A discrete damage zone model for mixed-mode delamination of composites under high-cycle fatigue

Abstract

A discrete damage zone model is developed to describe the mode-mix ratio and temperature dependent delamination of laminated composite materials under high cycle fatigue loading within the framework of the finite element method. In this approach, discrete nonlinear spring elements are placed at the finite element nodes of the laminate interface, and a combination of static and fatigue damage growth laws is used to define its constitutive behavior. The model is implemented in the commercial software Abaqus using the user element subroutine. The static damage model parameters are estimated from fracture mechanics principles, whereas the fatigue damage model parameters are calibrated by fitting the numerical results to published experimental data. A quadratic relation is proposed to describe the non-monotonic variation of fatigue damage model parameters with mode-mix ratio. Next, an Arrhenius relation is proposed for the temperature dependence of fatigue damage, in addition to the incorporation of the temperature dependence of critical fracture energy. The model is convergent upon mesh refinement; however, for accurate prediction the mesh size used for model calibration should be sufficiently small. The model predicted fatigue crack growth rates are in agreement with those obtained from a quadratic relation for the Paris law parameters for variable mode mix conditions, thus verifying the approach. While the model captures the temperature effects on delamination for mode I and 50 % mode II, our prediction deviates from experiments for pure mode II, since the corresponding damage mechanism entirely changes with temperature.

Appendices

Appendix 1: Relation between $\rho$ and $\Psi$

The mixed mode bending test was proposed by Reeder and Crews (1990), shown in Fig. 14.

Fig. 14

The geometry and boundary conditions for mixed mode bending (MMB) test

A concentrated load $P$ is applied through the rigid lever, and the forces transferred to the beam obtained from equilibrium are,

$$\begin{aligned} P_{1}=P\dfrac{c}{L},~~~P_{2}=P\dfrac{c+L}{L} \end{aligned}$$

(46)

This mixed mode bending (MMB) system can be considered as the superposition of Mode I and Mode II (Reeder and Crews 1990), as shown in Fig. 15. Due to the y-axis symmetry of the mode II four point end notch fracture (ENF) test, it is sufficient to consider one-half of the beam as a cantilever (Robinson et al. 2005; Williams 1988)). Now, following Eqs. (38) and (41), we can write the strain energy release rate under Mode I and Mode II as,

$$\begin{aligned} G_{\mathrm {I}}=\left( \dfrac{3c-L}{4L} \right) ^{2} \dfrac{P^2 a_{0}^{2}}{WEI} \end{aligned}$$

(47)

$$\begin{aligned} G_{\mathrm {II}}=\left( \dfrac{c+L}{4L} \right) ^{2} \dfrac{3P^2 a_{0}^2}{4WEI} \end{aligned}$$

(48)

Substituting the above two expressions for $G_{I}$ and $G_{II}$ into Eq. 26 we get the following relation for the mode ratio $\Psi$:

$$\begin{aligned} \Psi =\dfrac{G_{\mathrm {II}}}{G_{\mathrm {I}}+G_{\mathrm {II}}}=\dfrac{3\left( \dfrac{c}{L}+1 \right) ^2}{3\left( \dfrac{c}{L}+1 \right) ^2+4\left( 3\dfrac{c}{L}-1 \right) ^2}. \end{aligned}$$

(49)

An alternative mixed mode test is configured as shown in Fig. 16, which is equivalent to the MMB test shown in Fig. 14. The forces applied on the upper and lower arm are obtained from the superposition of Mode I and Mode II, given as,

$$\begin{aligned} P_{\mathrm {u}}=P\dfrac{c+L}{4L}+P\dfrac{3c-L}{4L} \end{aligned}$$

(50)

$$\begin{aligned} P_{\mathrm {d}}=P\dfrac{c+L}{4L}-P\dfrac{3c-L}{4L} \end{aligned}$$

(51)

The ratio between these two applied forces is defined as $\rho$, written as,

$$\begin{aligned} \rho =\dfrac{P_{\mathrm {d}}}{P_{\mathrm {u}}}=\dfrac{\left( \dfrac{c}{L}+1 \right) -\left( 3\dfrac{c}{L}-1 \right) }{\left( \dfrac{c}{L}+1 \right) +\left( 3\dfrac{c}{L}-1 \right) }=\dfrac{1-\dfrac{\left( 3\dfrac{c}{L}-1 \right) }{\left( \dfrac{c}{L}+1 \right) }}{1+\dfrac{\left( 3\dfrac{c}{L}-1 \right) }{\left( \dfrac{c}{L}+1 \right) }}. \end{aligned}$$

(52)

From Eq. 49 we get,

$$\begin{aligned} \left( \Psi ^{-1}-1\right) = \dfrac{4\left( 3\dfrac{c}{L}-1 \right) ^2}{3\left( \dfrac{c}{L}+1 \right) ^2}. \end{aligned}$$

(53)

Substituting the above equation in Eq. 52 we obtain the following relationship for the force ratio $\rho$ in terms of the mode ratio $\Psi$:

$$\begin{aligned} \rho =\dfrac{1-\dfrac{\sqrt{3}}{2} (\Psi ^{-1}-1)^{1/2}}{1+\dfrac{\sqrt{3}}{2} (\Psi ^{-1}-1)^{1/2}} \end{aligned}$$

(54)

Fig. 15

Superposition analysis of the mixed mode bending (MMB) system [Redrawn from Reeder and Crews (1990)]

Fig. 16

Geometry and boundary conditions for an alternative mixed mode bending (MMB) test

Appendix 2: Relation between $K_{\mathrm {e}}^0$ and $\Psi$

Herein, we derive the relation used in Sect. 3.2 for calculating the undamaged effective stiffness $K_{\mathrm {e}}^0$ of a spring element under mixed mode conditions. Let us consider a mixed-mode condition when the normal and tangential separations are $\delta _{\mathrm {n}}^*$ and $\delta _{\mathrm {t}}^*$, respectively. Under this deformation state, let the equivalent separation reach the critical value, that is,

$$\begin{aligned} (\delta ^{{\mathrm {cr}}}_{{\mathrm {e}}})^2 = (\delta _{\mathrm {t}}^*)^2 + (\delta _{\mathrm {n}}^*)^2 \end{aligned}$$

(55)

Let us consider the case when $\delta _{\mathrm {n}}^*<\delta ^{{\mathrm {cr}}}_{\mathrm {n}}$ and $\delta _{\mathrm {t}}^*<\delta ^{{\mathrm {cr}}}_{\mathrm {t}}$, that is, the loading condition falls within the elastic zones of the normal and tangential damage laws, respectively. Now, the strain energy release rates under Mode I and Mode II are given by,

$$\begin{aligned} G_{\mathrm {I}} = \dfrac{1}{2} K_{\mathrm {n}}^0(\delta _{\mathrm {n}}^*)^2 ~~ {\mathrm {and}} ~~G_{\mathrm {II}} = \dfrac{1}{2} K_{\mathrm {t}}^0(\delta _{\mathrm {t}}^*)^2. \end{aligned}$$

(56)

where $K_{n}^{0}$ and $K_{t}^{0}$ are the undamaged normal and tangential stiffnesses of the spring element. In terms of the equivalent stiffness and separations, the total strain energy release under this mixed mode condition is given by,

$$\begin{aligned} G = \dfrac{1}{2} K_{\mathrm {e}}^0(\delta ^{{\mathrm {cr}}}_{{\mathrm {e}}})^{2} \end{aligned}$$

(57)

Now, the mixed mode ratio is defined as,

$$\begin{aligned}&\!\!\! \Psi = \dfrac{G_{\mathrm {II}}}{G} = \dfrac{K_{\mathrm {t}}^0(\delta _{\mathrm {t}}^*)^2}{K_{\mathrm {e}}^0(\delta ^{{\mathrm {cr}}}_{{\mathrm {e}}})^2},\quad \mathrm{and}\nonumber \\&\!\!\! 1 - \Psi = \dfrac{G_{\mathrm {I}}}{G} = \dfrac{K_{\mathrm {n}}^0(\delta _{\mathrm {n}}^*)^2 }{K_{\mathrm {e}}^0(\delta ^{{\mathrm {cr}}}_{{\mathrm {e}}})^2}; \end{aligned}$$

(58)

Rearranging the above two expressions we get,

$$\begin{aligned} (\delta _{\mathrm {t}}^*)^2&= \left( \dfrac{\Psi }{K_{\mathrm {t}}^0} \right) K_{\mathrm {e}}^0(\delta ^{{\mathrm {cr}}}_{{\mathrm {e}}})^{2}; ~~{\mathrm {and}}~~ \nonumber \\ (\delta _{\mathrm {n}}^*)^2&= \left( \dfrac{1 - \Psi }{K_{\mathrm {n}}^0} \right) K_{\mathrm {e}}^0(\delta ^{{\mathrm {cr}}}_{{\mathrm {e}}})^{2}; \end{aligned}$$

(59)

Substituting the above relations into Eq. (55) and simplifying, we get the relation,

$$\begin{aligned} \dfrac{1}{K_{\mathrm {e}}^0} = \dfrac{\Psi }{K_{\mathrm {t}}^0} + \dfrac{1 - \Psi }{K_{\mathrm {n}}^0} \end{aligned}$$

(60)

The above relation can be rearranged to obtain Eq. (25). Clearly, under pure Mode I conditions ($\Psi = 0$), we have $K_{\mathrm {e}}^0= K_{\mathrm {n}}^0$ and under pure Mode II conditions ($\Psi = 1$), we have $K_{\mathrm {e}}^0= K_{\mathrm {t}}^0$.