Finite-form solution for anti-plane problem of nanoscale crack

A finite-form solution for the problem of an elastic body weakened by a straight nano-crack and disturbed by a screw dislocation in the mode-III deformation is proposed. The boundary equations incorporating the surface effects are established and reduced to single one via the integration over the crack boundary in the physical region. The cracked physical region is then mapped onto an image region with circular hole. By using Muskhelishvili (Some basic problems of the mathematical theory of elasticity, Noordhoff, Groningen, 1953) method and Cauchy's integral formula, the reduced boundary equation is further converted into a first-order differential equation. Consequently, the solution of the differential equation is mapped back to the physical region and the complex potential representing the displacement is formulated in finite form. The results show that the stresses at the crack tip are finite and the stress singularity is eliminated completely by the incorporation of the surface effects. The non-singularity of the stresses at the crack tip derived from the present article radically differs from the square-root singularity obtained from the classic theory of the elastic fracture and, however, is reasonable from the physical essence. The numerical results shows that when the surface effects are incorporated, the magnitudes of the stresses and the image forces near the crack tip are reduced, which are strongly dependent on the crack size. The corresponding results approach those from the classic theory as the crack length increases. The results by the present solution are also compared with those from other solutions.