Current theoretical approaches to collective behavior of dislocations

Plastic deformation is a highly dissipative process that induces a variety of patterns such as the cell structure in multislip conditions, the matrix structure and the persistent slip bands in cyclic deformation, as also the static and propagating bands in constant strain rate conditions. The diversity and the complexity of these patterns with length scales ranging from nanometers all the way to millimeters level, and time scales ranging from picoseconds to a few hours, pose serious challenges for modeling the collective behavior of dislocations. While a large body of knowledge has accumulated on the mechanics of dislocations and their interactions for a long time, describing such patterns has been slow mainly due to lack of methods to deal with the collective behavior of dislocations. The purpose of this review is to present the rich variety of dislocation patterns observed in different deformation conditions along with the recent advances in modeling using borrowed techniques traditionally used in condensed matter physics. These can be classified as statistical and dynamical approaches. The review begins with a summary of different types of patterns and their characterization. Appropriate background material is provided both in terms of basic dislocation mechanisms and theoretical methods. The latter includes the Langevin and distribution function approaches, and a host of standard dynamical methods such as the Ginzburg–Landau approach, methods of characterization of chaos and slow manifold analysis. Statistical models for the cell structure and persistent slip bands are based on Langevin dynamics and distribution function theoretic approaches. Of the dynamical models, the first set addresses the slowly emerging matrix structure and persistent slip bands. The second set of models is devoted to the study of a type of propagative instability called the Portevin–Le Chatelier effect. Generic features of the instability addressed include bistability, negative strain rate sensitivity of the flow stress, different types of bands, the dynamics and statistics of stress drops, and their characterization. Three different models all of which are dynamical in nature are discussed. While these models are quite different with regard to their frameworks, what they seek to describe and the levels of sophistication undertaken, these models capture a good variety of the observed features. The review ends with a summary and outlook