# Prediction of the Crack Front Shape of the Corner Interface Crack
## 2 Problem Formulation
 \tag{3}\]  
Figure 1: The three-dimensional bimaterial angle with a crack at the interface (**a**), for different shapes of the crack front: (**b**) concave, (**c**) triangular, and (**d**) convex. These angular functions depend on the elastic characteristics of the bimaterial combination via the parameter \(\varepsilon\), called the oscillatory index, which is defined as [1]:  
\[\varepsilon=\frac{1}{2\pi}\ln\biggl{(}\frac{1-\beta}{1+\beta}\biggr{)} \tag{4}\]  
where \(\beta\) is one of the two Dundurs parameters [30]:  
\[\alpha=\frac{\mu_{j}(\kappa_{i}+1)-\mu_{i}(\kappa_{j}+1)}{\mu_{j}(\kappa_{i}+1) +\mu_{i}(\kappa_{j}+1)}\quad\beta=\frac{\mu_{j}(\kappa_{i}-1)-\mu_{i}(\kappa_{ j}-1)}{\mu_{j}(\kappa_{i}+1)+\mu_{i}(\kappa_{j}+1)}, \tag{5}\]  
where: \(E_{i}\)--is the Young elasticity modulus, \(\mu_{i}\)--the shear modulus and \(\
u_{i}\)--the Poisson's ratio of material \(i\), \(\kappa_{i}=3-4\
u_{i}\) for the plane strain and \(\kappa_{i}=(3-\
u_{i})/(1+\
u_{i})\) for the plane stress, \(i\), \(j\) again stand for the corresponding interface parameters. In tackling the problem of a crack situated at an interface, achieving a comprehensive solution hinges on the precise knowledge of the individual stress intensity factors, namely \(K_{I}\), \(K_{II}\), and \(K_{III}\). Alternatively, understanding the energy release rate \(G\) along with the corresponding phase angles \(\psi\) and \(\phi\) also provides an equivalent and comprehensive depiction of the solution. The significance of these parameters lies in their ability to encapsulate the mechanics of crack propagation and behavior at the interface. Stress intensity factors are fundamental indicators of the stress field near the crack tip, crucial for evaluating the crack stability and growth potential. The energy release rate \(G\) is a key measure of the energy made available for crack advancement, providing a deeper understanding of the fracture process. Additionally, the phase angles \(\psi\) and \(\phi\) are pivotal to characterize the mode mixity, essentially quantifying the contribution of different loading modes to crack propagation. Together, these parameters constitute a vital framework for describing and predicting crack behavior at the interface, aiding engineers and researchers in devising effective crack prevention, control, and material design strategies in various applications. The energy release rate in the event of an interfacial crack is defined as [2]:  
\[G=\frac{1}{\cosh^{2}(\pi\varepsilon)}\cdot\frac{\left|K\right|^{2}}{E_{*}}+ \frac{K_{III}^{2}}{2\mu_{*}} \tag{6}\]  
where  
\[\frac{2}{E_{*}}=\frac{1}{\overline{E}_{1}}+\frac{1}{\overline{E}_{2}},\quad \quad\frac{2}{\mu_{*}}=\frac{1}{\mu_{1}}+\frac{1}{\mu_{2}}, \tag{7}\]  
with \(\overline{E}_{i}=E_{i}/(1-\
u_{i}^{2})\) for the plane strain state and \(\overline{E}_{i}=E_{i}\) for the plane stress state, while the indices \(1\) and \(2\) are referring to properties of the material above and below the interface, respectively. In the case of the interface crack, \(\varepsilon\
eq 0\), influences of the tensile and shear loads in the vicinity of the crack tip are not separated. To measure the relative dependence of the shear and normal forces (namely the Mode II to Mode I loads), it is necessary to determine the characteristic length \(L\). For the oscillatory field, the mixed mode (\(\psi\) and \(\phi\)) in the \(K\) space is being defined as, Figure 2 [3]:  
\[\mathrm{tg}\psi=\frac{\mathrm{Im}(KL^{it})}{\mathrm{Re}(KL^{it})},\quad\quad \cos\phi=\frac{K_{III}}{\sqrt{\left|K\right|^{2}+K_{III}^{2}}}\,. \tag{8}\]The selection of the characteristic length \(L\) is somewhat discretionary, yet for a given material setup it may remain constant and uninfluenced by the dimensions and form of the specimen. An optimal choice for \(L\) involves considering the length of the elastic region relative to the overall sample dimensions, as elucidated in reference [1].