Scripta Materialia, Vol. 35, No. 2, pp. 279-284, 1996 Elsevier Science Ltd Copyright 0 1996 Acta Metallurgica Inc. Printed in the USA. All rights resewed 1359-6462/96 $12.00 + .OO PII S1359-6462(96)00131-5 .A STOCHASTIC MODEL FOR DISLOCATION DENSITY EVOLUTION H. Braasch’, Y. Estrin’ and Y. BrCchet2 ‘Department of Mechanical and Materials Engineering, The University of Western Australia, Nedlands, WA 6907, Australia *Laboratoire de Thermodynamique et Physico-Chimie Metallurgiques, Institut National Polytechnique de Grenoble, B.P. 75,38402 Saint Martin d’Heres Cedex, France (Received January 10, 1996) (Accepted March 12, 1996) 1. Introduction A common trend in constitutive modelling of deformation behaviour of metallic materials is to utilise unified viscoplastic models. While many of them take an entirely or partly phenomenological approach, a very promising group of models base the constitutive equations on the development of dislocation densities. This was first attempted by Kocks [l] in 1976. In recent years, this concept has been further refined and successfully applied to a wide range of deformation mechanisms. Grain size and particle effects have been included [2]. The model has been further extended to take short-lived transients into account [3]. Re:cently, Estrin et al. [4] have developed a version of the model which describes cyclic deformation. All these models rely on the average dislocation densities as internal variables. Elements of stochusticity in modelling of plasticity have been introduced by Sandstrom and Lagneborg for dynamic recrystallisation [5] and recently on a more general basis by Hahner [6]. In this paper, we propose an approach similar in spirit to that of Sandstrom and Lagneborg, to account for the stochastic nature of the problem. The main goal is to investigate the evolution of the dislocation density distribution in the process of plastic deformation. A general mathematical formulation based on the Langevin approach leading to the Fokker-Planck equation [7] for the dislocation density distribution will be given. Our consideration will be confined to the case of sufftciently large plastic strains when dislocations are arranged in a cellular structure [8]. Using the principle of similitude [9], which relates the cell size to the dislocation spacing via a scaling law, .the cell size distribution will be calculated as a function of strain. The results will be compared with literature data [IO]. 2. The Deterministic Variant of the Model As mentioned above, the model is based on the notion of cellular dislocation structure [8]. The deformation stage preceding the formation of a well-developed cell structure is not included. While the interior of a cell is considered to contain only a low dislocation density, the cell walls are regarded as containing a high density of (relatively) immobile dislocations. Mobile dislocations generated in the cell walls travel through a cell nearly freely and become immobile when they reach the opposite wall thus 279