FREE VIBRATIONS OF A LAYER OF MICROPOLAR CONTINUUM

An asymptotic method is proposed to investigate the free vibrations of a layer of micropolar continuum. In this method the displacements, the microrotations and the frequencies are sought as power series of the dimensionless wavenumber ε{lunate}, where ε{lunate} = π × layer thickness/wavelength. By substituting the expansions in the displacement equations of motion and the boundary conditions, and by collecting terms of the same order ε{lunate}n, a system of coupled, second-order, inhomogeneous ordinary differential equations is obtained, with the thickness variable as independent variable, and with associated boundary conditions. Subsequent integration yields the coefficients of ε{lunate}n for the field variables and the frequencies for all modes and for the whole range of frequencies, in a range of dimensionless wavenumbers 0 < ε{lunate} < ε{lunate}* < 1, where ε{lunate}* increases as more terms are retained in the expansions. The results display the order of the dynamic coupling between the displacements and the microrotations, and between the thickness-shear motions and the thickness-stretch motions, for both the mode shapes and the frequencies. © 1969.