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.