Dislocation Modelling of Localized Plasticity in Persistent Slip Bands

We present models of localized plastic deformation inside Persistent Slip Band channels. First, we investigate the interaction between screw dislocations as they pass one another inside channel walls in copper. The model shows the mechanisms of dislocation bowing, dipole formation and binding, and finally dipole destruction as screw dislocations pass one another. The mechanism of (dipole passing) is assessed and interpreted in terms of the fatigue saturation stress. We also present results for the effects of the wall dipole structure on the dipole passing mechanism. Introduction A common ingredient in the models of fatigue crack nucleation is the existence of irreversible slip, caused by the progressive interaction of Persistent Slip Bands (PSBs) with the material surface resulting in concentration of the plastic displacement. A widely accepted model of PSBs is the Essman-Gösele-Mughrabi (EGM) extrusion theory [1], where they propose a semi-quantitative theory of the evolution of the surface profile of PSBs in fatigued metals on the basis of bulk dislocation processes. The essential features of this theory is the annihilation of pairs of both screw and edge dislocations, which leads to irreversible slip. Moreover, this theory suggests that extrusions form in a rapid manner as a result of the combined effect of dislocation glide and annihilation of close dipoles. Due to this latter effect, the mean glide plane becomes slightly inclined to the crystallographic glide plane, and edge dislocations of opposite sign are deposited on the two PSB-matrix interfaces. There are a number of factors that contribute to the saturation fatigue stress (or fatigue limit) in pure metals, and these are [2, 3]: (i) the stress required to allow two screw dislocations of opposite signs on parallel slip planes to pass one another, (ii) the stress required for bowing of screw dislocations in between PSB walls, and (iii) the long range internal stress field resulting from edge dislocation dipolar walls. To understand the influence of these mechanisms on the fatigue limit, a number of papers have been published on the glide of dislocations in PSBs [3 9]. Grosskreutz and Mughrabi [4] made an approximate estimate for the passing stress as a linear superposition of the screw dipole passing stress and the critical Orowan bowing stress for screw dislocations gliding inside dipolar PSB walls. This estimate was then adopted 23 by Brown [5, 6], who studied the problem of two rigid screw dislocations, of opposite sign, passing one another and depositing edge dislocations on the two diploar walls that confine them. Brown [5] deduced that the passing stress must be equal to the dipole passing stress, and that the two should be added together. Mughrabi and Pschenitzka [7] modified this conclusion by considering two elliptical screw dislocations confined between the walls. They predicted that the passing stress is not a direct addition of the bowing stress and the dipole passing stress but instead is about 20% larger than either one. Thus, they predicted that the linear superposition gives about 70-80% overestimate (i.e. only about 20% at most of the Orowan bowing stress adds to the passing stress). In response, Brown [3] presented a modified estimate in which the passing stress is computed as the sum of the bowing stress and the passing stress reduced from that for infinite straight screws. In this work, Brown concludes that about 50% of the Orowan bowing stress adds to the passing stress (40-70% when suitable parameters are chosen). Because of this wide range of conclusions obtained from analytical models of the fatigue saturation stress, the need for computer models based on numerical techniques is essential for accurate predictions. Discrete Dislocation Dynamics (DDD) has been developed in the last decade for fundamental descriptions of plasticity and fracture at the meso-scale. The approach relies on direct numerical simulations of the collective motion of dislocation ensembles without ad hoc assumptions by direct numerical solution of the equations of motion. This approach has been successfully used in many applications at the nanoand micro-scales [10 13]. In this paper, we explore, through the use of the DDD method, the effect of the shape of the screw dislocations as they simultaneously bow between the walls and pass one another on the passing stress. In addition, the effect of the long range internal stress field resulting from edge dislocation dipolar walls on the passing stress is investigated. Recently, Schwarz and Mughrabi [8] and Křǐst’an and Kratochvil [9] used the DDD method to provide additional insightinto the PSB problem. Although, the details of the present model are different from those of references [8, 9], the results will be shown to be comparable. In the work of Schwarz and Mughrabi [8], dislocations are confined in a channel and are modelled as short straight dislocations. In the study of Křǐst’an and Kratochvil [9], dislocations are modelled as planar flexible curves confined by an infinitely long edge dipole on each side (i.e the PSB wall is represented by a single dipole on each side). Also, for their calculation of the bowing stress, they use the line tension approximation. In the present investigation, we model the screw dislocations as curved parametric segments [14]. The screw dislocations are confined either by an infinitely long edge dipole on each side, or by a more realistic case of a high density dipolar wall. Self-forces are computed following the method of Gavazza and Barnett [15], as discussed in reference [13]. In this paper, first we first present details of our model. We then we present the numerical results of the fatigue saturation stress stress and investigate the interaction between screw dislocations as they pass one another inside PSB channels in copper. We then discuss the role of additional factors on the passing stress including the internal stress resulting form the resistance to plastic flow of the walls being much greater than that of the inter-wall material, the slip plane spacing, and the channel width.