**1. Introduction**

Mechanical failure of materials occurs by crack initiation and growth. Griffith [1] provided the first crack growth analysis using the change in the strain energy of the cracked body in comparison to uncracked bodies, in generating new surfaces by crack initiation and the growth process. In his classical analysis, interestingly, the crack initiation and its growth are considered the same. Most metallic materials, however, yield by plastic flow before cracks form and grow. In addition, if the material deforms uniformly across the specimen cross-section, then cracks may not be initiated. The specimen may fail by the necking process. Because cracks are high energy defects, they need localized stress concentrations for their initiation and growth. Simple necking to failure can be seen in some superplastic materials [2].

Most engineering materials, however, are polycrystalline with distributed grain sizes and orientations. This results in inhomogeneous deformations across the specimen cross-section, leading to stress concentrations. The origin of the famous Hall–Petch equation that describes the grain size dependence for deformation and fracture is intrinsically related to the inhomogeneous deformations [3]. In addition, surface grains are likely to deform first due to reduced constraints near the surface. In the following, we examine the crack initiation and growth process in crystalline materials to extract some of the basic concepts involved. Discrete dislocation analysis [4] and elastic-plastic fracture mechanics methods [5] are used to understand how localized deformation and stress concentration help in the formation and growth of an incipient crack leading to specimen failure.
