Johnson-Cook Plasticity Model (LAW2)

In this law the material behaves as linear elastic when the equivalent stress is lower than the yield stress.

For higher value of stress, the material behavior is plastic. This law is valid for brick, shell, truss and beam elements. The relation between describing stress during plastic deformation is given in a closed form:(1) σ = ( a + b ε p n ) ( 1 + c 1 n ε ˙ ε ˙ 0 ) ( 1 T * m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCcqGH9aqpdaqadaqaaiaadggacqGHRaWkcaWGIbGaeqyT du2aa0baaSqaaiaadchaaeaacaWGUbaaaaGccaGLOaGaayzkaaWaae WaaeaacaaIXaGaey4kaSIaam4yaiaaigdaciGGUbWaaSaaaeaacuaH 1oqzgaGaaaqaaiqbew7aLzaacaWaaSbaaSqaaiaaicdaaeqaaaaaaO GaayjkaiaawMcaamaabmaabaGaaGymaiabgkHiTiaadsfadaahaaWc beqaamaaBaaameaacaGGQaaabeaaliaad2gaaaaakiaawIcacaGLPa aaaaa@53D4@
Where,
σ
Flow stress (Elastic + Plastic Components)
ε p
Plastic strain (True strain)
a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@
Yield stress
b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@
Hardening modulus
n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@
Hardening exponent
c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@
Strain rate coefficient
ε ˙
Strain rate
ε ˙ 0
Reference strain rate
m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@
Temperature exponent
T * = T298 T melt 298 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aaciGGubWaaWbaaSqabeaaciGGQaaaaOGaeyypa0ZaaSaaaeaaciGG ubGaeyOeI0IaaGOmaiaaiMdacaaI4aaabaGaciivamaaBaaaleaaca WGTbGaamyzaiaadYgacaWG0baabeaakiabgkHiTiaaikdacaaI5aGa aGioaaaaaaa@47BA@
Tmelt
Melting temperature in Kelvin degrees. The adiabatic conditions are assumed for temperature computation:(2) T = T i + E int ρ C ρ ( V o l u m e ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aaciGGubGaeyypa0JaciivamaaBaaaleaacaWGPbaabeaakiabgUca RmaalaaabaGaamyramaaBaaaleaaciGGPbGaaiOBaiaacshaaeqaaa GcbaGaeqyWdiNaam4qamaaBaaaleaacqaHbpGCaeqaaOWaaeWaaeaa caWGwbGaam4BaiaadYgacaWG1bGaamyBaiaadwgaaiaawIcacaGLPa aaaaaaaa@4D1D@
Where,
ρ C p
Specific heat per unit of volume
T i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aaciGGubWaaSbaaSqaaiaadMgaaeqaaaaa@3ACB@
Initial temperature (in degrees Kelvin)
E int MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaWGfb WaaSbaaSqaaiGacMgacaGGUbGaaiiDaaqabaaaaa@3A1F@
Internal energy
Two optional additional inputs are:
σ max 0
Maximum flow stress
ε max
Plastic strain at rupture
Figure 1 shows a typical stress-strain curve in the plastic region. When the maximum stress is reached during computation, the stress remains constant and material undergoes deformation until the maximum plastic strain. Element rupture occurs if the plastic strain is larger than ε max . If the element is a shell, the ruptured element is deleted. If the element is a solid element, the ruptured element has its deviatoric stress tensor permanently set to zero, but the element is not deleted. Therefore, the material rupture is modeled without any damage effect.


Figure 1. Stress - Plastic Strain Curve

Chard in this material law is same like in /MAT/LAW44. For more details on Chard, refer to Cowper-Symonds Plasticity Model (LAW44).

Strain Rate Definition

Regarding to the plastification method used, the strain rate expression is different. If the progressive plastification method is used (i.e. integration points through the thickness for thin-walled structured), the strain rate is:(3) dε dt =max( d ε x dt , d ε y dt ,2 d dt ( ε xy ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaWcaaqaaiaadsgacqaH1oqzaeaacaWGKbGaamiDaaaacqGH9aqp ciGGTbGaaiyyaiaacIhadaqadaqaamaalaaabaGaamizaiabew7aLn aaBaaaleaacaWG4baabeaaaOqaaiaadsgacaWG0baaaiaacYcadaWc aaqaaiaadsgacqaH1oqzdaWgaaWcbaGaamyEaaqabaaakeaacaWGKb GaamiDaaaacaGGSaGaaGOmamaalaaabaGaamizaaqaaiaadsgacaWG 0baaamaabmaabaGaeqyTdu2aaSbaaSqaaiaadIhacaWG5baabeaaaO GaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@587F@ (4) ε = xy 1 2 γ xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaH1oqzcaWLa8UaaGzaVpaaBeaaleaacaWG4bGaamyEaaqabaGc cqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiabeo7aNjaaygW7da WgbaWcbaGaamiEaiaadMhaaeqaaaaa@47A8@
With global plastification method:(5) d ε d t = ( d E i / d t σ V M ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaWcaaqaaiaadsgacqaH1oqzaeaacaWGKbGaamiDaaaacqGH9aqp daqadaqaamaalaaabaGaamizaiaadweadaWgaaWcbaGaamyAaaqaba GccaGGVaGaamizaiaadshaaeaacqaHdpWCdaWgaaWcbaGaamOvaiaa d2eaaeqaaaaaaOGaayjkaiaawMcaaaaa@490A@

Where, E i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaWGfb WaaSbaaSqaaiaadMgaaeqaaaaa@3834@ is the internal energy.

For solid elements, the maximum value of the strain rate components is used:(6) ε ˙ =max( ε ˙ x , ε ˙ y , ε ˙ z ,2 ε ˙ xy ,2 ε ˙ yz ,2 ε ˙ xz ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacuaH1oqzgaGaaiabg2da9iGac2gacaGGHbGaaiiEamaabmaabaGa fqyTduMbaiaadaWgaaWcbaGaamiEaaqabaGccaGGSaGaaGzaVlaayk W7cuaH1oqzgaGaamaaBaaaleaacaWG5baabeaakiaacYcacuaH1oqz gaGaamaaBaaaleaacaWG6baabeaakiaacYcacaaIYaGafqyTduMbai aadaWgaaWcbaGaamiEaiaadMhaaeqaaOGaaiilaiaaikdacuaH1oqz gaGaamaaBaaaleaacaWG5bGaamOEaaqabaGccaGGSaGaaGOmaiqbew 7aLzaacaWaaSbaaSqaaiaadIhacaWG6baabeaaaOGaayjkaiaawMca aaaa@5CF5@

Strain Rate Filtering

The strain rates exhibit very high frequency vibrations which are not physical. The strain rate filtering option will enable to damp those oscillations and; therefore obtain more physical strain rate values.

If there is no strain rate filtering, the equivalent strain rate is the maximum value of the strain rate components:(7) ε ˙ e q = max ( ε ˙ x ε ˙ y ε ˙ z 2 ε ˙ x y 2 ε ˙ y z 2 ε ˙ X Z )
For thin-walled structures, the equivalent strain is computed by the following approach. If ε is the main component of strain tensor, the kinematic assumptions of thin-walled structures allows to decompose the in-plane strain into membrane and flexural deformations:(8) ε = K Z = ε m
Then, the expression of internal energy can by written as:(9) E i = t 2 t 2 σ ε d z = t 2 t 2 E ε 2 d z = t 2 t 2 E ( κ z + ε m ) 2 d z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGfbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0Zaa8qCaeaacqaH dpWCcqaH1oqzcaaMi8UaamizaiaadQhacqGH9aqpaSqaaiabgkHiTm aalaaabaGaamiDaaqaaiaaikdaaaaabaWaaSaaaeaacaWG0baabaGa aGOmaaaaa0Gaey4kIipakmaapehabaGaamyraiabew7aLnaaCaaale qabaGaaGOmaaaakiaayIW7caWGKbGaamOEaiabg2da9aWcbaGaeyOe I0YaaSaaaeaacaWG0baabaGaaGOmaaaaaeaadaWcaaqaaiaadshaae aacaaIYaaaaaqdcqGHRiI8aOWaa8qCaeaacaWGfbWaaeWaaeaacqaH 6oWAcaWG6bGaey4kaSIaeqyTdu2aaSbaaSqaaiaad2gaaeqaaaGcca GLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaamizaiaadQhaaSqa aiabgkHiTmaalaaabaGaamiDaaqaaiaaikdaaaaabaWaaSaaaeaaca WG0baabaGaaGOmaaaaa0Gaey4kIipaaaa@6BDB@
Therefore:(10) E i = t 2 t 2 E ( κ 2 z 2 + ε m 2 + 2 κ ε m z ) d z = E ( 1 3 κ 2 z 3 + ε m 2 z + κ ε m z 2 ) t 2 t 2
The expression can be simplified to:(11) E i =E[ 1 12 κ 2 t 3 + ε m 2 t ]=E ε eq 2 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGfbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0Jaamyramaadmaa baWaaSaaaeaacaaIXaaabaGaaGymaiaaikdaaaGaeqOUdS2aaWbaaS qabeaacaaIYaaaaOGaamiDamaaCaaaleqabaGaaG4maaaakiabgUca Riabew7aLnaaDaaaleaacaWGTbaabaGaaGOmaaaakiaadshaaiaawU facaGLDbaacqGH9aqpcaWGfbGaeqyTdu2aa0baaSqaaiaadwgacaWG XbaabaGaaGOmaaaakiaaygW7caaMi8UaamiDaaaa@5520@ (12) ε e q = 1 12 κ 2 t 2 + ε m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaH1oqzdaWgaaWcbaGaamyzaiaadghaaeqaaOGaeyypa0ZaaOaa aeaadaWcaaqaaiaaigdaaeaacaaIXaGaaGOmaaaacqaH6oWAdaahaa WcbeqaaiaaikdaaaGccaWG0bWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaeqyTdu2aa0baaSqaaiaad2gaaeaacaaIYaaaaaqabaaaaa@48E1@
The expression of the strain rate is derived from Equation 8:(13) ε ˙ = K ˙ Z + ε ˙ m
Admitting the assumption that the strain rate is proportional to the strain, i.e.:(14) ε ˙ m = α ε m (15) K ˙ = α K
Therefore:(16) ε ˙ = α ε
Referring to Equation 12, it can be seen that an equivalent strain rate can be defined using a similar expression to the equivalent strain:(17) ε ˙ e q = α ε e q (18) ε ˙ e q = 1 12 κ 2 t 2 + ε ˙ m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacuaH1oqzgaGaamaaBaaaleaacaWGLbGaamyCaaqabaGccqGH9aqp daGcaaqaamaalaaabaGaaGymaaqaaiaaigdacaaIYaaaaiabeQ7aRn aaCaaaleqabaGaaGOmaaaakiaadshadaahaaWcbeqaaiaaikdaaaGc cqGHRaWkcuaH1oqzgaGaamaaDaaaleaacaWGTbaabaGaaGOmaaaaae qaaaaa@48F3@
For solid elements, the strain rate is computed using the maximum element stretch:(19) ε ˙ e q = λ ˙ max
The strain rate at integration point, i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaWGPb aaaa@373E@ in /ANIM/TENS/EPSDOT/i ( 1 < i < n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIXaGaeyipaWJaamyAaiabgYda8iaad6gaaiaawIcacaGLPaaaaaa@3C23@ is calculated by:(20) ε ˙ i = ε ˙ m 1 2 ( ( 2 i 1 ) n 1 ) t ε ˙ b
Where,
ε ˙ m
Membrane strain rate /ANIM/TENS/EPSDOT/MEMB
ε ˙ b
Bending strain rate /ANIM/TENS/EPSDOT/BEND.
The strain rate in upper and lower layers is computed by:(21) ε ˙ u = ε m + 1 2 t ε ˙ b
/ANIM/TENS/EPSDOT/UPPER(22) ε ˙ 1 = ε ˙ m 1 2 t ε ˙ b

/ANIM/TENS/EPSDOT/LOWER

The strain rate is filtered by using:(23) ε ˙ f ( t ) = a ε ˙ ( t ) + ( 1 a ) ε ˙ f ( t 1 )
Where,
a=2 π  dt F c u t
dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaqGKb GaaeiDaaaa@382E@
Time interval
Fcut
Cutting frequency
ε ˙ f
Filtered strain rate

Strain Rate Filtering Example

An example of material characterization for a simple tensile test RD-E: 1100 Tensile Test is given in Radioss Example Guide. For the same example, a strain rate filtering allows to remove high frequency vibrations and obtain smoothed the results. This is shown in Figure 2 and Figure 3 where the cut frequency Fcut = 10 KHz is used.


Figure 2. Force Comparison


Figure 3. First Principal Strain Rate Comparison (max = 10%)