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 thick or thin stiff outer layers. In both cases, the value of the cut-off frequency corresponding to the lowest antisymmetric 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.

Indentation of a thin elastic film attached through an interlayer to a rigid support is studied. Because the common interpretations of depth-sensing indentation tests are not applicable to such structured coatings, usually various approximating functions are employed to take into account influence of the interlayer. Contrary to the common approaches, a strict mathematical approach is applied here to study the problems under consideration assuming that the thickness of the two-layer structure is much less than characteristic dimension of the region of contact between the indenter and the coating. A simple derivation of asymptotic relations for displacements and stresses is presented. It is shown that often the leading term approximation to the non-adhesive contact problems is equivalent to contact problem for a Winkler-Fuss elastic foundation with an effective elastic constant. Because the contact between the indenter and the film at nanoscale may be greatly affected by adhesion, the adhesive problem for these bilayer coatings is studied in the framework of the JKR (Johnson, Kendall, and Roberts) theory of adhesion. Assuming the indenter shape near the tip has some deviation from its nominal shape and using the leading term approximation of the layered coatings, the explicit expressions are derived for the values of the pull-off force and for the corresponding critical contact radius of adhesive contact region.