Failure criteria

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 :

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 (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

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 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.

[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.