Fundamental dislocation solution for an infinite plate with application to related crack problems

A fundamental solution for the three-dimensional stress field induced in an infinite plate by an infinitesimal dislocation loop in the interior of the plate is presented here. The solution is derived from the associated Green's function for the same geometry, which is, in turn, found by employing an image method and Muki's formulation for an axisymmetric elastic body. The solution obtained falls naturally into three parts: the first part is singular, and corresponds to the solution for a full space; the second part is regular, and represents the image of the first part to account for the presence of the upper surface of the plate; the third part is also regular, and gives the correction term to maintain the lower surface of the plate free of tractions. The first two terms are expressed in closed form, whilst the third term is expressed in Hankel integral form. Convergence of the integrals is ensured by an asymptotic analysis. The fundamental dislocation solution found is then employed to analyze the growth of planar cracks in a plate, where the cracks are modelled by a continuous distribution of infinitesimal dislocation loops over the crack faces, i.e., the eigenstrain procedure.