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ULiege - Aerospace & Mechanical Engineering

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# 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 [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 -