A fundamental solution was presented for the three-dimensional stress field which was produced in an infinite plate by an infinitesimal dislocation loop in the interior of the plate. The solution was derived from the associated Green's function for the same geometry which, in turn, was found by employing an image method and the Muki formulation for an axisymmetrical elastic body. The solution fell naturally into three parts. The first part was singular, and corresponded to the solution for a full space. The second part was regular, and represented an image of the first part which accounted for the presence of the upper surface of the plate. The third part was also regular, and supplied a correction term which maintained the lower surface of the plate free of tractions. The first two terms were expressed in closed form, while the third term was expressed in Hankel integral form. The convergence of the integrals was ensured by an asymptotic analysis. The fundamental dislocation solution which was found was then used to analyze the growth of planar cracks in a plate, where the cracks were modelled as a continuous distribution of infinitesimal dislocation loops over the crack faces.

A Fundamental Dislocation Solution for an Infinite Plate with Application to Related Crack Problems. D.N.Dai, D.A.Hills: International Journal of Solids and Structures, 1997, 34[21], 2755-70