Composite Failure Model

In Radioss the following composite failure models may be used to describe composite material failure.

/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

In HASHIN failure, two primary failure modes are considered.

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 $σ_{11}$ $σ_{11}$ will not be considered in this mode.

Fibers in uni-directional lamina model ¹ are only in one direction and in fabric lamina model are in two directions.

	Uni-directional Lamina Model	Fabric Lamina Model
Damage criteria	If $D$ $D$ =1, then failure. If $0 \leq D < 1$ $0 \leq D < 1$ , $D$ $D$ then no failure. With $D = M a x (F_{1}, F_{2}, F_{3}, F_{4})$ $D = M a x (F_{1}, F_{2}, F_{3}, F_{4})$	If $D$ $D$ =1, then failure. If $0 \leq D < 1$ $0 \leq D < 1$ , $D$ $D$ then no failure. With $D = M a x (F_{1}, F_{2}, F_{3}, F_{4})$ $D = M a x (F_{1}, F_{2}, F_{3}, F_{4})$
Tensile/shear fiber mode	$F_{1} = {(\frac{⟨ σ_{11} ⟩}{σ_{1}^{t}})}^{2} + (\frac{σ_{12}^{2} + σ_{13}^{2}}{σ_{12}^{f}^{2}})$ $F_{1} = {(\frac{〈 σ_{11} 〉}{σ_{1}^{t}})}^{2} + (\frac{σ_{12}^{2} + σ_{13}^{2}}{σ_{12}^{f}^{2}})$	$F_{1} = {(\frac{⟨ σ_{11} ⟩}{σ_{1}^{t}})}^{2} + (\frac{σ_{12}^{2} + σ_{13}^{2}}{σ_{a}^{f}^{2}})$ $F_{1} = {(\frac{〈 σ_{11} 〉}{σ_{1}^{t}})}^{2} + (\frac{σ_{12}^{2} + σ_{13}^{2}}{σ_{a}^{f}^{2}})$ $F_{2} = {(\frac{⟨ σ_{22} ⟩}{σ_{2}^{t}})}^{2} + (\frac{σ_{12}^{2} + σ_{23}^{2}}{σ_{b}^{f}^{2}})$ $F_{2} = {(\frac{〈 σ_{22} 〉}{σ_{2}^{t}})}^{2} + (\frac{σ_{12}^{2} + σ_{23}^{2}}{σ_{b}^{f}^{2}})$ With, $σ_{a}^{f} = σ_{12}^{f}, σ_{b}^{f} = σ_{12}^{f} \frac{σ_{2}^{t}}{σ_{1}^{t}}$ $σ_{a}^{f} = σ_{12}^{f} , σ_{b}^{f} = σ_{12}^{f} \frac{σ_{2}^{t}}{σ_{1}^{t}}$
Compression fiber mode	$F_{2} = {(\frac{⟨ σ_{a} ⟩}{σ_{1}^{c}})}^{2}$ $F_{2} = {(\frac{〈 σ_{a} 〉}{σ_{1}^{c}})}^{2}$ With $σ_{a} = - σ_{11} + ⟨ - \frac{σ_{22} + σ_{33}}{2} ⟩$ $σ_{a} = - σ_{11} + 〈 - \frac{σ_{22} + σ_{33}}{2} 〉$	$F_{3} = {(\frac{⟨ σ_{a} ⟩}{σ_{1}^{c}})}^{2}$ $F_{3} = {(\frac{〈 σ_{a} 〉}{σ_{1}^{c}})}^{2}$ With $σ_{a} = - σ_{11} + ⟨ - σ_{33} ⟩$ $σ_{a} = - σ_{11} + 〈 - σ_{33} 〉$ $F_{4} = {(\frac{⟨ σ_{b} ⟩}{σ_{2}^{c}})}^{2}$ $F_{4} = {(\frac{〈 σ_{b} 〉}{σ_{2}^{c}})}^{2}$ With $σ_{b} = - σ_{22} + ⟨ - σ_{33} ⟩$ $σ_{b} = - σ_{22} + 〈 - σ_{33} 〉$
Crush mode	$F_{3} = {(\frac{⟨ p ⟩}{σ_{c}})}^{2}$ $F_{3} = {(\frac{〈 p 〉}{σ_{c}})}^{2}$ With $p = - \frac{σ_{11} + σ_{22} + σ_{33}}{3}$ $p = - \frac{σ_{11} + σ_{22} + σ_{33}}{3}$	$F_{5} = {(\frac{⟨ p ⟩}{σ_{c}})}^{2}$ $F_{5} = {(\frac{〈 p 〉}{σ_{c}})}^{2}$ With $p = - \frac{σ_{11} + σ_{22} + σ_{33}}{3}$ $p = - \frac{σ_{11} + σ_{22} + σ_{33}}{3}$
Shear failure matrix mode		$F_{6} = {(\frac{σ_{12}}{σ_{12}^{m}})}^{2}$ $F_{6} = {(\frac{σ_{12}}{σ_{12}^{m}})}^{2}$
Failure matrix mode	$F_{4} = {(\frac{⟨ σ_{22} ⟩}{σ_{2}^{t}})}^{2} + {(\frac{σ_{23}}{S_{23}})}^{2} + {(\frac{σ_{12}}{S_{12}})}^{2}$ $F_{4} = {(\frac{〈 σ_{22} 〉}{σ_{2}^{t}})}^{2} + {(\frac{σ_{23}}{S_{23}})}^{2} + {(\frac{σ_{12}}{S_{12}})}^{2}$ Where, $\begin{matrix} S_{12} = σ_{12}^{m} + ⟨ - σ_{22} ⟩ tan ϕ S_{23} = σ_{23}^{m} + ⟨ - σ_{22} ⟩ tan ϕ \end{matrix}$ $\begin{array}{l} S_{12} = σ_{12}^{m} + 〈 - σ_{22} 〉 \tan ϕ \\ S_{23} = σ_{23}^{m} + 〈 - σ_{22} 〉 \tan ϕ \end{array}$
Delamination mode	$F_{5} = S_{d e l}^{2} ⎡ ⎣ {(\frac{⟨ σ_{33} ⟩}{σ_{3}^{t}})}^{2} + {(\frac{σ_{23}}{{˜ S}_{23}})}^{2} + {(\frac{σ_{13}}{S_{13}})}^{2} ⎤ ⎦$ $F_{5} = S_{d e l}^{2} [{(\frac{〈 σ_{33} 〉}{σ_{3}^{t}})}^{2} + {(\frac{σ_{23}}{{\tilde{S}}_{23}})}^{2} + {(\frac{σ_{13}}{S_{13}})}^{2}]$ Where, $\begin{matrix} S_{13} = σ_{13}^{m} + ⟨ - σ_{33} ⟩ tan ϕ {˜ S}_{23} = σ_{23}^{m} + ⟨ - σ_{33} ⟩ tan ϕ \end{matrix}$ $\begin{array}{l} S_{13} = σ_{13}^{m} + 〈 - σ_{33} 〉 \tan ϕ \\ {\tilde{S}}_{23} = σ_{23}^{m} + 〈 - σ_{33} 〉 \tan ϕ \end{array}$	$F_{7} = S_{d e l}^{2} ⎡ ⎣ {(\frac{⟨ σ_{33} ⟩}{σ_{3}^{t}})}^{2} + {(\frac{σ_{23}}{S_{23}})}^{2} + {(\frac{σ_{13}}{S_{13}})}^{2} ⎤ ⎦$ $F_{7} = S_{d e l}^{2} [{(\frac{〈 σ_{33} 〉}{σ_{3}^{t}})}^{2} + {(\frac{σ_{23}}{S_{23}})}^{2} + {(\frac{σ_{13}}{S_{13}})}^{2}]$ Where, $\begin{matrix} S_{13} = σ_{13}^{m} + ⟨ - σ_{33} ⟩ tan ϕ {˜ S}_{23} = σ_{23}^{m} + ⟨ - σ_{33} ⟩ tan ϕ \end{matrix}$ $\begin{array}{l} S_{13} = σ_{13}^{m} + 〈 - σ_{33} 〉 \tan ϕ \\ {\tilde{S}}_{23} = σ_{23}^{m} + 〈 - σ_{33} 〉 \tan ϕ \end{array}$

Note:

⟨ a ⟩ = {\begin{matrix} a i f a > 0 0 i f a < 0 \end{matrix}

$〈 a 〉 = {\begin{matrix} a i f a > 0 \\ 0 i f a < 0 \end{matrix}$

In /FAIL/HASHIN, material strength $σ_{1}^{t}, σ_{2}^{t}, σ_{3}^{t}, σ_{1}^{c}, σ_{2}^{c}$ $σ_{1}^{t}, σ_{2}^{t}, σ_{3}^{t}, σ_{1}^{c}, σ_{2}^{c}$ are derived from tension/compression test for composite.

Crush strength $σ_{c}$ $σ_{c}$ and fiber shear strength $σ_{12}^{f}$ $σ_{12}^{f}$ may be obtained from a quasi-static punch shear test (QS-PST). ⁶ Crush strength $σ_{c}$ $σ_{c}$ from the span to punch ratio (SPR) =0 and fiber shear strength $σ_{12}^{f}$ $σ_{12}^{f}$ from SPR=1.1.

$ϕ$ $ϕ$ 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.

The shear strength is assumed to be proportional to compression stress and is computed as:(1)

S_{12} = σ_{12}^{m} + ⟨ - σ_{22} ⟩ tan ϕ

$S_{12} = σ_{12}^{m} + 〈 - σ_{22} 〉 \tan ϕ$

Friction angle

ϕ

$ϕ$ may be fitted with an Off-Axis compression test with different angles

θ

$θ$ (for example,

30^{\circ}, 45^{\circ}, 60^{\circ}

$30^{\circ}, 45^{\circ}, 60^{\circ}$ , …).

$σ_{12}^{m}, σ_{13}^{m}, σ_{23}^{m}$ $σ_{12}^{m}, σ_{13}^{m}, σ_{23}^{m}$ may be derived from a matrix shear test in three directions.

$S_{d e l}$ $S_{d e l}$ 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

In Puck failure, two types of failure are considered.

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 $D$ $D$ =1, then failure. If $0 \leq D < 1$ $0 \leq D < 1$ $D$ $D$ , then no failure. With $D = M a x (e_{f} (t e n s i l e), e_{f} (c o m p r e s s i o n), e_{f} (M o d e A), e_{f} (M o d e B), e_{f} (M o d e C))$ $D = M a x (e_{f} (t e n s i l e), e_{f} (c o m p r e s s i o n), e_{f} (M o d e A), e_{f} (M o d e B), e_{f} (M o d e C))$
Fiber fraction failure	Tensile fiber failure mode: $σ_{11} > 0$ $σ_{11} > 0$ $e_{f} (t e n s i l e) = \frac{σ_{11}}{σ_{1}^{t}}$ $e_{f} (t e n s i l e) = \frac{σ_{11}}{σ_{1}^{t}}$
Fiber fraction failure	Compressive fiber failure mode: $σ_{11} < 0$ $σ_{11} < 0$ $e_{f} (c o m p r e s s i o n) = \frac{\| σ_{11} \|}{σ_{1}^{c}}$ $e_{f} (c o m p r e s s i o n) = \frac{\| σ_{11} \|}{σ_{1}^{c}}$
Inter fiber failure (IFF) ²	Mode A, if $σ_{22} > 0$ $σ_{22} > 0$ : Figure 4. $e_{f} (M o d e A) = \frac{1}{{¯ σ}_{12}} ⎡ ⎢ ⎣ \sqrt{{(\frac{{¯ σ}_{12}}{σ_{2}^{t}} - p_{12}^{+})}^{2} σ_{22}^{2} + σ_{12}^{2}} + p_{12}^{+} σ_{22} ⎤ ⎥ ⎦$ $e_{f} (M o d e A) = \frac{1}{{\bar{σ}}_{12}} [\sqrt{{(\frac{{\bar{σ}}_{12}}{σ_{2}^{t}} - p_{12}^{+})}^{2} σ_{22}^{2} + σ_{12}^{2}} + p_{12}^{+} σ_{22}]$
	Mode C, if $σ_{22} < 0$ $σ_{22} < 0$ : Figure 5. $e_{f} (M o d e C) = ⎡ ⎣ {(\frac{σ_{12}}{2 (1 + p_{22}^{-}) {¯ σ}_{12}})}^{2} + {(\frac{σ_{22}}{σ_{2}^{c}})}^{2} ⎤ ⎦ (\frac{σ_{2}^{c}}{- σ_{22}})$ $e_{f} (M o d e C) = [{(\frac{σ_{12}}{2 (1 + p_{22}^{-}) {\bar{σ}}_{12}})}^{2} + {(\frac{σ_{22}}{σ_{2}^{c}})}^{2}] (\frac{σ_{2}^{c}}{- σ_{22}})$
	Mode B: Figure 6. $e_{f} (M o d e B) = \frac{1}{{¯ σ}_{12}} (\sqrt{σ_{12}^{2} + {(p_{12}^{-} σ_{_{22}})}^{2}} + p_{12}^{-} σ_{_{22}})$ $e_{f} (M o d e B) = \frac{1}{{\bar{σ}}_{12}} (\sqrt{σ_{12}^{2} + {(p_{12}^{-} σ_{_{22}})}^{2}} + p_{12}^{-} σ_{_{22}})$

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 $σ_{1}^{t}, σ_{1}^{c}$ $σ_{1}^{t}, σ_{1}^{c}$ from tension and compression composite tests in the fiber direction.

For inter fiber failure, you could obtain strengths $σ_{2}^{t}, σ_{2}^{c}$ $σ_{2}^{t}, σ_{2}^{c}$ from tension and compression composite tests in the transverse fiber direction.

You could obtain shear strength ${¯ σ}_{12}$ ${\bar{σ}}_{12}$ with a pure shear test ( $σ_{2} = σ_{1} = 0$ $σ_{2} = σ_{1} = 0$ ).

With $σ_{2}^{t}, σ_{2}^{c}, {¯ σ}_{12}$ $σ_{2}^{t}, σ_{2}^{c}, {\bar{σ}}_{12}$ , then $p_{22}^{-}$ $p_{22}^{-}$ and $p_{12}^{-}$ $p_{12}^{-}$ for Mode C and Mode B may be determined.

With $σ_{2}^{t}, {¯ σ}_{12}$ $σ_{2}^{t}, {\bar{σ}}_{12}$ and additional tension-shear tests in the transverse fiber direction, $p_{12}^{+}$ $p_{12}^{+}$ may be determined. The additional tension-shear test in transverse fiber direction could take equal tension-shear (by $σ_{22} = σ_{12}$ $σ_{22} = σ_{12}$ ) loading.

Then you may derive the fracture curve in

σ_{22} - σ_{12}

$σ_{22} - σ_{12}$ plane, as shown below.

For $p_{12}^{+}, p_{12}^{-}, p_{22}^{-}$ $p_{12}^{+}, p_{12}^{-}, p_{22}^{-}$ parameters ³. For carbon fiber composite, use $p_{12}^{+} = 0.35, p_{12}^{-} = 0.3, p_{22}^{-} = 0.2$ $p_{12}^{+} = 0.35, p_{12}^{-} = 0.3, p_{22}^{-} = 0.2$ and for glass fiber composite, use $p_{12}^{+} = 0.3, p_{12}^{-} = 0.25, p_{22}^{-} = 0.2$ $p_{12}^{+} = 0.3, p_{12}^{-} = 0.25, p_{22}^{-} = 0.2$ .

/FAIL/LAD_DAMA

/FAIL/LAD_DAMA is used to describe delamination between composite layers (damage propagation in the matrix). Assume that layers are connected through a virtual interface (contact).

For example, if composite is under loading, shown below, the traction

σ

$σ$ and displacement in direction 3

δ

$δ$ is as shown in curve.

The area under the traction versus displacement curve is the absorbed energy by delamination, which is also termed the strain energy of the damage interface. The failure is governed by this strain energy is described below, considering 3 Modes of delamination:(2)

E_{D} = \frac{1}{2} [\frac{{⟨ σ_{33} ⟩}^{2}}{K_{3} (1 - d_{3})} + \frac{{⟨ - σ_{33} ⟩}^{2}}{K_{3}} + \frac{σ_{32}^{2}}{K_{2} (1 - d_{2})} + \frac{σ_{31}^{2}}{K_{1} (1 - d_{1})}]

$E_{D} = \frac{1}{2} [\frac{{〈 σ_{33} 〉}^{2}}{K_{3} (1 - d_{3})} + \frac{{〈 - σ_{33} 〉}^{2}}{K_{3}} + \frac{σ_{32}^{2}}{K_{2} (1 - d_{2})} + \frac{σ_{31}^{2}}{K_{1} (1 - d_{1})}]$

Where,

σ_{33}, σ_{32}, σ_{31}

$σ_{33}, σ_{32}, σ_{31}$ are the stresses shown below in three modes of delamination behavior.

With strain energy of delamination

E_{D}

$E_{D}$ , the thermodynamic force (contact force of virtual interface), also termed damage energy release rates may be calculated for these three modes:

Model I (DCB specimen ⁵)
$Y_{d_{3}} = {\frac{\partial E_{D}}{\partial d_{3}} ∣ ∣ ∣}_{σ = c s t} = \frac{1}{2} \frac{{⟨ σ_{33} ⟩}^{2}}{K_{3} {(1 - d_{3})}^{2}}$ $Y_{d_{3}} = {\frac{\partial E_{D}}{\partial d_{3}} |}_{σ = c s t} = \frac{1}{2} \frac{{〈 σ_{33} 〉}^{2}}{K_{3} {(1 - d_{3})}^{2}}$
Model II (ENF specimen ⁵)
$Y_{d_{2}} = {\frac{\partial E_{D}}{\partial d_{2}} ∣ ∣ ∣}_{σ = c s t} = \frac{1}{2} \frac{σ_{32}^{2}}{K_{2} {(1 - d_{2})}^{2}}$ $Y_{d_{2}} = {\frac{\partial E_{D}}{\partial d_{2}} |}_{σ = c s t} = \frac{1}{2} \frac{σ_{32}^{2}}{K_{2} {(1 - d_{2})}^{2}}$
Model III
$Y_{d_{1}} = {\frac{\partial E_{D}}{\partial d_{1}} ∣ ∣ ∣}_{σ = c s t} = \frac{1}{2} \frac{σ_{31}^{2}}{K_{1} {(1 - d_{1})}^{2}}$ $Y_{d_{1}} = {\frac{\partial E_{D}}{\partial d_{1}} |}_{σ = c s t} = \frac{1}{2} \frac{σ_{31}^{2}}{K_{1} {(1 - d_{1})}^{2}}$

Where,

K_{3}, K_{2}, K_{1}

$K_{3}, K_{2}, K_{1}$ are the stiffnesses of the virtual interface, also called interlaminar stiffness. These values may be computed as:(3)

K_{3} = \frac{2 E_{33}}{t}

$K_{3} = \frac{2 E_{33}}{t}$

K_{2} = \frac{2 G_{23}}{t}

$K_{2} = \frac{2 G_{23}}{t}$

K_{1} = \frac{2 G_{13}}{t}

$K_{1} = \frac{2 G_{13}}{t}$

Where,

$t$ $t$: Thickness of the virtual interface. It may be assumed to be 1/5 layer thickness.
$G_{13}$ $G_{13}$ , $G_{23}$ $G_{23}$ , $E_{33}$ $E_{33}$: From upper or lower layer.
$d_{i}$ $d_{i}$: (with $i$ $i$ =1,2,3), the damage variable.; It has a range from 0~1. It starts to accumulate once the composite reaches $Y_{0}$ $Y_{0}$ .

An example of Mode I, during traction in direction 3, at the beginning

d_{3}

$d_{3}$ always remains 0 until thermodynamic force

Y_{d_{3}}

$Y_{d_{3}}$ reaches

Y_{0}

$Y_{0}$ (left figure).

After $Y_{0}$ $Y_{0}$ is reached, the damage variable starts to increase and when it reaches 1, $d_{3} = 1$ $d_{3} = 1$ (thermodynamic force $Y_{d_{3}}$ $Y_{d_{3}}$ at this point then becomes the critical damage $Y_{c}$ $Y_{c}$ ). The composite could be considered as fully delaminated and may be deleted immediately or the stress may be reduced. In Radioss, the option $τ_{max}$ $τ_{\max}$ is used to simulate exponential function stress reduction nd the stress at $Y_{c}$ $Y_{c}$ is $σ_{d} (t_{r})$ $σ_{d} (t_{r})$ (Stress Decrease in Damage).

The relation with thermodynamic force

Y_{d_{i}}

$Y_{d_{i}}$ and

d_{i}

$d_{i}$ is:

If $d \geq 1$ $d \geq 1$ , then take $d = 1$ $d = 1$
If $d < 1$ $d < 1$ , then $d$ $d$ is function of $Y$ $Y$ (damage evaluation law):(4) $d = w (Y) = \frac{⟨ \sqrt{Y} - \sqrt{Y_{0}} ⟩}{\sqrt{Y_{c}} - \sqrt{Y_{0}}}$ $d = w (Y) = \frac{〈 \sqrt{Y} - \sqrt{Y_{0}} 〉}{\sqrt{Y_{c}} - \sqrt{Y_{0}}}$
$Y = Y_{d_{3}} + γ_{1} Y_{d_{1}} + γ_{2} Y_{d_{2}}$ $Y = Y_{d_{3}} + γ_{1} Y_{d_{1}} + γ_{2} Y_{d_{2}}$ with $Y_{d_{i}} |_{t} = sup Y_{d_{i}} |_{τ \leq t}$ $Y_{d_{i}} |_{t} = \sup Y_{d_{i}} |_{τ \leq t}$

Here, $γ_{1}, γ_{2}$ $γ_{1}, γ_{2}$ are scale factors to consider two other delamination modes. This may be validated with experiments (DCB and ENF specimen test ⁵).

In the example of Mode I,

γ_{1}, γ_{2}

$γ_{1}, γ_{2}$ may be 0 as this is pure delamination in direction 3 and then

Y = Y_{d_{3}}

$Y = Y_{d_{3}}$ , the relation of

Y_{d_{3}}

$Y_{d_{3}}$ and

d_{3}

$d_{3}$ is:

How quickly will the damage variable increase? The damage velocity

˙ d

$\dot{d}$ (also called damage evaluation law) is computed as:

If $d = 1$ $d = 1$ , then $˙ d = c o n s t .$ $\dot{d} = c o n s t .$
If $d < 1$ $d < 1$ , then $˙ d = \frac{k}{a} [1 - exp (- a ⟨ w (Y) - d ⟩)]$ $\dot{d} = \frac{k}{a} [1 - \exp (- a 〈 w (Y) - d 〉)]$

\frac{k}{a}

$\frac{k}{a}$ is the maximum damage rate, which means minimum duration of the failure phenomenon. Its reciprocal,

\frac{a}{k}

$\frac{a}{k}$ is called the characteristic time, which may be obtained with a one-dimension tensile test. ⁷

Through tensile sample with different stress to find the minimum time of composite damage

Δ t

$Δ t$ , the

σ - Δ t

$σ - Δ t$ curve is vertical asymptote corresponding to the characteristic time

\frac{a}{k}

$\frac{a}{k}$ .

Parameters

a

$a$ and

k

$k$ govern the damage evaluation law. For example, with constant parameter

a

$a$ (here

a = 1

$a = 1$ ), decreasing values of

k

$k$ leads to more brittle failure of the composite.

With constant parameter

k

$k$ (here

k = 1

$k = 1$ ), increasing values of

a

$a$ leads to more brittle failure of the composite.

/FAIL/CHANG

In Chang-Chang failure, two primary failure modes are considered.

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.

This failure criteria is used only for shell elements.

Damage criteria		If $D = 1$ $D = 1$ , then failure. If $0 \leq D < 1$ $0 \leq D < 1$ $D$ $D$ , then no failure. With $D = M a x (e_{f}^{2}, e_{c}^{2}, e_{m}^{2}, e_{d}^{2})$ $D = M a x (e_{f}^{2}, e_{c}^{2}, e_{m}^{2}, e_{d}^{2})$ .
Fiber breakage	Tensile fiber mode $σ_{11} > 0$ $σ_{11} > 0$	$e_{f}^{2} = {(\frac{σ_{11}}{σ_{1}^{t}})}^{2} + β {(\frac{σ_{12}}{{¯ σ}_{12}})}^{2}$ $e_{f}^{2} = {(\frac{σ_{11}}{σ_{1}^{t}})}^{2} + β {(\frac{σ_{12}}{{\bar{σ}}_{12}})}^{2}$
Fiber breakage	Compression fiber mode $σ_{11} < 0$ $σ_{11} < 0$	$e_{c}^{2} = {(\frac{σ_{11}}{σ_{1}^{c}})}^{2}$ $e_{c}^{2} = {(\frac{σ_{11}}{σ_{1}^{c}})}^{2}$
Matrix cracking	Tensile matrix mode $σ_{22} > 0$ $σ_{22} > 0$	$e_{m}^{2} = {(\frac{σ_{22}}{σ_{2}^{t}})}^{2} + {(\frac{σ_{12}}{{¯ σ}_{12}})}^{2}$ $e_{m}^{2} = {(\frac{σ_{22}}{σ_{2}^{t}})}^{2} + {(\frac{σ_{12}}{{\bar{σ}}_{12}})}^{2}$
Matrix cracking	Compressive matrix mode $σ_{22} < 0$ $σ_{22} < 0$	$e_{d}^{2} = {(\frac{σ_{22}}{2 {¯ σ}_{12}})}^{2} + [{(\frac{σ_{2}^{c}}{2 {¯ σ}_{12}})}^{2} - 1] \frac{σ_{22}}{σ_{2}^{c}} + {(\frac{σ_{12}}{{¯ σ}_{12}})}^{2}$ $e_{d}^{2} = {(\frac{σ_{22}}{2 {\bar{σ}}_{12}})}^{2} + [{(\frac{σ_{2}^{c}}{2 {\bar{σ}}_{12}})}^{2} - 1] \frac{σ_{22}}{σ_{2}^{c}} + {(\frac{σ_{12}}{{\bar{σ}}_{12}})}^{2}$

Where,

direction 1: Fiber direction.
$σ_{1}^{t}, σ_{1}^{c}$ $σ_{1}^{t}, σ_{1}^{c}$: Fiber tensile/compressive strength.
$σ_{2}^{t}, σ_{2}^{c}$ $σ_{2}^{t}, σ_{2}^{c}$: Matrix strength.; Tensile or compressive loading in direction 2 (transverse to direction 1).
${¯ σ}_{12}$ ${\bar{σ}}_{12}$: Shear strength in composite ply plane.
$β$ $β$: Shear scale factor, which can be determined experimentally.

Stress Decrease in Damage

After reaching the damage criteria:

HASHIN:
$D = M a x (F_{1}, F_{2}, F_{3}, F_{4}) \geq 1$ $D = M a x (F_{1}, F_{2}, F_{3}, F_{4}) \geq 1$
PUCK:
$D = M a x (e_{f} (t e n s i l e), e_{f} (c o m p r e s s i o n), e_{f} (M o d e A), e_{f} (M o d e B), e_{f} (M o d e C)) \geq 1$ $D = M a x (e_{f} (t e n s i l e), e_{f} (c o m p r e s s i o n), e_{f} (M o d e A), e_{f} (M o d e B), e_{f} (M o d e C)) \geq 1$
LAD_DAMA:
$d \geq 1$ $d \geq 1$
CHANG:
$D = M a x (e_{f}^{2}, e_{c}^{2}, e_{m}^{2}, e_{d}^{2}) \geq 1$ $D = M a x (e_{f}^{2}, e_{c}^{2}, e_{m}^{2}, e_{d}^{2}) \geq 1$

Stresses start decreasing and decrease gradually by using an exponential function to avoid numerical instabilities.(5)

$\begin{array}{l} σ (t) = σ_{d} (t_{r}) \cdot f (t) \\ = σ_{d} (t_{r}) \cdot \exp (- \frac{t - t_{r}}{τ_{\max}}) \end{array}$

with, $t \geq t_{r}$

The option

$τ_{\max}$ controls how gradually the stress is decreased in damage.

Where,

$σ_{d} (t_{r})$: Stress components when damage is reached $D \geq 1$ .
$t_{r}$: Time of $σ_{d} (t_{r})$ .
$τ_{\max}$: Time of dynamic relaxation.; The higher the value of $τ_{\max}$ , the slower stress decreases during damage. Normally, it takes 10~20 time step.

References

Hashin, Z., "Failure Criteria for Unidirectional Fiber Composites," Journal of Applied Mechanics, Vol. 47, 1980, pp. 329-334.

A. Puck, J. Kopp, and M. Knops., “Failure analysis of FRP laminates by means of physically based phenomenological models”. Composites Science Technology, 62. pp. 1633-1662. 2002.

A. Puck, J. Kopp, and M. Knops. “Guidelines for the determination of the parameters in Puck's action plane strength criterion”. Composites Science Technology 62. pp. 371-378. 2002.

L. Gornet, “Finite Element Damage Prediction of Composite Structures”.

Ladevèze, P., Allix, O., Douchin, B., Lévêque, D., “A Computational Method for damage Intensity Prediction in a Laminated Composite Structure”, Computational mechanics—New Trends and Applications In: Idelsohn, S., Oñate E., and Dvorkin E., (eds.) CIMNE, Barcelona, Spain (1998).

Gama B.A., Gillespie J.W., Punch Shear Behavior of Composites at Low and High Rates[M]// Fracture of Nano and Engineering Materials and Structures. Springer Netherlands, 2006.

Allix, O. & Deü, Jean-François. (1997). Delay-damage modeling for fracture prediction of laminated composites under dynamic loading. Engineering Transactions. 45. 29-46.