The presently available elastic continuum theories of lattice defects are reviewed. After introducing a few elementary concepts and the basic equations of elasticity the Eshelby’s theory of misfitting inclusions and inhomogeneities is outlined. Kovács’ result that any lattice defect can be described by a surface distribution of elastic dipoles is described. The generalization of the isotropic continuum approach to anisotropic models and to Eringen’s isotropic but non-local model is discussed. Kröner’s theroy (where a defect is viewed as a lack of strain compatibility in the medium) and the elastic field equations (formulated in a way analogous to Maxwell’s field equations of magnetostatics) are described. The concept of the dislocation density tensor is introduced and the utility of higher-order dislocation density correlation tensors is discussed. The beautiful theory of the affine differential geometry of stationary lattice defects developed by Kondo and Kröner is outlined. Kosevich’s attempt to include dynamics in the elastic field equations is described. Wadati’s quantum field theory of extended objects is mentioned qualitatively. Some potential areas of research are identified.

A gauge theory of defects in an elastic continuum is developed after providing the necessary background in continuum elasticity and gauge theories. The gauge group is the three-dimensional (3D) Euclidean group [semi-direct product of the translation group T (3) with the rotation group SO (3)]. We obtainboth dislocations and disclinations by breaking of the translational invariance. Breaking of the rotational invariance is shownnot to lead to any interesting effects in a linear analysis. These results are shown to be consistent with the topological analysis which is briefly discussed at the end of the paper. Any defect given by the present theory acquires acore which removes the singularity of the stress field at the origin. The stress field agrees with the continuum result asymptotically, as is expected. Geometrical aspects of the deformed state of condensed matter are also briefly touched upon.

An analytic expression for the force between two parallel screw dislocations, derived earlier on the basis of the gauge theory of dislocations, has been used to investigate the static distribution of a given numberN of parallel screw dislocations confined between two immobile dislocation obstacles. It is shown that in the limit of a continuous distribution of dislocations the equilibrium condition leads to a Fredholm integral equation of first type which does not admit any nontrivial solution. Implication of this result is discussed. For a finite number of dislocations, the ratio (η) of the obstacle separation to the core radius is an important parameter governing the nature of solution of the discrete equation. It is found that for a givenN, there is a critical valueη_c below which there does not exist any solution.