Long-Wave Anti-Plane Motion in a Pre-Stressed Compressible Elastic Laminate with One Fixed and One Free Face

Abstract

In this paper, long-wave anti-plane shear motion in a multilayered laminate composed of pre-stressed compressible elastic layers is investigated. The layers of the laminate are perfectly bonded, while a fixed-free boundary condition is prescribed on the outer faces of the laminate. The solution of the model is determined analytically via the propagator matrix and numerically through the asymptotic approach. Moreover, the numerical results featuring harmonic curves are presented graphically, together with an asymptotic long-wave analysis of the vibration modes. As a special case of materials, linear isotropic with one shear modulus is considered. A polynomial long-wave low-frequency approximation of the related dispersion relation is also studied. It governs dispersion curves including the lowest harmonic. It is revealed that a low-frequency mode exists in both the two- and three-layered laminates, which are taken as prototypical structures. Lastly, comparisons between the exact and approximate asymptotic results are presented, and excellent agreement is observed.