Evolution of Fractal Structures in Dislocation Ensembles during Plastic Deformation A. Vinogradov, 1, * I. S. Yasnikov, 1 and Y. Estrin 2 1 Laboratory for the Physics of Strength of Materials and Intelligent Diagnostic Systems, Togliatti State University, Togliatti 445667, Russia 2 Centre for Advanced Hybrid Materials, Department of Materials Engineering, Monash University, Clayton VIC 3800, Australia (Received 8 February 2012; published 15 May 2012) Based on the irreversible thermodynamics approach to dislocation plasticity of metals, a simple description of the dislocation density evolution and strain hardening was suggested. An analytical expression for the fractal dimension (FD) of a cellular (or tangled) dislocation structure evolving in the course of plastic deformation was obtained on the basis of the dislocation model proposed. This makes it possible to trace the variation of FD of the dislocation cell structure with strain by just measuring the macroscopic stress-strain curve. The FD behavior predicted in this way showed good agreement with the experimentally measured FD evolution at different stages of deformation of a Ni single crystal and a Cu polycrystal. One new result following from the present model is that the FD of the bulk dislocation structure in a deforming metal peaks at a certain strain close to the onset of necking. The significance of fractal analysis as an informative index to follow the spatial evolution of dislocation structures approaching the critical state is highlighted. DOI: 10.1103/PhysRevLett.108.205504 PACS numbers: 62.20.F, 05.45.Df, 05.70.Ln, 81.40.Ef Being a highly dissipative process, plastic deformation occurs in a broad variety of patterns, which may be differ- ent with regard to their inner length scale or, by contrast, exhibit a scale-free behavior [1]. Characterizing and predicting patterning in the dislocation population that accompanies plastic deformation of crystalline solids is a particularly challenging problem. Fractal analysis is a potent tool to account for multiscale features and relate the macroscopic properties of a material to its dislocation structure. Plastic deformation in a metallic sample produces a characteristic relief on its surface. It is clear intuitively that the features of the surface relief are affected by the microstructure evolving within the bulk, so that fractal dimension (FD) of the surface topography must be related to that of the bulk dislocation structure. Accordingly, both bulk and surface measurements have been used to deter- mine the evolution of FD in literature. Zaiser et al. [2] have measured the Hausdorf FD from the surface profile quan- tified by atomic force microscopy (AFM) and scanning white-light interferometry (SWLI) in copper polycrystals deformed up to 25%. With both techniques used, the FD was found to increase from 2.0 to 2.3 and then to saturate at this level. These results correlate well with the findings by transmission electron microscopy (TEM) [3] that self- affine cellular dislocation patterns occur at various length scales in the bulk of a deforming crystal. The observed FDs (box and gap) were found to show an increase, while no apparent plateau region could be found at large strain. The evolution of self-affine surface roughness during early stages of deformation was also studied on KCl single crystals [4]. It was found that during the easy glide (stage I) deformation the FD shows a slight tendency to grow, followed by a rapid rise in stage II of strain harden- ing. Finally, by using AFM profiling of deformed surface of a nickel single crystal Meissner et al. [5] observed a similar behavior of FD in stages I and II. Overall, the available experimental data indicate that (i) both the deformation-induced surface patterns and the underlying dislocation structures in the bulk are self-affine, (ii) the FD of the dislocation pattern changes slightly during stage I of strain hardening in single crystals and tends to increase sharply at the onset of pronounced strain hardening in stage II and (iii) the FD tends to saturate with strain, although genuine saturation could not be established conclusively. To add to this uncertainty, the results by Kuznetsov et al. [6] suggest that the FD value peaks just before fracture. Most of the currently available results cover early deformation stages and do not extend to the point of necking, which is one of the critical points of interest on the stress-strain curve. The present study aims at shedding more light on the evolution of the FD, particularly by extending the strain range investigated to the necking point. An impor- tant ingredient of the intended FD analysis is a reliable description of the dislocation density evolution. We shall develop such a description using an irreversible thermody- namics approach. The rationale behind the earlier research, seeking insight in the evolution of the dislocation structure through FD measurements, also applies to the present study. While it is not our goal here to resolve the question of whether the quantities measured at the surface can represent the behavior of the bulk, some of the results reported below do support this hypothesis. PRL 108, 205504 (2012) PHYSICAL REVIEW LETTERS week ending 18 MAY 2012 0031-9007= 12=108(20)=205504(5) 205504-1 Ó 2012 American Physical Society