Hierarchical models of plasticity: dislocation nucleation and interaction

Abstract

Abstract.One avenue being pursued in the development of dislocation-based models of plasticity is the explicit simulation of the dynamics of dislocations, based on the recognition that such dislocations are the carriers of plasticity. The construction of models of dislocation dynamics requires insights into both the nucleation and interaction of dislocations, many of the details of which fall outside the domain of validity of the linear theory of elasticity. The aim of the present paper is to show how preliminary steps have been made to elucidate the mechanisms of dislocation nucleation and interaction, and to illustrate how such information can be imported into explicit models of dislocation dynamics. This effort reflects, in part, the research program of Gilles Canova, to whom the present volume is dedicated and to whom the authors dedicate this paper. Hierarchical modelling in plasticityPlasticity has attracted attention from a number of different quarters, ranging from engineers strictly interested in a proper accounting of the deformation history of a material to those studying the physics of exotic materials, such as the vortex lattice in high-temperature superconductors. The traditional discipline of single-crystal plasticity has been built around a series of postulated phenomenological constitutive assumptions in which laws for initial yield and subsequent hardening are chosen in accordance with experimental observations while maintaining a reasonable level of computational tractability [1]. Such models explicitly acknowledge the underlying existence of dislocations only through the geometry of crystalline slip in which it is assumed that slip can only occur on certain planes and in certain directions.On the other hand, a longstanding ambition of the theory of dislocations has been the construction of a theory of plasticity built upon insights into the fundamental properties of dislocations [2]. Despite the difficulties of constructing convincing analytic models involving the motions of many dislocations, one class of recent computational model has attempted to simulate plastic deformation directly on the basis of dislocation dynamics [3][4][5][6][7]. Such calculations represent an intriguing first attempt at an explicit construction of models of plasticity from the ground up, in which the governing defects themselves give rise to the mechanical properties with recourse to far fewer phenomenological intermediaries.The theses set forth in the present paper reflect an approach that was elucidated by Gilles Canova, to whom the present volume is dedicated. The argument is that despite the desire to explicitly simulate the plasticity of materials on the basis of dislocation theory alone, there are a number of features of dislocation physics that remain uncertain, including a variety of nonlinear 0965-0393/99/050769+12$30.00

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“…As the moving and forest dislocations intersect, they form jogs or junctions of varying strengths [27,28,4,29,30,31,32,33,34] which, provided the junction is sufficiently short, may be idealized as point obstacles. Moving dislocations are pinned down by the forest dislocations and require a certain elevation of the applied resolved shear stress in order to bow out and bypass the pinning obstacles.…”

“…The interaction between primary and secondary dislocations may result in a variety of reaction products, including jogs and junctions [30,31,27,32,24,28,4,34,29]. Experimental estimates of junction strengths have been given by Franciosi and Zaoui [35] for the twelve slip systems belonging to the family of {111} planes and [110] directions in fcc crystals, and by Franciosi [36] for the twenty-four systems of types {211}…”

“…The strength of some of these interactions has recently been computed using atomistic and continuum models [27,28,4,29]. Tang et al have numerically estimated the average strength of dislocation junctions for Nb and Ta crystals [24].…”

“…The present approach is aligned with the current divide and conquer paradigm in micromechanics (see, e. g., [1,2,3,4,5,6]. This paradigm first identifies and models the controlling unit process at microscopic scale.…”

Error Control for Molecular Statics Problems

“…Numerical simulations to illustrate the performance of the error estimator and adaptive strategy will be performed on the nanoindentation problem suggested by Tadmor et al [15,17,19]. This example is actually provided as a model example accompanying the open source software package [10].…”