# Prediction of the Crack Front Shape of the Corner Interface Crack
## 2 Problem Formulation  
During this research, we performed an in-depth analysis of the propagation behavior of an interface corner crack that initiates from a right angle within the joint plane connecting the two plates made of the dissimilar materials. The focus was on comprehensively investigating three distinct shapes that the crack front may assume, as illustrated in Figure 1.  
The central objective of this study was to unravel the intricate mechanics governing the propagation of a crack at this critical interface. Various geometric parameters were considered, such as the crack length denoted by \(a\), within the cylindrical coordinate system (\(r\), \(\theta\), \(\gamma\)) to precisely describe the crack's spatial parameters. Within the complex domain of stress analysis near the crack tip, positioned at the interface between these dissimilar materials, the stress field reveals itself as a composite of two fundamental types of fields. The first is the coupled oscillatory field, characterized by the complex stress intensity factor denoted as \(K\), represented as a combination of \(K_{1}\) and \(iK_{2}\). The second is the non-oscillatory stress field, characterized by the real stress intensity factor, \(K_{III}\). This intricate stress field near the interface crack tip is fundamental to comprehending crack behavior and propagation in this heterogeneous material junction. The stress field in the vicinity of the interface crack tip has the form [2]:\[\sigma_{ij}=\frac{1}{\sqrt{2\pi\tau}}\Big{[}\text{Re}\left(Kr^{ik}\right)\Sigma_{a \beta}^{I}(\theta,\epsilon)+\text{Im}\left(Kr^{ik}\right)\Sigma_{a\beta}^{II}( \theta,\epsilon)+K_{III}\Sigma_{a\beta}^{III}(\theta)\Big{]} \tag{1}\]  
where: \(\Sigma_{a\beta}^{I,II,III}(\theta,\epsilon)\) are the angular functions, which correspond to tensile forces, in-plane shear and anti-plane shear across the interface, respectively. For the material above the interface, material 1, functions \(\Sigma_{a\beta}^{III}(\theta)\) have the form [4]:  
\[ \begin{split}\Sigma_{rr}^{I}(\theta)&=-\frac{she(\pi -\theta)}{che\pi}\cos\frac{3\theta}{2}+\frac{e^{-\epsilon(\pi-\theta)}}{che\pi }\cos\frac{\theta}{2}(1+\sin^{2}\frac{\theta}{2}+\varepsilon\sin\theta)\\ \Sigma_{\theta\theta}^{I}(\theta)&=\frac{she(\pi- \theta)}{che\pi}\cos\frac{3\theta}{2}+\frac{e^{-\epsilon(\pi-\theta)}}{che\pi }\cos\frac{\theta}{2}(\cos^{2}\frac{\theta}{2}-\varepsilon\sin\theta)\\ \Sigma_{r\theta}^{I}(\theta)&=\frac{she(\pi-\theta) }{che\pi}\sin\frac{3\theta}{2}+\frac{e^{-\epsilon(\pi-\theta)}}{che\pi\pi} \sin\frac{\theta}{2}(\cos^{2}\frac{\theta}{2}-\varepsilon\sin\theta)\\ \Sigma_{r\theta}^{II}(\theta)&=-\frac{che(\pi- \theta)}{che\pi}\sin\frac{3\theta}{2}-\frac{e^{-\epsilon(\pi-\theta)}}{che\pi \gamma}\sin\frac{\theta}{2}(1+\cos^{2}\frac{\theta}{2}-\varepsilon\sin\theta) \\ \Sigma_{\theta\theta}^{II}(\theta)&=-\frac{che(\pi- \theta)}{che\pi}\sin\frac{3\theta}{2}-\frac{e^{-\epsilon(\pi-\theta)}}{che\pi }\sin\frac{\theta}{2}(\sin^{2}\frac{\theta}{2}+\varepsilon\sin\theta)\\ \Sigma_{r\theta}^{II}(\theta)&=\frac{che(\pi- \theta)}{che\pi\pi}\cos\frac{3\theta}{2}+\frac{e^{-\epsilon(\pi-\theta)}}{che \pi\tau}\cos\frac{\theta}{2}(\sin^{2}\frac{\theta}{2}+\varepsilon\sin\theta) \\ \Sigma_{r\theta}^{I,II}(\theta)&=\Sigma_{z\theta}^{ I,II}(\theta)=0,\quad\Sigma_{zz}^{I,II}(\theta)=\
u\left(\Sigma_{rr}^{I,II}( \theta)+\Sigma_{\theta\theta}^{I,II}(\theta)\right)\.\end{split}  \tag{2}\]  
For material 2 below the interface angular functions \(\Sigma_{a\beta}^{I,II}(\theta)\) have the forms analogous to those in Equation (2), one just has to simply change \(-\pi\) to \(\pi\) and vice versa, everywhere. The mode III functions \(\Sigma_{a\beta}^{III}(\theta)\) are the same as for a homogeneous solid [4]:  
\[ \begin{split}\Sigma_{rr}^{III}(\theta)&=\Sigma_{ \theta\theta}^{III}(\theta)=\Sigma_{r\theta}^{III}(\theta)=\Sigma_{z^{2}}^{III}( \theta)=0,\\ \Sigma_{rz}^{III}(\theta)&=\sin\frac{\theta}{2},\quad \Sigma_{\theta z}^{III}(\theta)=\cos\frac{\theta}{2}.\end{split}