/MAT/LAW48 (ZHAO)

Block Format Keyword This law describes the Zhao material law used to model an elasto-plastic strain rate dependent materials. The law is applicable only for solids and shells.

The global plasticity option for shells (N=0 in shell property keyword) is not available in the actual version.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW48/mat_ID/unit_ID or /MAT/ZHAO/mat_ID/unit_ID
mat_title
ρ i
E ν
A B n Chard σ max
C D m EI k
ε ˙ 0 Fcut
ε p m a x ε t 1 ε t 2

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 ]
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 Plasticity maximum stress.

Default = 1030 (Real)

[ Pa ]
C Relative strain rate coefficient.

Default = 1.0 (Real)

[ Pa ]
D Strain rate plasticity factor.

Default = 0.0 (Real)

m Relative strain rate exponent.

Default = 1.0 (Real)

EI Strain rate coefficient.

Default = 0.0 (Real)

[ Pa ]
k Strain rate exponent.

Default = 1.0 (Real)

ε ˙ 0 Reference strain rate.

(Real)

[ 1 s ]
Fcut Cutoff frequency for strain rate filtering.

Default = 0.0 (Real)

[Hz]
ε p m a x Failure plastic strain.

Default = 1030 (Real)

ε t 1 Tensile failure strain 1.

Default = 1030 (Real)

ε t 2 Tensile failure strain 2.

Default = 1030 (Real)

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/LAW48/1/1
metal
#              RHO_I 
                .008   
#                  E                  nu
              200000                  .3
#                  A                   B                   n               Chard             sig_max
                 145                 550                 .42                   1                   0
#                  C                   D                   m                  E1                   k
                  35                  47                  .3                 185                  .3
#         eps_rate_0                Fcut
                 .05                   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 stress-strain function is based on the formula published by Zhao:
    σ=( A+B ε p n )+( CD ε p m )ln ε ˙ ε ˙ 0 + E 1 ε ˙ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0ZaaeWaaeaacaWGbbGaey4kaSIaamOqaiabew7aLnaaBaaaleaa caWGWbaabeaakmaaCaaaleqabaGaamOBaaaaaOGaayjkaiaawMcaai abgUcaRmaabmaabaGaam4qaiabgkHiTiaadseacqaH1oqzdaWgaaWc baGaamiCaaqabaGcdaahaaWcbeqaaiaad2gaaaaakiaawIcacaGLPa aacqGHflY1ciGGSbGaaiOBamaalaaabaGafqyTduMbaiaaaeaacuaH 1oqzgaGaamaaBaaaleaacaaIWaaabeaaaaGccqGHRaWkcaWGfbWaaS baaSqaaiaaigdaaeqaaOGafqyTduMbaiaadaahaaWcbeqaaiaadUga aaaaaa@5790@
    Where,
    ε p
    Plastic strain
    ε ˙
    Strain rate
  2. Except for the strain rate formulation, the plasticity curve is strictly identical to a Johnson-Cook model:
    Figure 1.

    mat_law48

    However, compared to Johnson-Cook, the Zhao law allows a better approximation of a nonlinear strain rate dependent behavior.

  3. Yield stress should be strictly positive.
  4. The hardening exponent n must be less than 1.
    Figure 2.

    clip0079
  5. The iso-kinematic hardening parameter is defined as:
    • If Chard = 0, hardening is a full isotropic model
    • If Chard = 1, hardening uses the kinematic Prager-Ziegler model
    • If 0 < Chard < 1, hardening is interpolated between the two models
  6. If ε ˙ ε ˙ 0 , the term ( C D ε p m ) ln ε ˙ ε ˙ 0 = 0 , and Equation 1 becomes:
    σ = ( A + B ε p n ) + E 1 ε ˙ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0ZaaeWaaeaacaWGbbGaey4kaSIaamOqaiabew7aLnaaBaaaleaa caWGWbaabeaakmaaCaaaleqabaGaamOBaaaaaOGaayjkaiaawMcaai abgUcaRiaadweadaWgaaWcbaGaaGymaaqabaGccuaH1oqzgaGaamaa CaaaleqabaGaam4Aaaaaaaa@461E@
  7. The strain rate filtering is used to smooth strain rate. It is only available for shell and solid elements.
  8. When ε p reaches ε max in one integration point, then based on the element type:
    • 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.
  9. If ε 1 > ε t 1 ( ε 1 is the largest principal strain), the stress is reduced as:
    σ n + 1 = σ n ( ε t 2 ε 1 ε t 2 ε t 1 )
  10. If ε 1 > ε t 2 , the stress is reduced to 0 (but the element is not deleted).