaaigdaa8aabeaakmaaCaaaleqabaWd biaaikdaaaGccqGHsislcqaHepaDpaWaa0baaSqaa8qacaWG5baapa qaa8qacaaIYaaaaaaa@4FE2@

      a 1 = A 1 F + A 1 H ε p l MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyya8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpcaWG bbWdamaaBaaaleaapeGaaGymaiaadAeaa8aabeaak8qacqGHRaWkca WGbbWdamaaBaaaleaapeGaaGymaiaadIeaa8aabeaak8qacqaH1oqz paWaaSbaaSqaa8qacaWGWbGaamiBaaWdaeqaaaaa@436F@ and a 2 = A 2F + A 2H ε pl MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyya8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH9aqpcaWG bbWdamaaBaaaleaapeGaaGOmaiaadAeaa8aabeaak8qacqGHRaWkca WGbbWdamaaBaaaleaapeGaaGOmaiaadIeaa8aabeaak8qacqaH1oqz paWaaSbaaSqaa8qacaWGWbGaamiBaaWdaeqaaaaa@4372@

    • Iform = 2: von Mises formulation:(2)
      f=  J 2 + A 2F 3 I 1 + 3 2 A 1F A 2F τ 0 2 τ y 2 + A 1F 2 A 2F τ 0 2 4 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzaiabg2da9iaacckacaWGkbWdamaaBaaaleaapeGaaGOmaaWd aeqaaOWdbiabgUcaRmaalaaapaqaa8qacaWGbbWdamaaBaaaleaape GaaGOmaiaadAeaa8aabeaaaOqaa8qacaaIZaaaaiaadMeapaWaaSba aSqaa8qacaaIXaaapaqabaGcpeGaey4kaSYaaSaaa8aabaWdbmaaka aapaqaa8qacaaIZaaaleqaaaGcpaqaa8qacaaIYaaaamaalaaapaqa a8qacaWGbbWdamaaBaaaleaapeGaaGymaiaadAeaa8aabeaaaOqaa8 qacaWGbbWdamaaBaaaleaapeGaaGOmaiaadAeaa8aabeaaaaGcpeGa eqiXdq3damaaBaaaleaapeGaaGimaaWdaeqaaOWaaWbaaSqabeaape GaaGOmaaaakiabgkHiTmaabmaapaqaa8qacqaHepaDpaWaa0baaSqa a8qacaWG5baapaqaa8qacaaIYaaaaOGaey4kaSYaaSaaa8aabaWdbi aadgeapaWaa0baaSqaa8qacaaIXaGaamOraaWdaeaapeGaaGOmaaaa aOWdaeaapeGaamyqa8aadaWgaaWcbaWdbiaaikdacaWGgbaapaqaba aaaOWdbmaalaaapaqaa8qacqaHepaDpaWaa0baaSqaa8qacaaIWaaa paqaa8qacaaIYaaaaaGcpaqaa8qacaaI0aaaaaGaayjkaiaawMcaaa aa@60DF@

    These 2 functions are written in terms of the damaged stress tensor: σ d = σ / ( 1 D ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamizaaWdaeqaaOWdbiabg2da9iab eo8aZjaac+cadaqadaWdaeaapeGaaGymaiabgkHiTiaadseaaiaawI cacaGLPaaaaaa@40C1@

    Where, D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiraaaa@36D5@ represents the isotropic damage.

  2. Plastic potential is expressed as:(3)
    f * =  J 2 + A S 3 I 1 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaahaaWcbeqaa8qacaGGQaaaaOGaeyypa0JaaiiOaiaa dQeapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaey4kaSYaaSaaa8 aabaWdbiaadgeapaWaaSbaaSqaa8qacaWGtbaapaqabaaakeaapeGa aG4maaaacaWGjbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWaaWbaaS qabeaapeGaaGOmaaaaaaa@42E7@
  3. Yield stress is rate dependent:
    • Table_ID ≠ 0, the yield stress is tabulated.
    • Table_ID = 0, it is analytic.
    (4)
    τ y =( τ 0 +R )g( ε ˙ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3damaaBaaaleaapeGaamyEaaWdaeqaaOWdbiabg2da9maa bmaapaqaa8qacqaHepaDpaWaaSbaaSqaa8qacaaIWaaapaqabaGcpe Gaey4kaSIaamOuaaGaayjkaiaawMcaaiaadEgadaqadaWdaeaapeGa fqyTdu2dayaacaaapeGaayjkaiaawMcaaaaa@4500@
    Where, R=Q( 1exp( β ε pl ) )+H ε pl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOuaiabg2da9iaadgfadaqadaWdaeaapeGaaGymaiabgkHiTiaa bwgacaqG4bGaaeiCamaabmaapaqaa8qacqGHsislcqaHYoGycqaH1o qzpaWaaSbaaSqaa8qacaWGWbGaamiBaaWdaeqaaaGcpeGaayjkaiaa wMcaaaGaayjkaiaawMcaaiabgUcaRiaadIeacqaH1oqzpaWaaSbaaS qaa8qacaWGWbGaamiBaaWdaeqaaaaa@4CB2@ .(5)
    g( ε ˙ )=1+C[ ln( ε ˙ ε ˙ ref )ln( ε ˙ ε ˙ max ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4zamaabmaapaqaa8qacuaH1oqzpaGbaiaaa8qacaGLOaGaayzk aaGaeyypa0JaaGymaiabgUcaRiaadoeadaWadaWdaeaapeGaamiBai aad6gadaqadaWdaeaapeWaaSaaa8aabaWdbiqbew7aL9aagaGaaaqa a8qacuaH1oqzpaGbaiaadaWgaaWcbaWdbiaadkhacaWGLbGaamOzaa WdaeqaaaaaaOWdbiaawIcacaGLPaaacqGHsislcaWGSbGaamOBamaa bmaapaqaa8qadaWcaaWdaeaapeGafqyTdu2dayaacaaabaWdbiqbew 7aL9aagaGaamaaBaaaleaapeGaamyBaiaadggacaWG4baapaqabaaa aaGcpeGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@55D5@
  4. Damage initiation and rupture are function of triaxiality σ * = σ m σ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaCaaaleqabaWdbiaacQcaaaGccqGH9aqpdaWcaaWd aeaapeGaeq4Wdm3damaaBaaaleaapeGaamyBaaWdaeqaaaGcbaWdbi qbeo8aZ9aagaqeaaaaaaa@3F1B@ with σ m = I 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamyBaaWdaeqaaOWdbiabg2da9maa laaapaqaa8qacaWGjbWdamaaBaaaleaapeGaaGymaaWdaeqaaaGcba Wdbiaaiodaaaaaaa@3D24@ and σ ¯ e q = 3 J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gafq4Wdm3dayaaraWaaSbaaSqaa8qacaWGLbGaamyCaaWdaeqaaOWd biabg2da9maakaaapaqaa8qacaaIZaGaamOsa8aadaWgaaWcbaWdbi aaikdaa8aabeaaa8qabeaaaaa@3E22@ .(6)
    D ˙ = n ε p l ε c ε f ε c n 1 ε ˙ p l ε f ε c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gabmira8aagaGaa8qacqGH9aqpcaWGUbWaaSaaa8aabaWdbiabew7a L9aadaWgaaWcbaWdbiaadchacaWGSbaapaqabaGcpeGaeyOeI0Iaeq yTdu2damaaBaaaleaapeGaam4yaaWdaeqaaaGcbaWdbiabew7aL9aa daWgaaWcbaWdbiaadAgaa8aabeaak8qacqGHsislcqaH1oqzpaWaaS baaSqaa8qacaWGJbaapaqabaaaaOWaaWbaaSqabeaapeGaamOBaiab gkHiTiaaigdaaaGcdaWcaaWdaeaapeGafqyTdu2dayaacaWaaSbaaS qaa8qacaWGWbGaamiBaaWdaeqaaaGcbaWdbiabew7aL9aadaWgaaWc baWdbiaadAgaa8aabeaak8qacqGHsislcqaH1oqzpaWaaSbaaSqaa8 qacaWGJbaapaqabaaaaaaa@55F3@
    (7)
    ε c = D 1 c + D 2 c exp D t r x σ * 1 + D J C l n ε ˙ ε ˙ r e f MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaBaaaleaapeGaam4yaaWdaeqaaOWdbiabg2da9maa dmaapaqaa8qacaWGebWdamaaBaaaleaapeGaaGymaiaadogaa8aabe aak8qacqGHRaWkcaWGebWdamaaBaaaleaapeGaaGOmaiaadogaa8aa beaak8qacaqGLbGaaeiEaiaabchadaqadaWdaeaapeGaamira8aada WgaaWcbaWdbiaadshacaWGYbGaamiEaaWdaeqaaOWdbiabeo8aZ9aa daahaaWcbeqaa8qacaGGQaaaaaGccaGLOaGaayzkaaaacaGLBbGaay zxaaWaaeWaa8aabaWdbiaaigdacqGHRaWkcaWGebWdamaaBaaaleaa peGaamOsaiaadoeaa8aabeaak8qacaWGSbGaamOBamaabmaapaqaa8 qadaWcaaWdaeaapeGafqyTdu2dayaacaaabaWdbiqbew7aL9aagaGa amaaBaaaleaapeGaamOCaiaadwgacaWGMbaapaqabaaaaaGcpeGaay jkaiaawMcaaaGaayjkaiaawMcaaaaa@5EC8@
    (8)
    ε f = D 1f + D 2f exp D trx σ * 1+ D JC ln ε ˙ ε ˙ ref MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaBaaaleaapeGaamOzaaWdaeqaaOWdbiabg2da9maa dmaapaqaa8qacaWGebWdamaaBaaaleaapeGaaGymaiaadAgaa8aabe aak8qacqGHRaWkcaWGebWdamaaBaaaleaapeGaaGOmaiaadAgaa8aa beaak8qacaqGLbGaaeiEaiaabchadaqadaWdaeaapeGaamira8aada WgaaWcbaWdbiaadshacaWGYbGaamiEaaWdaeqaaOWdbiabeo8aZ9aa daahaaWcbeqaa8qacaGGQaaaaaGccaGLOaGaayzkaaaacaGLBbGaay zxaaWaaeWaa8aabaWdbiaaigdacqGHRaWkcaWGebWdamaaBaaaleaa peGaamOsaiaadoeaa8aabeaak8qacaWGSbGaamOBamaabmaapaqaa8 qadaWcaaWdaeaapeGafqyTdu2dayaacaaabaWdbiqbew7aL9aagaGa amaaBaaaleaapeGaamOCaiaadwgacaWGMbaapaqabaaaaaGcpeGaay jkaiaawMcaaaGaayjkaiaawMcaaaaa@5ED1@