Modeling fracture properties for heterogeneous materials using J-Integral

The main objective of this paper is to analyze the stability of existing cracks in two-dimensional materials, we applied the principles of fracture mechanics to calculate the stress intensity factor KI using the integral J .
The stress intensity factor is a common parameter in fracture mechanics, this parameter helps to predict whether a specific crack causes the structure to fail. When this calculated value becomes equal to the material's critical fracture toughness, KIc (a property of the material), catastrophic failure usually occurs.
The determination of the stress intensity factor directly starting from the local state in crack tip is problematic, because the stresses are singular there. For this reason, more indirect energy-based methods are attractive. In this work, KI is calculated using the J-integral and the energy release rate. In addition, the crack growth rate and the number of cycles required to propagate the crack over a certain distance are calculated