# An Elastic Interface Model for the Delamination of Bending-Extension Coupled Laminates
## 5 Results
### Effects of Interface Stiffness  
Figure 7 shows a schematic of the double cantilever beam specimen loaded by uneven bending moments (DCB-UBM) test [63]. Here, this particular test configuration was used to illustrate the effects of interface stiffness, i.e., of the values of the interface elastic constants, \(k_{x}\) and \(k_{z}\), on the solution according to the proposed model.  
The same laminated specimens considered in the examples of Section 5.1 were again analyzed, but subjected to different loading conditions. The upper sublaminate was loaded by a unit bending couple, \(M_{1}=1\) N m, while the lower sublaminate was loaded by a variable bending couple, \(M_{2}\). Figure 8 shows the dimensionless Mode I and Mode II contributions to the energy release rate, \(\gamma_{\Pi}=\mathcal{G}_{\Pi}/\mathcal{G}\) and \(\gamma_{\Pi}=\mathcal{G}_{\Pi}/\mathcal{G}\), as functions of the bending moment ratio, \(M_{2}/M_{1}\).  
For the UD//UD specimen (Figure 8a), the modal contributions to \(\mathcal{G}\) turned out to be independent of the interface stiffness. This result was due to the mid-plane symmetry of the specimen and the independence of the crack tip bending moments on the delamination length. In this case, also the mode mixity depended uniquely on the bending moment ratio, \(M_{2}/M_{1}\). In particular, it can be noted how pure Mode I fracture conditions (\(\gamma_{\Pi}=0\)) were obtained for \(M_{2}=-M_{1}\), while pure Mode II fracture conditions (\(\gamma_{1}=0\)) were obtained for \(M_{2}=M_{1}\). Values of \(M_{2}/M_{1}\) greater than one also corresponded to pure Mode II because of the incompressibility constraint introduced for \(\mathcal{G}_{\Pi}\) in Equation (44).  
Figure 6: UD//MD specimen: (**a**) interfacial stresses, (**b**) axial forces, (**c**) shear forces, and (**d**) bending moments in the crack tip neighborhood.  
Figure 7: Double cantilever beam specimen with uneven bending moments.  
Figure 8: Dimensionless modal contributions to energy release rate vs. sublaminate bending moment ratio: (**a**) UD//UD specimen, (**b**) UD//CP specimen, and (**c**) UD//MD specimen.  
Instead, for both the UD//CP (Figure 8b) and UD//MD specimens (Figure 8c), the modal contributions to \(\mathcal{G}\) depended on the values of the interface elastic constants, \(k_{x}\) and \(k_{z}\). For each modal contribution, the plots show five curves with different shades of blue: from lighter to darker, and these correspond to \(k_{x}=k_{z}=10^{2},10^{3},10^{4},\) and \(10^{5}\) N/mm\({}^{3}\); the darkest curves represent the limit case \(k_{x}=k_{z}\rightarrow+\infty\), which corresponds to a rigid interface model [51]. For the specimens considered, also the mode mixity and the conditions for pure fracture modes depended on the interface stiffness.  
The range of values adopted for \(k_{x}\) and \(k_{z}\) in the previous plots may look quite wide. Indeed, this range corresponds to real values that may be obtained when performing a compliance calibration with experimental results or numerical simulations [62]. The results obtained suggest that the elastic interface model is potentially capable of highly-accurate prediction and interpretation of experimental results, provided that the appropriate values of the interface elastic constants are selected. To this aim, it would be useful to validate the theoretical predictions obtained with experimental results, such as those obtained by [PERSON] et al. [64; 65]. Such a comparison, similar to the one carried out by [PERSON] et al. to validate different mixed-mode partition theories [66; 67], will be the subject of future work.