Modeling microbending of thin films through discrete dislocation dynamics, continuum dislocation theory, and gradient pl
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Daniel Weygand and Christian Motz Karlsruher Institut für Technologie, Institute for Applied Materials, 76131 Karlsruhe, Germany
Nikolaos Nikitas and Michael Zaiser Institute for Materials and Processes, The University of Edinburgh, Edinburgh EH9 3JL, United Kingdom (Received 28 June 2011; accepted 26 September 2011)
Constitutive models that describe crystal microplasticity in a continuum framework can be envisaged as average representations of the dynamics of dislocation systems. Thus, their performance needs to be assessed not only by their ability to correctly represent stress–strain characteristics on the specimen scale but also by their ability to correctly represent the evolution of internal stress and strain patterns. Three-dimensional discrete dislocation dynamics (3D DDD) simulations provide complete knowledge of this evolution, and averages over ensembles of statistically equivalent simulations can therefore be used to assess the performance of continuum models. In this study, we consider the bending of a freestanding thin film. From a continuum mechanics point of view, this is a one-dimensional (1D) problem as stress and strain fields vary only in one dimension. From a dislocation plasticity point of view, on the other hand, the spatial degrees of freedom associated with the bending and piling up of dislocations are essential. We compare the results of 3D DDD simulations with those obtained from a simple 1D gradient plasticity model and a more complex dislocation-based continuum model. Both models correctly reproduce the nontrivial strain patterns predicted by 3D DDD for the microbending problem. I. INTRODUCTION
From a conceptual point of view, plasticity of onedimensional (1D) materials is a paradoxical notion. Plastic deformation implies change in neighborhood relations between atoms or molecules, which necessarily implies that their motion occurs in more than one spatial dimension. In crystalline solids, plastic deformation occurs by the stress-driven motion of interacting dislocation lines, which is an inherently three-dimensional (3D) process. However, there are many phenomena of recent interest, for instance, in the deformation of nanowires1 or thin films2 (a main component in Microelectromechanical systems), which can from a continuum mechanics point of view be considered as quasi-1D problems. In this study, we investigate, taking the example of microbending, how the inherently 3D dynamics of discrete dislocations maps onto 1D continuum models of the same microdeformation process. Every constitutive model for crystal plastic deformation can be envisaged as an average representation a)
Address all correspondence to this author. e-mail: [email protected] This author was an editor of this focus issue during the review and decision stage. For the JMR policy on review and publication of manuscripts authored by editors, please refer to http://www.mrs. org/jmr-editor-manuscripts/. DOI: 10.1557/jmr.2011.390 612
J. Mater. Res., Vol. 27, No. 3, Feb 14, 2012
of dislocation dynamics. Three-dimension
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