Zhao Plasticity Model (LAW48)

The elasto-plastic behavior of material with strain rate dependence is given by Zhao formula: 1 2
図 1.
σ = ( A + B ε p n ) + ( C − D ε p m )   .   In   ε ¨ ε ˙ 0 + E ε ˙ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCcqGH9aqpdaqadaqaaiaadgeacqGHRaWkcaWGcbGaeqyT du2aa0baaSqaaiaadchaaeaacaWGUbaaaaGccaGLOaGaayzkaaGaey 4kaSYaaeWaaeaacaWGdbGaeyOeI0Iaamiraiabew7aLnaaDaaaleaa caWGWbaabaGaamyBaaaaaOGaayjkaiaawMcaaiaaykW7caGGUaGaaG PaVlGacMeacaGGUbGaaGPaVpaalaaabaGafqyTduMbamaaaeaacuaH 1oqzgaGaamaaBaaaleaacaaIWaaabeaaaaGccqGHRaWkcaWGfbGafq yTduMbaiaadaahaaWcbeqaaiaadUgaaaaaaa@5BFA@
Where,
ε p
Plastic strain
ε ˙
Strain rate
A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaWGPb aaaa@373E@
Yield stress
B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaWGPb aaaa@373E@
Hardening parameter
n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaWGPb aaaa@373E@
Hardening exponent
C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaWGPb aaaa@373E@
Relative strain rate coefficient
D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaWGPb aaaa@373E@
Strain rate plasticity factor
m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaWGPb aaaa@373E@
Relative strain rate exponent
E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaWGPb aaaa@373E@
Strain rate coefficient
k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaWGPb aaaa@373E@
Strain rate exponent

In the case of material without strain rate effect, the hardening curve given by 式 1 is identical to those of Johnson-Cook. However, Zhao law allows a better approximation of strain rate dependent materials by introducing a nonlinear dependency.

As described for Johnson-Cook law, a strain rate filtering can be introduced to smooth the results. The plastic flow with isotropic or kinematic hardening can be modeled as described in Cowper-Symonds Plasticity Model (LAW44). The material failure happens when the plastic strain reaches a maximum value as in Johnson-Cook model. However, two tensile strain limits are defined to reduce stress when rupture starts:
図 2.
σ n + 1 = σ n     ( ε t 2 − ε 1 ε t 2 − ε t 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaeyyp a0Jaeq4Wdm3aaSbaaSqaaiaad6gaaeqaaOGaaGjcVlaaykW7daqada qaamaalaaabaGaeqyTdu2aaSbaaSqaaiaadshacaaIYaaabeaakiab gkHiTiabew7aLnaaBaaaleaacaaIXaaabeaaaOqaaiabew7aLnaaBa aaleaacaWG0bGaaGOmaaqabaGccqGHsislcqaH1oqzdaWgbaWcbaGa amiDaiaaigdaaeqaaaaaaOGaayjkaiaawMcaaaaa@552F@
Where,
ε 1
Largest principal strain
ε t 1 and ε t 2
Rupture strain limits

If ε 1 > ε f 1 , the stress is reduced by 式 2. When ε 1 > ε t 2 the stress is reduced to zero.

1 Zhao Han, A Constitutive Model for Metals over a Large Range of Strain Rates, Materials Science & Engineering, A230, 1997.
2 Zhao Han and Gerard Gary, The Testing and Behavior Modelling of Sheet Metals at Strain Rates from 10.e-4 to 10e+4 s-1, Materials Science & Engineering" A207, 1996.