Continuously Dislocated Elastic Bodies with a Neo-Hookean Like Energy Subjected to Anti-plane Shear

In this paper, we consider a materially uniform but inhomogeneous body and we are interested in three particular cases of inhomogeneities corresponding to three distinct distributions of dislocations. The field of defects enters the equilibrium equations through the components of the tensor field describing the relaxation procedure. We examine what form should these components take in order for the material to admit states of anti-plane shear. The results obtained in this paper hold for a class of materials that obey a specific form for the stored energy function. In the special case of no dislocations, this class falls under the well known class of Neo-Hookean materials.