Abstract
© 2018 authors. Published by the American Physical Society. This paper studies the relationship between the atomistic representation of crystalline dislocations as Kanzaki forces and the continuum representation of dislocations as Burridge-Knopoff (BK) force distributions. We first derive a complete theory of the BK force representation of dislocations in an anisotropic linear elastic continuum, showcasing a number of fundamental features found when dislocations are represented as distributions of body forces in defect-free continuum media. We then build, within the harmonic approximation, the Kanzaki force representation of dislocations in atomistic lattice models. We rigorously show that in the long-wave limit, the Kanzaki force representation converges to the continuum BK representation. We therefore justify employing the Kanzaki forces as source terms in continuum theories of dislocations. We do this by establishing a methodology to compute the Kanzaki forces of dislocations via the force constant matrix of the material's perfect lattice. We use it to study a model of a screw dislocation in bcc tungsten, where we show the existence of two distinct Kanzaki force terms: the slip Kanzaki forces, which we show directly correspond with the BK forces implied by a Volterra dislocation; and the core Kanzaki forces, which are computed from the relaxed dislocation structure, and serve to model all core effects not captured by the Volterra dislocation. We build a multipolar field expansion of both the core and the slip Kanzaki forces, showing that the dislocation core is agreeable to correction via the multipolar field expansion of the core Kanzaki forces.
Highlights
Dislocations are crystalline defects defined by an atomic disregistry: the atoms at either side of the slip surface [1] lie misaligned with respect to their equilibrium positions
Both the Burridge-Knopoff and the Kanzaki forces are applied over a defect-free system, and both act as source terms in their respective continuum and atomistic formalisms; neither represent explicitly applied forces but, rather, the forces one ought to apply in the perfect continuum or in the perfect crystalline lattice in order to generate the topology of a dislocation
We have explored in detail how crystalline dislocations are represented as force distributions in the continuum invoking the Burridge-Knopoff representation theorem [20]
Summary
Dislocations are crystalline defects defined by an atomic disregistry: the atoms at either side of the slip surface [1] lie misaligned with respect to their equilibrium (perfect lattice) positions. One may model the dislocation in a perfect lattice instead, and attain the same elastic fields and elastic energy; this is achieved by applying a set of fictitious forces on the atoms, the magnitude and distribution of which is such that it ensures that the elastic fields due to these forces match one to one those of the dislocation qua topological defect These forces are the so-called Kanzaki forces [28], and are commonly employed in the modeling of point defects in solids [29,30,31], where they originate [28]. In order to do so, we exploit the Burridge-Knopoff (1964) theorem [20], which enables representing any elastic source with an equivalent distribution of body forces applied on an undisturbed solid This stands in direct analogy to the concept of Kanzaki forces, as those are applied on the defect-free solid (the perfect lattice) to generate the defect.
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