A four-dimensional formulation of defect dynamics and some of its consequences

Abstract

A 4-dimensional formulation of the equations of defect dynamics is given through use of concepts from the exterior calculus. The formulation provides a clear separation of the three basic effects; geometric responses of bodies to loadings, evolution of dislocations, and evolution of disclinations. The theory is shown to admit a 45-fold gauge group and a system of natural gauge conditions whereby the elastic and the plastic parts of the responses may be disentangled. The gauge conditions also lead to both spatial and temporial nonlocality. Self-equilibrating dislocations and disclinations (i.e. those for which there is no plastic distortion or velocity) are shown to exist and conditions for their elimination are given. A complete analogy is established between defect dynamics and co-occupying systems of electromagnetic fields with both electric and magnetic charges and currents. Analogous magnetic charges and currents are shown to vanish only if there are no disclinations present. The gauge group structure and the electrodynamic analogy provide expectations of parallels with Yang-Mills type unified gauge theories. A constitutive theory is obtained through use of the practices of non-equilibrium thermodynamics, one example of which leads to a natural implementation of a yield criteria, an incremental law of plastic distortion, and a proof of Drucker's postulate.