The paper is concerned with the study of free anti-plane shear vibrations of an ultra-thin plate coated with a nanofilm or with a free upper surface possessing surface energy with inhomogeneous surface shear modulus and density. The boundary condition at the upper surface is considered within the framework of the Gurtin–Murdoch theory of surface elasticity, while the lower surface is assumed to be fixed or free with no surface energy in the latter case. Introducing the ratio of the plate thickness to the wavelength as a small parameter and applying the complex Wentzel–Kramers–Brillouin (WKB) method, we construct a formal asymptotic solution of the boundary value problem in the form of a function decaying away from the assumed “weakest line” on the upper face. It was revealed that there are two different types of localized eigenmodes. The profile of the first-type mode is the Gaussian curve with small non-harmonic oscillation in the amplitude and with the center where the inhomogeneous surface density attains the local extremum. The second type of mode is associated with the exotic case of a negative surface shear modulus, which is allowed by the Gurtin–Murdoch model with the surface elastic constants reported in the literature for real elastic materials. This mode is approximated by sinusoid/cosinusoid inscribed into the Gaussian curve, with small non-harmonic oscillations in the amplitude and with the center at the line where the surface shear modulus and the surface density have a local extremum. The main results of this paper are explicit relations for eigenfrequencies and graphical plots in dimensionless form, which can be used to model free localized vibrations of nanoplates made of different materials.
Nath et al. (Wed,) studied this question.