A deformed medium including a defect field and differential forms

We consider basic equations for a deformed medium including a defect field on the basis of differential forms. To make our analysis, we extend three basic equations: (I) an incompatibility equation; (II) the Peach-Köhler equation; (III) the Navier equation based on the Hodge duality of the deformed medium. By combining two exterior differential operators, we derive (I) an incompatibility equation that extends the compatibility equation to include a defect field. The Hodge dual of the incompatibility equation becomes a generalized stress function, which includes previously derived stress functions such as Beltrami's, Morera's, Maxwell's and Airy's stress functions. By applying homotopy operators, we extend (II) the Peach-Köhler equation to include disclinations. In this case, we can define the basic quantities of stress space by analogy with the monopole theory. By combining exterior differential operators and star operators, we extend (III) the Navier equation to include a defect field. In this analysis, we define a Navier operator that is related to the Laplace operator through Hodge duality. We consider gauge conditions for a defect field based on the differential geometry of a deformed medium. This suggests a duality between yielding and fatigue fractures. The gauge condition in strain space-time is interpreted as basic relations in polycrystalline plastic deformation.