*2.1. Discrete Dislocation Models*

Figure 2 shows schematics of discrete dislocation models. The cracks are modeled using crack dislocations, meaning the Burgers vectors of the dislocations can be variable and need not correspond to the translation vectors of the crystals. The crack is packed with a sufficient number of dislocations and their equilibrium positions are then found under the imposed applied tensile stress. The equilibrium position corresponds to the zero-force on the dislocations. Therefore, it corresponds to the traction-free condition at the crack surfaces. The total energy of the system involves an algebraic sum of the self-energies, mutual interaction energies, the work done by the applied stress in moving the dislocation from the origin (center of the crack) to their equilibrium positions, and the surface energy in creating the two crack surfaces. For glide dislocations, one has to consider also the work done against the lattice frictional stress. The total energy reaches a peak that corresponds to the Griffith condition. Beyond the peak, the total energy decreases with the crack accelerating as it grows, contributing to specimen failure. The calculations are valid if, for the elastic crack, the results match with that of the Griffith crack.

The crack can emit crystal-lattice dislocations on the slip plane (Burgers Vectors of these correspond to the crystal in question) that make an angle with the crack plane (cleavage plane) [17–19]. These dislocations move against the lattice frictional stress (which can be equated to the yield stress of the material) depending on the relative ratio of frictional stress to the applied stress (μ). If the frictional stress reaches zero, the emitted dislocations can go to infinity. On the other hand, if the stress is infinite, the crack reduces to an elastic crack. Thus, a range of material behavior can be obtained by changing this relative ratio, μ.
