# Prediction of the Crack Front Shape of the Corner Interface Crack
## 3 Results and Discussion  
In this investigation, our focus lies on understanding the crack propagation, a phenomenon primarily dictated by the energy release rate. All the simulations were executed in the _Mathematica(r)_ programming package. Figure 4 portrays the normalized energy release rate, a crucial parameter offering profound insights into the crack propagation process, for all three crack front shapes under consideration. The interface under scrutiny constitutes Al\({}_{2}\)O\({}_{3}\)/steel, characterized by specific material properties. These include \(E_{1}=375\) GPa and \(\
u_{1}=0.27\) for one material, and \(E_{2}=216\) GPa and \(\
u_{2}=0.28\) for the other, as documented in reference [24]. Additionally, the material combination is defined by Dundurs parameters \(\alpha=0.2663\) and \(\beta=0.0779\), while the oscillatory index is denoted by \(\varepsilon=-0.025\). The statement presented in [29], that the most stable shape of the crack front is the triangular one, is confirmed for this material config  
Figure 4: Normalized energy release rate for the three shapes of the crack front for the crack at the Al\({}_{2}\)O\({}_{3}\)/steel interface in terms of the angle \(\theta\). Examining Figure 5 provides valuable insights into the crack propagation dynamics. Notably, the quarter-circle-shaped crack exhibits swift progression near the free surface zone but considerably slower advancement in the central region. Conversely, this trend is inverted for concave and triangular cracks, where the central front advances at a faster pace compared to the edge areas. These observations collectively suggest that the triangular shape of the corner crack front presents a higher degree of stability. To substantiate these findings, two additional material combinations were subjected to testing, involving the propagation of corner cracks with three different front shapes: concave, straight, and convex. In the case of crack propagation illustrated in Figure 6, the bimaterial combination (grey cast iron/PMMA) encompasses distinct material properties: \(E_{1}\) = 130 GPa, \(\
u_{1}\) = 0.22, \(E_{2}\) = 2.4 GPa, \(\
u_{2}\) = 0.33. Additionally, this setup incorporates Dundurs parameters \(\alpha\) = 0.9613 and \(\beta\) = 0.2419, along with an oscillatory index \(\varepsilon\) = \(-\)0.079. These parameters define the material behavior crucial for understanding crack propagation, and their selection is based on extensive experimentation and analysis. By examining these combinations, we gain a comprehensive understanding of how material properties interact with crack geometry, shedding light on stability and progression mechanisms in interface cracks. Figure 5: Presentation of the corner crack front shape for the crack at the Al\({}_{2}\)O\({}_{3}\)/steel interface. (**a**) Concave, (**b**) triangular, (**c**) convex. Figure 6: The propagation of the interfacial corner crack for the bimaterial combination grey cast iron/PMMA. (**a**) Concave crack front, (**b**) triangular crack front, (**c**) convex crack front. The bimaterial combination (silicon/epoxy) in the crack propagation case presented in Figure 7, possesses the following properties \(E_{1}=128\) GPa, \(\
u_{1}=0.28\), \(E_{2}=12\) GPa, \(\
u_{2}=0.33\), with Dundurs parameters \(\alpha=0.8232\) and \(\beta=0.2043\) and the oscillatory index \(\varepsilon=-0.066\). Both Figures 6 and 7 provide compelling visual evidence demonstrating distinct patterns in crack propagation concerning different crack front shapes. In the case of the concave crack front shape, the periphery or edge areas exhibit a more rapid propagation compared to the central region of the front. Conversely, when dealing with the convex shape of the crack front this trend is notably reversed. This observation sheds the light on the intricate interplay between the crack geometry and a propagation dynamics. The crack front shape evidently plays a crucial role in influencing how the crack propagates within a material.