For an anisotropic, linearly elastic crystal, Mura29 derived a line integral expression for the elastic dis­tortion of a dislocation loop, as

uij(x) E/»kCpqmnb?

where nk is the unit tangent vector of the dislocation loop line L, d/ is the dislocation line element, jh is the permutation tensor, Cijk/ is the fourth-order elas­tic constants tensor, Gy,/(x — x0) = dGy (x — x’)/dxi, and Gij(x — x0) are the Green’s tensor functions, which correspond to displacement components along the xi direction at point x due to a unit point force in the xj direction applied at point x0 in an infinite medium.

It can be seen that the elastic distortion eqn [10] involves derivatives of the Green’s functions, which need special consideration. For general anisotropic solids, analytical expressions for Gij, k are not available. However, these functions can be expressed in an integral form (see e. g., Barnett,3 Willis,3 Bacon

et a/.,32 and Mura33), as

Gij, k(x x ) [—?kNi^- (k)D—1 (k)

Ck

+ kk Clpmq (rpkq + kprq )Nil (k)Njm(k)D 2(k)]df [11]

where r = x — x0, r = r/r, k is the unit vector on the plane normal to r, the integral is taken around the unit circle Ck on the plane normal to r, and Nj(k) and D(k) are the adjoint matrix and the determinant of the second-order tensor Cikjlkkkl, respectively.

At a point P on a dislocation line L, the external force per unit length is obtained by the Peach — Koehler formula FA = (b-oA) x t, where sA is the sum of the stress from an applied load and that arising from internal sources at P, b is the Burgers vector of the dislocation, and t is the unit tangent vector of the element dl. For the special, yet important case of a

-o— 3-Nodes ■o…. 4-Nodes -o-— 5-Nodes
L/4 L/4 L/4 L/4 Figure 2 Nodal displacements for the first time step of an initially straight segment.
with only two linear and equal segments connected at point A, as shown in Figure 2. We will compute the shape of the line, advancing it from its initial straight configuration to a curved position. Under these sim­plifications, the variation in Gibbs free energy 8G for any one of the two segments is given by
1 ft dr |ds| 0
V 8r |ds
B	[15]
8G
Now, we expand the virtual displacement and veloc- ity in only two shape functions: C1 = u, C2 = 1 — u. Thus
drk = dqtkC,	[16]
Vk = q, k,t C, [17]
Since we allow the displacement to be only in a direction normal to the dislocation line (y direction), we drop the subscript k as well. For arbitrary varia- tions of 8qik, the following equation is applicable to any of the two segments.
At x (Pk + fs)C, |ds| = — B
DqmCmCi | ds|	[18:
Equation [18] can be explicitly integrated over a short time interval At. The resistivity matrix elements are defined by g, m = BjC, Cm|ds|, and the force vector 0 1 elements by f = J C, x (fpK + fs) | ds | . With these definitions, we have the following (2 x 2) algebraic system for each of the two elements: Aqmgim = f x At [19]

For any one linear element, the line equation can be determined by

And the resistivity matrix can be simplified as [21]

Furthermore, as a result of the shear stress t and the absence of self-forces during the first time step only, the distributed applied force vector reads

tbi 1 "2 1

Since the dislocation line is divided into two equal segments, we can now assemble the force vector, stiff­ness matrix, and displacement vector in the global coordinates, and arrive at the following equation for the global nodal displacements AQ:

F1

+ At 0

F3

An important point to note here is that, at the two fixed ends, we know the boundary conditions but the reaction forces needed to satisfy overall equilibrium are unknown. These reactions act on the fixed obsta­cles at L and R, and are important in determining the overall stability of the configuration (e. g., if they exceed a critical value, the obstacle is destroyed, and the line is released). If AQ = AQ = 0 at both fixed ends, we can easily solve for the nodal displacement AQ = 3 lbAr and for the unknown reaction forces at the two ends: F1 = F3 = 1|rbl. If we divide the disloca­tion line into more equal segments, the size ofthe matrix equation expands, but the nodal displacements and reaction forces can be calculated similarly. Results of analytical solutions for successively larger number of nodes on the dislocation segment are shown in Figure 2.

In the following section, we present several applica­tions of the computational method to irradiated mate­rials, where modeling defect behavior has contributed to a greater understanding of radiation effects.