The theory of solution hardening

Abstract

The glide plane of a dislocation contains a concentration c of obstacles, which we take to be attractive potential wells each made of four parabolic arcs of width w, exerting a maximum attractive force f_m. If the line tension of the dislocation is T, the nature of its motion is controlled by the parameter β = f _m/4Tcw ². When β is large the motion occurs in the Friedel limit. A length of dislocation line interacting with three consecutive obstacles A, B, C breaks away from B and moves until it interacts with a fourth obstacle D. The portions of dislocation line outside the segment AC do not move appreciably during this process. If the Burgers vector of the dislocation is 6, the critical resolved shear stress τ_c is given in this limit by bτ_c = (f _m ³c/2T)^1/2. In the Labusch limit, where β is small, we find, in agreement with Labusch, that the motion of a dislocation segment through a well is then adiabatic and non-dissipative. The segment of dislocation which moves coherently is in contact with many obstacles. Labusch assumes that the obstacles are statistically aggregated into clusters, and shows that, if the clusters are large enough, one obtains a flow stress roughly β^−1/6 times that predicted by Friedel. We believe that his argument is not self-consistent, because the dislocation then interacts coherently with only three consecutive clusters. By using an earlier statistical idea, due to Mott, and applying it to a simple model due to Lowell, we give an analysis which clarifies the physical processes occurring. While the advance of a segment of dislocation which is interacting with a large number n of obstacles is reversible, it does not occur under zero force, but under a force of order ± n ^1/2/_m. In regions where this force opposes the motion of the dislocation, it determines the flow stress; in regions where this force aids the motion, both this force and the force on the dislocation produced by the external stress lead to an acceleration of the dislocation and hence to dissipation.