| Figure 2.1: | Edge dislocation in a cubic crystal. |
The presence of one dislocation displaces the atoms in the crystal from their perfect lattice sites and the resulting distortion produces a stress field in the crystal around the dislocation. The dislocation is therefore a source of internal strain and stress fields which increase the internal energy of the crystal. For example, consider the edge dislocation in Figure 2.1. The region above the slip plane contains the extra half-plane forced between the normal lattice planes. Consequently, the region above the slip plane is in compression, while the region below is in tension. The stresses and strains produced by dislocations in the bulk of the crystal are sufficiently small for linear elasticity theory to be applied to quantify them. An understanding of elasticity theory is a necessary prerequisite for the development of a quantitative theory for dislocations.
In this chapter the parts of elasticity theory which are essential for later derivations are reviewed. Section 2.2 is mainly based on the second chapter of the book of Hirth and Lothe [23] and it is integrated with parts from the book of Hull and Bacon [32]. For the complete description of linear elasticity, the reader is referred to the text by Love [45] on which much of the reviews in the above mentioned books are based. The texts of Landau and Lifshitz [40], and Sokolnikoff [70] are useful for supplementary study. A good discussion of the limitations of linear elasticity is found in a review article by Eshelby [13]. In addition, the reader is referred to the text by Balluffi [4] for the application of elasticity theory to lattice defects. The procedure to rotate the stiffness and compliance tensors reported in Section 2.3 is derived from the text by Ting [74].
