Here we give a brief review of the principles of stress, strain, and isotropic elasticity that will be needed in the study of defects. Readers familiar with this elementary material can skip on to the next chapter. The review is given here as a reference that will be used from time to time in the remainder of this book. A more thorough treatment of this subject can be found in many elasticity textbooks, such as [10, 11].

Stress

Stress is a measure of the intensity of force transmitted through a surface separating different parts of a body. The basic definition of stress is force per unit area. There are two types of stress: axial and shear, as shown in Fig. 2.1a, b, respectively. The axial stress is σ = P/A, where the force P is perpendicular to the surface area A. In other words, the force P acts along the surface normal. Hence the axial stress is also called the normal stress. The shear stress is τ = P/A, where the shear force P is parallel to the surface area A. The dimension of stress is f/l 2, where f is the dimension of force and l is the dimension of length. The unit of stress is N/m2= Pa (pascal).

Stress as a second-rank tensor

To completely specify the stress state at a point, we need to consider a small cube around this point and specify the traction forces per unit area on all faces of this cube. The edges of the cube are chosen to be parallel to the axes of a given coordinate system. The positive faces of the cube are defined as the three faces whose outward normal vectors are along the positive x, y, and z axes, respectively. Because the size of the cube is vanishingly small, it suffices to specify the forces on the positive faces of the cube. The forces on the negative faces must be opposite to the forces on the corresponding positive faces.

The main purpose of this chapter is to introduce the geometrical properties of dislocations, the rules governing dislocation reactions, and the directions of dislocation motion in response to applied stress. The goal is to develop an intuitive understanding of the basic behaviors of dislocations without obtaining their stress field (which is the subject of the next chapter).

We start with Section 8.1 on why dislocations are necessary for plastic deformation of crystals. In Section 8.2, we introduce Volterra dislocations in an elastic continuum, and then describe the differences between them and dislocations in a crystal. In Section 8.3, we define the Burgers vector of a dislocation, and describe the geometric rule for Burgers vectors that must be satisfied when dislocations react. Section 8.4 shows which direction a dislocation should move on its glide plane under an applied stress. It also introduces cross-slip and climb, as alternative modes of dislocation motion. Section 8.5 describes where crystal dislocations come from. Severalmechanisms are presented in which the motion of existing dislocations can lead to multiplication, i.e. an increase of total dislocation length.

Role of dislocations in plastic deformation

We begin our study of dislocations by first thinking about plastic deformation in crystals – a problem that first led to the concept of crystal dislocations in the early 1930s. Although one's common experience with plastic deformation usually involves the continuous bending or stretching of a soft metal wire, the fundamental mechanism of plastic deformation is a shear process, as shown in Fig. 8.1.

The crystal, represented as a rectangular box, is plastically deformed in tension by sequential slip on various crystal planes. The bold lines indicate the active slip plane for that particular strain increment. Notice that the cumulative effect of these events is to make the crystal permanently longer and narrower. So themacroscopic shape change associated with ordinary tensile deformation is actually the cumulative effect of a large number of shear events. This can be confirmed by observing the surface of a plastically deformed metal crystal under an optical microscope, which reveals lots of surface steps, called slip traces. These are the intersection lines between slip planes and the sample surface.