calculation is the displacement field in a crystal con­taining a dislocation loop, which can be expressed

Consider now the virtual motion of the disloca­tion loop. The mechanical power during the virtual motion is composed of two parts: (1) change in the elastic energy stored in the medium upon loop motion under the influence of its own stress (i. e., the change in the loop self-energy); (2) the work done on moving the loop as a result of the action of external and internal stresses, excluding the stress contribution ofthe loop itself. These two components constitute the work of Peach-Koehler.27 The main idea of DD is to derive approximate equations of motion from the principle of virtual power dissipa­tion of the second law of thermodynamics,15 by finding virtual Peach-Koehler forces that would result in the simultaneous displacement of all dislo­cation loops in the crystal. A major simplification is that this many-body problem is reduced to the single loop problem. In this simplification, instead of moving all the loops simultaneously, they are moved sequentially, with the motion of each one against the collective field of all other loops. The approach is reminiscent of the single-electron simplification of the many-electron problem in quantum mechanics.

If the material is assumed to be elastically isotro­pic and infinite, a great reduction in the level of required computations ensues. First, surface integrals can be replaced by line integrals along the disloca­tion. Second, Green’s functions and their derivatives have analytical solutions. Thus, the starting point in most DD simulations so far is a description of the elastic field of dislocation loops of arbitrary shapes by line integrals of the form proposed in de Wit26 as

dk [2]

where m and n are the shear modulus and Poisson’s ratio, respectively, b is the Burgers vector with Car­tesian components b,, and the vector potential Ak(R) = EijkX, Sj/[R(R T R • s)] satisfies the differen­tial equation CpikA^p(R) = XjRT3, where s is an arbi­trary unit vector. The radius vector R connects a source point on the loop to a field point, as shown in Figure 1, with Cartesian components R,, succes­sive partial derivatives R, jk. .., and magnitude R. The line integrals are carried out along a closed contour C defining the dislocation loop of differential arc length dl of components dlk.

41 nEkmn(R, ijm dijR, ppm)

[3]

segments. Reproduced from Ghoniem, N. M.; Huang, J.; Wang, Z. Philos. Mag. Lett. 2001, 82(2), 55-63.

where m and n are the shear modulus and Poisson’s ratio, respectively. The line integral is discretized, and the stress field of dislocation ensembles is obtained by a summation process over line segments, as shown in Figure 1.

Once the parametric curve for the dislocation segment is mapped onto the scalar interval {o 2 [0,1]}, the stress field everywhere is obtained as a fast numerical quadrature sum.13 The Peach-Koehler force exerted on any other dislocation segment can be obtained from the total stress field (external and internal) at the segment as27

FPK = s • b x t [4]