We have noted in the preceding chapters that the dislocation movement under applied stress creates plastic strain (permanent deformation) in crystals. This is dif­ferent from the elastic deformation under applied stress where the atomic bonds are stretched without the help of a dislocation. It is not simple to relate plastic strain to applied stress as it depends on temperature, strain rate, and micro­structural factors. However, Orowan developed a simple expression, known as Oro — wan’s equation, relating the macroscopic strain rate to the microscopic parameters such as dislocation density, velocity, and Burgers vector. This derivation is based on the recognition of the fact that when a single dislocation moves, it creates a dis­placement of the magnitude equal to the Burgers vector (b).

Consider a crystal of volume HLD containing a number of straight edge disloca­tions (for the sake of simplicity, dislocation type is assumed to be edge), as shown in Figure 4.5. Now let us consider an applied shear stress causing a dislocation move by di. We know that if the dislocation moves the distance D along the Burgers vector, it produces b displacement. Therefore, it can be said that the contribution of the dislocation moving by di distance to the displacement is (b/D) x di. The dis­placement generated by a single dislocation is pretty small. To produce plastic strain of engineering significance, there are a large number of dislocations which

when move produce the total displacement A given by

b N

A dj.

D j=i

where N is the total number of dislocations that have moved in the crystal volume. The macroscopic plastic shear strain (y) is given by

(4.4)

Equation (4.4) can be further simplified by defining the average distance x trav­eled by a dislocation as

x 1 N

X = N -1 d>’

Hence, by replacing Eq. (4.5) into Eq. (4.4), we obtain

The expression NL/HLD gives the total dislocation line length (NL) per unit vol­ume (HLD), that is, mobile dislocation density. Let us denote the mobile dislocation density by gm. Note that the term gm is not the total dislocation density as the immobile dislocation does not contribute to the plastic strain. Therefore, Eq. (4.6) can be rewritten as

C = Qmbx.

Equation (4.7) can be further expressed in terms of shear strain rate by taking differential of both sides of the equation with respect to time t.

. dc, dx

C = d = rmb d = PmbVd’

where Vd is the average velocity of dislocations. The equation is universal in nature and also applicable for climb of edge dislocations. Furthermore, it can be universally applied to screw dislocations and mixed dislocations. Equation (4.8) is a nice exam­ple of relating macroscopic behavior of a material described by a set of microscopic parameters. The Burgers vector (b) is determined by crystal structure of the material. The other two quantities gm and Vd are the parameters that depend on several other factors such as stress, temperature, prior processing history, and so on.

4.1.4