# Fatigue Characteristics Analysis of Carbon Fiber Laminates with Multiple Initial Cracks
## 2 Methods  
### Paris-XFEM Method  
This paper utilizes the Paris-XFEM technique for fatigue modeling, forecasting the service life of carbon fiber composite laminates riddled with multiple cracks. By incorporating the Paris law, which delineates the velocity of crack propagation due to fatigue, and using the Extended Finite Element Method (XFEM) to model crack propagation that is independent of the underlying mesh [18], this method offers a robust solution for predicting fatigue behavior. The Paris formula [19] is articulated in detail as follows:  
\[\mathrm{da}/\mathrm{dN}\ =\ \mathrm{C}(\Delta\mathrm{K})^{\mathrm{m}} \tag{1}\]  
where \(a\) denotes the crack depth; \(N\) indicates the cycle count of fatigue stress; \(\mathrm{da}/\mathrm{dN}\) is the velocity of crack extension; \(\Delta K\) represents the amplitude of the stress intensity factor due to fatigue stress; and \(C\) and \(\mathrm{m}\) are material-specific parameters that characterize the fatigue crack growth behavior. The fatigue life is expressible through the cumulative count of fatigue cycles. Integration of the Paris equation yields the correlation between crack length and fatigue cycle count, thereby determining the lifespan of fatigue crack propagation. \[\mathrm{N}\ =\ \int_{a_{0}}^{a_{c}}\frac{\mathrm{da}}{C(\Delta\mathrm{K})^{ \mathrm{m}}} \tag{2}\]  
where \(a_{0}\) represents the initial size of the crack, and \(a_{c}\) indicates the critical size at which the crack becomes critical. In pursuit of enhanced applicability, ABAQUS has adopted an energy release rate-based fatigue crack propagation methodology, which meticulously delineates the thresholds for energy release rates. Illustrated in Figure 3, the crack maintains a state of dormancy and refrains from expanding when the rate of energy liberation at the crack's edge is beneath the minimum critical value \(G_{thresh}\). Upon surpassing this minimum threshold \(G_{thresh}\), the crack initiates growth, with its rate of expansion incrementally escalating. When the energy discharge rate at the fracture tip goes over the highest threshold \(G_{pl}\), the rate of crack growth precipitously increases, culminating in the eventual failure of the structure. In the realm of numerical simulation for expansion of cracks under cyclic stress with ABAQUS, the progression of a crack is not exclusively dictated by the comparison of the velocity of energy release at the fracture's tip to the predetermined critical threshold, as delineated in Equation (3). It is also subject to the impact of the relative release rate [20]. Figure 3: Paris fatigue expansion curve. Consequently, the initiation expansion of fatigue-induced cracks can be delineated with Equation (4) as the foundational criterion. \[\left\{ \begin{array}{l}\frac{\mathrm{G_{Ultrash}}}{\mathrm{G_{CC}}}\ =\ 0.01\\ \frac{\mathrm{C_{Pal}}}{\mathrm{G_{CC}}}\ =\ 0.85\end{array} \right. \tag{3}\]  
\[\mathrm{f}\ =\ \frac{\mathrm{N}}{\mathrm{c_{1}}\mathrm{\Delta G^{\mathrm{c_{2}}}}} \geq 1.0 \tag{4}\]  
where the pivotal energy liberation thresholds \(G_{c}\) in mixed-mode fracture scenarios are characterized by specific values, with \(c_{1}\) and \(c_{2}\) serving as material parameters, conventionally assigned the values of \(0.5\) and \(-0.1\)[21], respectively. The term \(\mathrm{\Delta G}\) denotes the disparity between the peak \(G_{max}\) and the trough \(G_{min}\) of the rate of energy liberation during the entirety of cyclic loading. Accordingly, the formula governing the crack growth rate within ABAQUS, employing the Paris-XFEM approach, is hereby presented. \[\mathrm{da/dN}\ =\ \mathrm{c_{3}}(\mathrm{\Delta G})^{\mathrm{c_{4}}} \tag{5}\]  
where the parameters \(c_{3}\) and \(c_{4}\) are material-specific constants that are ascertained through experimental means. The interplay between the stress concentration factor and the rate of energy liberation, a concept deeply rooted in the principles of fracture mechanics, unfolds as follows:  
\[\mathrm{G_{C}}\ =\ \frac{\mathrm{K_{C}^{2}}}{\mathrm{E^{\prime}}},\ \ \ \mathrm{E^{\prime}}\ =\ \left\{