Composite Failure Model
 /FAIL/HASHIN
 /FAIL/PUCK
 /FAIL/LAD_DAMA
 /FAIL/CHANG
A composite material consists of two different materials (matrix and reinforcement fiber). Each material has a different failure behavior. In Radioss it is possible to use different failure models for matrix and fiber in one composite element (for elements with property TYPE11, TYPE16, TYPE17, TYPE51, PCOMPP or TYPE22). For example, you may use /FAIL/HASHIN for fiber failure, /FAIL/PUCK for matrix failure and /FAIL/LAD_DAMA for delamination between layers or plies (if there is more than one layer or plies defined for the composite).
Besides the above typical composite failure models, /FAIL/FLD (used for isotropic brittle composite materials in layers(plies) as in, glass), /FAIL/ENERGY, /FAIL/TBUTCHER and /FAIL/TENSSTRAIN may also be used to describe failure for composite layers(plies).
/FAIL/HASHIN
 Fiber mode: composite fails, due to fiber rupture in tension or fiber buckling in compression. So, in /FAIL/HASHIN, tensile/shear fiber mode, compression fiber mode and crush mode are the fiber modes. If direction 1 is the fiber direction, then plane 23 is the predominant failure plane for fiber mode.
 Matrix mode: composite fails, due to matrix cracking from the fiber. Failure matrix mode (or shear failure matrix mode) and delamination mode are both matrix modes. The failure plane for matrix mode is parallel to the fiber, and stress $${\sigma}_{11}$$ will not be considered in this mode.
Unidirectional Lamina Model  Fabric Lamina Model  

Damage criteria  If $$D$$=1, then failure. If $0\le D<1$, $$D$$ then no failure. With $$D=Max\left({F}_{1},{F}_{2},{F}_{3},{F}_{4}\right)$$ 
If $$D$$=1, then failure. If $0\le D<1$, $$D$$ then no failure. With $$D=Max\left({F}_{1},{F}_{2},{F}_{3},{F}_{4}\right)$$ 
Tensile/shear fiber mode  $${F}_{1}={\left(\frac{\langle {\sigma}_{11}\rangle}{{\sigma}_{1}^{t}}\right)}^{2}+\left(\frac{{\sigma}_{12}^{2}+{\sigma}_{13}^{2}}{{\sigma}_{12}^{f}{}^{2}}\right)$$  $${F}_{1}={\left(\frac{\langle {\sigma}_{11}\rangle}{{\sigma}_{1}^{t}}\right)}^{2}+\left(\frac{{\sigma}_{12}^{2}+{\sigma}_{13}^{2}}{{\sigma}_{a}^{f}{}^{2}}\right)$$ $${F}_{2}={\left(\frac{\langle {\sigma}_{22}\rangle}{{\sigma}_{2}^{t}}\right)}^{2}+\left(\frac{{\sigma}_{12}^{2}+{\sigma}_{23}^{2}}{{\sigma}_{b}^{f}{}^{2}}\right)$$ With, ${\sigma}_{a}^{f}={\sigma}_{12}^{f}\text{}\text{\hspace{0.17em}},\text{\hspace{0.05em}}\text{\hspace{1em}}\text{}{\sigma}_{b}^{f}={\sigma}_{12}^{f}\frac{{\sigma}_{2}^{t}}{{\sigma}_{1}^{t}}$ 
Compression fiber mode  $${F}_{2}={\left(\frac{\langle {\sigma}_{a}\rangle}{{\sigma}_{1}^{c}}\right)}^{2}\text{\hspace{0.17em}}$$ With $$\text{\hspace{0.17em}}{\sigma}_{a}={\sigma}_{11}+\text{}\langle \frac{{\sigma}_{22}+{\sigma}_{33}}{2}\rangle $$ 
$${F}_{3}={\left(\frac{\langle {\sigma}_{a}\rangle}{{\sigma}_{1}^{c}}\right)}^{2}$$ With $${\sigma}_{a}={\sigma}_{11}+\text{}\langle {\sigma}_{33}\rangle $$ $${F}_{4}={\left(\frac{\langle {\sigma}_{b}\rangle}{{\sigma}_{2}^{c}}\right)}^{2}$$ With $${\sigma}_{b}={\sigma}_{22}+\text{}\langle {\sigma}_{33}\rangle $$ 
Crush mode  $${F}_{3}={\left(\frac{\langle p\rangle}{{\sigma}_{c}}\right)}^{2}$$ With $$p=\frac{{\sigma}_{11}+{\sigma}_{22}+{\sigma}_{33}}{3}$$ 
$${F}_{5}={\left(\frac{\langle p\rangle}{{\sigma}_{c}}\right)}^{2}$$ With $$p=\frac{{\sigma}_{11}+{\sigma}_{22}+{\sigma}_{33}}{3}$$ 
Shear failure matrix mode  $${F}_{6}={\left(\frac{{\sigma}_{12}}{{\sigma}_{12}^{m}}\right)}^{2}$$  
Failure matrix mode  $${F}_{4}={\left(\frac{\langle {\sigma}_{22}\rangle}{{\sigma}_{2}^{t}}\right)}^{2}+{\left(\frac{{\sigma}_{23}}{{S}_{23}}\right)}^{2}+{\left(\frac{{\sigma}_{12}}{{S}_{12}}\right)}^{2}$$ Where, $$\begin{array}{l}{S}_{12}={\sigma}_{12}^{m}+\langle {\sigma}_{22}\rangle \mathrm{tan}\varphi \\ {S}_{23}={\sigma}_{23}^{m}+\langle {\sigma}_{22}\rangle \mathrm{tan}\varphi \end{array}$$ 

Delamination mode  $${F}_{5}={S}_{del}^{2}\left[{\left(\frac{\langle {\sigma}_{33}\rangle}{{\sigma}_{3}^{t}}\right)}^{2}+{\left(\frac{{\sigma}_{23}}{{\tilde{S}}_{23}}\right)}^{2}+{\left(\frac{{\sigma}_{13}}{{S}_{13}}\right)}^{2}\right]$$ Where, $$\begin{array}{l}{S}_{13}={\sigma}_{13}^{m}+\langle {\sigma}_{33}\rangle \mathrm{tan}\varphi \\ {\tilde{S}}_{23}={\sigma}_{23}^{m}+\langle {\sigma}_{33}\rangle \mathrm{tan}\varphi \end{array}$$ 
$${F}_{7}={S}_{del}^{2}\left[{\left(\frac{\langle {\sigma}_{33}\rangle}{{\sigma}_{3}^{t}}\right)}^{2}+{\left(\frac{{\sigma}_{23}}{{S}_{23}}\right)}^{2}+{\left(\frac{{\sigma}_{13}}{{S}_{13}}\right)}^{2}\right]$$ Where, $$\begin{array}{l}{S}_{13}={\sigma}_{13}^{m}+\langle {\sigma}_{33}\rangle \mathrm{tan}\varphi \\ {\tilde{S}}_{23}={\sigma}_{23}^{m}+\langle {\sigma}_{33}\rangle \mathrm{tan}\varphi \end{array}$$ 
In /FAIL/HASHIN, material strength $${\sigma}_{1}^{t},{\sigma}_{2}^{t},{\sigma}_{3}^{t},{\sigma}_{1}^{c},{\sigma}_{2}^{c}$$ are derived from tension/compression test for composite.
Crush strength $${\sigma}_{c}$$ and fiber shear strength $${\sigma}_{12}^{f}$$ may be obtained from a quasistatic punch shear test (QSPST). ^{6} Crush strength $${\sigma}_{c}$$ from the span to punch ratio (SPR) =0 and fiber shear strength $${\sigma}_{12}^{f}$$ from SPR=1.1.
$$\varphi $$ is the Coulomb friction angle. It is observed that composite shear strength is higher if the composite is also under compression (rather than under tension). This is due to the friction between matrix and fiber.
$${\sigma}_{12}^{m},{\sigma}_{13}^{m},{\sigma}_{23}^{m}$$ may be derived from a matrix shear test in three directions.
$${S}_{del}$$ is the scale factor for delamination criteria. It may be fitted with composite delamination experimental datain order for delamination failure to correlate with the damage area in experiment.
/FAIL/PUCK
 Fiber fracture: composite fails, due to the fiber reaching the tensile or compression strength limit.
 Inter fiber failure (IFF): composite fails, due to the fiber matrix cracking.
Damage criteria  If $$D$$=1, then failure. If $0\le D<1$ $$D$$, then no failure. With $$D=Max\left({e}_{f}(tensile),{e}_{f}(compression),{e}_{f}(ModeA),{e}_{f}(ModeB),{e}_{f}(ModeC)\right)$$ 
Fiber fraction failure  Tensile fiber failure mode: $${\sigma}_{11}>0$$
$${e}_{f}\left(tensile\right)=\frac{{\sigma}_{11}}{{\sigma}_{1}^{t}}$$ 
Compressive fiber failure mode: $${\sigma}_{11}<0$$ $${e}_{f}\left(compression\right)=\frac{\left{\sigma}_{11}\right}{{\sigma}_{1}^{c}}$$ 

Inter fiber failure (IFF) ^{2}  Mode A, if $${\sigma}_{22}>0$$: $${e}_{f}\left(ModeA\right)=\frac{1}{{\overline{\sigma}}_{12}}\left[\sqrt{{\left(\frac{{\overline{\sigma}}_{12}}{{\sigma}_{2}^{t}}{p}_{12}^{+}\right)}^{2}{\sigma}_{22}{}^{2}+{\sigma}_{12}{}^{2}}+{p}_{12}^{+}{\sigma}_{22}\right]$$ 
Mode C, if $${\sigma}_{22}<0$$: $${e}_{f}\left(ModeC\right)=\left[{\left(\frac{{\sigma}_{12}}{2(1+{p}_{22}^{}){\overline{\sigma}}_{12}}\right)}^{2}+{\left(\frac{{\sigma}_{22}}{{\sigma}_{2}^{c}}\right)}^{2}\right]\left(\frac{{\sigma}_{2}^{c}}{{\sigma}_{22}}\right)$$ 

Mode B: $${e}_{f}\left(ModeB\right)=\frac{1}{{\overline{\sigma}}_{12}}\left(\sqrt{{\sigma}_{12}^{2}+{\left({p}_{12}^{}{\sigma}_{{}_{22}}\right)}^{2}}+{p}_{12}^{}{\sigma}_{{}_{22}}\right)$$ 
In inter fiber failure, Mode A shows failure under tension in transverse fiber direction (90 degrees to fiber direction), and in this case, shear loading could reduce the failure limit.
If under compression in transverse fiber direction, at first increasing compression will increase composite shear loading (Mode B). If compression continues to increase, then shear loading will decrease (Mode C).
Input Parameters
For fiber fracture failure, you could obtain fiber strengths $${\sigma}_{1}^{t},{\sigma}_{1}^{c}$$ from tension and compression composite tests in the fiber direction.
For inter fiber failure, you could obtain strengths $${\sigma}_{2}^{t},{\sigma}_{2}^{c}$$ from tension and compression composite tests in the transverse fiber direction.
You could obtain shear strength $${\overline{\sigma}}_{12}$$ with a pure shear test ($${\sigma}_{2}={\sigma}_{1}=0$$).
With $${\sigma}_{2}^{t},{\sigma}_{2}^{c},{\overline{\sigma}}_{12}$$, then $${p}_{22}^{}$$ and $${p}_{12}^{}$$ for Mode C and Mode B may be determined.
With $${\sigma}_{2}^{t},{\overline{\sigma}}_{12}$$ and additional tensionshear tests in the transverse fiber direction, $${p}_{12}^{+}$$ may be determined. The additional tensionshear test in transverse fiber direction could take equal tensionshear (by $${\sigma}_{22}={\sigma}_{12}$$) loading.
For $${p}_{12}^{+},{p}_{12}^{},{p}_{22}^{}$$ parameters ^{3}. For carbon fiber composite, use $${p}_{12}^{+}=0.35,{p}_{12}^{}=0.3,{p}_{22}^{}=0.2$$ and for glass fiber composite, use $${p}_{12}^{+}=0.3,{p}_{12}^{}=0.25,{p}_{22}^{}=0.2$$.
/FAIL/LAD_DAMA
 Model I (DCB specimen
^{5})
$${Y}_{{d}_{3}}={\frac{\partial {E}_{D}}{\partial {d}_{3}}}_{\sigma =cst}=\frac{1}{2}\frac{{\langle {\sigma}_{33}\rangle}^{2}}{{K}_{3}{\left(1{d}_{3}\right)}^{2}}$$
 Model II (ENF
specimen ^{5})
$${Y}_{{d}_{2}}={\frac{\partial {E}_{D}}{\partial {d}_{2}}}_{\sigma =cst}=\frac{1}{2}\frac{{\sigma}_{32}{}^{2}}{{K}_{2}{\left(1{d}_{2}\right)}^{2}}$$
 Model III
$${Y}_{{d}_{1}}={\frac{\partial {E}_{D}}{\partial {d}_{1}}}_{\sigma =cst}=\frac{1}{2}\frac{{\sigma}_{31}{}^{2}}{{K}_{1}{\left(1{d}_{1}\right)}^{2}}$$
 $$t$$
 Thickness of the virtual interface. It may be assumed to be 1/5 layer thickness.
 $${G}_{13}$$, $${G}_{23}$$, $${E}_{33}$$
 From upper or lower layer.
 $${d}_{i}$$
 (with $$i$$=1,2,3), the damage variable.
After $${Y}_{0}$$ is reached, the damage variable starts to increase and when it reaches 1, $${d}_{3}=1$$ (thermodynamic force $${Y}_{{d}_{3}}$$ at this point then becomes the critical damage $${Y}_{c}$$). The composite could be considered as fully delaminated and may be deleted immediately or the stress may be reduced. In Radioss, the option $${\tau}_{\mathrm{max}}$$ is used to simulate exponential function stress reduction nd the stress at $${Y}_{c}$$ is ${\sigma}_{d}({t}_{r})$ (Stress Decrease in Damage).
 If $$d\ge 1$$, then take $$d=1$$
 If $$d<1$$, then $$d$$ is function of $$Y$$ (damage evaluation law):
(4) $$d=w(Y)=\frac{\langle \sqrt{Y}\sqrt{{Y}_{0}}\rangle}{\sqrt{{Y}_{c}}\sqrt{{Y}_{0}}}$$ $$Y={Y}_{{d}_{3}}+{\gamma}_{1}{Y}_{{d}_{1}}+{\gamma}_{2}{Y}_{{d}_{2}}$$ with $${Y}_{{d}_{i}}{}_{t}=\mathrm{sup}{Y}_{{d}_{i}}{}_{\tau \le t}$$
Here, $${\gamma}_{1},{\gamma}_{2}$$ are scale factors to consider two other delamination modes. This may be validated with experiments (DCB and ENF specimen test ^{5}).
 If $$d=1$$, then $\dot{d}=const.$
 If $$d<1$$, then $\dot{d}=\frac{k}{a}\left[1\mathrm{exp}\left(a\langle w\left(Y\right)d\rangle \right)\right]$
/FAIL/CHANG
 Fiber mode: composite fails, due to fiber rupture in tension or fiber buckling in compression.
 Matrix mode: composite fails, due to matrix failure under tension or compression.
Damage criteria  If $$D=1$$, then failure. If $0\le D<1$ $$D$$, then no failure. With $$D=Max\left({e}_{f}{}^{2},{e}_{c}{}^{2},{e}_{m}{}^{2},{e}_{d}{}^{2}\right)$$. 

Fiber breakage  Tensile fiber mode ${\sigma}_{11}>0$  $${e}_{f}{}^{2}={\left(\frac{{\sigma}_{11}}{{\sigma}_{1}^{t}}\right)}^{2}+\beta {\left(\frac{{\sigma}_{12}}{{\overline{\sigma}}_{12}}\right)}^{2}$$ 
Compression fiber mode ${\sigma}_{11}<0$  $${e}_{c}{}^{2}={\left(\frac{{\sigma}_{11}}{{\sigma}_{1}^{c}}\right)}^{2}$$  
Matrix cracking  Tensile matrix mode ${\sigma}_{22}>0$  $${e}_{m}{}^{2}={\left(\frac{{\sigma}_{22}}{{\sigma}_{2}^{t}}\right)}^{2}+\text{\hspace{0.17em}}\text{}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\left(\frac{{\sigma}_{12}}{{\overline{\sigma}}_{12}}\right)}^{2}$$ 
Compressive matrix mode ${\sigma}_{22}<0$  $${e}_{d}{}^{2}={\left(\frac{{\sigma}_{22}}{2{\overline{\sigma}}_{12}}\right)}^{2}+\left[{\left(\frac{{\sigma}_{2}^{c}}{2{\overline{\sigma}}_{12}}\right)}^{2}1\right]\frac{{\sigma}_{22}}{{\sigma}_{2}^{c}}+{\left(\frac{{\sigma}_{12}}{{\overline{\sigma}}_{12}}\right)}^{2}$$ 
 direction 1
 Fiber direction.
 $${\sigma}_{1}^{t},{\sigma}_{1}^{c}$$
 Fiber tensile/compressive strength.
 $${\sigma}_{2}^{t},{\sigma}_{2}^{c}$$
 Matrix strength.
 $${\overline{\sigma}}_{12}$$
 Shear strength in composite ply plane.
 $\beta $
 Shear scale factor, which can be determined experimentally.
Stress Decrease in Damage
 HASHIN:
$$D=Max\left({F}_{1},{F}_{2},{F}_{3},{F}_{4}\right)\ge 1$$
 PUCK:
$$D=Max\left({e}_{f}(tensile),{e}_{f}(compression),{e}_{f}(ModeA),{e}_{f}(ModeB),{e}_{f}(ModeC)\right)\ge 1$$
 LAD_DAMA:
$$d\ge 1$$
 CHANG:
$$D=Max\left({e}_{f}{}^{2},{e}_{c}{}^{2},{e}_{m}{}^{2},{e}_{d}{}^{2}\right)\ge 1$$
with, $$\text{\hspace{0.17em}}\text{\hspace{0.05em}}t\ge {t}_{r}$$
 $${\sigma}_{d}\left({t}_{r}\right)$$
 Stress components when damage is reached $D\ge 1$.
 $${t}_{r}$$
 Time of $${\sigma}_{d}\left({t}_{r}\right)$$.
 ${\tau}_{\mathrm{max}}$
 Time of dynamic relaxation.