**2. Crack Growth Analysis**

It is important to look at the problem from the basics. Figure 1 gives a schematic illustration describing the kinetics of crack initiation and the growth process, from an elastic (Griffith) crack to elastic-plastic crack. The problem is discussed both from the point of continuum mechanics principles and from the discrete dislocation modeling.

**Figure 1.** Analysis of crack initiation and growth for elastic to elastic-plastic cracks: (**a**) crack configurations (**b**) variation of total energy with crack length (**c**) log of stress vs. log of crack length for elastic and elastic-plastic cracks.

In Figure 1a, Griffith's elastic crack [1] is described. The total energy of the system involves the change in the elastic energy due to the presence of a crack under applied stress σ**y**, work done by the applied stress, and work expended in creating two new surfaces. The total energy reaches a peak, where crack length longer than the critical size **ac** will expand continuously with the reduction in the total energy, Figure 1b. The energy gradient provides the crack-tip driving force. If the energy to nucleate crystal dislocations from the crack tip is lower than the energy needed for the crack to expand further as an elastic crack, then the crack undergoes plastic relaxation, causing a reduction in the total energy of the system.

Starting from Bilby, Cottrell, and Swindon [6], continuum dislocation models are used to characterize the growth of a crack with the dislocation-plasticity. For a given crack size, the crack and associated plastic zone are analyzed using continuum dislocations. The conditions for a critical crack size for its continuous expansion are determined as a function of applied stress and lattice friction stress. In this model for mathematical simplicity, each crack with its plastic zone is treated separately. Hence, inherently the history dependence of a growing incipient crack with its plastic zone was not considered. A similar approach has been adopted by Orawan [7]. There have been many attempts to characterize tensile elastic-plastic cracks with crystal plasticity on inclined slip planes [8–10]. For a valid reason, the problem becomes mathematically intractable, and there are many attempted numerical solutions to the problem.

Figure 1a also shows the continuously expanding elastic-plastic crack. The crack growth alternates between glide and cleavage modes of crack growth. The total energy of such a crack also reaches a maximum as a function of crack length, Figure 1b. Analytically it is difficult to formulate the growth of such a crack. Figure 1c shows the log(stress) vs. log(crack length) plot for both elastic and continuous elastic-plastic cracks. For an elastic crack, which is the same as the Griffith crack, the slope is 0.5, depicting square root singularity of stress with crack length. For the continuous elastic-plastic crack, the slope is less than 0.5, and depends on the relative ratio of friction stress (or ~yield stress) to applied stress, μ.

Since it is difficult to formulate the tensile elastic-plastic cracks analytically, Marcinkoski and his group [4,11–13] analyzed the problem using discrete dislocations. Here, we present some recent results using such models [14], correlated with the results derived from continuum elastic-plastic fracture mechanics calculations [15,16], and also bring in our modified Kitagawa–Takahashi diagram [15] to extract some basic physical principles. In comparing the continuum models with the dislocation models, caution is exercised by recognizing that the scale of applications is different. Dislocation models are at the micron size level, while the continuum models are more relevant at the continuum level. Nevertheless, the fundamental concepts remain the same, as will be shown here.
