SMALL AMPLITUDE VIBRATIONS OF AN INTERNALLY CONSTRAINED ELASTIC LAYER

Small amplitude vibrations, in the form of infinitesimal harmonic wave s: in an incompressible elastic layer are considered. When the layer i s additionally subject to the constraint of restricted shear and the a ssociated preferred planes are parallel to the free surface, it is sho wn that the order of the governing equations of motion is reduced. The implication is that traction free boundary conditions at the upper an d lower surfaces of the plate cannot be satisfied. Unlike previous stu dies, involving fibre inextensibility, it seems that no simple physica l interpretations is possible. In an attempt to resolve this anomaly a nd 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 solu tions above a certain wave speed, this usually occurring in the low wa venumber regime. It is verified analytically that no low wavenumber ph ase speed limit exists. It is postulated that the reason for this rath er unusual behaviour is attributable to the rate of shearing. In the h igh wavenumber regime asymptotic expansions are obtained which give ph ase speed as a function of wavenumber and harmonic number. These expan sions are shown to provide excellent agreement with the numerical solu tion over a remarkably large wavenumber region. (C) 1998 Academic Pres s Limited.