RuptureCriterion manages various failure criteria. Two parameters are common in all laws. First, the critical value C of a variable above which the element is broken. Second, the type of failure : the element is broken if the criterion is verified on one integration point (ONEBROKEN), on all of them (ALLBROKEN), or in average (MEANBROKEN) over the element.

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.

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 $$ C = \int_0^{\overline{\varepsilon}^p} \frac{\sigma_1}{\overline{\sigma}} d\overline{\varepsilon}^p$$ Brozzo criterion $$ C = \int_0^{\overline{\varepsilon}^p} \frac{2\sigma_1}{3(\sigma_1-p)} d\overline{\varepsilon}^p$$ Ayada criterion $$ C = \int_0^{\overline{\varepsilon}^p} \frac{p}{\overline{\sigma}} d\overline{\varepsilon}^p$$ Rice and Tracey criterion $$ C = \int_0^{\overline{\varepsilon}^p} \exp\left(\frac{3}{2} \frac{p}{\overline{\sigma}}\right) d\overline{\varepsilon}^p$$ Parameters

Bao-Wierzbicki criterion [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:

$$ \varepsilon^{f} = \left\{ \begin{array}{ll} \infty &\mbox{if } \dfrac{p}{J_2}\leq-\dfrac{1}{3}\\ P_1 \left(\dfrac{p}{J_2} + \dfrac{1}{3}\right)^{P_2} &\mbox{if } -\dfrac{1}{3}<\dfrac{p}{J_2}\leq 0\\ P_3 \left(\dfrac{p}{J_2}\right)^2 + P_4 \dfrac{p}{J_2} + P_5 &\mbox{if } 0<\dfrac{p}{J_2}<0.4\\ \exp\left(P_6\dfrac{p}{J_2}\right) &\mbox{if } \dfrac{p}{J_2}>0.4 \end{array} \right. $$

Hancock and Mackenzie criterion [2]. The critical plastic strain $\varepsilon^{f}$ is defined as:

$$ \varepsilon^{f} = D_1 + D_2 \exp\left(D_3\frac{p}{J_2}\right) $$

Johnson and Cook criterion [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:

$$ \varepsilon^{f} = \left(D_1 + D_2 \exp\left(D_3\dfrac{p}{J_2}\right)\right) \left(1 + D_4\ln\dfrac{\dot\varepsilon^{pl}}{\dot\varepsilon_0}\right) \left(1 + D_5 \dfrac{T-T_{room}}{T_{melt}-T_{room}}\right) $$

Lemaitre criterion [4]. The element is broken if the variable C, defined below, reaches a critical value:

$$ C = \int_0^{\varepsilon^{pl}}\left(\frac{2}{3}\left(1+\nu\right) + 3\left(1-2\nu\right)\left(\frac{p}{J_2}\right)^2\right)d\varepsilon^{pl} $$

Goijaerts criterion [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:

$$ \left<x\right> = \dfrac{1}{2} \left(x + \left|x\right|\right) $$

Maximum Principal Strain criterion [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.

Bai and Wierzbicki rupture criterion [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: $$\overline{\varepsilon}^p_f (\eta,\overline{\theta}) = \left[ \frac{1}{2}\left( D_1e^{-D_2\eta}+D_5e^{-D_6\eta} \right)-D_3e^{-D_4\eta} \right]\overline{\theta}^2 + \frac{1}{2}\left( D_1e^{- D_2\eta}-D_5e^{-D_6\eta} \right)\overline{\theta}+D_3e^{-D_4\eta}$$

Lou, Yoon and Huh rupture criterion [8]. 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 $\overline{\varepsilon}^p_f$ is defined as: $$ \overline{\varepsilon}^p_f = c_3\left( \frac{2}{\sqrt{L^2+3}} \right)^{-c_1} \left( \left\langle \frac{1}{1+C} \left[ \eta+\frac{3-L}{3\sqrt{L^2+3}}+C \right] \right\rangle \right)^{-c_2} $$

with, $$ L = \frac{3 \tan\left( \theta \right) - \sqrt{3}}{\tan \left( \theta \right) + \sqrt{3}} $$ where $c_1$, $c_2$ and $c_3$ 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.

[1] 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] 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] 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] Lemaitre J. A Course on Damage Mechanics. Springer-Verlag Berlin Heidelberg, 1992.

[5] 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] 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] 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.