/MAT/LAW44 (COWPER)

Block Format Keyword The Cowper-Symonds law models an elasto-plastic material. The basic principle is the same as the standard Johnson-Cook model; the only difference between the two laws lies in the expression for strain rate effect on flow stress.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW44/mat_ID/unit_ID or /MAT/COWPER/mat_ID/unit_ID
mat_title
ρ i                
E ν            
a b n Chard σ max 0
c p ICC Fsmooth Fcut   VP
ε p m a x εt1 εt2        
Optional line for defining a yield stress function
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
fct_IDy   Fscaley            

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 ]
E Young's modulus.

(Real)

[ Pa ]
ν Poisson's ratio.

(Real)

 
a Plasticity yield stress.

(Real)

[ Pa ]
b Plasticity hardening parameter.

(Real)

[ Pa ]
n Plasticity hardening exponent.

Default = 1.0 (Real)

 
Chard Plasticity Iso-kinematic hardening factor.
= 0
Hardening is full isotropic model.
= 1
Hardening uses the kinematic Prager-Ziegler model.
= between 0 and 1
Hardening is interpolated between the two models.

Default = 0.0 (Real)

 
σ max 0 Plasticity maximum stress.

Default = 1020 (Real)

[ Pa ]
c Strain rate coefficient.
= 0 (Default)
No strain rate effect.

(Real)

[ 1 s ]
p Strain rate exponent.

Default = 1.0 (Real)

 
ICC Strain rate computation flag. 6
= 0 (Default)
Set to 1.
= 1
Strain rate effect on σ max .
= 2
No strain rate effect on σ max .

(Integer)

 
Fsmooth Smooth strain rate option flag.
= 0 (Default)
No strain rate smoothing.
= 1
Strain rate smoothing active.

(Integer)

 
Fcut Cutoff frequency for strain rate filtering.

Default = 1030 (Real)

[Hz]
VP Formulation for rate effects.
= 0
Set to 2.
= 1
Plastic strain rate.
= 2 (Default)
Total strain rate.
= 3
Deviatoric strain rate.

(Integer)

 
ε p m a x Failure plastic strain.

Default = 1020 (Real)

 
ε t 1 Tensile failure strain 1.

Default = 1020 (Real)

 
ε t 2 Tensile failure strain 2.

Default = 2x1020 (Real)

 
fct_IDy Yield stress function identifier.

(Integer)

 
Fscaley Scale factor for ordinate (stress) in fct_IDy

Default = 1.0 (Real)

[ Pa ]

Example (Metal)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                   g                  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/COWPER/1/1
metal
#              RHO_I
               .0078                   
#                  E                  nu
               20500                  .3
#                  a                   b                   n              C_hard          SIGMA_max0
                  50                 100                  .5                   1                  90
#                  c                   p       ICC   Fsmooth               F_cut
                 100                   5         1         0                   0
#            EPS_max              EPS_t1              EPS_t2
                   0                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

  1. The yield stress can be defined by the three stress coefficients (a, b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36DD@ , and n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36DD@ ), a function fct_IDy, or a combination of both. The stress is then scaled by the Cowper-Symonds strain rate coefficient.
    • If fct_IDy is defined (> 0), a=0 and VP=1:(1)
      σ = f c t _ I D y * F s c a l e y + ( a + b ε p n ) ( ε ˙ c ) 1 p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCcq GH9aqpcaWGMbGaam4yaiaadshacaGGFbGaamysaiaadseadaWgaaWc baGaamyEaaqabaGccaGGQaaeaaaaaaaaa8qacaWGgbGaam4Caiaado gacaWGHbGaamiBaiaadwgapaWaaSbaaSqaa8qacaWG5baapaqabaGc cqGHRaWkdaqadaqaaiaadggacqGHRaWkcaWGIbGaeqyTdu2aaSbaaS qaaiaadchaaeqaaOWaaWbaaSqabeaacaWGUbaaaaGccaGLOaGaayzk aaWaaeWaaeaadaWcaaqaaiqbew7aLzaacaaabaGaam4yaaaaaiaawI cacaGLPaaadaahaaWcbeqaamaalaaabaGaaGymaaqaaiaadchaaaaa aaaa@5671@
    • If fct_IDy is defined (> 0) and a > 0:(2)
      σ = f c t _ I D y * F s c a l e y * ( 1 + ( ε ˙ c ) 1 p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCcq GH9aqpcaWGMbGaam4yaiaadshacaGGFbGaamysaiaadseadaWgaaWc baGaamyEaaqabaGccaGGQaaeaaaaaaaaa8qacaWGgbGaam4Caiaado gacaWGHbGaamiBaiaadwgapaWaaSbaaSqaa8qacaWG5baapaqabaGc caGGQaWaaeWaaeaacaaIXaGaey4kaSYaaeWaaeaadaWcaaqaaiqbew 7aLzaacaaabaGaam4yaaaaaiaawIcacaGLPaaadaahaaWcbeqaamaa laaabaGaaGymaaqaaiaadchaaaaaaaGccaGLOaGaayzkaaaaaa@5139@
    • If fct_IDy is not defined (= 0):(3)
      σ = ( a + b ε p n ) ( 1 + ( ε ˙ c ) 1 p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCcq GH9aqpdaqadaqaaiaadggacqGHRaWkcaWGIbGaeqyTdu2aaSbaaSqa aiaadchaaeqaaOWaaWbaaSqabeaacaWGUbaaaaGccaGLOaGaayzkaa WaaeWaaeaacaaIXaGaey4kaSYaaeWaaeaadaWcaaqaaiqbew7aLzaa caaabaGaam4yaaaaaiaawIcacaGLPaaadaahaaWcbeqaamaalaaaba GaaGymaaqaaiaadchaaaaaaaGccaGLOaGaayzkaaaaaa@4AA9@
    Where,
    ε p
    Plastic strain.
    ε ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH1oqzga Gaaaaa@380E@
    Plastic strain rate for VP =1.
    Total strain rate for VP =2.
    Deviatoric strain rate for VP =3.
  2. The law is compatible with truss, beam, shell, and solid elements.
  3. Yield stress should be strictly positive.
  4. The hardening exponent n must be less than 1.

    clip0050
    Figure 1.
  5. The strain rate filtering is used to smooth strain rates, with the following:
    • If VP = 1, the strain-rate filtering is set by default and the cutoff frequency is automatically computed by Radioss according to time step value. Fcut and Fsmooth are ignored.
    • If VP = 2 or 3, and:
      • Fsmooth = 0 + Fcut = 0.0, the strain-rate filtering is turned off;
      • Fsmooth = 1 + Fcut = 0.0, the strain-rate filtering uses a cutoff frequency which is automatically computed by Radioss according to time step value (as for VP = 1);
      • Fcut0, Fsmooth is automatically set to 1 and the strain-rate filtering uses the cutoff frequency provided by the user.
  6. ICC is a flag of the strain rate effect on material maximum stress σ max :

    law_plaszeril
    σ = σ y ( 1 + ( ε ˙ c ) 1 / p ) σ = σ y ( 1 + ( ε ˙ c ) 1 / p )
    σ max = σ max 0 ( 1 + ( ε ˙ c ) 1 / p ) σ max = σ max 0
    Figure 2.
  7. When ε p reaches ε p m a x in one integration point, then based on the element type:
    • Truss and Beam elements: The element is deleted
    • Shell elements: The corresponding shell element is deleted
    • Solid elements: The deviatoric stress of the corresponding integral point is permanently set to 0; however, the solid element is not deleted
  8. If ε 1 > ε t 1 ( ε 1 is the largest principal strain), the stress is reduced as:(4)
    σ n + 1 = σ n ( ε t 2 ε 1 ε t 2 ε t 1 )
  9. If ε 1 > ε t 2 , the stress is reduced to 0 (but the element is not deleted).