## 2 Proposed Mixed Mode I/II Test Specim for Adhesive Bounded Joints  
Figure 1 shows the bi-material test configuration proposed herein for investigating the mixed mode I/II fracture in adhesive joints. The specimen is called bi-material inclined notch short bend beam (BISBB). Two parts that are attached by an adhesive material create a rectangular shape as the overall configuration of the BISBB specimen with length \(L\) and width \(W\). The width of adhesive is d and an edge crack with length \(a\) is assumed to be introduced in the middle of adhesive material. The specimen is loaded using a conventional three-point bend loading fixture in which the loading support distance is 2S. The state of mode I and mode II mixities can be changed in this specimen by inclining the interface or adhesive plane with respect to the upper loading direction shown by \(\alpha\) angle in Figure 1. Indeed, by increasing the angle \(\alpha\) from zero, the contribution of mode II or in-plane sliding deformation in the failure behavior of bi-material assembly increases and therefore the footprint of both crack opening and crack in-plane sliding deformations can be seen in the failure process of bi-material specimen joined by the adhesive material. Compared to many of the previously used adhesive bounded joints, the proposed BISBB specimen has a very simple geometry and testing setup and requires very little amount and size of adhesive and the adherent materials. The short beam like specimen containing inclined crack and  
loaded by symmetric three-point bend loading was investigated recently by [PERSON] and coworkers [11; 12; 13; 14; 15; 16; 17; 18; 19] for assessing mixed mode I/II fracture of different materials. The state of stress in the vicinity of crack tip in the short bend beam specimen can be influenced by the geometrical and loading parameters (such as crack length (\(a\)), crack inclination angle (\(\alpha\)) and the loading span (\(2S\))). The left and right materials creating the BISBB specimen can be either similar or dissimilar. Here, alumina ceramic and aluminum metal alloy (Al) were considered for joining by an epoxy adhesive material to manufacture the proposed BISBB specimen. Table 1 illustrates the mechanical properties of the selected materials. According to [PERSON] [21], the polar stress fields in the vicinity of crack tip can be written as:  
\[\sigma_{rr}=\frac{1}{\sqrt{2\pi\,r}}\cos\frac{\theta}{2}\bigg{[}K_{\rm I} \bigg{(}1+\sin^{2}\frac{\theta}{2}\bigg{)}+\frac{3}{2}\,K_{\rm II}\,\sin\theta -2K_{\rm II}\tan\frac{\theta}{2}\bigg{]}+T\,\cos^{2}\theta\,+O\Big{(}r^{\frac{ 1}{2}}\Big{)}\]  
\[\sigma_{\theta\theta}=\frac{1}{\sqrt{2\pi\,r}}\cos\frac{\theta}{2}\bigg{[}K_{ \rm I}\cos^{2}\frac{\theta}{2}-\frac{3}{2}\,K_{\rm II}\,\sin\theta\bigg{]}+T \,\sin^{2}\theta\,+O\Big{(}r^{\frac{1}{2}}\Big{)} \tag{1}\]  
\[\sigma_{r\theta}=\frac{1}{\sqrt{2\pi\,r}}\cos\frac{\theta}{2}\big{[}K_{\rm I} \sin\theta+\,K_{\rm II}\,(3\,\cos\theta-1)\big{]}-T\,\sin\theta\,\cos\theta\,+ O\Big{(}r^{\frac{1}{2}}\Big{)}\]  
where \(r\) and \(\theta\) are the crack tip coordinate located at the crack tip, and \(K_{\rm I}\) and \(K_{\rm II}\) are related to the singular terms of in this series expansion are the stress intensity factors. The first \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Properties** & **Alumina** & **Aluminum Alloy (Al)** & **Epoxy Adhesive** \\ \hline Elastic modulus \(E\) (GPa) & 300 & 70 & 2.84 \\ \hline Poisson’s ratio \(\
u\) & 0.21 & 0.33 & 0.35 \\ \hline Fracture toughness \(K_{\rm IC}\) (MPa\(\sur\)m) & 3.5 & 29 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of parts utilized for manufacturing the BISBB specimen [10; 20]. Figure 1: Geometry and loading configuration of bi-material fracture test geometry joined by an adhesive and containing an inclined edge crack of length a.  
higher order term is \(T\)-stress which is constant and independent from the crack tip distance. O (\(t^{0.5}\)) describes the higher order terms.