Analysis of Vibrating Micropolar Plate in Contact With a Fluid

Abstract

Micropolar theory constitutes extension of the classical field theories. It is based on the idea that every particles of the material can make both micro rotation and volumetric micro elongation in addition to the bulk deformation. Since this theory includes the effects of micro structure which could affect the overall behaviour of the medium, it reflects the physical realities much better than the classical theory for the engineering materials. In the micropolar theory, the material points are considered to possess orientations. A material point carrying three rigid directors introduces one extra degree of freedom over the classical theory. This is because in micropolar continuum, a point is endowed with three rigid directors only. A material point is then equipped with the degrees of freedom for rigid rotations, in addition to the classical translational degrees of freedom. In fact, the micropolar covers the results of the classical continuum mechanics. The micropolar theory recently takes attentions in fluid mechanics and mathematicians and engineers are implementing this theory in various theoretical and practical applications. In this paper the fluid-structure analysis of a vibrating micropolar plate in contact with a fluid is considered. The fluid is contained in a cube which all faces except for one of the lateral faces are rigid. The only non-rigid lateral face is made of a flexible micropolar plate and therefore, interacts with the fluid. An analytical approach is utilized to investigate the vibration characteristics of the aforementioned fluid-structure problem. The fluid is non-viscous and incompressible. Duplicate Chebyshev series, multiplied by boundary functions are used as admissible functions and the frequency equations of the micropolar plate are obtained by the use of Chebyshev-Ritz method. Also the vibration analysis of the plates modeled by micropolar theory has been done. This analysis shows that some additional frequencies due to the micropolarity of the plate appears among the values of the frequencies obtained in the classical theory of elasticity, as expected. These new frequencies are called micro-rotational waves. We also observed that when the micropolar material constants vanish, these additional frequencies disappear and only the classical frequencies remain. Specially, we observed that these additional frequencies are more sensitive to the change of the micro elastic constants than the classical frequencies. The frequencies and mode shapes of the coupled fluid structure interaction problem are obtained in the present study based on the micropolar and classical modeling. The numerical results for the problem are compared with those obtained by the analytical method for their differences and to confirm the proposed method. The microrotatinal wave frequencies and mode shapes are also developed. The results show that the natural frequencies and mode shapes for the transverse vibrations of the problem are in good agreement with the classical one and our knowledge from the physical nature of the problem.