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.

Our study of grain boundaries to this point has focused on their geometry and special misorientations that lead to periodic patterns in the GB structure.We now turn to another important aspect of grain boundaries: their energies and possible elastic fields. A planar grain boundary usually does not have a long range stress field by itself. However, certain grain boundaries contain periodic dislocation arrays as part of their structures. In such cases, there is an appreciable stress field around the GB at distances comparable to the inter-dislocation spacing in the GB. The GB model based on dislocation arrays, combined with the theory of coincidence site and DSC lattices, provides a way to understand the GB energy as a function of its misorientation angle.

We have seen that the GB energy as a function of the misorientation angle has a complex structure, as shown in Fig. 13.2 and Fig. 13.8. Nonetheless, such plots suggest a classification of grain boundaries broadly into three types: singular, vicinal, and general [129]. The singular GBs correspond to the sharp minima on the energy plots. Their misorientations usually correspond to low-Σ CSLs. The singular GBs are usually special in other properties as well, such as mobility and point defect segregation. The vicinal GBs have both misorientation and GB plane direction sufficiently close to the singular GBs, and they can be considered as singular GBs superimposed with one or more GB dislocation arrays. The spacing between the nearest dislocations in the array reduces as the misorientation deviates further away from that of the singular GB. The general GBs are those boundaries that are sufficiently different from the singular GBs that the dislocation array is no longer a useful model as the necessary dislocation density would be so large that the dislocation cores would overlap.

In this classification of GBs, the case of zero misorientation and its vicinal range deserves extra attention. On the one hand, when the misorientation angle θ equals zero, the grain boundary disappears and the GB energy is zero, because the two crystals are perfectly aligned with each other.

A qualitative understanding of the behaviors of point defects can be established by considering atoms as hard spheres packed together to form the crystal. Crude as the hard sphere model may seem, it can be used to explain many of the observations made about point defects. In Section 4.1, we define the hard sphere radius of an atom and show its influence on the site preference of solute atoms. In Section 4.2, we use the hard sphere model to show the type of the distortions (spherically symmetric or not) in the host crystal around a solute atom. This allows us to explain why certain solutes have a much stronger solid solution hardening effect than others.

We then need to go beyond the hard sphere model in order to be more quantitative. In Section 4.3, we define the Seitz radius, which is more useful than the hard sphere radius for keeping track of the volume occupied by atoms of different kinds in solid solutions. We will see that atoms often appear to take on a different radius as a solute atom in another crystal compared to the radius it takes in its own crystal. In Section 4.4, we apply elasticity theory to predict the elastic fields around a solute atom. For simplicity, the size of the point defect is shrunk to zero and is modeled as force dipoles acting on a point in an elastic medium. In Section 4.5, a more realistic model is developed, in which the solute atom is modeled as an elastic sphere to be inserted into a hole inside an elastic medium. Elastic fields arise because the initial size of the sphere is larger than the initial size of the hole. Even though many atomistic and electronic details concerning point defects are ignored, the models developed in this chapter are increasingly more quantitative and can be used to explain a large number of behaviors of point defects.

Hard sphere model

Hard sphere radius

It is common to treat atoms in a crystal as undeformable spheres and to calculate the atomic sizes from the lattice parameters (measured using X-ray diffraction).We call this the hard sphere approach.

In our treatment of dislocations thus far, we have avoided the dislocation core. For example, in Volterra's dislocation model, the stress–strain fields diverge on the dislocation line, so that a cylindrical region of material is usually removed around the dislocation line to avoid the singularity. In the line tension model, the dislocation is modeled as a string that carries a line energy per unit length, but is otherwise featureless. In Chapter 11, we have seen that perfect dislocations in close-packed metals tend to dissociate into partial dislocations, but the partial dislocations were still treated as Volterra's dislocation lines. In reality, every (perfect or partial) crystal dislocation has a core region, which possesses a specific atomistic structure, called the core structure. The core structure is determined by non-linear interatomic interactions and the crystal structure, and, in turn, strongly influences the energetics and mobility of the dislocations. In this chapter, we discuss typical dislocation core structures and their effects on dislocation properties in several crystal structures.

In Section 12.1, we start our discussion with the classical Peierls–Nabarro (PN) model, which was the first physical model for the dislocation core and naturally predicts that the dislocation core should have a finite width. In Section 12.2, we generalize the original PN model to account for the presence of stacking faults in FCC metals. Consistent with the hard sphere model in Chapter 11, the generalized PN model also predicts dissociation of perfect dislocations into partials, except that each partial now has a finite width.

For crystals whose structures are sufficiently different from close-packed, hard spheres are no longer a good model for the atoms. Nonetheless, the geometry of the stacking of atomic layers is still useful for understanding the dislocation core structures in these crystals, as discussed in Section 12.3 (diamond cubic crystals) and Section 12.4 (BCC crystals). Finally, in Section 12.5 we discuss the interaction between dislocations and point defects, which usually leads to segregation of point defects around the dislocation core.

Peierls–Nabarro model

The classical model by Peierls and Nabarro [111, 112] considers the spreading of the dislocation over the glide plane.

Having discussed the elastic field around a single point defect, we now apply the thermodynamics principles (Chapter 3) to obtain the equilibrium concentration of point defects in crystals under a given temperature and pressure. The fundamental principle used repeatedly is that the Gibbs free energy of the crystal is minimized when the point defects reach the equilibrium concentration.

We start by discussing the equilibrium concentration of extrinsic point defects, i.e. substitutional and interstitial solutes, in Section 5.1. The approach is then applied, in Section 5.2, to vacancies, which are intrinsic point defects. In Section 5.3 we discuss the experimental methods to measure the equilibrium concentration and thermodynamic properties of vacancies, and compare the experimental data with theoretical estimates. The chemical potential of point defects is defined in Section 5.4.

Equilibrium concentration of solutes

We consider a dilute substitutional solution of B atoms in an A matrix. Let NA (which is fixed) be the number of A atoms in the system and let NB be the number of B atoms dissolved in the A-rich crystal. The total number of atomic sites in the A-rich crystal is N = NA + NB. Thus χ = NB/N is the fraction of atomic sites where the “wrong” kind of atom is located. χ is also the molar fraction of B atoms in the crystal.We follow the regular solution/quasi-chemical approach in which the formation energy of the point defect is dominated by the energies of the chemical bonds associated with the impurity defect (quasi-chemical, Eq. (5.5)) and where the mixing entropy is that for an ideal solution (regular solution, Eq. (5.19)).

Let the A-rich crystal be in contact with a large B crystal, which acts as an infinite supply of B atoms. For simplicity, we only allow B atoms to enter the A-rich crystal as solutes, but forbid A atoms to enter the B crystal as solutes. Each time a B atom is dissolved in the lattice, the A atom it replaces takes up a site at the A/B interface and extends the A lattice by one atomic volume, as shown in Fig. 5.1a. Note that the number of A atoms NA is conserved, while the number of B solute atoms NB and the total number of atomic sites N for the A-rich crystal are not conserved.