Dislocation Pile-Ups against a Locked Dislocation of a Different Burgers Vector

Abstract

The classical analysis of Eshelby, Frank, and Nabarro of a linear dislocation pile-up is generalized to include the case in which the locked dislocation may have a Burgers vector of mb, where b is the Burgers vector of free dislocations and m is a positive real number. The equilibrium positions of (n−1) free dislocations piled up against the locked dislocation under a uniform applied stress are given by the roots of the Laguerre polynomial Ln−1(2m−1). Simple expressions for the distance between the locked and nearest free dislocation, x1, the length of the pile-up, L, and the stress at its tip, σtip, are obtained. Increasing m will increase x1 and decrease σtip, while L is only slightly extended. For large n the stress field within a certain distance range around the tip is found to be independent of m. Based on the Petch model of yielding it is shown that increasing m increases the Hall-Petch slope by a factor of (m)1/2. The effect of m on the coalescence of leading dislocations leads to a higher-fracture stress if m is increased.