The example of a line defect is dislocation, and it has a very important role to play in plastic deformation of crystalline materials. In this section, we introduce the con­cept of dislocations; however, the majority of the dislocation theories will be

discussed in detail in Chapter 4. Dislocations are not equilibrium defects like point defects because the associated energy is higher than the increase in the enthalpy. They are generally created during solidification, cooling, and mechanical working, and sometimes just by handling. Hence, they are introduced in the crystal in a non­equilibrium way due to the action of mechanical stresses, thermal stresses, collapse of vacancies, or during the precipitate growth and different other events such as exposure to high-energy radiation.

Orowan, Taylor, and Polanyi first conceptualized crystal dislocations during 1930s without directly observing them. The direct evidence of the presence of dis­locations was obtained later (during 1950s) using X-ray topography and transmis­sion electron microscopy (TEM) techniques. Some other indirect techniques (such as etch pit method and decoration method) were also used, but none was viable in comparison to TEM (Figure 2.30).

Under normal conditions, plastic deformation occurs through the relative shear­ing of two crystal parts on a particular plane, called slip plane, along certain crystal­lographic direction (recall close-packed planes and directions in Section 2.1). It would have taken a lot of energy to create the deformation if the atoms needed to jump all at the same time. It has been estimated from the theories of rigid body shear and some other simple assumptions that the shear stress required to initiate plastic deformation (shear yield stress) in a crystal should be ~G/2p, where G is the shear modulus of the crystal. This leads to a huge yield strength number for real metals, which is physically never observed. That is why it led to the belief that the deformation remains localized in a narrow region and propagates through the

crystal in a wave-like fashion (Figure 2.31a). Thus, there are some defects that aid in the plastic deformation such that the stress required to initiate plastic deforma­tion becomes quite less. Let us try the case of aluminum. Aluminum has a shear modulus of ~27 GPa; thus, a shear yield stress of ~4.3 GPa. But we know that the shear yield stress of pure aluminum is in the range of 3-10 MPa. There is a differ­ence of several orders of magnitude between the predicted and observed values. This dichotomy can be solved by the inclusion of dislocations in the conversation. Dislocations can be defined as the line defect (AB) that demarcates the unslipped (D) and slipped (C) regions of a crystal (Figure 2.31b).

The dislocation line perpendicular to the slip direction is called edge or Orowan- Taylor dislocation (Figure 2.32a), and that parallel to the slip direction is called screw or Burgers dislocation (Figure 2.32b). But most dislocations remain in a mixed con­figuration as the dislocation line is typically curved. An important characteristic of dislocation is Burgers vector (b) that represents the unit slip distance and is always along the slip direction. An edge dislocation is illustrated by inserting an extra half­plane of atoms, thus creating a large disturbance in the atomic configuration in a region just below the extra half-plane. If the half-plane is above the slip plane, the dislocation is called a positive edge dislocation (represented by?), and if it is situated below the slip plane, it is called a negative edge dislocation (represented by T). On the other hand, the situation of screw dislocation is little different. The screw disloca­tion moves in a single surface helicoid, much like a spiral staircase. If we look

down on the dislocation and the helix appears to advance in a clockwise circuit, the dislocation is called a right-handed screw dislocation (or positive screw), and if it is anti­clockwise, it is called a left-handed screw dislocation (or negative screw).

The Burgers vector can be found out by constructing a Burgers circuit around the dislocation. Burgers circuit is any atom-to-atom path taken in a crystal that forms a closed path, while the circuit passes through the good part of the crystal. In the presence of a dislocation, the vector needed to close the circuit is the Burgers vec­tor, as illustrated for a schematic edge dislocation configuration in Figure 2.33a. Construction of Burgers circuit for a screw dislocation is shown in Figure 2.33b. Dislocations can end at the crystal surface, internal interfaces (grain boundaries), and so on, but never within the grain unless it forms a node (the sum of the dislo­cation Burgers vectors is zero at the node, Sb = 0) or a closed loop.

A pure edge dislocation can glide or slip perpendicular to its line vector (t) on the slip plane. As the edge dislocation line is normal to its Burgers vector, it remains confined to a specific plane along with its Burgers vector, and hence its glide is limited to a specific plane that contains both the Burgers and line vectors. But it can move vertically leaving the slip plane via a process known as climb. The climb process requires addition or subtraction of atoms from the end edge

mechanism. Since the climb process requires diffusion, the process is likely only at a higher homologous temperature, and it is generally slower than the glide. Because the climb process requires new atoms coming into or going out of the region, the local mass is not conserved, and that is why it is called nonconservative movement (unlike glide that is called conservative movement). If the dislocation moves vertically upward, the process is called positive climb (Figure 2.34a); if the dislocation moves vertically down, it is called negative climb. On the other hand, if a screw dislocation having the Burgers vector is parallel to its line vector, it does not have a preferred slip plane and thus the glide of screw dislocation is much less restricted. It is worth noting that screw dislocation cannot climb, but it can leave its slip plane by a process known as cross-slip or cross-glide (Figure 2.34b). The seg­ments of z and x are the screw components of the dislocation loop shown in Figure 2.34b, and thus are able to cross-slip. Table 2.4 summarizes some interest­ing features of the edge and screw dislocations in a comparative way. More discus­sion on this topic will be initiated in Chapter 4.

An important aspect of dislocation microstructure is the dislocation density (gd) so that the number of dislocations in a given volume can be quantified to be used in a variety of relations describing mechanical behavior of crystalline materials. Thus,

dislocation density is primarily defined as “the total line length of dislocations per unit volume.” Hence, the unit of dislocation density would be cm cm~3, that is, cm~2. Based on the derived unit, there is another way to define dislocation density: the number of dislocation lines that intersect a unit area. Carefully prepared crystal tends to have a low dislocation density, ~102cm~2. Some single crystal whiskers can be made nearly free of any dislocation. On the other hand, heavily deformed metals (cold worked) may contain dislocation density in the range of 1010­1012 cm~2 or more, whereas an annealed crystal may contain 106-108 cm~2.

2.2.3