Zero approximation of micropolar prismatic bodies via Legendre polynomial expansion: dispersion and wave phenomena

Abstract

The dynamic behaviour of a micropolar elastic prismatic body is investigated using the zero approximation of the three-dimensional Cosserat continuum, reducing the governing equations across the thickness. The resulting system decouples into independent subsystems, yielding six dispersion branches, including micro-rotational modes with material-dependent cutoff frequencies. The framework is applied to a finite simply supported square plate under harmonic loading, and analytical predictions are verified by direct numerical evaluation of the derived closed-form solutions. A parametric study reveals a pronounced size effect: thinner plates exhibit enhanced micro-rotational response, absent in classical plate theories. Equivalent fourth-order wave equations for transverse displacement and micro-rotation highlight three advantages over classical Timoshenko-type theory: the emergence of a micropolar length scale, an analytical shear correction factor derived from material constants, and well-posedness in the incompressible limit. The proposed formulation provides a compact yet physically rich approach for predicting size-dependent wave phenomena in microstructured materials and metamaterials. These hyperbolic equations emerge at the zero approximation, whereas the classical analogue requires the first approximation.