Part VI – June 1968 - Papers - Dislocation Reactions in Anisotropic Bcc Metals

Expressions are obtained for the energy changes associated with the reaction of (a& (111) slip dislocations on intersecting (110)planes in anisotropic bcc metals. An energy criterion for assessing the likelihood of dissociation of the products of such reactions is also presented. It is found that the "burrier reactions" which form a(100) dislocations at the intersection of two active {110) slip planes are more energetically favorable in metals which exhibit a high value of Zener's anisotropy factor, A, than those which have a low value. The results are presented in a form which permits the stacking fault energy to be obtained from a measurement of the separation between par-tials in a dissociated configuration. However, until accurate calculations or measurements of the stacking fault energies involved are available, it is not possible to assess the physical importance of dissociated dislocations. In a recent paper,' the energy changes associated with several types of reactions between two slip dislocations, (a/2)(111){110), in bcc structures were calculated.* Isotropic elasticity and the approxima- tion v = -3- were employed. The purpose of this work is to present calculations of the energy changes for many of the same reactions using anisotropic elasticity. The problem of dissociation of a(100) and a(110) dislocations is also considered, and maximum fault energies for which dissociation will be energetically favorable are calculated for several bcc metals. Two general types of reactions are considered; those for which the reactant (a/2)(111) dislocations have long-range attractive forces and those for which the reverse is true. An example of the former is: (a/2)[lll] + (a/2)[lll]-a[l00] while the latter are typified by: (a/2)[lll] + (a/2)[111] -a[011] Only reactants lying in different slip planes are considered; therefore, the products must lie along (111) or (100) directions, which are the intersection of two {llO} planes. It will be assumed that the reactants and products are infinitely long parallel dislocations, since in this case the energy change associated with the reactions is a maximum.' THEORY The self-energy per unit length of a straight mixed dislocation in an anisotropic medium can be written? where b is the Burgers vector, K is an appropriate combination of the single-crystal elastic constants, and R and ro are, respectively, outer and inner cut-off radii of the elastic solution. The energy given by Eq. [I] does not account for any variation of the core energy with orientation. This could be manifested by an orientation dependence of the core radius or, equivalently, the Peierls width, of the dislocation. However, the energy contribution due to this source is expected to be small, and current models of the dislocation core are not sufficiently accurate to justify such a refinement. It has already been shown that for the isotropic case the energy contributions due to nonzero tractions across the cores of the reactants and products exactly cancel one another in the reaction.' Accordingly, it will be assumed that this contribution to the total energy change in the anisotropic case is small. In the subsequent discussion it is also assumed that the core radii of the reactant and product dislocation are the same and that, where stacking faults are formed, the faulted region is bounded by the centers of the partials. Consequently only changes in elastic energy due to the reactions will be considered. When the dislocation is parallel to either the (111) or the (100) directions, K may be written:375 K = (Ke sin2 a + Ks cos2 a) [2] where K, and Ks are the combination of elastic constants corresponding to an edge and screw dislocation lying along the same direction as the mixed dislocation, and a is the angle between the direction tangent to the dislocation line and the Burgers vector. Eq. [2] should not be confused with the isotropic approximation to the variation in energy with line Orientation.6 It should be noted that the essentially isotropic expression for K is a result of the characteristic symmetry of the (111) and (100) directions and is not, in general, valid for other dislocation directions in anisotropic cubic metals. The energy* change for a reaction in which the re- actant and product dislocations are parallel perfect dislocations can be written: where Ep and E, refer to the self-energies of the products and reactants, respectively. For dislocations parallel to (100) and (111) directions, Eq. [3] becomes: