Comparison of closure approximations for continuous dislocation dynamics
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Comparison of closure approximations for continuous dislocation dynamics
Mehran Monavari, Michael Zaiser and Stefan Sandfeld Institute for Materials Simulation (WW8), Friedrich-Alexander-University Erlangen-N¨urnberg, Dr.-Mack-Str. 77, 90762 F¨urth, Germany
ABSTRACT We discuss methods to describe the evolution of dislocation systems in terms of a limited number of continuous field variables while correctly representing the kinematics of systems of flexible and connected lines. We show that a satisfactory continuum representation may be obtained in terms of only four variables. We discuss the consequences of different approximations needed to formulate a closed set of equations for these variables and propose a benchmark problem to assess the performance of the resulting models. We demonstrate that best results are obtained by using the maximum entropy formalism to arrive at an optimal estimate for the dislocation orientation distribution based on its lowest-order angular moments. INTRODUCTION Continuum modeling of dislocation systems has evolved along two largely independent lines. The first line originates from the continuum theory of dislocations and internal stresses of Kr¨oner and Nye [1,2]. This theory formulates geometrically rigorous relationships between the dislocation microstructure, the plastic distortion and the associated internal stress fields. It is based on the dislocation density tensor α = curlβpl which is defined as the curl of the plastic distortion. This theory can be directly applied to describe dislocation microstructure evolution and plastic flow in cases where all dislocations are geometrically necessary, i.e., they can be envisaged as contour lines of the plastic shear strain γ on the respective slip systems. In many plasticity models, however, the plastic strain is resolved only on a coarse grained scale larger than the spacing between individual dislocations. The ”statistically stored” dislocations associated with the averaged out features are then no longer represented by the dislocation density tensor. Unfortunately, they still contribute both to plastic flow and to work hardening. We are thus faced with the problem how they can be incorporated into a continuum theory. The accumulation of statistically stored dislocations, but not the dynamics of geometrically necessary ones, is the main concern of a second line of continuum models which originates from the work of Gilman [3] and Kocks [4]. Here, the dislocation microstructure is characterized by a scalar density ρ of dimension length−2 . For this density, evolution equations are formulated and combined with phenomenological relations which relate the scalar density of dislocations to the flow stress and to the plastic flow rate to arrive at plasticity models. In recent years, several attempts have been made to unify both strands of continuum modeling and formulate models which can capture the combined evolution of ”statistically stored” and ”geometrically necessary” dislocation densities [5-10]. However, few attempts have been made t
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