# An Elastic Interface Model for the Delamination of Bending-Extension Coupled Laminates
## 5 Results  
### Examples  
By way of illustration, we applied the developed solution to the analysis of the DCB test (Figure 3). Laminated specimens with the following geometric sizes were considered: \(L=100\) mm, \(B=25\) mm, and \(H=3.2\) mm. A delamination length \(a=25\) mm and a reference load \(P=100\) N were used in all of the following calculations. Boundary conditions (12) and (13) were specialized to the present case by taking the crack tip internal forces as \(N_{1}^{A}=N_{2}^{A}=0\), \(Q_{1}^{A}=-Q_{2}^{A}=P\), and \(M_{1}^{A}=-M_{2}^{A}=Pa\) and assuming a CTE length \(\ell=L-a\). Three laminated specimens with different stacking sequences were ideally built as follows. We started from a typical carbon-epoxy fiber-reinforced lamina of thickness \(t_{p}=0.2\) mm and elastic moduli shown in Table 1[61]. Based on this, three stacking sequences were designed corresponding to uni-directional (UD), cross-ply (CP), and multi-directional (MD) sublaminates (Table 2). Lastly, the DCB test specimens were obtained by variously assembling the sublaminates one over the other:  
\(\bullet\) UD//UD  
\(\bullet\) UD//CP  
\(\bullet\) UD//MD  
where the double slash (//) represents the position of the delamination crack. The sublaminate plate stiffness matrices, calculated as explained in Appendix A, are reported in Table 3. It can be noted that no bending-extension coupling was exhibited by the UD sublaminate (all terms \(B_{ij}=0\)), while bending-extension coupling was shown by both the CP (terms \(B_{xx}=-B_{yy}\
eq 0\)) and MD sublaminates (all terms \(B_{ij}\
eq 0\), except \(B_{xs}=B_{ys}=0\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(E_{1}\) & \(E_{2}=E_{3}\) & \(\
u_{12}=\
u_{13}\) & \(\
u_{23}\) & \(G_{12}=G_{13}\) & \(G_{23}\) \\ (MPa) & (MPa) & – & – & (MPa) & (MPa) \\ \hline
109,000 & 8819 & 0.342 & 0.380 & 4315 & 3200 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Lamina elastic moduli. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Sublaminate**} & **Stacking Sequence** \\ \hline UD & \([0^{\circ}8]\) \\ \hline CP & \([0^{\circ}/90^{\circ}/0^{\circ}2/90^{\circ}3/0^{\circ}]\) \\ \hline MD & \([45^{\circ}/-45^{\circ}2/90^{\circ}/45^{\circ}2/-45^{\circ}/0^{\circ}]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Sublaminate stacking sequences. Figure 3: Double cantilever beam test. The sublaminate equivalent beam stiffnesses, calculated via Equations (A10) and (A19), are given in Table 4. It can be noted that the UD subliminate had the highest extensional and bending stiffnesses, A and D, while the MD subliminate had the highest bending-extension coupling stiffness, B. The shear stiffness, C, showed little variations among the chosen stacking sequences. A crucial point for the successful implementation of the present model is the appropriate choice of the values of the elastic interface constants. These can be assigned by conducting a compliance calibration with experimental tests or numerical simulations, if available [62]. In a previous study on asymmetric DCB test specimens [22], compliance calibration with finite element simulations gave:  
\[k_{x}\approx 16.2\frac{G_{13}}{H}\quad\text{and}\quad k_{z}\approx 6.9\frac{E_{3} }{H}, \tag{48}\]  
which, in the present case, yielded \(k_{x}=21\),845 N/mm\({}^{3}\) and \(k_{z}=19\),016 N/mm\({}^{3}\). Such values were adopted for the present examples. Figure 4 shows the plots of the interfacial stresses and internal forces as functions of the \(x\)-coordinate in the neighborhood of the crack tip for the UD//UD specimen. This specimen was symmetric; hence, the unbalance parameter was \(\beta_{0}=0\), and the analytical solution for the balanced case applied (see Section 3.2). For the DCB test, the specimen response also corresponded to pure Mode I fracture conditions: actually, Figure 4a shows that the shear stresses were identically null, while the normal stresses attained a peak value at the crack tip cross-section. Figure 4b shows that also the subliminate axial forces were null.