A discussion of the structure and behaviour of dipole walls in cyclic plasticity

Abstract

Assuming that cross-slip by thermally activated migration of jogs can cause annihilation of screw dislocation dipoles without macroscopic crystallographic confinement of cross-slip to the cross-slip plane, an attempt is made to re-derive earlier equations for the saturation stress and the plastic strain amplitude in persistent slip bands. These equations had been based on the assumption that cross-slip could occur only on a cross-slip plane making an obtuse angle with the slip plane, an assumption which limits the mean free path of screw dislocations. The key new assumption is that the walls of edge dislocation dipoles which dominate the structure of persistent slip bands are penetrable obstacles, which increases the mean free paths of the mobile dislocations. Agreement with experiment is obtained if the penetration probability in cyclic saturation is on average one third, a value for which there is a simple rationalization. Estimates can be made of the wall width, which is independent of temperature, in agreement with recent observations by Tippelt et al. However, the main unresolved difficulty is the role of the very fine dipoles, particularly the faulted dipoles, in the walls. A further weakness in the theory is that it ignores the cutting of dipoles by the cross-slipping screw dislocations. Despite these problems, the distribution of dipole heights can be worked out and is found to be in reasonable agreeement with experiment.

Acknowledgements

I am indebted to Professor C. Holste for an invitation to attend a conference in Dresden, where these ideas were first presented. I am particularly indebted to Dr A. Schwab, who discovered a crucial error in the original version of the paper, and whose work points the way to an understanding of the observed cycle-to-cycle variation in the persistent slip band displacements in terms of a statistical distribution of the penetration probability. I am grateful to Professor H. Mughrabi and to an anonymous referee for critical comments, and also to Robinson College and the Cavendish Laboratory for support.

Notes

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The equality of bowing stress and passing stress was first postulated by Mughrabi (1981) although he did not consider the saturation stress to be equal to the sum of the two, but rather to include a term from internal stress.

See especially the latest review by Mughrabi and Ungar (2002). Pedersen (2002) has recently proposed that penetration of the wall is caused by local dipole flipping. However, the notion that the wall structure is deformable, with a definite yield stress at which dislocations penetrate it, is quite different from the picture of a statistically penetrable wall proposed here.

This is true regardless of whether the dipoles are of random sign, that is have the half-planes pointing randomly inwards and outwards, or whether they are biased towards vacancy type (half-planes outwards), as observed by Antonopoulos et al. (1976) and Tippelt et al. (1997).

See figure 10 of this paper and also the related discussion given in the text.

Note that, although the average value is unchanged, the distribution of dislocations within the wall is changed. In particular, with a probability of one third, two planes are left without dislocations and, with a probability of two thirds, one is left with two dislocations that are part of a multipole. The multipole will be eliminated on the reverse stroke and can form the new source of slip dislocations (Neumann 1975). Further work is required to understand the statistical distribution of the penetration probability, although its average value must be given by equation (10).

Essmann and Rapp (1973), Essmann and Mughrabi (1979) and Mughrabi and Essmann (1979) made extensive use of this equation with an interdislocation spacing of 1.6 nm, equal to their proposed height for spontaneous annihilation of an edge dislocation dipole, in place of d ₁₁₁. This produces better agreement with the measured dislocation densities but is based upon the notion of a quasi-uniform array of dislocations within the wall, and not upon an array of dipoles spaced much more widely than their height.

A simple calculation is as follows: assume that the resistivity of a dislocation is proportional to the square of its Burgers vector, and that the resistivity of a perfect dislocation is

and that of stacking fault is

(Watts 1989, Niewczas et al. 2001). Then we find that the ratio of the resistivity of a faulted dipole of 1 nm height to that of an unfaulted dipole is about one third, and that the stacking fault contributes only about one third to the reduced resistivity of the faulted dipole. Thus the density of such dipoles might be underestimated by a factor of 3.

The limit of long screw dislocations can be derived thus: choose one dislocation, with B breakthroughs, L to the left and R to the right. The probability of finding another of opposite sign is (1 − p)² p ^B /2. Of these, only half will have L links left and R links right, giving the limit found from equation (17) for B finite but large, and p nearly unity. There is another method: consider one channel between walls, and choose one long dislocation of B breakthroughs which ends in that channel. Half the other dislocations ending in the channel will go off in the same direction as the chosen direction, and half will have opposite sign. The chance that the distant end of the second dislocation lies in the same distant channel as the end of the first dislocation is (1 − p)² p ^B /4, giving the same result.

In fact,

, which means that in a ladder structure with a random distribution of penetration probabilities, about one quarter of the screw dislocations are mutually annihilated.

The density of dipoles at the discontinuity is determined by the normalization only.

The temperature independence of the wall width may also be seen in images of persistent slip bands in Mg reported by Kwadjo and Brown (1978) where the wall spacing increases by a factor of 4 between 77 and 300 K but the wall width stays constant at about 0.5 µm. In Cu, the wall width seems to be the same, about 0.2 µm, at 77 and 300 K, but the images given by Basinski et al. (1980) suggest that at 4.2 K the walls are thinner. It seems a fair conclusion that the wall width in all materials is independent of temperature, except perhaps at the lowest temperatures.

† These data are analysed by L. M. Brown in ‘Brief rejoinder’ in the same volume.