Analytic solution for linear dislocation arrays in an arbitrary stress field

Abstract

Applying the continuum approximation, the behaviour of a dislocation pile-up in an arbitrary periodic stress field σ( x) is examined. The distribution function, aswell as the length of pile-up the individual dislocation positions and stresses outside the pile-up that results from it, can be derived in an analytical way. The main result of this work shows that the dislocation distribution depends not only on the kind of σ( x) function but also additionally on the position of σ( x) relative to the pile-up. Therefore, in the case where σ( x) is given by a single arbitrary periodic stress field, the phase shift ϕ x of the stress amplitude of σ( x) with respect to the center of pile-up appears as a parameter only. Further, the conditions for splitting of pile-ups depends also on the wavelength and stress amplitude of the stress field additionally from ϕ x . The consequences of these results are discussed and compared with earlier results of an asymmetric periodic as well as uniform stress field.