====== Failure criteria ====== ===== RuptureCriterion ===== === Description === ''RuptureCriterion'' manages various failure criteria. The critical value //C// (''RUPT_CRIT_VALUE'') of a variable above which the element is broken. The type of failure (''RUPT_TYPE_CRIT'') are defined in the table below : ^ Name ^ Description ^ | ''NOBREAK'' | Compute the criterion, but never break any element | | ''ONEBROKEN'' | Break an element when ONE integration point override the critical value | | ''ALLBROKEN'' | Break an element when ALL the integration points override the critical value | | ''MEANBROKEN'' | Break an element when the averaged value over the integration points override the critical value | === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ |Critical value | ''RUPT_CRIT_VALUE'' | - | |Type of failure | ''RUPT_TYPE_CRIT'' | - | ===== IFRuptureCriterion ===== === Description === The element is broken if an ''InternalField'' reaches a critical value. The critical ''InternalField'' is defined with the following command, which must be added when defining the criterion: rc.setInternalField(IF_EPL) for a criterion based on a critical value of the equivalent plastic strain. ===== OneParameterRuptureCriterion ===== === Description === Four simple rupture criteria are gathered in one single family. In order to selected one of the criteria the parameter ''RUPT_OP_LAW'' (only parameter in this criterion) need to be defined as: ''COCKROFT'', ''BROZZO'', ''AYADA'' or ''RICE''. Then, the element is broken if the variable C reaches a critical value, which is defined in each case as: //Cockroft and Latham criterion (dimensional Value) : ''COCKROFT2''// $$ C = \int_0^{\overline{\varepsilon}^p} \sigma_1 d\overline{\varepsilon}^p$$ //Cockroft and Latham criterion (adimensional value) : ''COCKROFT''// $$ C = \int_0^{\overline{\varepsilon}^p} \frac{\sigma_1}{\overline{\sigma}} d\overline{\varepsilon}^p$$ //Brozzo criterion : ''BROZZO''// $$ C = \int_0^{\overline{\varepsilon}^p} \frac{2\sigma_1}{3(\sigma_1-p)} d\overline{\varepsilon}^p$$ //Ayada criterion : ''AYADA'' // $$ C = \int_0^{\overline{\varepsilon}^p} \frac{p}{\overline{\sigma}} d\overline{\varepsilon}^p$$ //Rice and Tracey criterion : ''RICE''// $$ C = \int_0^{\overline{\varepsilon}^p} \exp\left(\frac{3}{2} \frac{p}{\overline{\sigma}}\right) d\overline{\varepsilon}^p$$ **Parameters** ^ Name ^ Metafor Code ^ Dependency ^ |Criterion | ''RUPT_OP_LAW'' | - | ===== BaoRuptureCriterion ===== === Description === Bao-Wierzbicki criterion [[doc:user:elements:volumes:rupturecritere#References|[1]]]. The element is broken if the variable //C//, defined below, reaches a critical value: $$ C = \int_0^{\varepsilon^{pl}}\dfrac{d\varepsilon^{pl}}{\varepsilon^{f}} $$ where $\varepsilon^{f}$ is defined as: $$ \varepsilon^{f} = \left\{ \begin{array}{ll} \infty &\mbox{if } \dfrac{p}{J_2}\leq-\dfrac{1}{3}\\ P_1 \left(\dfrac{p}{J_2} + \dfrac{1}{3}\right)^{P_2} &\mbox{if } -\dfrac{1}{3}<\dfrac{p}{J_2}\leq 0\\ P_3 \left(\dfrac{p}{J_2}\right)^2 + P_4 \dfrac{p}{J_2} + P_5 &\mbox{if } 0<\dfrac{p}{J_2}<0.4\\ \exp\left(P_6\dfrac{p}{J_2}\right) &\mbox{if } \dfrac{p}{J_2}>0.4 \end{array} \right. $$ === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ |$P_1$ | ''RUPT_BAO_P1'' | - | |$P_2$ | ''RUPT_BAO_P2'' | - | |$P_3$ | ''RUPT_BAO_P3'' | - | |$P_4$ | ''RUPT_BAO_P4'' | - | |$P_5$ | ''RUPT_BAO_P5'' | - | |$P_6$ | ''RUPT_BAO_P6'' | - | ===== HancockMackenzieRuptureCriterion ===== === Description === Hancock and Mackenzie criterion [[doc:user:elements:volumes:rupturecritere#References|[2]]]. The critical plastic strain $\varepsilon^{f}$ is defined as: $$ \varepsilon^{f} = D_1 + D_2 \exp\left(D_3\frac{p}{J_2}\right) $$ === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ |$D_1$ | ''RUPT_HANCOCK_D1'' | - | |$D_2$ | ''RUPT_HANCOCK_D2'' | - | |$D_3$ | ''RUPT_HANCOCK_D3'' | - | ===== JohnsonCookRuptureCriterion ===== === Description === Johnson and Cook criterion [[doc:user:elements:volumes:rupturecritere#References|[3]]]. The element is broken if the variable //C//, defined below, reaches a critical value: $$ C = \int_0^{\varepsilon^{pl}}\dfrac{d\varepsilon^{pl}}{\varepsilon^{f}} $$ where $\varepsilon^{f}$ is defined as: $$ \varepsilon^{f} = \left(D_1 + D_2 \exp\left(D_3\dfrac{p}{J_2}\right)\right) \left(1 + D_4\ln\dfrac{\dot\varepsilon^{pl}}{\dot\varepsilon_0}\right) \left(1 + D_5 \dfrac{T-T_{room}}{T_{melt}-T_{room}}\right) $$ === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ |$D_1$ | ''RUPT_JC_D1'' | - | |$D_2$ | ''RUPT_JC_D2'' | - | |$D_3$ | ''RUPT_JC_D3'' | - | |$D_4$ | ''RUPT_JC_D4'' | - | |$D_5$ | ''RUPT_JC_D5'' | - | |$\dot\varepsilon_0$ | ''RUPT_JC_EPSP0'' | - | |Room temperature $T_{room}$ | ''RUPT_JC_TROOM'' | - | |Melting temperature $T_{melt}$ | ''RUPT_JC_TMELT'' | - | ===== LemaitreRuptureCriterion ===== === Description === Lemaitre criterion [[doc:user:elements:volumes:rupturecritere#References|[4]]]. The element is broken if the variable //C//, defined below, reaches a critical value: $$ C = \int_0^{\varepsilon^{pl}}\left(\frac{2}{3}\left(1+\nu\right) + 3\left(1-2\nu\right)\left(\frac{p}{J_2}\right)^2\right)d\varepsilon^{pl} $$ === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ |$\nu$ | ''RUPT_LEMAITRE_NU'' | - | ===== GoijaertsRuptureCriterion ===== === Description === Goijaerts criterion [[doc:user:elements:volumes:rupturecritere#References|[5]]]. The element is broken if //W//, whose evolution law is defined below, reaches 1. $$ \dot W = \dfrac{1}{C} \left<1+A\dfrac{p}{J_2}\right> \left(\varepsilon^{pl}\right)^{B} \dot\varepsilon^{pl} $$ where brackets are MacCaulay brackets: $$ \left = \dfrac{1}{2} \left(x + \left|x\right|\right) $$ === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ |$A $ | ''RUPT_GOIJAERTS_A'' | - | |$B $ | ''RUPT_GOIJAERTS_B'' | - | |$C $ | ''RUPT_GOIJAERTS_C'' | - | ===== MaximumPrincipalStrainRuptureCriterion===== === Description === Maximum Principal Strain criterion [[doc:user:elements:volumes:rupturecritere#References|[6]]]. Element failure is detected differently whether the element is globally under tension of compression. It is broken if: $ \| \epsilon_{I} \|> $ ''RUPT_MPSTRAIN_TL'' if $ \epsilon_{I}\ $ + $ \epsilon_{II}\ $ + $ \epsilon_{III}\ $ > 0 $ \| \epsilon_{III}\| > $ ''RUPT_MPSTRAIN_CL'' if $ \epsilon_{I}\ $ + $ \epsilon_{II}\ $ + $ \epsilon_{III}\ $ < 0 where $\epsilon_{I} $, $ \epsilon_{II} $ and $ \epsilon_{III} $ are principal strains in decreasing order. === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ |$A $ | ''RUPT_MPSTRAIN_CL'' | - | |$B $ | ''RUPT_MPSTRAIN_TL'' | - | ===== BaiRuptureCriterion ===== === Description === Bai and Wierzbicki rupture criterion [[doc:user:elements:volumes:rupturecritere#References|[7]]]. The element is broken if the variable C, defined below, reaches a critical value: $$ C = \int_0^{\overline{\varepsilon}^p}\dfrac{d\overline{\varepsilon}^{p}}{\overline{\varepsilon}^p_f (\eta,\overline{\theta})} $$ where $\overline{\varepsilon}^p_f (\eta,\overline{\theta})$ is defined as: $$\overline{\varepsilon}^p_f (\eta,\overline{\theta}) = \left[ \frac{1}{2}\left( D_1e^{-D_2\eta}+D_5e^{-D_6\eta} \right)-D_3e^{-D_4\eta} \right]\overline{\theta}^2 + \frac{1}{2}\left( D_1e^{- D_2\eta}-D_5e^{-D_6\eta} \right)\overline{\theta}+D_3e^{-D_4\eta}$$ === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ |$D_1$ | ''RUPT_BAI_D1'' | - | |$D_2$ | ''RUPT_BAI_D2'' | - | |$D_3$ | ''RUPT_BAI_D3'' | - | |$D_4$ | ''RUPT_BAI_D4'' | - | |$D_5$ | ''RUPT_BAI_D5'' | - | |$D_6$ | ''RUPT_BAI_D6'' | - | |$\eta_{cutoff}$ | ''RUPT_BAI_CUTOFF'' | - | ===== LouRuptureCriterion ===== === Description === Lou, Yoon and Huh rupture criterion [[doc:user:elements:volumes:rupturecritere#References|[8]]]. The element is broken if the variable K, defined below, reaches a critical value: $$ K = \int_0^{\overline{\varepsilon}^p}\dfrac{d\overline{\varepsilon}^{p}}{\overline{\varepsilon}^p_f (\eta,\overline{\theta})} $$ where $\overline{\varepsilon}^p_f$ is defined as: $$ \overline{\varepsilon}^p_f = D_3\left( \frac{2}{\sqrt{L^2+3}} \right)^{-D_1} \left( \left\langle \frac{1}{1+C} \left[ \eta+\frac{3-L}{3\sqrt{L^2+3}}+C \right] \right\rangle \right)^{-D_2} $$ with, $$ L = \frac{3 \tan\left( \theta \right) - \sqrt{3}}{\tan \left( \theta \right) + \sqrt{3}} $$ where $D_1$, $D_2$ and $D_3$ are material parameters. $L$ corresponds to an alternative definition of the Lode angle and the $\left\langle \bullet \right\rangle$ symbol denotes the MacAuley brackets. === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ |$D_1$ | ''RUPT_LOU_D1'' | - | |$D_2$ | ''RUPT_LOU_D2'' | - | |$D_3$ | ''RUPT_LOU_D3'' | - | |$C$ | ''RUPT_LOU_C'' | - | ===== References ===== [1] [[http://www.sciencedirect.com/science/article/pii/S0020740304000360|Bao Y, Wierzbicki T. On fracture locus in the equivalent strain and stress triaxiality space. International Journal of Mechanical Sciences 2004;46:81-98.]] [2] [[http://www.sciencedirect.com/science/article/pii/0022509676900247|Hancock JW, Mackenzie AC. On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states. Journal of the Mechanics and Physics of Solids 1976;24:147-160.]] [3] [[http://wbldb.lievers.net/10134084.html|Johnson GR, Cook WH. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In: 7th International Symposium on Ballistics. The Hague: The Netherlands, 1983; 541-547.]] [4] [[http://www.springer.com/us/book/9783662027615|Lemaitre J. A Course on Damage Mechanics. Springer-Verlag Berlin Heidelberg, 1992.]] [5] [[http://manufacturingscience.asmedigitalcollection.asme.org/article.aspx?articleid=1437060|Goijaerts AM, Govaert LE, Baaijens FPT. Prediction of ductile fracture in metal blanking. Journal of Manufacturing Science and Engineering 2000;122:476-483.]] [6] [7] [[http://www.sciencedirect.com/science/article/pii/S0749641907001246|Bai I, Wierzbicki T. A new model of metal plasticity and fracture with pressure and Lode dependence. International Journal of Plasticity 2008;24:1071-1096.]] [8] [[http://www.sciencedirect.com/science/article/pii/S0749641913001617|Lou Y, Yoon JW, Huh H. Modeling of shear ductile fracture considering a changeable cut-off value for stress triaxiality. International Journal of Plasticity 2014;54:56-80.]]