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Abstract
This paper presents an extension of mathematical static model to dynamic problems of micropolar elastic plates, recently developed by the authors. The dynamic model is based on the generalization of Hellinger-Prange-Reissner (HPR) variational principle for the linearized micropolar (Cosserat) elastodynamics. The vibration model incorporates high accuracy assumptions of the micropolar plate deformation. The computations predict additional natural frequencies, related with the material microstructure. These results are consistent with the size-effect principle known from the micropolar plate deformation. The classic Mindlin-Reissner plate resonance frequencies appear as a limiting case for homogeneous materials with no microstructure.
Highlights
Classical theory of elasticity ignores the size effects of the particles and their mutual rotational interactions, considering the material particles to have only three degrees of freedom that represent their macrodisplacements
This paper presents a mathematical model for the vibration of micropolar elastic plates
This model is based on the proposed generalization of Hellinger-Prange-Reissner (HPR) variational principle for the linearized micropolar (Cosserat) elastodynamics
Summary
Classical theory of elasticity ignores the size effects of the particles and their mutual rotational interactions, considering the material particles to have only three degrees of freedom that represent their macrodisplacements. The first theory of elasticity that took into account the microstructure of the material was developed in 1909 by Cosserat brothers They presented the equations of local balance of momenta for stress and couple stress, and the expressions for surface tractions and couples [13]. We reformulate a generalization of Hellinger-Prange-Reissner (HPR) variational principle [23] for elastodynamics of micropolar materials, and using our assumptions, we postulate the variational principle for the Cosserat plate dynamics. This principle allows us to obtain dynamic equilibrium equations and constitutive relations. We present our preliminary study of the influence of plate size effect on the natural frequencies in comparison with Mindlin-Reissner plates and perform the computations for different levels of the asymmetric microstructure
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