Asymptotic analysis of an anti-plane dynamic problem for a three-layered strongly inhomogeneous laminate

Anti-plane dynamic shear of a strongly inhomogeneous dynamic laminate with traction-free faces is analysed. Two types of contrast are considered, including those for composite structures with stiff thick or thin outer layers. In both cases, the value of the cut-off frequency corresponding to the lowest anti-symmetric vibration mode tends to zero. For this mode the shortened dispersion relations and the associated formulae for displacement and stresses are obtained. The latter motivate the choice of appropriate settings supporting the limiting forms of the original anti-plane problem. The asymptotic equation derived for a three-layered plate with thick faces is valid over the whole low-frequency range, whereas the range of validity of its counterpart for another type of contrast is restricted to a narrow vicinity of the cut-off frequency