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In the present work, a novel mathematical scheme is developed to investigate the transverse natural vibration of composite materials with complex interfaces, which belong to the extreme mechanical category. Two kinds of interfaces, as examples, are used to demonstrate the mathematical scheme. One is the triangular wave interface that has non-differentiable points in its interface description function, and the other is the square wave interface that has discontinuity points in its interface description function. The non-differentiable points or the discontinuity points in the description interface functions can pose great challenge for traditional mathematical/mechanical treatments dealing with the vibration problem. Governing equations of the composites are derived according to the generalized Hamiltonian principle. For the piecewise interface function of the triangular/square wave interface, as it is not a continuous and derivative function, the Fourier expansion method with finite terms is adopted to approximate the piecewise function in the calculations. The proposed iterative scheme can quickly find the natural frequencies of the composites, with the help of the Rayleigh quotient and boundary functions. The obtained natural frequencies are compared with those obtained from the finite element method. The effects of interface geometrical properties (the amplitude and the number of waves of the interface) on the natural frequencies are investigated systematically, and we show a plausible way to tune the natural frequencies of the composites by changing the interface geometries.

A General Algorithm on the Natural Vibration Analysis of Composite Beams with Arbitrary Complex Interfaces