Anti-symmetric waves in pre-stressed imperfectly bonded incompressible elastic layered composites

Abstract

The effect of an imperfect interface on the dispersive behavior of in-plane time-harmonic symmetric waves in a pre-stressed incompressible symmetric layered composite, was analyzed recently by Leungvichcharoen and Wijeyewickrema (2003). In the present paper the corresponding case for time harmonic anti-symmetric waves is considered. The bi-material composite consists of incompressible isotropic elastic materials. The imperfect interface is simulated by a shear-spring type resistance model, which can also accommodate the extreme cases of perfectly bonded and fully slipping interfaces. The dispersion relation is obtained by formulating the incremental boundary-value problem and using the propagator matrix technique. The dispersion relations for anti-symmetric and symmetric waves differ from each other only through the elements of the propagator matrix associated with the inner layer. The behavior of the dispersion curves for anti-symmetric waves is for the most part similar to that of symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress for perfectly bonded and imperfect interface cases, a finite phase speed may exist only for the fundamental mode while other higher modes have an infinite phase speed. However, for a fully slipping interface in the low wavenumber region it may be possible for both the fundamental mode and the next lowest mode to have finite phase speeds. For the higher modes which have infinite phase speeds in the low wavenumber region an expression to determine the cut-off frequencies is obtained. At the high wavenumber limit, the phase speeds of the fundamental mode and the higher modes tend to the phase speeds of the surface wave or the interfacial wave or the limiting phase speed of the composite. The bifurcation equation obtained from the dispersion relation yields neutral curves that separate the stable and unstable regions associated with the fundamental mode or the next lowest mode. Numerical examples of dispersion curves are presented, where when the material has to be prescribed either Mooney–Rivlin material or Varga material is assumed. The effect of imperfect interfaces on anti-symmetric waves is clearly evident in the numerical results.