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Three-Dimensional Continuum Dislocation Dynamics Simulations of Dislocation Structure Evolution in Bending of a Micro-Beam Alireza Ebrahimi ¹, Thomas Hochrainer ¹ 1

Universität Bremen, Am Biologischen Garten 2, 28359 Bremen, Germany

Contact e-mail: [email protected]

ABSTRACT A persistent challenge in multi-scale modeling of materials is the prediction of plastic materials behavior based on the evolution of the dislocation state. An important step towards a dislocation based continuum description was recently achieved with the so called continuum dislocation dynamics (CDD). CDD captures the kinematics of moving curved dislocations in flux-type evolution equations for dislocation density variables, coupled to the stress field via average dislocation velocity-laws based on the Peach-Koehler force. The lowest order closure of CDD employs three internal variables per slip system, namely the total dislocation density, the classical dislocation density tensor and a so called curvature density. In the current work we present a three-dimensional implementation of the lowest order CDD theory as a materials sub-routine for Abaqus® in conjunction with the crystal plasticity framework DAMASK. We simulate bending of a micro-beam and qualitatively compare the plastic shear and the dislocation distribution on a given slip system to results from the literature. The CDD simulations reproduce a zone of reduced plastic shear close to the surfaces and dislocation pile-ups towards the center of the beam, which have been similarly observed in discrete dislocation simulations. INTRODUCTION We use the term continuum dislocation dynamics (CDD) for single crystal plasticity laws which are solely based on dislocation densities and their flux-type evolution equations. The basis for such theories was laid by the introduction of the dislocation densities tensor, α , by Kröner [1] and Nye [2]. In the small deformation framework the Kröner-Nye tensor derives as the curl of the plastic distortion tensor β pl , i.e. α curl β pl . The Kröner-Nye tensor is well known to account only for a fraction of all dislocations, the so-called geometrically necessary dislocations (GNDs). For small scale plasticity, if all dislocations may be modeled as GNDs, CDD theories have been developed, e.g., by Mura [3], Acharya [4] and Sedlacek [5]. Truly averaged theories which take into account all dislocations have only recently been presented with the CDD theory developed by Hochrainer and co-workers [6,7,8]. The most simple such CDD theory is based on only three internal state variables, namely the total dislocation density U , the classical dislocation

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density tensor α , and the curvature density q [7,8]. Numerical implementations regarding kinematics of CDD were, e.g., presented in [9,10]. In the current contribution we