Method of Images Solution for an Edge Dislocation and a Circular Cavity in Crystalline Solids

Abstract: Mechanics of defects in solids across awide span of length scales is commonly formulated using thedislocations theory. This paper revisits the classical problemof interaction between an elastic edge dislocation and acircular cavity. A heuristic, yet, mechanistic approach is takento obtain the stress solution to this problem. The approach usescomplex variable theory of elasticity, along with method ofimages. For this purpose, a definition and formulation ofelastic dipole singularities similar to dipole charges inelectrostatics is developed. It is shown that an imagedislocation with Burger’s vector of the same strength as thereal dislocation but in opposite direction, as well as a set offour singularities including a dislocation dipole, amoment-dilatation dipole, and two centers of dilatation wouldestablish a circular, traction-free boundary in an infiniteelastic medium. Adding a Volterra dislocation to thefinite-length edge dislocation from this study would recover therelated problem of interaction between an infinite-length edgedislocation and circular cavity. The interesting analogy betweenthe considered elastic problem and the electrostatic problem ofinteraction between a line electric charge and a cylindricalconductor is discussed.