Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Calculations involving pile ups of dislocations, both analytical and numerical, using either discrete dislocations or continuous distribution of dislocations of infinitesimal Burgers vectors, are reviewed in the light of their effects on the relation between yield or flow stress and grain size. The limitations of the pileup models are discussed and some nonpileup theories of yielding are critically reviewed also. More critical experiments are still needed to reveal the fundamental mechanicm of yielding.

The role of dislocations in the flow stress grain size relationships

Hardening by spinodal decomposition

The statistical problem of hardening arises when we consider the motion of a dislocation, a flux line in a Type II superconductor, or a Bloch wall, through a crystal containing defects which act as pinning centres. After discussing the general idea of a back stress, and introducing some dimensionless parameters which arise, we consider the influence of interactions between dislocations or flux lines as they move through the hardened crystal. The theories of Mott and Friedel for the mechanical hardening produced by point obstacles, and the estimates of Mott and Nabarro and of Riddhagni and Asimow for the hardening by diffuse obstacles, are then outlined. A fuller account is given of Labusch's method of distribution functions in its applications to diffuse and to localized obstacles. Finally, we enquire which of the results lie within the scope of second order perturbation theory.

The statistical problem of hardening

Based on the theory of continuously distributed dislocations, the equilibrium equations for both edge‐and screw‐dislocation pile‐ups in materials composed of soft and hard phases are formulated in terms of a singular integral equation. The integral equation is solved exactly by using the Wiener‐Hopf technique with Mellin transforms.The dislocation distribution function is found in explicit form. It is shown that the number of dislocations in the piled‐up array can be determined directly from the Mellin transform of the distribution function without its inverse transform.

Dislocation Pile‐up in Two‐Phase Materials

Approximate solutions for the distribution of dislocation arrays under nonuniform stresses are given. Three basic types of stress fields are investigated: The finite power series, the simple exponential and simple trigonometric functions.A number of quantitative examples are given. It is found that the number of dislocations in a piled-up array is particularly sensitive to the relative magnitude of the stress near the tail of the array.

Y. T. Chou

Papers

Linear Dislocations in Nonuniform Stress Fields