/MAT/LAW106 (JCOOK_ALM)

Block Format Keyword This law represents an isotropic elasto-plastic material using the Johnson-Cook material model. This model expresses material stress as a function of strain and temperature.

This law is not compatible with an EOS. The dependence between pressure and volumetric strain is linear. A built-in failure criterion, based on the maximum plastic strain is available. This material law is compatible with solid elements only.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW106/mat_ID/unit_ID or /MAT/JCOOK_ALM/mat_ID/unit_ID
mat_title
ρ i ρ 0            
E ν fct_ID1 fct_ID2 fct_ID3      
a b n ε p m a x σ max
Pmin   Nmax Tol    
        m Tmelt Tmax
ρ 0 C p     Tr    

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 ]
ρ 0 Reference density used in EOS (equation of state).

Default = ρ 0 = ρ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCda WgaaWcbaGaaGimaaqabaGccqGH9aqpcqaHbpGCdaWgaaWcbaGaamyA aaqabaaaaa@3CEF@ (Real)

[ kg m 3 ]
E If fct_ID1 = 0: Young's modulus.

If fct_ID1 ≠ 0: Ordinate scale factor of fct_ID1 and fct_ID2.

(Real)

[ Pa ]
ν If fct_ID3 = 0: Poisson's ratio.

If fct_ID3 ≠ 0: Ordinate scale factor of fct_ID3.

(Real)

 
fct_ID1 Function identifier defining Young’s modulus versus temperature when heating.

(Integer)

 
fct_ID2 Function identifier defining Young’s modulus versus temperature when cooling.

(Integer)

 
fct_ID3 Function identifier defining Poisson’s ratio versus temperature.

(Integer)

 
a Yield stress.

(Real)

[ Pa ]
b Plastic hardening parameter.

(Real)

[ Pa ]
n Plastic hardening exponent.

Default = 1 (Real)

 
ε p m a x Failure plastic strain.

Default = 1030 (Real)

 
σ max Maximum stress.

Default = 1030 (Real)

[ Pa ]
Pmin Pressure cutoff (< 0).

Default = -1030 (Real)

[ Pa ]
Nmax Maximum number of iterations to compute plastic strains.

Default = 1 (Integer)

 
Tol Tolerance.

Default = 10-7 (Real)

 
m Temperature exponent.

Default = 1.0 (Real)

 
Tmelt Melting temperature.
= 0
No temperature effect.

Default = 1030 (Real)

[ K ]
Tmax For T > Tmax: m = 1 is used.

Default = 1030 (Real)

[ K ]
ρ 0 C p Specific heat per unit volume.

(Real)

[ J m 3 K ]
Tr Reference temperature.

Default = 300K (Real)

[ K ]

Example (Metal)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW106/1/1
Metal
#              RHO_I
                8E-9                   0
#                  E                  Nu   fct_ID1   fct_ID2   fct_ID3
              200000                 0.3         4         5         6
#                  a                   b                   n             EPS_max            SIG_max0
                 400                 500                  .5                   0                   0
#               Pmin                NMAX                 TOL     
                   0                   0                   0
#                                                          m              T_melt               T_max
                                                           3                2500                3000
#              RhoCP                                                        Tref
                 3.5                                                         298
/HEAT/MAT/1
#                 T0             RHO0_CP                  AS                  BS     IFORM
                 298                 3.5                  20                   0         1
#                 T1                  AL                  BL               EFRAC
                2500                  20                   0                  .9
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/4
Young modulus factor versus temperature during heating
#                  X                   Y
                   0                   1
                 300                   1
                1500                  .1
                2000                  .1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/5
Young modulus factor versus temperature during cooling
#                  X                   Y
                   0                   1
                 300                   1
                1500                  .1
                2000                  .1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/6
Poisson's Ratio factor versus temperature
#                  X                   Y
                   0                   1
                 300                   1
                1500                 1.5
                2000                 1.5
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Comments

  1. In this model, the material behavior is elastic-plastic and the yield stress is calculated as:(1)
    σ = ( a + b ε p n ) ( 1 ( T ) m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCcq GH9aqpdaqadaqaaiaadggacqGHRaWkcaWGIbGaeqyTdu2aaSbaaSqa aiaadchaaeqaaOWaaWbaaSqabeaacaWGUbaaaaGccaGLOaGaayzkaa WaaeWaaeaacaaIXaGaeyOeI0IaaiikaiaadsfadaahaaWcbeqaaiab gEHiQaaakiaacMcadaahaaWcbeqaaiaad2gaaaaakiaawIcacaGLPa aaaaa@490D@
    Where,(2)
    T * = T - T r T m e l t - T r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaCa aaleqabaGaamOkaaaakiaad2dadaWcaaqaaiaadsfacaWGTaGaamiv amaaBaaaleaacaWGYbaabeaaaOqaaiaadsfadaWgaaWcbaGaamyBai aadwgacaWGSbGaamiDaaqabaGccaWGTaGaamivamaaBaaaleaacaWG Ybaabeaaaaaaaa@439B@
    Where,
    ε p
    Equivalent plastic strain
    T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaaaa@36CF@
    Temperature
    T r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGYbaabeaaaaa@37F2@
    Reference temperature
    T m e l t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGTbGaamyzaiaadYgacaWG0baabeaaaaa@3AC1@
    Melting temperature

    The material behaves as a linear-elastic material when the equivalent stress is lower than the yield stress.

    When /HEAT/MAT (with Iform =1) references this material model, the values of Tr and Tmelt defined in this card will be overwritten by the corresponding T0 and Tmelt defined in /HEAT/MAT.

    When the temperature is not initialized using /HEAT/MAT or /INITEMP, the reference temperature (Tr) is also the initial temperature.

  2. The plastic yield stress should always be greater than zero. To model pure elastic behavior, the plastic yield stres,s a can be set to 1030.
  3. When ε p reaches the value of ε p m a x (for tension, compression or shear), in one integration point, then the deviatoric stress of the corresponding integration point is permanently set to 0; however, the solid element is not deleted.
  4. The plastic hardening exponent must be n 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgs MiJkaaigdaaaa@395A@ .
  5. The hydrostatic pressure is linearly proportional to volumetric strain:(3)
    P = K μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0Jaam4saiabeY7aTbaa@3ABF@
    Where,
    K = E 3 ( 1 2 v ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4saiabg2da9maalaaabaGaamyraaqaaiaaiodadaqadaqaaiaa igdacqGHsislcaaIYaGaamODaaGaayjkaiaawMcaaaaaaaa@4101@
    Bulk modulus
    μ = ρ ρ 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8akY=xipgYlh9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaeqiVd0Maeyypa0ZaaSaaaeaacqaHbpGCaeaacqaHbpGCdaWgaaWc baGaaGimaaqabaaaaOGaeyOeI0IaaGymaaaa@42BA@
    Volumetric strain
  6. This material can be used with the material options /HEAT/MAT and /VISC.