Mechanics of dislocations and metadislocations in quasicrystals and their approximants: power invariance and balance

We discuss the mechanics of a single dislocation in quasicrystals. We account for dislocation's core by means of a non-constant line energy. Above all, we focus attention on the origin of the balance equations involving the pertinent bulk and line actions. We derive all balances, including those of configurational actions along the dislocation line,from a unique invariance requirement for the so-called relative power; it is an axiom that furnishes also the action-reaction principle for the standard traction and the phason one, as well as the standard Cauchy theorem and, in addition, the existence of phason stress and self-action. The theoretical scheme presented here includes the description of the large core of metadislocations in quasicrystal's approximants (complex metallic alloys). Each metadislocation features thousands of atoms per unit thickness in the direct vicinity of the core, a circumstance suggesting to consider it as a sort of micro-rod, a view adopted in the present paper. In fact, from an abstract point of view, the results presented here describe the motion of a rod within a body with vectorial microstructure.