Brittle Damage: Johnson-Cook Plasticity Model (LAW27)

Johnson-Cook plasticity model is presented in Johnson-Cook Plasticity Model (LAW2). For shell applications, a simple damage model can be associated to this law to take into account the brittle failure. The crack propagation occurs in the plan of shell in the case of mono-layer property and through the thickness if a multi-layer property is defined (図 1).
図 1. Damage Affected Material


The elastic-plastic behavior of the material is defined by Johnson-Cook model. However, the stress-strain curve for the material incorporates a last part related to damage phase as shown in 図 2. The damage parameters are:
ε t 1
Tensile rupture strain in direction 1
ε m 1
Maximum strain in direction 1
dmax1
Maximum damage in direction 1
ε f 1
Maximum strain for element deletion in direction 1
The element is removed if one layer of element reaches the failure tensile strain, ε f 1 . The nominal and effective stresses developed in an element are related by:
図 2.
σ n = σ e f f ( 1 − d )
Where,
0 < d < 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaqGWa GaaeiiaiaabYdacaqGGaGaaeizaiaabccacaqG8aGaaeiiaiaabgda aaa@3CA8@
Damage factor
The strains and the stresses in each direction are given by:
図 3.
ε 1 = σ 1 ( 1 − d 1 ) E − ν σ 2 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaH1oqzdaWgaaWcbaGaaGymaaqabaGccqGH9aqpdaWcaaqaaiab eo8aZnaaBaaaleaacaaIXaaabeaaaOqaamaabmaabaGaaGymaiabgk HiTiaadsgadaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacaWG fbaaaiabgkHiTmaalaaabaGaeqyVd4Maeq4Wdm3aaSbaaSqaaiaaik daaeqaaaGcbaGaamyraaaaaaa@4B42@
図 4.
ε 2 = σ 2 E − ν σ 1 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaH1oqzdaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiab eo8aZnaaBaaaleaacaaIYaaabeaaaOqaaiaadweaaaGaeyOeI0YaaS aaaeaacqaH9oGBcqaHdpWCdaWgaaWcbaGaaGymaaqabaaakeaacaWG fbaaaaaa@4638@
図 5.
γ 12 = σ 12 ( 1 − d 1 ) G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHZoWzdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0ZaaSaa aeaacqaHdpWCdaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaWaaeWaae aacaaIXaGaeyOeI0IaamizamaaBaaaleaacaaIXaaabeaaaOGaayjk aiaawMcaaiaadEeaaaaaaa@4688@
図 6.
σ 1 = E ( 1 − d 1 ) [ 1 − ( 1 − d 1 ) ν 2 ] ( ε 1 + ν ε 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaWgaaWcbaGaaGymaaqabaGccqGH9aqpdaWcaaqaaiaa dweadaqadaqaaiaaigdacqGHsislcaWGKbWaaSbaaSqaaiaaigdaae qaaaGccaGLOaGaayzkaaaabaWaamWaaeaacaaIXaGaeyOeI0YaaeWa aeaacaaIXaGaeyOeI0IaamizamaaBaaaleaacaaIXaaabeaaaOGaay jkaiaawMcaaiabe27aUnaaCaaaleqabaGaaGOmaaaaaOGaay5waiaa w2faaaaadaqadaqaaiabew7aLnaaBaaaleaacaaIXaaabeaakiabgU caRiabe27aUjabew7aLnaaBaaaleaacaaIYaaabeaaaOGaayjkaiaa wMcaaaaa@571A@
図 7.
σ 2 = E [ 1 − ( 1 − d 1 ) ν 2 ] ( ε 2 + ( 1 − d 1 ) ν ε 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCdaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiaa dweaaeaadaWadaqaaiaaigdacqGHsisldaqadaqaaiaaigdacqGHsi slcaWGKbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaeqyV d42aaWbaaSqabeaacaaIYaaaaaGccaGLBbGaayzxaaaaamaabmaaba GaeqyTdu2aaSbaaSqaaiaaikdaaeqaaOGaey4kaSYaaeWaaeaacaaI XaGaeyOeI0IaamizamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawM caaiabe27aUjabew7aLnaaBaaaleaacaaIXaaabeaaaOGaayjkaiaa wMcaaaaa@571B@

The conditions for these equations are:

0 < d < 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgY da8iaadsgacqGH8aapcaaIXaaaaa@3A5C@

ε = ε t ; d = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaqGKb Gaaeiiaiaab2dacaqGGaGaaeimaaaa@39F0@

ε = ε m ; d = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaq1acaqGKb Gaaeiiaiaab2dacaqGGaGaaeimaaaa@39F0@

A linear damage model is used to compute the damage factor in function of material strain.
図 8.
d= ε− ε t ε m − ε t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGKbGaeyypa0ZaaSaaaeaacqaH1oqzcqGHsislcqaH1oqzdaWg aaWcbaGaamiDaaqabaaakeaacqaH1oqzdaWgaaWcbaGaamyBaaqaba GccqGHsislcqaH1oqzdaWgaaWcbaGaamiDaaqabaaaaaaa@46C8@
The stress-strain curve is then modified to take into account the damage by 式 1. Therefore:
図 9.
σ=E ε m −ε ε m − ε t ( ε− ε t p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHdpWCcqGH9aqpcaWGfbWaaSaaaeaacqaH1oqzdaWgaaWcbaGa amyBaaqabaGccqGHsislcqaH1oqzaeaacqaH1oqzdaWgaaWcbaGaam yBaaqabaGccqGHsislcqaH1oqzdaWgaaWcbaGaamiDaaqabaaaaOWa aeWaaeaacqaH1oqzcqGHsislcqaH1oqzdaqhaaWcbaGaamiDaaqaai aadchaaaaakiaawIcacaGLPaaaaaa@5058@
The softening condition is given by:
図 10.
ε m − ε t ≤ ε t − ε t p
The mathematical approach described here can be applied to the modeling of rivets. Predit law in Radioss allows achievement of this end by a simple model where for the elastic-plastic behavior a Johnson-Cook model or a tabulated law (LAW36) may be used.
図 11. Stress-strain Curve for Damage Affected Material