# Prediction of the Crack Front Shape of the Corner Interface Crack
## 2 Problem Formulation
This study delves into the progressive advancement of the interface crack originating from a right-angled corner, following the plane of conjunction between two distinct material plates. The investigation spans across diverse crack front geometries, highlighting this progression in Figure 3. This analytical exploration enhances comprehension of crack propagation dynamics along the interface and its relationship with various crack front shapes. In Figure 3, a comprehensive view of the interface crack front is presented, displaying its intriguing variability in shape--concave, straight (triangular), or convex. The crack length, denoted as \(a\), is a fundamental parameter in this representation, while \(r\) and \(\theta\) serve as essential polar coordinates that aid in characterizing the crack's geometric intricacies. To delve into the mechanics of crack propagation, we adopt the modified Paris law--a valuable tool for analyzing the dynamic advancement of the crack. This law allows us to discern a critical threshold where the stress intensity factor \(K_{l}\) surpasses the intrinsic material property, the fracture toughness \(K_{lc}\). Beyond this threshold, the crack experiences an unstable growth phase, significantly impacting the material's structural integrity. The crack growth equation, a pivotal component of our analysis, delineates the nuanced relationship between the incremental extension of the crack, \(\Delta a\), and the corresponding increase in the number of loading cycles, \(\Delta N\). The pioneering work of [PERSON] and [PERSON] in 1963 has laid the foundation for our understanding, demonstrating that the stress intensity factor variation accurately characterizes the subcritical crack propagation. This extends our understanding from rapid or critical fractures to the subcritical realm, which is particularly crucial in fatigue loading conditions [31]. This knowledge forms a cornerstone for comprehending interface crack behavior and its progression under cyclic  
Figure 3: Different shapes of the crack front for the corner interfacial crack: (**a**) concave, (**b**) triangular, (**c**) convex. Figure 2: Mode mixity defined as solid angles in the \(K\) space: (**a**) for \(\varepsilon=0\), (**b**) for \(\varepsilon\
eq 0\). loading, shedding light on the complex interplay of mechanical forces and material characteristics. They also established that the crack propagation rate is the linear function of the stress intensity factor variation in the logarithmic form, i.e., [31; 32; 33; 34]:  
\[\frac{da}{dN}=C(\Delta K)^{m}, \tag{9}\]  
where the crack length \(a\) changes from the initial to the critical value that leads to fracture, while \(N\) is the number of the load cycles, \(C\) and \(m\) are the material constants and \(\Delta K=K_{max}-K_{min}\) is the variation of the stress intensity factor, namely the difference between its values at the maximum and minimum loads. The interface crack growth, in the mixed mode conditions, can be described as [32]:  
\[\varepsilon=\frac{1}{2\pi}\ln\biggl{(}\frac{1-\beta}{1+\beta}\biggr{)} \tag{4}\]0  
Considering the difficulties in calculating the phase angles \(\psi\) and \(\phi\), defined by Equation (8), it is assumed here that the interface corner crack growth, in the mixed mode conditions, can be represented by the simple function of the energy release rate variation [32]:  
\[\varepsilon=\frac{1}{2\pi}\ln\biggl{(}\frac{1-\beta}{1+\beta}\biggr{)} \tag{4}\]1  
where, for the further calculations, the constants \(C\) and \(m\) are taken to be \(C=4\times 10^{-13}\) J/m\({}^{2}\) and \(m=6\).