Analysis of the Antiplane Problem with an Embedded Zero Thickness Layer Described by the Gurtin-Murdoch Model

The antiplane problem of an infinite isotropic elastic medium subjected to a far-field load and containing a zero thickness layer of arbitrary shape described by the Gurtin-Murdoch model is considered. It is shown that, under the antiplane assumptions, the governing equations of the complete Gurtin-Murdoch model are inconsistent for non-zero surface tension. For the case of vanishing surface tension, the analytical integral representations for the elastic fields and the dimensionless parameter that governs the problem are introduced. The solution of the problem is reduced to the solution of the hypersingular integral equation written in terms of elastic stress of the layer. For the case of a layer along a straight segment, theoretical analysis of the hypersingular equation is performed and asymptotic behavior of the elastic fields near the tips is studied. The appropriate numerical solution techniques are discussed and several numerical results are presented. Additionally, it is demonstrated that the problem under study is closely related to the specific case of the well-known problem of a thin and stiff elastic inhomogeneity embedded into a homogeneous elastic medium.