Materials Science and Engineering A272 (1999) 443 – 454 Dislocation dynamics and work hardening of fractal dislocation cell structures Peter Ha ¨hner a, *, Michael Zaiser b a TU Braunschweig, Institut fu ¨r Metallphysik und Nukleare Festko ¨rperphysik, Mendelssohnstr. 3, D-39106 Braunschweig, Germany b Max -Planck -Institut fu ¨r Metallforschung, Heisenbergstr. 1, D-70569 Stuttgart, Germany Received 24 February 1999; received in revised form 21 July 1999 Abstract The dislocation dynamics during multiple slip deformation is formulated in terms of a simple stochastic model for the evolution of the densities of mobile and immobile dislocations. Randomness results in modified effective dislocation multiplication and reaction rates which account for the topology of the evolving microstructure. Depending on the intensity of local strain-rate fluctuations, two types of solution can be distinguished: (1) At low noise levels homogeneous dislocation structures develop which are described by a single characteristic length scale, i.e. the mean dislocation spacing. This is the case of b.c.c. metals deformed at low temperature. (2) Above a critical noise level self-similar dislocation cell patterns are found which are characterized by a lower cut-off length, i.e. the minimum dislocation spacing in the cell walls, and scale invariance beyond that cut-off. This case refers to rate-insensitive f.c.c. metals, where fractal dislocation structures have been identified recently [P. Ha ¨hner, K. Bay, M. Zaiser, Phys. Rev. Lett. 81 (1998) 2470]. The model yields critical deformation conditions for fractal dislocation patterning and enables one to establish relations between the evolution of the fractal dimension of the cell structure, the strain-hardening behaviour, and the underlying dislocation dynamics. This is achieved without postulating a priori that the dislocation microstructure be heterogeneous. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Stochastic dislocation dynamics; Fractals; Dislocation cell structures; Work hardening www.elsevier.com/locate/msea 1. Introduction In recent years materials science has witnessed re- newed interest in the metal physical fundamentals of plastic deformation and dislocation dynamics of pure metals and simple alloy systems. It is becoming increas- ingly clear that a comprehensive understanding of the mechanical properties of materials cannot be estab- lished without going into the dynamics of defects and deformation-induced microstructures which exhibit an intricate diversity. In general, the problem consists in bridging the scales from dislocation microstructures to the macroscopic mechanical behaviour, while in prac- tice this becomes a difficult exercise due to the predom- inant importance of the mesoscopic scale, that is the micrometer range where dislocations may self-organize collectively and spontaneously form dislocation patterns. The collective nature of the dislocation dynamical processes that are relevant on the meso- scopic scale makes it difficult to single out simple mechanisms that may be cast in a mathematically man- ageable form. For lack of an established theoretical framework various approaches have been proposed to cope with the observed dislocation patterning phenom- ena [1]. Depending on the materials and the deformation conditions qualitatively different types of dislocation patterns are observed. In particular, one distinguishes (1) homogeneous structures that are adequately charac- terized by the average dislocation spacing; (2) periodic or quasi-periodic structures where dislocation-rich and dislocation-poor regions alternate in a more or less regular way, and (3) self-similar or fractal structures [2 – 4] which, within certain limits, are scale invariant due to the presence of multiple length scales, i.e. the absence of a characteristic scale. * Corresponding author. Tel.: +49-531-3917959; fax: +49-531- 3915129. E-mail address: p.haehner@tu-bs.de (P. Ha ¨hner) 0921-5093/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII:S0921-5093(99)00527-4