Analogies between continuum dislocation theory, continuum mechanics and fluid mechanics

Abstract

Continuum Dislocation Theory (CDT) relates gradients of plastic deformation in crystals with the presence of geometrically necessary dislocations. Interestingly, CDT shows striking analogies to other branches of continuum mechanics. The present contribution demonstrates this on two essential kinematical quantities which reflect tensorial dislocation properties: the (resultant) Burgers vector and the dislocation density tensor. First, the limiting process for the (resultant) Burgers vector from an integral to a local quantity is performed analogously to the limiting process from the force vector to the traction vector. By evaluating the balance of forces on a tetrahedral volume element, Cauchy found his famous formula relating traction vector and stress tensor. It is shown how this procedure may be adopted to a continuously dislocated tetrahedron. Here, the conservation of Burger's vector implicates the introduction of the dislocation density tensor. Second, analogies between the plastic flow of a continuously dislocated solid and the liquid flow of a fluid are highlighted: the resultant Burgers vector of a dislocation ensemble plays a similar role as the (resultant) circulation of a vortex tube. Moreover, both vortices within flowing fluids and dislocations within deforming solids induce discontinuities in the velocity field and the plastic distortion field, respectively. Beyond the analogies, some peculiar properties of the dislocation density tensor are presented as well.