## 3 Numerical Analyses and Results  
A finite element (FE) model of BISBB sample was created in the ABAQUS software. The overall dimensions of the specimen were: \(L=54\) mm, \(W=15\) mm, \(t=5\) mm and \(d=0.4\) mm. The crack length (\(a\)), loading span (\(S\)) and crack inclination angle (\(\alpha\)) were considered as variables. Figure 2 shows the mesh patterns of adherent and adhesive parts and the crack tip region in the modeled BISBB specimen. A total number of 4600 CPE8 (i.e., 8-node biquadratic plane strain quadrilateral) elements were used for simulating the finite element model of specimen. In order to investigate the influence of adherent type on the fracture parameters, four permutations for modeling the left and right materials in the BISBB specimen were considered as (Aluminum- Adhesive- Alumina, Alumina- Adhesive-Aluminum, Aluminum- Adhesive-Alumina, and Alumina- Adhesive-Alumina) as shown in Figure 3a-d, respectively. A vertical reference load \(P=100\) N was applied to the created finite element models and the mechanical properties of each part given in Table 1 were assigned to the models. Due to high stresses induced in the crack tip and also between the interface of adhesive and adherents, very fine elements and meshes were utilized. Using the J-integral method built in the ABAQUS code, the fracture parameters were determined for different \(a\)/\(W\), \(S\)/\(W\) and \(\alpha\) values. Figures 4 and 5 show the variations of mode I and mode II geometry factors (\(Y_{\rm I}\) and \(Y_{\rm II}\)) with crack inclination angle (\(\alpha\)) for different \(\alpha\)/W and \(S\)/W values. It is seen that by increasing the crack angle relative to the vertical direction, the contribution of mode I decreases and conversely mode II component increases. Indeed, the SIF ratio (\(K_{\rm I}\)/\(K_{\rm II}\)) approaches zero showing switching the state of crack tip deformation from pure opening tensile mode (for \(\alpha=0^{\circ}\) in which \(Y_{\rm II}=0\)) towards pure shear and in plane sliding mode (i.e., \(Y_{\rm I}=0\)) at higher crack inclination angles. It is obvious from Figures 4 and 5 that the magnitudes of both mode I and mode II geometry factors increase by increasing the crack length ratio and loading support distance ratio. The maximum value of \(Y_{\rm I}\) occurs at \(\alpha=0^{\circ}\) but a maximum of \(Y_{\rm II}\) is seen typically at \(\alpha=10^{\circ}\). The highest sensitivity of mode I geometry factor (\(Y_{\rm I}\)) and mode II geometry factor (\(Y_{\rm II}\)) to the crack length ratio and span ratio (\(\alpha\)/\(W\) and \(S\)/\(W\)) is seen at maximum values of geometry factors (i.e., \(\alpha=0^{\circ}\) for mode I geometry factor and \(\alpha=10^{\circ}\) for mode II geometry factor). For both geometry factors it is  
Figure 3: Four different permutations considered for the FE analyses of the BISBB specimen. (**a**) Al-Adhesive-Alumina, (**b**) Alumina-Adhesive-Al, (**c**) Al-Adhesive-Al, (**d**) Alumina-Adhesive-Alumina. Figure 2: Finite element model of BISBB specimen. seen also from Figures 4 and 5 that the sensitivity of data to _a/W_ and _S/W_ becomes smaller by increasing the crack inclination angle. Although pure mode I condition occurs at \(\alpha=0^{\circ}\) for any desired _a/W_ and _S/W_ ratios, the corresponding condition for pure mode loading depends on the geometrical parameters, loading conditions and adherent materials. The corresponding value of mode II crack inclination angle (\(\alpha_{\rm II}\)) determined from the finite element analyses of the BISBB specimen is listed in Table 2. According to this Table, for any given ratios of _a/W_ and _S/W_, pure mode II crack angle varies typically between \(30^{\circ}\) and \(55^{\circ}\). This angle decreases by increasing _a/W_ ratio. A parameter called mixity parameter, defined as \(M^{\rm e}=\frac{2}{\pi}\tan^{-1}\left(K_{\rm I}/K_{\rm II}\right)\), is used herein for assessment of the fraction and contribution of both modes I and II on fracture behavior of the BISBB specimen.