Localized Modes of Anti-plane Shear Vibrations of an Elastic Ultrathin Layer with a Free Upper Face Having Imperfections

The paper deals with the localized modes of free anti-plane shear vibrations of an elastic isotropic ultrathin layer with free upper face having small imperfections and rigidly clamped bottom face. The constitutive boundary-conditions taking into account the surface shear stresses and inertia effects within the Gurtin-Murdoch theory of the surface elasticity are assumed at the upper face. Using the asymptotic WKB-method, a solution of the wave equation is constructed in the form of a function decaying far away from a straight line at which the layer thickness reaches a local maximum. From the consideration of the first two approximations, relations for natural frequencies and a parameter characterizing the localization rate are explicitly derived. Introducing ratios of the surface shear modulus and the surface density to their counterparts in the bulk, called as the internal characteristic dimensions of the free surface or coating, and assuming that they are values of the same order as the layer maximal thickness, we revealed that the incorporation of the surface inertia effect in the model results in decreasing the natural frequencies and increasing the area, where high-frequency vibrations are localized, while the influence of the surface stresses on the natural frequencies turned out to be weak.