Long wave asymptotic integration in incompressible transversely isotropic elastic structures

A two-dimensional theory is developed for the motion of incompressible transversely isotropic layered structures in the vicinity of their cut-off frequencies. The dynamic asymptotic stress-strain-state is determined in terms of the long wave amplitude by direct asymptotic integration. Leading order (and refined) governing equations are obtained for the long wave amplitude. At both orders these are shown to be asymptotically consistent with the full three-dimensional theory. The leading order governing equation is observed to show possible wave-like behavior for certain material classes, this being connected to the possible existence of negative group velocity in the long wave regime.