# Effect of Cohesive Properties on Low-Velocity Impact Simulations of Woven Composite Shells
## 2 Finite Element Models
### Interlaminar Damage Modelling  
Among the various damage modes observed in composite laminates subjected to low-velocity impact loads, the debonding of adjacent layers stands out as one of the most significant. In this way, a S-BCM was incorporated in the FE models to account for delamination. This model has features that are comparable to cohesive elements and it is particularly designed for extremely thin interfaces. Moreover, the cohesive behaviour is classified as a surface interaction attribute and is controlled by a bilinear traction-separation \((\tau-\delta)\) constitutive model, which is illustrated in Figure 2 for the pure normal and shear responses, and for the expected mixed-mode response. Notice that the subscripts \(n,s\), and \(t\) correspond to the normal, first, and second shear directions, respectively, and that the superscripts \(0\) and \(f\) to the onset and fully delaminated stages, respectively. The interface is defined with an initial elastic response with a slope defined by the normal \(k_{n}\) and shear \(k_{s,t}\) cohesive stiffnesses, and which is maintained until the failure criterion condition is met. The stress-based quadratic failure criterion shown in Equation (1) is used to determine the damage onset of the interface material:  
\[\left(\frac{\langle\ \tau_{n}\rangle}{\tau_{n}^{0}}\right)^{2}+\left(\frac{ \tau_{s}}{\tau_{s}^{0}}\right)^{2}+\left(\frac{\tau_{t}}{\tau_{t}^{0}}\right) ^{2}=1 \tag{8}\]  
where, in the context of the present study, the variables \(\tau_{n},\tau_{s}\) and \(\tau_{t}\) symbolise the interface normal and shear tractions, while \(\tau_{n}^{0},\tau_{s}^{0}\), and \(\tau_{t}^{0}\) represent the respective peak contact stress values associated with these tractions. The Macaulay brackets \(\langle\ \rangle\) indicate that the compressive stress does not contribute to damage. When the quadratic interaction function involving the nominal stress ratios is equal to one, interlaminar damage is presumed to start occurring. Equation (9) represents the softening response of the cohesive surface where \(d\) is the damage coefficient. \[\tau_{i}=(1-d)k_{i}\delta_{i}\ \ with\ i=n,s,t \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **Symbol** & **Units** & **Value** \\ \hline Mass density & \(\rho\) & kg/m\({}^{3}\) & 1900 \\ \hline \multirow{4}{*}{Stiffness properties} & \(E_{1}^{+,-}=E_{2}^{+,-}\) & GPa & 21.9 \\  & \(E_{3}\) & GPa & 8.6 \\  & \(G_{12}\) & GPa & 3.4 \\  & \(G_{13}\) & GPa & 2.4 \\  & \(\
u_{12}\) & - & 0.14 \\ \hline \multirow{3}{*}{Strength properties} & \(X_{1}^{+}=X_{2}^{+}\) & MPa & 250 \\  & \(X_{1}^{-}=X_{2}^{-}\) & MPa & 200 \\  & \(X_{12}\) & MPa & 40 \\ \hline Fracture toughness & \(G_{f}^{1,2}\) & N/mm & 4.5 \\ \hline \multirow{3}{*}{Shear plasticity} & \(d_{12}^{max}\) & - & 1 \\  & \(\tilde{v}_{y0}\) & MPa & 25 \\ \cline{1-1}  & \(C\) & - & 800 \\ \cline{1-1}  & \(p\) & - & 0.552 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Intralaminar properties defined for each layer of the composite shell [16]. From this point on, the energy \(G_{c}\) that was dissipated during the damage process and that corresponds to the region below the \(\tau-\delta\) curves controls the damage progression. Its value is obtained implementing the Benzeggagh and Kenane (B-K) damage evolution criterion under mixed-mode loading [28]:  
\[G_{c}=G_{Ic}+(G_{IIc}+G_{Ic})\bigg{(}\frac{G_{II}+G_{III}}{G_{I}+G_{II}+G_{III}} \bigg{)}^{\eta} \tag{10}\]  
where the parameters \(G_{I}\), \(G_{II}\), and \(G_{III}\) signify the strain energy release rates associated with mode I, II, and III, respectively. On the other hand, \(G_{Ic}\) and \(G_{IIc}\) denote the critical strain energy release rates, and \(\eta\) represents a material coefficient of the B-K equation. Notice that with this constitutive model, damage evolution depicts the deterioration of the cohesive stiffness. In contrast, damage evolution for cohesive elements describes the deterioration of the material stiffness.