Cracked elastic layer with surface elasticity under antiplane shear loading

A mode-III crack embedded in a homogeneous isotropic elastic layer of nanoscale finite thickness is studied in this article. The classical elasticity incorporating surface elasticity is employed to reduce a nonclassical mixed boundary value problem, where the layer interior obeys the traditional constitutive relation and the surfaces of the layer and the crack are dominated by the surface constitutive relation. Using the Fourier transform, we convert the problem to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces. By expanding the out-of-plane displacement as series of Chebyshev polynomials, the Galerkin method is invoked to reduce the singular integro-differential equation with Cauchy kernel to a set of algebraic linear equations for the unknown coefficients. An approximate solution is determined, and the influences of surface elasticity on the elastic field and stress intensity factor are examined and displayed graphically. It is shown that surface elasticity decreases the bulk stress and its intensity factor near the crack tips for positive surface shear modulus and gives rise to an opposite trend for a negative surface shear modulus.