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Table of Contents
Failure criteria
RuptureCriterion
Description
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.
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 $$ C = \int_0^{\overline{\varepsilon}^p} \frac{\sigma_1}{\overline{\sigma}} d\overline{\varepsilon}^p$$ 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
Name | Metafor Code | Dependency |
---|---|---|
Criterion | RUPT_OP_LAW | - |
BaoRuptureCriterion
Description
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:
$$ \varepsilon^{f} = \left\{ \begin{array}{ll} \infty &\mbox{if } \dfrac{p}{J_2}\leq-\dfrac{1}{3}\\ P_1 \left(\dfrac{p}{J_2} + \dfrac{1}{3}\right)^{P_2} &\mbox{if } -\dfrac{1}{3}<\dfrac{p}{J_2}\leq 0\\ P_3 \left(\dfrac{p}{J_2}\right)^2 + P_4 \dfrac{p}{J_2} + P_5 &\mbox{if } 0<\dfrac{p}{J_2}<0.4\\ \exp\left(P_6\dfrac{p}{J_2}\right) &\mbox{if } \dfrac{p}{J_2}>0.4 \end{array} \right. $$
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 [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 [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 [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 [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:
$$ \left<x\right> = \dfrac{1}{2} \left(x + \left|x\right|\right) $$
Parameters
Name | Metafor Code | Dependency |
---|---|---|
$A $ | RUPT_GOIJAERTS_A | - |
$B $ | RUPT_GOIJAERTS_B | - |
$C $ | RUPT_GOIJAERTS_C | - |
MaximumPrincipalStrainRuptureCriterion
Description
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.
Parameters
Name | Metafor Code | Dependency |
---|---|---|
$A $ | RUPT_MPSTRAIN_CL | - |
$B $ | RUPT_MPSTRAIN_TL | - |
BaiRuptureCriterion
Description
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: $$\overline{\varepsilon}^p_f (\eta,\overline{\theta}) = \left[ \frac{1}{2}\left( D_1e^{-D_2\eta}+D_5e^{-D_6\eta} \right)-D_3e^{-D_4\eta} \right]\overline{\theta}^2 + \frac{1}{2}\left( D_1e^{- D_2\eta}-D_5e^{-D_6\eta} \right)\overline{\theta}+D_3e^{-D_4\eta}$$
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 [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 | - |