Application of a 3D-Continuum Theory of Dislocations to a Problem of Constrained Plastic Flow: Microbending of a Thin Fi
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Application of a 3D-Continuum Theory of Dislocations to a Problem of Constrained Plastic Flow: Microbending of a Thin Film Stefan Sandfeld1,3, Dr. Thomas Hochrainer1,2 and Prof. Michael Zaiser3 1 Institut für Zuverlässigkeit von Bauteilen und Systemen, Universität Karlsruhe (TH), Kaiserstr.12, 76131 Karlsruhe, Germany 2 Fraunhofer-Institut für Werkstoffmechanik IWM, Wöhlerstr. 11, 79108 Freiburg, Germany 3 The University of Edinburgh, Institute for Materials and Processes, The Kings Buildings, Sanderson Building, Edinburgh EH9 3JL, United Kingdom ABSTRACT The advancing miniaturisation of e.g. microelectronic devices leads to an increasing interest in physically motivated continuum theories of plasticity in small volumes. Such theories need to be based on the averaged dynamics of dislocations. Preserving the line-like character of these defects, however, posed serious problems for the development of dislocation-based continuum theories, while continuum theories based on scalar dislocation densities necessarily stay on a phenomenological level. Within this work we apply a dislocation-based continuum theory, which is based on a physically meaningful averaging of dislocation lines, to the benchmark problem of bending of a free-standing thin film. INTRODUCTION Already about half a century ago Kröner [1], Nye [2], Bilby and co-workers [3] and Kondo [4] introduced independently continuum theories of dislocations. Those were based – with slightly different formulations and accents – on a dislocation density tensor, characterising the dislocation state of a crystal. However, already then it was obvious that the dislocation density tensor can only partially describe the defect state of a crystal. The averaged dislocation density tensor measures the 'geometrically necessary' dislocations. Nonetheless, the classical theory is complete in the case when dislocations form smooth bundles of nearly parallel lines with uniform line orientation (as e.g. in [5],[6]). Hochrainer [7] recently introduced an Extended Continuum Theory (abbreviated by ECT), which contains the classical continuum theory as a special case, but which is furthermore applicable to very general dislocation configurations. We will give a brief introduction of this theory before applying it to the case of micro-bending of a thin film; for a more thorough introduction and derivation of the ECT the reader is referred to [7] and [8]. THEORY The starting point of ECT is to discriminate dislocation line segments by their line orientation and to average over the so-called lifts of the dislocation lines. Since we consider dislocation glide only we can define the lifted curve in a configuration space where each point (r, φ ) con-
tains the spatial point r and the orientation defined as the angle φ between the lines’ tangent vector and an arbitrary reference vector. To describe the kinematics of single lines within this space we have to introduce the notion of the generalized line direction L and the generalized velocity V, which denote the tangent to the lif
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