Ductile Damage Model for Porous Materials (LAW52)

The Gurson constitutive law 1 models progressive microrupture through void nucleation and growth. It is dedicated to high strain rate elasto-viscoplastic porous metals. A coupled damage mechanical model for strain rate dependent voided material is used. The material undergoes several phases in the damage process as described in 図 1.


図 1. Damage Process for Visco-elastic-plastic Voided Materials
The constitutive law takes into account the void growth, nucleation and coalescence under dynamic loading. The evolution of the damage is represented by the void volume fraction, defined by:(1)
f = V a V m V a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGMbGaeyypa0ZaaSaaaeaacaWGwbWaaSbaaSqaaiaadggaaeqa aOGaeyOeI0IaamOvamaaBaaaleaacaWGTbaabeaaaOqaaiaadAfada WgaaWcbaGaamyyaaqabaaaaaaa@41AC@
Where,
V a , V m
Respectively, are the elementary apparent volume of the material and the corresponding elementary volume of the matrix.
The rate of increase of the void volume fraction is given by:(2)
f = f g + f n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGMbGaeyypa0JaamOzamaaBaaaleaacaWGNbaabeaakiabgUca RiaadAgadaWgaaWcbaGaamOBaaqabaaaaa@3FC1@
The growth rate of voids is calculated by:(3)
f g = ( 1 f ) T r a c e [ D p ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGMbWaaSbaaSqaaiaadEgaaeqaaOGaeyypa0ZaaeWaaeaacaaI XaGaeyOeI0IaamOzaaGaayjkaiaawMcaaiaadsfacaWGYbGaamyyai aadogacaWGLbWaamWaaeaacaWGebWaaWbaaSqabeaacaWGWbaaaaGc caGLBbGaayzxaaaaaa@4875@
Where, T r a c e [ D p ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiaadk hacaWGHbGaam4yaiaadwgadaWadaqaaiaadseadaahaaWcbeqaaiaa dchaaaaakiaawUfacaGLDbaaaaa@3E65@ is the trace of the macroscopic plastic strain rate tensor. The nucleation rate of voids is given by:(4)
f ˙ n = f N S N 2 π e 1 2 ( ε M ε N S N ) 2 ε ˙ M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaaca WaaSbaaSqaaiaad6gaaeqaaOGaeyypa0ZaaSaaaeaacaWGMbWaaSba aSqaaiaad6eaaeqaaaGcbaGaam4uamaaBaaaleaacaWGobaabeaakm aakaaabaGaaGOmaiabec8aWbWcbeaaaaGccaWGLbWaaWbaaSqabeaa cqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaabaWaaSaaae aacqaH1oqzdaWgaaadbaGaamytaaqabaWccqGHsislcqaH1oqzdaWg aaadbaGaamOtaaqabaaaleaacaWGtbWaaSbaaWqaaiaad6eaaeqaaa aaaSGaayjkaiaawMcaamaaCaaameqabaGaaGOmaaaaaaGccuaH1oqz gaGaamaaBaaaleaacaWGnbaabeaaaaa@5095@
Where,
f N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGobaabeaaaaa@37E0@
Nucleated void volume fraction
S N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGobaabeaaaaa@37E0@
Gaussian standard deviation
ε N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaad6eaaeqaaaaa@389C@
Nucleated effective plastic strain
ε M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaad6eaaeqaaaaa@389C@
Admissible plastic strain
The viscoplastic flow of the porous material is described by:(5)
{ Ω e v p = σ e q 2 σ M 2 + 2 q 1 f cosh ( 3 2 q 2 σ m σ M ) ( 1 + q 3 f 2 ) i f σ m > 0 Ω e v p = σ e q 2 σ M 2 + 2 q 1 f ( 1 + q 3 f 2 ) i f σ m 0
Where,
σ e q
von Mises is effective stress
σ M
Admissible elasto-viscoplastic stress
σ m
Hydrostatic stress
f * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCa aaleqabaGaaiOkaaaaaaa@37BC@
Specific coalescence function which can be written as:
(6)
{ Ω e v p = σ e q 2 σ M 2 + 2 q 1 f * cosh ( 3 2 q 2 σ m σ M ) ( 1 + q 3 f * 2 ) if σ m > 0 Ω e v p = σ e q 2 σ M 2 + 2 q 1 f * ( 1 + q 3 f * 2 ) if σ m 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaGabaabaeqabaGaeuyQdC1aaSbaaSqaaiaadwgacaWG2bGaamiC aaqabaGccqGH9aqpdaWcaaqaaiabeo8aZnaaDaaaleaacaWGLbGaam yCaaqaaiaaikdaaaaakeaacqaHdpWCdaqhaaWcbaGaamytaaqaaiaa ikdaaaaaaOGaey4kaSIaaGOmaiaadghadaWgaaWcbaGaaGymaaqaba GccaWGMbWaaWbaaSqabeaacaGGQaaaaOGaci4yaiaac+gacaGGZbGa aiiAamaabmaabaWaaSaaaeaacaaIZaaabaGaaGOmaaaacaWGXbWaaS baaSqaaiaaikdaaeqaaOWaaSaaaeaacqaHdpWCdaWgaaWcbaGaamyB aaqabaaakeaacqaHdpWCdaWgaaWcbaGaamytaaqabaaaaaGccaGLOa GaayzkaaGaeyOeI0YaaeWaaeaacaaIXaGaey4kaSIaamyCamaaBaaa leaacaaIZaaabeaakiaadAgadaahaaWcbeqaaiaacQcacaaIYaaaaa GccaGLOaGaayzkaaGaaGjbVlaaysW7caqGPbGaaeOzaiaaysW7cqaH dpWCdaWgaaWcbaGaamyBaaqabaGccqGH+aGpcaaIWaaabaGaeuyQdC 1aaSbaaSqaaiaadwgacaWG2bGaamiCaaqabaGccqGH9aqpdaWcaaqa aiabeo8aZnaaDaaaleaacaWGLbGaamyCaaqaaiaaikdaaaaakeaacq aHdpWCdaqhaaWcbaGaamytaaqaaiaaikdaaaaaaOGaey4kaSIaaGOm aiaadghadaWgaaWcbaGaaGymaaqabaGccaWGMbWaaWbaaSqabeaaca GGQaaaaOGaeyOeI0YaaeWaaeaacaaIXaGaey4kaSIaamyCamaaBaaa leaacaaIZaaabeaakiaadAgadaahaaWcbeqaaiaacQcacaaIYaaaaa GccaGLOaGaayzkaaGaaGjbVlaaysW7caqGPbGaaeOzaiaaysW7cqaH dpWCdaWgaaWcbaGaamyBaaqabaGccqGHKjYOcaaIWaaaaiaawUhaaa aa@96D4@
Where,
f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGJbaabeaaaaa@37F5@
Critical void volume fraction at coalescence
f F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGgbaabeaaaaa@37D8@
Critical void volume fraction at ductile fracture
f u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWG1baabeaaaaa@3807@
Corresponding value of the coalescence function f u = 1 q 1 , f * ( f F ) = f u
The variation of the specific coalescence function is shown in 図 2.


図 2. Variation of Specific Coalescence Function
The admissible plastic strain rate is computed as:(7)
ε ˙ M = σ : D p ( 1 f ) σ M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacuaH1oqzgaGaamaaBaaaleaacaWGnbaabeaakiabg2da9maalaaa baGaeq4WdmNaaiOoaiaadseadaahaaWcbeqaaiaadchaaaaakeaada qadaqaaiaaigdacqGHsislcaWGMbaacaGLOaGaayzkaaGaeq4Wdm3a aSbaaSqaaiaad2eaaeqaaaaaaaa@47F8@
Where,
σ
Cauchy stress tensor
σ M
Admissible plastic stress
D p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCa aaleqabaGaamiCaaaaaaa@37E1@
Macroscopic plastic strain rate tensor which can be written in the case of the associated plasticity as:(8)
D p = λ ˙ Ω e v p σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGebWaaWbaaSqabeaacaWGWbaaaOGaeyypa0Jafq4UdWMbaiaa daWcaaqaaiabgkGi2kabfM6axnaaBaaaleaacaWGLbGaamODaiaadc haaeqaaaGcbaGaeyOaIyRaeq4Wdmhaaaaa@46CC@
with Ω e v p the yield surface envelope. The viscoplastic multiplier is deduced from the consistency condition:(9)
Ω e v p = Ω ˙ e v p = 0
From this last expression we deduce that:(10)
λ ˙ = Ω e v p Ω e v p 2 : C e : Ω e v p σ Ω e v p σ M σ M ε M A 2 Ω e v p f [ ( 1 f ) Ω e v p σ : I + A 1 A 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacuaH7oaBgaGaaiabg2da9maalaaabaGaaeyQdmaaBaaaleaacaWG LbGaamODaiaadchaaeqaaaGcbaWaaSaaaeaacqGHciITcaqGPoWaaS baaSqaaiaadwgacaWG2bGaamiCaaqabaaakeaacaaIYaGaeyOaIyla aiaacQdacaWGdbWaaWbaaSqabeaacaWGLbaaaOGaaiOoamaalaaaba GaeyOaIyRaaeyQdmaaBaaaleaacaWGLbGaamODaiaadchaaeqaaaGc baGaeyOaIyRaeq4WdmhaaiabgkHiTmaalaaabaGaeyOaIyRaaeyQdm aaBaaaleaacaWGLbGaamODaiaadchaaeqaaaGcbaGaeyOaIyRaeq4W dm3aaSbaaSqaaiaad2eaaeqaaaaakmaalaaabaGaeyOaIyRaeq4Wdm 3aaSbaaSqaaiaad2eaaeqaaaGcbaGaeyOaIyRaeqyTdu2aaSbaaSqa aiaad2eaaeqaaaaakiaadgeadaWgaaWcbaGaaGOmaaqabaGccqGHsi sldaWcaaqaaiabgkGi2kaabM6adaWgaaWcbaGaamyzaiaadAhacaWG WbaabeaaaOqaaiabgkGi2kaadAgaaaWaamWaaeaadaqadaqaaiaaig dacqGHsislcaWGMbaacaGLOaGaayzkaaWaaSaaaeaacqGHciITcaqG PoWaaSbaaSqaaiaadwgacaWG2bGaamiCaaqabaaakeaacqGHciITcq aHdpWCaaGaaiOoaiaadMeacqGHRaWkcaWGbbWaaSbaaSqaaiaaigda aeqaaOGaamyqamaaBaaaleaacaaIYaaabeaaaOGaay5waiaaw2faaa aaaaa@86A0@
Where,(11)
A 2 = σ : δ Ω e v p δ σ ( 1 f ) σ M ; A 1 = f N S N 2 π e 1 2 ( ε M ε N S N ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa aaleaacaaIYaaabeaakiabg2da9maalaaabaGaeq4WdmNaaiOoamaa laaabaGaeqiTdqMaeuyQdC1aaSbaaSqaaiaadwgacaWG2bGaamiCaa qabaaakeaacqaH0oazcqaHdpWCaaaabaGaaiikaiaaigdacqGHsisl caWGMbGaaiykaiabeo8aZnaaBaaaleaacaWGnbaabeaaaaGccaGG7a GaamyqamaaBaaaleaacaaIXaaabeaakiabg2da9maalaaabaGaamOz amaaBaaaleaacaWGobaabeaaaOqaaiaadofadaWgaaWcbaGaamOtaa qabaGcdaGcaaqaaiaaikdacqaHapaCaSqabaaaaOGaamyzamaaCaaa leqabaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaam aalaaabaGaeqyTdu2aaSbaaWqaaiaad2eaaeqaaSGaeyOeI0IaeqyT du2aaSbaaWqaaiaad6eaaeqaaaWcbaGaam4uamaaBaaameaacaWGob aabeaaaaaaliaawIcacaGLPaaadaahaaadbeqaaiaaikdaaaaaaaaa @63F7@
1 Gurson A. L. 「Continuum theory of ductile rupture by void nucleation and growth: Part I - Yield criteria and flow rules for porous ductile media」, Journal of Engineering Materials and Technology, Vol. 99, 2-15, 1977.