# Failure Models¶

MatEditor allows you to define the failure material properties. The supported properties are listed below.

• Glass
• Cockcroft
• Connect
• Extended Mohr-Coulomb
• Energy
• Fabric
• Forming Limit Diagram
• Hashin
• Hosford-Coulomb
• Johnson-Cook
• Mullins Effect
• NXT
• Orthotropic Strain
• Puck
• Tuler-Butcher
• Tensile Strain
• Wierzbicki
• Wilkins

The failure strain is described by two parabolic functions that user input.

## Cockcroft¶

A nonlinear stress-strain based failure criterion with linear damage accumulation.

$C_0 = \int _0 ^{\bar{\epsilon}_f} max(\sigma_1, 0) \cdot d\bar{\epsilon}$

where $$\epsilon_1$$ is the first principal tension stress, $$\bar{\epsilon}$$ is the equivalent strain.

## Extended Mohr-Coulomb¶

The failure criteria is calculated as:

$D = \sum \dfrac{\Delta \bar{\epsilon}_p}{\bar{\epsilon}_{p,fail}}$

where effective failure strain is

$\bar{\epsilon}_{p,fail} = b \cdot (1+c)^{\frac{1}{n}} \cdot \{[\dfrac{1}{2}((f_1-f_2)^a + (f_2-f_3)^a + (f_1 - f_3)^a )]^{\frac{1}{a}} + c(2\eta+f_1+f_3) \}^{-\frac{1}{n}}$

the coefficient b is computed as

$b = b_0[1+\gamma ln(\dfrac{\dot{\bar{\epsilon}}_p}{\dot{\bar{\epsilon}}_0})] \quad if\, \dot{\bar{\epsilon}}_p > \dot{\bar{\epsilon}}_0$

or

$b = b_0 \quad if\, \dot{\bar{\epsilon}}_p \le \dot{\bar{\epsilon}}_0$

## Energy¶

The damage is defined as

$D = \dfrac{E-E_1}{E_2 - E_1}$

where the energy density is the current internal energy of the element divided by the current element volume.

## Fabric¶

The failure and damage is defined independently in each direction ($$i$$=1,2)

$D_i = \dfrac{\epsilon_i - \epsilon_{fi}}{\epsilon_{ri} - \epsilon_{fi}}$

where $$\epsilon_i \ge \epsilon_{fi}$$.

## Hashin¶

This model can be used for the composite materials.

The damage factor is calculated as

$D = Max(F_1,F_2,F_3, F_4, F_5) \quad for\quad uni-directional\, lamina\, model$
$D = Max(F_1,F_2,F_3, F_4, F_5, F_6, F_7) \quad for\quad fabric\, lamina\, model$

#### For the uni-directional lamina model:¶

Tensile/shear fiber mode:

$F_1 = (\dfrac{\langle\sigma_{11}\rangle}{\sigma_1^t})^2 + (\dfrac{\sigma_{12}^2 + \sigma_{13}^2}{{\sigma_{12}^f}^2})$

Compression fiber mode:

$F_2 = (\dfrac{\langle \sigma_a \rangle}{ \sigma_1^c})^2$

with $$\sigma_{\alpha} = -\sigma_{11}+\langle -\dfrac{\sigma_{22}+\sigma_{33}}{2} \rangle$$.

Crush mode:

$F_3 = (\dfrac{\langle p \rangle}{\sigma_c})^2$

with $$p=-\dfrac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3}$$.

Failure matrix mode:

$F_4 = (\dfrac{\langle \sigma_{22} \rangle}{\sigma_2^t})^2 + (\dfrac{\sigma_{23}}{S_{23}})^2 + (\dfrac{\sigma_{12}}{S_{12}})^2$

Delamination mode:

$F_5 = S^2_{del}[(\dfrac{\langle \sigma_{33} \rangle}{\sigma^t_2})^2 + (\dfrac{\sigma_{23}}{\tilde{S}_{23}})^2 + (\dfrac{\sigma_{12}}{S_{12}})^2 ]$

#### For the fabirc lamina model:¶

Tensile/shear fiber mode

$F_1 = (\dfrac{\langle\sigma_{11}\rangle}{\sigma_1^t})^2 + (\dfrac{\sigma_{12}^2 + \sigma_{13}^2}{{\sigma_{a}^f}^2})$
$F_2 = (\dfrac{\langle\sigma_{22}\rangle}{\sigma_2^t})^2 + (\dfrac{\sigma_{12}^2 + \sigma_{23}^2}{{\sigma_{b}^f}^2})$

Compression fiber mode:

$F_3 = (\dfrac{\langle \sigma_a \rangle}{ \sigma_1^c})^2$
$F_4 = (\dfrac{\langle \sigma_b \rangle}{ \sigma_2^c})^2$

Crush mode:

$F_5 = (\dfrac{\langle p \rangle}{\sigma_c})^2$

Shear failure matrix mode:

$F_6 = (\dfrac{\sigma_12}{\sigma_12^m})^2$

Matrix failure mode:

$F_7 = S^2_{del}[(\dfrac{\langle \sigma_{33} \rangle}{\sigma^t_3})^2 + (\dfrac{\sigma_{23}}{\tilde{S}_{23}})^2 + (\dfrac{\sigma_{12}}{S_{12}})^2 ]$

## Hosford-Coulomb¶

The failure strain is described y the Hosford-Coulomb function.

The damage is defined as

$D = \sum \dfrac{\Delta \bar{\epsilon}_p} {\bar{\epsilon}^{pr}_{HC}(\eta) }$

where the strain is calcualted as

$\bar{\epsilon}^{pr}_{HC}(\eta, \theta) = b(1+c)^{\frac{1}{n_f}} \{[\dfrac{1}{2}((f_1-f_2)^a + (f_2-f_3)^a + (f_1-f_3)^a)]^{\frac{1}{a}} + c(a\eta + f_1 +f_2) \}^{\frac{1}{n_f}}$

## Johnson-Cook¶

The failure strain is calculated by the constutitive relation:

$\epsilon_f = [D_1+D_2exp(D_3\sigma^*)] [1+D_4 ln(\dot{\epsilon}^*)] (1 + D_5 T^*)$

The damage factor is defined as

$D = \sum \dfrac{\Delta \epsilon_p}{\epsilon_f}$

This is the Ladeveze failure model for delamination (interlaminar fracture). The damage parameters are defined as

$Y_{d_3} = \dfrac{\partial E_D}{\partial d_3} \vert _{\sigma=cst}=\dfrac{1}{2} \dfrac{\langle\sigma_{33}\rangle^2}{K_3(1-d_3)^2} \quad Mode\,I$
$Y_{d_2} = \dfrac{\partial E_D}{\partial d_2} \vert _{\sigma=cst}=\dfrac{1}{2} \dfrac{\langle\sigma_{32}\rangle^2}{K_2(1-d_2)^2} \quad Mode\,II$
$Y_{d_1} = \dfrac{\partial E_D}{\partial d_1} \vert _{\sigma=cst}=\dfrac{1}{2} \dfrac{\langle\sigma_{31}\rangle^2}{K_1(1-d_1)^2} \quad Mode\,III$

The damage value can be

$D = \dfrac{k}{a}[1- exp(-a\langle w(Y)-d\rangle)]$

## Mullins Effect¶

$\sigma = \eta dev(\sigma) - pI$

where dev($$\sigma$$) is the deviatoric part of the stress, $$p$$ is the hydrostatic pressure. The damage factor $$\eta$$ is given as

$\eta = 1 - \dfrac{1}{R} erf(\dfrac{W_{max}-W}{m+\beta W_{max}})$

where $$erf$$ is the Gauss error function.

## NXT¶

This model describes the forming limit baed on stresses. This failure is used for shell elements only.

An instability factor is defined as:

$\lambda_f=\dfrac{\sigma/h - (\sigma/h)_{SR}}{(\sigma/h)_{3D}-(\sigma/h)_{SR}} + 1$

The material is defined as free if $$0<\lambda_f<1$$, warning if $$1<\lambda_f<2$$, failure if $$\lambda_f \ge 2$$.

The failure strain is described by two parabolicfunctions calculated using curve fitting from user input failure strains.

## Orthotropic Strain¶

A damage factor is the maximum over time and is calculated for each direction and stress state via:

$d_ijl = \dfrac{\epsilon_{ijf\_l}}{\epsilon_{ijl}} \cdot \dfrac{\epsilon_{ijl}-\alpha\cdot\epsilon_{ijd\_l}}{\epsilon_{ijf\_l}-\epsilon_{ijd\_l}}$

where the direction is indicated by using the common $$ij$$ notation and loading state is either compression ($$l=c$$) or tension ($$l=t$$). The parameter $$\alpha=factor_{el}\cdot factor_{rate}$$.

The element size correction factor is :

$factor_{el} = Fscale_{el} \cdot f_{el} \dfrac{Size_{el}}{El_ref}$

where $$f_{el}$$ is the element size correction factor function, $$Size_{el}$$ is the characteristic element size.

The strain rate factor is

$factor_{rate} = f_{ijl}(\dfrac{\dot{\epsilon}_{ijl}}{\dot{\epsilon}_0})$

where $$f_ijl$$ is strain rate factor function, $$\dot{\epsilon}_{ijl}$$ is the current strain rate in direction ij and load case l, and $$\dot\epsilon_0$$ is the reference strate rate.

Generally, the damange for this model is

$D = Max(d_{ijl}) = Max(\dfrac{\epsilon_{ijf\_l}}{\epsilon_{ijl}} \cdot \dfrac{\epsilon_{ijl}-\alpha\cdot\epsilon_{ijd\_l}}{\epsilon_{ijf\_l}-\epsilon_{ijd\_l}})$

## Puck¶

This failure model can be applied for both solid and shell elements.

For the fiber fraction failure, the damage parameter $$e_f$$ is defined by

$e_f=\dfrac{\sigma_{11}}{\sigma_{1}^t} \quad for\, tensile$

or

$e_f=\dfrac{|\sigma_{11}|}{\sigma_{1}^c} \quad for\, compression$

For the inter fiber failure: the damage parameter $$e_f$$ is

$e_f=\dfrac{1}{\bar{\sigma}_{12}} [ \sqrt{(\dfrac{\bar{\sigma}_{12}}{\sigma_2^t} -p^+_{12})^2\sigma_{22}^2 + \sigma_{12}^2}+p^+_{12}\sigma_{22}] \quad for\, Mode\, A$

or

$e_f=[(\dfrac{\sigma_{12}}{2(1+p^-_{22})\bar{\sigma}_{12}})^2 + (\dfrac{\sigma_{22}}{\sigma_2^c})^2](\dfrac{\sigma^c_2}{-\sigma_{22}}) \quad for\, Mode\, C$

or

$e_f=\dfrac{1}{\bar{\sigma}_{12}} ( \sqrt{\sigma_{12}^2+(p^-_{12}\sigma_{22})^2}+p^-_{12}\sigma_{22}) \quad for\, Mode\, B$

when the damage parameter $$e_f \ge 1.0$$, the stresses are decreased by using an exponential function to avoid numerical instabilities.

The damage is defined by

$D = Max(e_f(tensile),e_f(compression), e_f(ModaA), e_f(ModeB), e_f(ModeC) )$

## Tuler-Butcher¶

An element fails once the damage is greater than specified critical damage value K. For ductile materials, the cumulative damage parameter is:

$D=\int_0^t{max(0, \sigma-\sigma_r)^{\lambda})dt}>K$

where $$\sigma_r$$ is initial fracture stress, $$\sigma$$ maximum principal stress, $$\lambda$$ is material constant, $$t$$ is the time when the element cracks, $$D$$ is the damage integral, $$K$$ is the critical value of the damage integral.

For brittle materials (shells only), the damage parameter is: $$\dot{D} = \dfrac{1}{K}(\sigma - \sigma_r)^a$$ $$\sigma_r=\sigma_0(1-D)^b$$ $$D=D+\dot{D}\Delta t$$

## Tensile Strain¶

This is a strain-based failure model that is compatible with both solid and shell elements. The damage is calculated by:

$D = \dfrac{\epsilon - \epsilon_{t1}}{\epsilon_{t2} - \epsilon_{t1}}$

where $$\epsilon$$ is either the quivlent strain or maximum principal tensile strain.

## Wierzbicki model¶

This model describes the Bao-Xue-Wierzbicki failure model. The damage is defined by

$D=\sum{\dfrac{\Delta\epsilon_{p}}{\bar{\epsilon}_f}}$

where the effective failure strain is

$\bar{\epsilon}_f =\{ \bar{\epsilon}_{max}n-[\bar{\epsilon}_{max}n - \bar{\epsilon}_{min}n](1-\bar{\xi}^m)^{\dfrac{1}{m}} \}^{\dfrac{1}{n}}$

where $$\bar{\epsilon}_{max} = C_1 e^{-1C_{2}\eta}$$, and $$\bar{\epsilon}_{min} = C_{3} e^{-1C_{4}\eta}$$.

For solid element, the parameters $$\bar{\xi}$$ and $$\bar{\eta}$$ are defined by the two options.

The option 1 (default) is : $$\bar{\xi}=\dfrac{\sigma_m}{\sigma_{VM}} \quad \bar{\eta}=\dfrac{27J_3}{2\sigma^3_{VM}}$$

The option 2 is: $$\bar{\xi}=\dfrac{\int_0^{\epsilon_p}\dfrac{\sigma_m}{\sigma_{VM}}d\epsilon_p}{\epsilon_p} \quad \bar{\eta}=\dfrac{\int_0^{\epsilon_p} \dfrac{27J_3}{2\sigma^3_{VM}} d\epsilon_p}{\epsilon_p}$$

For shell element, the parameters $$\bar{\xi}$$ and $$\bar{\eta}$$ are $$\bar{\xi}=\dfrac{\sigma_m}{\sigma_{VM}} \quad \bar{\eta}=-\dfrac{27}{2}\bar{\eta}(\bar{\eta}^2-\dfrac{1}{3})$$

where $$\sigma_m$$ is Hydrostatic stress, $$\sigma_{VM}$$ is von Mises stress, and $$J_3$$ is the third invariant deviatoric stress.

## Wilkins model¶

The cumulative damage is given by:

$D_c = \int W_1 W_2 d \bar{\epsilon_p}$

where $$W_1=(\dfrac{1}{1-\dfrac{P}{P_{lim}}})^{\alpha}$$, $$W_2=(2-A)^{\beta}$$, and hydro-pressure $$P=-\dfrac{1}{3}\sum_{j=1}^{3}\sigma_{jj}$$, $$A=max(\dfrac{s_2}{s_1}, \dfrac{s_2}{s_3})$$. $$s_1$$, $$s_2$$, $$s_3$$ are the deviatoric stresses, and $$s_1 \ge s_2 \ge s_3$$.