*2.2. Continuous Elastic-Plastic Crack*

By emitting crystal lattice dislocations, a growing elastic crack can reduce its energy. As the dislocations accumulate on the slip plane against the frictional stresses and mutual repulsive forces, a stage will be reached where the back stress from the emitted dislocations in the plastic zone prevents further emission of the dislocations at the crack tip. The configuration corresponds closely to that of the Bilby, Cottrell, and Swindon (BCS) crack other than the fact that the dislocations are on an inclined plane.

The continuous elastic-plastic crack, on the other hand, starts emitting crystal lattice dislocations as it grows. Hence, during the calculations, at each crack increment, the total energy is compared for dislocation emission vs. further crack extension as a cleavage crack. If the energy for emission is lower, then the crystal dislocation on the glide plane is allowed. The same test is again done for the emission of the second dislocation on the glide plane, in contrast to further crack extension as an elastic crack. For most cases, depending on the value of μ, the emission of the second dislocation is prevented due to the back stress from the previously emitted dislocation. Hence, the calculations are somewhat tedious and become intensive as the crack grows with changing glide to cleavage components depending on μ and the surface energy of the material. However, the case represents a more realistic situation with the plastic zone accumulating in the wake of the growing crack. The continuous elastic-plastic crack also

captures crack growth history. Thus, depending on the μ and γ (surface energy) values, the relative components of glide vs. cleavage components change. The total energy of the incipient crack increases with an increase in the length of the crack until it reaches a peak value. Further increase in crack size only reduces the total energy, resulting in the acceleration of the crack, contributing to the total failure, Figure 2b. The material can harden as the crack grows (thus changing the μ value), thereby altering the energy-crack length curve or contributing to crack growth toughness. Figure 1c shows the log of stress vs. the log of critical crack size (at the peck energy value). Calculations show that in the log–log coordinates, the stress vs. crack length for the continuous elastic-plastic crack follows a straight line but with the slope less that of the elastic Griffith crack, which is 0.5. The slope decreases with a decrease in μ. This is similar to the effect of the decrease of the yield stress of the material. Conversely, as the yield stress increases, the crack growth behavior approaches that of the elastic crack.

**Figure 2.** Discrete dislocation modeling of a continuously growing elastic-plastic crack. (**a**) Crack with crack and crystal dislocations with applied and lattice frictional forces. (**b**) Continuously expanding elastic-plastic crack with inbuilt history. Based on the total energy of the system, the crack can expand elastically or emit crystal dislocations. The crack mouth angle depends on the relative ratio of glide and cleavage planes. (**c**) Quasi-static growth of the crack under continuously decreasing stress holding the total energy constant after its first nucleation.

Figure 2c also depicts a thought experiment. After reaching the peak-energy for given stress as the energy starts decreasing, if the applied stress is reduced, then the energy has to again peak at a larger **ac** value. One can think of an infinitesimal change in the applied stress to keep the energy at the same peak level, without it increasing or decreasing. This process can be continued with a continuous decrease in the applied stress as the crack length slowly increases to maintain the growth in equilibrium. Since the total energy remains constant, such a crack grows at a quasi-steady state. For an elastic crack, the applied stress has to be reduced, maintaining the Griffith stress with the increasing crack length. Hence in the log–log plot, the stress vs. crack length line represents the quasi-steady crack growth condition for continuously decreasing stress. If the stress is higher than the Griffith line, then the growing crack accelerates. On the other hand, if the stress falls below the line, the growing crack is arrested. This forms the condition for the crack arrest of an incipient growing crack due to a

sharp decrease of applied or internal stresses that are contributing to the growth of a crack. It also leads to the Kitagawa–Takahashi type of diagram [20], as will be discussed below.
