SMALL AMPLITUDE VIBRATIONS OF AN INTERNALLY CONSTRAINED ELASTIC LAYER

Abstract

Small amplitude vibrations, in the form of infinitesimal harmonic waves, in an incompressible elastic layer are considered. When the layer is additionally subject to the constraint of restricted shear and the associated preferred planes are parallel to the free surface, it is shown that the order of the governing equations of motion is reduced. The implication is that traction free boundary conditions at the upper and lower surfaces of the plate cannot be satisfied. Unlike previous studies, involving fibre inextensibility, it seems that no simple physical interpretation is possible. In an attempt to resolve this anomaly and satisfy the boundary conditions, the constraint is relaxed slightly and the strain energy function expanded as a Taylor series. However, it is found that even in this case the dispersion relation has no solutions above a certain wave speed, this usually occurring in the low wavenumber regime. It is verified analytically that no low wavenumber phase speed limit exists. It is postulated that the reason for this rather unusual behaviour is attributable to the rate of shearing. In the high wavenumber regime asymptotic expansions are obtained which give phase speed as a function of wavenumber and harmonic number. These expansions are shown to provide excellent agreement with the numerical solution over a remarkably large wavenumber region.