/MAT/LAW109

Block Format Keyword Elasto-plastic material with isotropic von Mises yield criterion with plastic strain rate and temperature depending nonlinear hardening.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW109/mat_ID/unit_ID
mat_title
ρ i                
E ν            
Cp η Tref T0    
tab_ID_h tab_ID_t Xscale_h Yscale_h     Ismooth
tab_ID_ η Xscale_ η              

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 modulus.

(Real)

 [ Pa ]
ν Poisson’s ratio.

(Real)
Ismooth Choice of yield function interpolation versus strain rate.
= 1 (Default)
Linear interpolation.
= 2
Logarithmic interpolation (base 10).
= 3
Logarithmic interpolation (base n).

(Integer)
Cp Specific heat.

(Real)

 [ J kg⋅K ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada WcaaqaaiaabQeaaeaacaqGRbGaae4zaiabgwSixlaabUeaaaaacaGL BbGaayzxaaaaaa@3DB3@

Tref Reference temperature.

Default = 293K (Real)

[ K ]
T0 Initial temperature.

Default = Tref (Real)

 [ K ]
η Taylor-Quinney coefficient (fraction of plastic work converted to heat). Value between 0.0 and 1.0.

(Real)
tab_ID_ η (Optional) Table identifier defining scale factor for η depending on strain rate, temperature, and plastic strain. Value between 0.0 and 1.0.

(Integer Id)
tab_ID_h Table identifier for yield stress depending on effective plastic strain and strain rate.

(Integer)
Xscale_ η Abscissa scale factor (strain rate) for tab_ID_ η .

Default = 1.0 (Real)

 [ 1 s ]
Xscale_h Abscissa scale factor (strain rate) for tab_ID_h.

Default = 1.0 (Real)

 [ 1 s ]
Yscale_h Scale factor for ordinate (stress) for tab_ID_h.

Default = 1.0 (Real)

 [ Pa ]
tab_ID_t Table identifier for quasi-static yield stress depending on effective plastic strain and temperature.

(Integer Id)

Example (Aluminum)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/2275
unit_Mg_mm_s
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW109/18/2275
Aluminium
#        Init. dens.         
              7.8E-9
#                  E                  Nu
             70000.0                  .3         
#                 CP                 Eta                Tref                Tini
              0.45E9                0.95               293.0               293.0        
#  Tab_Yld  Tab_Temp              Xscale              Yscale                                 Ismooth
        25        26                 1.0                 1.0                                       1
#  tab_eta          xcsale_eta
        34                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/TABLE/1/25
Yld Functions : plastic strain + strain rate dependency
#DIMENSION
         2
#   FCT_ID                             X                                                     Scale_y
         2                           0.0                                                        1.0
         2                      100000.0                                                        1.35
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/TABLE/1/26
 Yld Functions (quasistatic): plastic strain + temperature dependency
#DIMENSION
         2
#   FCT_ID                             X                                                     Scale_y
         2                         293.0                                                        1.00
         2                        1000.0                                                        0.70
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/TABLE/1/34
taylor-quinney coef = f(strain rate, temp)
#DIMENSION
         2
#   FCT_ID                             X                                                     Scale_y
        35                           239                                                         1.0
        35                          1000                                                         0.9
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/35
taylor-quinney factor = f(strain.rate)
#                  X                   Y
               0.000                   0                                                            
               0.002                   0                                                            
                0.04                   1                                                            
           1000000.0                   1                                                            
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/2
ALU Stress-strain
          0.00000            310.0
          9.3E-04            330.8
          1.1E-03            334.5
          2.1E-03            339.9
          2.6E-03            340.9
          3.3E-03            342.3
          6.1E-03            344.7
          7.8E-03            346.0
          9.1E-03            347.1
          1.0E-02            348.7
          1.2E-02            350.7 
          1.4E-02            352.6 
          1.6E-02            354.0 
          1.8E-02            356.5 
          2.0E-02            358.7 
          3.0E-02            369.0 
          3.5E-02            373.5 
          1.0                410.0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

  1. Yield criterion using isotropic von Mises equivalent stress:(1)
    ϕ = σ V M − σ y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHvpGzcq GH9aqpcqaHdpWCdaWgaaWcbaGaamOvaiaad2eaaeqaaOGaeyOeI0Ia eq4Wdm3aaSbaaSqaaiaadMhaaeqaaaaa@41B3@
  2. Yield stress hardening defined by tabulated input as:(2)
    σ y = f h ε p , ε ˙ p f t ε p , T f t ε p , T r e f MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaamyEaaqabaGccqGH9aqpciGGMbWaaSbaaSqaaiaadIga aeqaaOWaaeWaaeaacqaH1oqzdaWgaaWcbaGaamiCaaqabaGccaGGSa GafqyTduMbaiaadaWgaaWcbaGaamiCaaqabaaakiaawIcacaGLPaaa daWcaaqaaiGacAgadaWgaaWcbaGaamiDaaqabaGcdaqadaqaaiabew 7aLnaaBaaaleaacaWGWbaabeaakiaacYcacaWGubaacaGLOaGaayzk aaaabaGaciOzamaaBaaaleaacaWG0baabeaakmaabmaabaGaeqyTdu 2aaSbaaSqaaiaadchaaeqaaOGaaiilaiaadsfadaWgaaWcbaGaamOC aiaadwgacaWGMbaabeaaaOGaayjkaiaawMcaaaaaaaa@5867@
    Where,
    f h ε p , ε ˙ p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaaS baaSqaaiaadIgaaeqaaOWaaeWaaeaacqaH1oqzdaWgaaWcbaGaamiC aaqabaGccaGGSaGafqyTduMbaiaadaWgaaWcbaGaamiCaaqabaaaki aawIcacaGLPaaaaaa@415A@
    Function table of yield stresses depending on plastic strain and plastic strain rate.
    f t ε p , T MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaaS baaSqaaiaadshaaeqaaOWaaeWaaeaacqaH1oqzdaWgaaWcbaGaamiC aaqabaGccaGGSaGaamivaaGaayjkaiaawMcaaaaa@3F64@
    Table ID of quasi-static yield function depending on plastic strain and temperature.
    T r e f MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaS baaSqaaiaadkhacaWGLbGaamOzaaqabaaaaa@3B36@
    Reference temperature. Corresponds to conditions during experimental tests.
  3. In adiabatic conditions, the temperature is updated using:(3)
    T = T 0 + η ⋅ f η ε p , ε ˙ p , T ρ C p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubGaey ypa0JaamivamaaBaaaleaacaaIWaaabeaakiabgUcaRmaalaaabaGa eq4TdGMaeyyXICTaciOzamaaBaaaleaacqaH3oaAaeqaaOWaaeWaae aacqaH1oqzdaWgaaWcbaGaamiCaaqabaGccaGGSaGafqyTduMbaiaa daWgaaWcbaGaamiCaaqabaGccaGGSaGaamivaaGaayjkaiaawMcaaa qaaiabeg8aYjaadoeacaWGWbaaaaaa@4FAF@

    Where, η is the constant Taylor-Quinney coefficient which may be modified by introducing scalar factor defined by function f η ε p , ε ˙ p , T MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaaS baaSqaaiabeE7aObqabaGcdaqadaqaaiabew7aLnaaBaaaleaacaWG WbaabeaakiaacYcacuaH1oqzgaGaamaaBaaaleaacaWGWbaabeaaki aacYcacaWGubaacaGLOaGaayzkaaaaaa@43A2@ .

    Otherwise, if /HEAT/MAT is present in the model, the temperature is imposed on all elements and cannot be updated using Equation 3.

    Function f η ε p , ε ˙ p , T MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaaS baaSqaaiabeE7aObqabaGcdaqadaqaaiabew7aLnaaBaaaleaacaWG WbaabeaakiaacYcacuaH1oqzgaGaamaaBaaaleaacaWGWbaabeaaki aacYcacaWGubaacaGLOaGaayzkaaaaaa@43A2@ may be one dimensional, two-dimensional, or three-dimensional, but the first abscissa is always strain rate and the second one may be only the temperature.