# A Numerical Investigation of Delamination Response of CNT/Epoxy Film Interleaved Composite
## 3 Numerical Modeling  
The cohesive element has constitutive behavior according to the traction-separation law, as shown in Figure 2. The traction-separation law includes the initial load, the increase in stress until the onset of damage, and the decrease in stress as damage advance of the cohesive elements. When a certain level of stress or displacement is reached, the strength decreases until complete delamination of the specimen occurs. These parts can be explained by cohesive strength and cohesive energy parameters that set the properties of the cohesive region. Cohesive strength is the maximum stress required for separation. Cohesive energy is the work per unit area required to separate interface. Cohesive strength is a peak stress required for separation. Cohesive energy is separation work per unit area. The appropriate values of the parameters of the traction-separation law were chosen to describe the damage and failure of the structure. In Figure 2a, \(\Gamma\) is the arbitrary path of the contour around the crack tip. The J-integral method, which represents a method for calculating the strain energy release rate along a contour, was employed to connect the cohesive theory with linear elastic fracture mechanics. \[J=\int_{\Gamma}Wdy-T\frac{\partial u}{\partial x}ds \tag{2}\]  
Figure 1: Experimental setting of ENF test. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Unidirectional CFRP (k/m\({}^{2}\))**} & \multicolumn{1}{c}{
\begin{tabular}{c} **Woven CFRP** \\ **(k/m\({}^{2}\))** \\ \end{tabular} } \\ \hline Baseline & 2.01 & 1.69 \\ \hline Epoxy film interleaved & 2.04 & 1.71 \\ \hline
1 wt\% CNT/epoxy film interleaved & 2.89 & 2.03 \\ \hline
3 wt\% CNT/epoxy film interleaved & 4.58 & 3.08 \\ \hline
5 wt\% CNT/epoxy film interleaved & 4.55 & 2.83 \\ \hline
7 wt\% CNT/epoxy film interleaved & 3.59 & 2.35 \\ \hline
9 wt\% CNT/epoxy film interleaved & 1.97 & 1.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average mode II interlaminar fracture toughness (\(G_{IIC}\)) values. where \(W\) is the strain energy density, \(T\) is the traction vector, \(u\) is the displacement vector, and \(ds\) is a length increment along the contour. Due to the relatively long crack length, \(dy\) becomes zero. Consequently, Equation (2) becomes  
\[J=-\int\limits_{\Gamma}\sigma(\delta)\frac{d\delta}{dx}dx=-\int\limits_{\Gamma} \frac{d}{dx}\left\{\int_{0}^{\delta}\sigma(\delta)d\delta\right\}dx=\int_{0}^{ \delta^{*}}\sigma(\delta)d\delta=W \tag{3}\]  
In this study, the bilinear traction-separation law is used for calculation. The initial loading and softening after damage initiation is linear in this behavior. As shown in Figure 2b, \(t_{N}\), and \(t_{T}\) are the normal and tangential components of the traction vector \(t\), respectively. \(t\) is the restraining stress which will increase and reach its maximum value, then begin to decrease, and eventually be reduced to zero with increasing separation. \(\delta_{N}\), and \(\delta_{T}\) are the normal and tangential components of the separation vector \(\delta\), respectively, at the point of the cohesive element interface. \(K_{N}\) and \(K_{t}\) are the initial normal and shear stiffness of the interface, respectively. Since strain energy is equivalent to the work done by external forces, \(\Gamma\) is equivalent to the area under the curve, so it can be expressed by the following equation (\(\Gamma_{II}=G_{IIc}\)). \[\Gamma=\int_{0}^{\delta}T(\delta)d\delta \tag{4}\]  
Using a 2D plane strain finite element model of commercial simulation tool ABAQUS, these parameters were applied to the cohesive zone model to simulate delamination of the interleave interface of CFRP laminate by the ENF test. To model the effect of the presence of a CNT/epoxy film interleave into an CFRP, the geometrical features of the interleave are linked to the global mechanical properties of the CFRP laminate.