Analytical Modeling of Vibration of Micropolar Plates


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.

Share and Cite:

Steinberg, L. and Kvasov, R. (2015) Analytical Modeling of Vibration of Micropolar Plates. Applied Mathematics, 6, 817-836. doi: 10.4236/am.2015.65077.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Lakes, R. (1986) Experimental Microelasticity of Two Porous Solids. International Journal of Solids and Structures, 22, 55-63.
[2] Lakes, R. (1995) Experimental Methods for Study of Cosserat Elastic Solids and Other Generalized Elastic Continua. In: Mühlhaus, H., Ed., Continuum Models for Materials with Microstructures, Wiley, New York, 1-22.
[3] Nowacki, W. (1986) Theory of Asymmetric Elasticity. Pergamon Press, Oxford, New York.
[4] Merkel, A., Tournat, V. and Gusev, V. (2011) Experimental Evidence of Rotational Elastic Waves in Granular Phononic Crystals. Physical Review Letters, 107, Article ID: 225502.
[5] Altenbach, H. and Eremeyev, V.A. (2010) Thin-Walled Structures Made of Foams. In: Cellular, H. and Materials, P., Eds., Cellular and Porous Materials in Structures and Processes, Vol. 521, Springer, Vienna, 167-242.
[6] Gerstle, W., Sau, N. and Aguilera, E. (2007) Micropolar Peridynamic Constitutive Model for Concrete.
[7] Chang, T., Lin, H., Wen-Tse, C. and Hsiao, J. (2006) Engineering Properties of Lightweight Aggregate Concrete Assessed by Stress Wave Propagation Methods. Cement and Concrete Composites, 28, 57-68.
[8] Kumar, R. (2000) Wave Propagation in Micropolar Viscoelastic Generalized Thermoelastic Solid. International Journal of Engineering Science, 38, 1377-1395.
[9] Anderson, W. and Lakes, R. (1994) Size Effects Due to Cosserat Elasticity and Surface Damage in Closed-Cell Polymenthacrylimide Foam. Journal of Materials Science, 29, 6413-6419.
[10] Purasinghe, R., Tan, S. and Rencis, J. (1989) Micropolar Elasticity Model the Stress Analysis of Human Bones. Proceedings of the 11th Annual International Conference, Seattle, 9-12 November 1989, 839-840.
[11] Gauthier, R.D. and Jahsman, W.E. (1975) A Quest for Micropolar Elastic Constants. Journal of Applied Mechanics, 42, 369-374.
[12] Gauthier, R. (1982) Experimental Investigations on Micropolar Media. Mechanics of Micropolar Media, 395-463.
[13] Cosserat, E. and Cosserat, F. (1909) Theorie des Corps déformables. Nature, 81, 67.
[14] Eringen, A.C. (1999) Microcontinuum Field Theories. Springer, New York.
[15] Eringen, A.C. (1967) Theory of Micropolar Plates. Journal of Applied Mathematics and Physics, 18, 12-31.
[16] Reissner, E. (1945) The Effect of Transverse Shear Deformation on the Bending of Elastic Plates. Journal of Applied Mechanics, 3, 69-77.
[17] Reissner, E. (1944) On the Theory of Elastic Plates. Journal of Mathematics and Physics, 23, 184-191.
[18] Reissner, E. (1985) Reflections on the Theory of Elastic Plates. Applied Mechanics Reviews, 38, 1453-1464.
[19] Steinberg, L. (2010) Deformation of Micropolar Plates of Moderate Thickness. Journal of Applied Mathematics and Mechanics, 6, 1-24.
[20] Steinberg, L. and Kvasov, R. (2012) Enhanced Mathematical Model for Cosserat Plate Bending. Thin-Walled Structures, 63, 51-62.
[21] Kvasov, R. and Steinberg, L. (2013) Numerical Modeling of Bending of Micropolar Plates. Thin-Walled Structures, 69, 67-78.
[22] Donnell, L.H. (1976) Beams Plates and Shells. McGraw-Hill, New York.
[23] Gurtin, M.E. (1972) The Linear Theory of Elasticity. In: Truesdell, C., Ed., Handbuch der Physik, Vol. VIa/2, Springer-Verlag, Berlin, 1-296.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.