Dynamics of a FGM thin elastic coating subject to anti-plane shear

A multimode long-wave asymptotic procedure is developed in the framework of a two-dimensional time-harmonic scalar problem for the dynamic anti-plane shear of a functionally graded (FGM) coating. The resulting reduced 1D equation explicitly depends on the frequency parameter extending previous results for homogeneous coatings. The associated dispersion relations provide accurate approximations of the exact spectrum near the vicinities of all cut-off frequencies and offer detailed insight into the dynamic response of the coating. The formulation is valid for a distributed loading and accurately captures long-wave, high-frequency dynamics across multiple modes, while short-wavelength effects remain outside its scope.