Characterizing short-range vs. long-range spatial correlations in dislocation distributions
Introduction
Dislocation-mediated plasticity is commonly considered as a regular process occurring smoothly in time and homogeneously in space, as suggested by the term “plastic flow” used to describe this plasticity. Conventional wisdom contends that fluctuations of dislocation activity are sufficiently small and independent of one another to add at random, over sufficiently large length and time scales, to a net smooth and homogeneous overall response. However, recent studies have offered evidence that, instead, plasticity is a scale-invariant phenomenon characterized by power law distributions of avalanche size, and by space and time coupling over several orders of magnitude [1], [2], [3], [4], [5], [6], [7]. In all of these studies, the long-range lattice distortion and internal stress field associated with the presence of dislocations are considered to be involved in the self-organization of the plastic activity. Dislocation distributions reflect in space the history and properties of the past plastic activity. Hence, characterizing post-mortem dislocation patterns yields information about the dynamic processes that led to these arrangements. In X-ray-transparent solids such as ice, bulk characterization of dislocation dynamics processes is therefore feasible via X-ray diffraction techniques. Earlier work in that field [8] reported scale-invariance of the axial dislocation distribution in ice single crystals deformed in torsion creep at moderate strain, with long-range spatial correlations also attributed to the lattice distortion and elastic internal stress fields. In contrast, unstrained samples showed no evidence of spatial correlation. From these results, one may erroneously infer that more strain and larger dislocation densities lead to enhanced long-distance correlations. It is shown in the following that such is not the case.
In the present paper, we report on ice single crystals deformed in torsion creep at moderate to large strains. We use the hard X-ray diffraction technique as a model approach to investigate the self-organization of dislocation distributions. We believe that such experiments are insightful in the context of dislocation self-organization for several reasons. First, torsion tests involve stress and strain gradients from the axis to the edge of the sample, and therefore their accommodation by “geometrically necessary” or “polar” dislocations, whose densities are associated with lattice distortion and internal stress fields. Using single crystals avoids introducing additional complexity due to dislocation–grain boundary interactions. Under terrestrial temperature and pressure conditions, ice has a hexagonal crystallographic structure. It is a particularly suitable material for the present purpose due to its strong anisotropy of slip. Indeed, ice deforms almost exclusively by glide of dislocations in the basal planes [9]. In particular, the torsion of single crystals whose crystallographic c-axis is oriented along the torsion axis is accommodated by screw dislocations, which leads to the emergence of strong elastic internal stresses [10], [11]. Unlike the applied stress field, the internal stress field has non-basal components. Therefore it favors the transport of dislocations along the c-axis by the occurrence of double cross-slip of screw dislocations through prismatic planes [10], [12]. Hence, there is the distinct possibility that the short-range interactions inherent to double cross-slip could also be involved in the self-organization of the plastic activity, complementing long-range stresses. Indeed, double cross-slip tends to disseminate slip activity along the c-axis, while the edge jogs formed in the prismatic planes during the process may serve as strong obstacles to dislocation motion. Such a distinct role of short-range correlations is likely to be effective at large strains, when dissemination of slip by cross-slip becomes significant. Note that the role of short-range correlations was suggested in recent work on dislocation patterning [13] and on the intermittency of plastic activity [10], [14], [15], [16]. Therefore, the objectives of the present paper are threefold: (i) to show that characterizing the role of short-range vs. long-range correlations in the self-organization of dislocation ensembles is possible from the experimental analysis of the spatial distribution of dislocation densities; (ii) to show the progressive prevalence of short-range over long-range correlations in dislocation self-organization as strain increases; and (iii) to retrieve the above trends in the dislocation distributions predicted by a heuristic field dislocation dynamics model featuring internal stresses and dislocation transport.
The outline of the paper is therefore as follows. Section 2 deals with the experimental procedure used to provide dislocation distributions through torsion tests and hard X-ray diffraction analysis. The tests are performed at different strain levels to follow the evolution of spatial correlations with strain. A statistical analysis and a multifractal analysis are used to accurately monitor possible variations in the correlation regime. The results and their multifractal analysis are presented in Section 3. Although the reader is referred to Ref. [17] for a general presentation of the method, the section offers a short primer on the multifractal techniques of data analysis. Section 4 is devoted to an interpretation of the results based on a field dislocation dynamics theory that accounts for both long-range correlations through internal stress field development, and short-range correlations through dislocation transport [18]. Here, the aim is to provide evidence that such generic features lead to behavior similar to our experimental results in a simple representative situation. A summary and conclusions complete the paper.
Section snippets
Torsion tests
Laboratory-grown single crystals were used for the torsion experiments. The initial dislocation density of these crystals, measured by hard X-ray diffraction analysis (see below) was less than . The orientation of the samples was chosen to align the torsion axis with the c-axis. Basal planes were therefore parallel to the permanent shear plane, within an accuracy of . A constant torsion torque was applied to all samples, leading to ice deformation by creep, while no load was applied in
Results
Fig. 4, Fig. 5, Fig. 6 show the distribution of the dislocation density along the sample axis in samples 1–3. The figures reveal the heterogeneous nature of the distribution and its increasing complexity as deformation proceeds. With the standard deviation, the fluctuations of the dislocation density about a mean value are roughly characterized by the relative standard deviation . A value close to zero of the latter suggests a rather uniform distribution with limited fluctuations, as
Interpretation
The autocorrelation function, the power spectrum density and the multifractal analysis of the dislocation density distributions suggest evolving characteristics of scale-invariance and compounded spatial correlation regimes as strain increases. Interpretation of these statistical trends is now sought through the various dislocation mechanisms involved during straining. The basal screw dislocations are primarily responsible for the accommodation of the torsion strain. They nucleate close to the
Summary and concluding remarks
Hard X-ray diffraction provides evidence of a strongly heterogeneous distribution of dislocation densities along the axis of cylindrical ice single crystals oriented for basal slip in torsion creep. At moderate strain, the dislocation distributions display long-distance correlations with scale-invariant character [8]. However, as strain increases, a trend to decreasing autocorrelation of the dislocation distributions is observed in the autocorrelation function and Fourier power spectrum. In
Acknowledgements
Fruitful discussions with Armand J. Beaudoin and Jérôme Weiss are gratefully acknowledged. J.C. and P.D. thank also ST2I Institute of Centre National de la Recherche Scientifique for funding support.
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