Higher order alignment tensors for continuum dislocation dynamics
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Higher order alignment tensors for continuum dislocation dynamics Thomas Hochrainer Universität Bremen, IW3, Am Biologischen Garten 2,28359 Bremen, Germany.
ABSTRACT Dislocation density based modeling of crystal plasticity remains one of the central challenges in multi scale materials modeling. A dislocation based theory requires sufficiently rich dislocation density measures which are capable of predicting their own evolution. Continuum dislocation dynamics is based on a higher dimensional dislocation density tensor comprised of two distribution functions on the space of local orientations, which are the density of dislocations per orientation and the density of dislocation curvature per orientation. We propose to expand these functions into series of symmetric tensors (alignment tensors), to be used in dislocation based theories without extra dimensions. The first two terms in the expansion of the density define the total dislocation density and the Kröner-Nye tensor. The first term in the expansion of the curvature density, the scalar total curvature density, turns out to be a conserved quantity; the integral of which corresponds to the total number of dislocations. The content of the next higher order tensors is discussed. INTRODUCTION Dislocation density based modeling of crystal plasticity remains one of the central challenges in multi scale materials modeling. The (re-)discovery of size effects in micro- and nano-plasticity during the 1990s into the early 2000s revived classical dislocation density theory. A central question in dislocation density modeling, however, remained unanswered so far: what is a sufficiently rich dislocation density measure capable of not only describing essential features of the dislocation state but also of predicting its own evolution self consistently. It is well known that the dislocation density (Kröner – Nye) tensor α will typically give a very incomplete picture of the dislocation state of a crystal (see e.g. [1]). Though α may be used to set up a continuum theory of dislocations in the special situation of smoothly aligned dislocations [2,3], no really averaged theory may be built upon the dislocation density tensor alone. The other well-known dislocation density measure is the scalar total dislocation density t . Strain rate dependent evolution equations for t are used successfully in modeling macroscopic stress strain curves [4]. But on a slip system level these laws are of limited use because such local evolution laws cannot consider dislocation fluxes. Various hybrid laws combining the flux-based evolution of the dislocation density tensor and the phenomenological evolution equations for the total dislocation density (then usually termed the density of statistically stored dislocations, SSD) have been proposed (see [5] for a recent review). However, aside from the problem of discriminating between GNDs and SSDs, decoupling SSD-evolution from dislocation fluxes seems hardly justifiable. The few theories actually considering fluxes of all dislocations are based on brea
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