×

Free vibrations of a polar body at elastic range. (English) Zbl 1115.74029

Summary: We study certain features of the equations governing time-harmonic free vibrations of a polar body in elastic range. The governing equations of micropolar elasticity are expressed in differential form, and then, the uniqueness of their solutions is investigated. The conditions sufficient for uniqueness are enumerated using the logarithmic convexity argument without any positive-definiteness assumptions of material elasticity. Applying a general principle of physics and modifying it through an involutory transformation, a unified variational principle is obtained that leads to all the governing equations of free vibrations as its Euler-Lagrange equations. The governing equations are alternatively expressed in terms of the operators related to kinetic and potential energies of the body. The basic properties of vibrations are studied, and a variational principle is given for Rayleigh quotient. As an application, we examine high-frequency vibrations of an elastic plate.

MSC:

74H45 Vibrations in dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74A35 Polar materials
74G65 Energy minimization in equilibrium problems in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cosserat, E. & F. Théorie des corps deformables, Herman et Fils, Paris, 1909.
[2] C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, Band III/3, Springer-Verlag, Berlin, 1965, pp. 1 – 602. · Zbl 1068.74002
[3] Gauthier, R.D., Experimental investigations on micropolar media, in: O. Brulin, R.K.T. Hsieh , Mechanics of micropolar media, World Scientific, Singapore,1982, pp. 395-463.
[4] Chen, Y.; Lee, J.D., Determining material constants in micromorphic theory through phonon dispersion relations, Int. J. Eng. Sci. 41(2003) 871-886.
[5] Isaak A. Kunin, Elastic media with microstructure. I, Springer Series in Solid-State Sciences, vol. 26, Springer-Verlag, Berlin-New York, 1982. One-dimensional models; Translated from the Russian. Isaak A. Kunin, Elastic media with microstructure. II, Springer Series in Solid-State Sciences, vol. 44, Springer-Verlag, Berlin, 1983. Three-dimensional models. · Zbl 0527.73002
[6] W. Nowacki, Theory of asymmetric elasticity, Pergamon Press, Oxford; PWN — Polish Scientific Publishers, Warsaw, 1986. Translated from the Polish by H. Zorski. · Zbl 0604.73020
[7] A. Cemal Eringen, Microcontinuum field theories. I. Foundations and solids, Springer-Verlag, New York, 1999. · Zbl 0953.74002
[8] G. Capriz, Continua with microstructure, Springer Tracts in Natural Philosophy, vol. 35, Springer-Verlag, New York, 1989. · Zbl 0676.73001
[9] Erofeyev, V.I., Wave processes in solid with microstructure, World Scientific, London, 2003. · Zbl 1056.74001
[10] Pabst, W., Micropolar materials, Ceramics-Silikaty 49(2005), no.3, 170-180.
[11] M. Shahinpoor and G. Ahmadi, Uniqueness in elastodynamics of Cosserat and micropolar media, Quart. Appl. Math. 31 (1973/74), 257 – 261. · Zbl 0277.73006
[12] Altay, G.; Dökmeci, M. C., Vibrations of 1-D/2-D micropolar elastic continua, ITU and BU, TR 7, November 2001.
[13] Reissner, E., A note on variational principles in elasticity, Int. J. Solids Struct. 1 (1965) 93-95.
[14] Dökmeci, M.C., Dynamic variational principles for discontinuous elastic fields, J. Ship Res. 23 (1979) 115-122.
[15] C. A. Felippa, Parametrized variational principles for micropolar elasticity, Rev. Internac. Métod. Numér. Cálc. Diseñ. Ingr. 8 (1992), no. 3, 267 – 281 (Spanish, with English and Spanish summaries). · Zbl 0772.73100
[16] P. Steinmann and E. Stein, A unifying treatise of variational principles for two types of micropolar continua, Acta Mech. 121 (1997), no. 1-4, 215 – 232. · Zbl 0878.73004 · doi:10.1007/BF01262533
[17] Janusz Dyszlewicz, Micropolar theory of elasticity, Lecture Notes in Applied and Computational Mechanics, vol. 15, Springer-Verlag, Berlin, 2004. · Zbl 1057.74003
[18] A. E. H. Love, A treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. Fourth Ed. · Zbl 0063.03651
[19] G. A. Altay and M. C. Dökmeci, Fundamental equations of certain electromagnetic-acoustic discontinuous fields in variational form, Contin. Mech. Thermodyn. 16 (2004), no. 1-2, 53 – 71. · Zbl 1100.74548 · doi:10.1007/s00161-003-0141-5
[20] R. J. Knops and L. E. Payne, Uniqueness theorems in linear elasticity, Springer-Verlag, New York-Berlin, 1971. Springer Tracts in Natural Philosophy, Vol. 19. · Zbl 0224.73016
[21] G. Aşkar Altay and M. Cengiz Dökmeci, A uniqueness theorem in Biot’s poroelasticity theory, Z. Angew. Math. Phys. 49 (1998), no. 5, 838 – 846. · Zbl 0911.73003 · doi:10.1007/s000330050124
[22] Altay, G.; Dökmeci, M.C., Fundamental variational equations of discontinuous thermopiezoelectric fields, Int. J. Eng. Sci. 34 (1996) 769-783. · Zbl 0899.73452
[23] C. Truesdell and R. Toupin, The classical field theories, Handbuch der Physik, Bd. III/1, Springer, Berlin, 1960, pp. 226 – 793; appendix, pp. 794 – 858. With an appendix on tensor fields by J. L. Ericksen.
[24] Lanczos, C., The variational principles of mechanics, Univ. Toronto Press, Toronto, 1964. · Zbl 0037.39901
[25] Dökmeci, M.C., Certain integral and differential types of variational principles in nonlinear piezoelectricity, IEEE Trans. Ultrason. Ferroelec. Freq. Cont. UFFC 35 (1988) 775-787.
[26] Yang, J.S.; Batra, R.C., Free vibrations of a piezoelectric body, J. Elasticity 34 (1994) 239-254. · Zbl 0812.73053
[27] Tiersten, H.F., Linear piezoelectric plate vibrations, Plenum Press, New York, 1969.
[28] J. S. Yang, Variational formulations for the vibration of a piezoelectric body, Quart. Appl. Math. 53 (1995), no. 1, 95 – 104. · Zbl 0816.73049 · doi:10.1090/qam/1315450
[29] J. S. Yang and X. Y. Wu, The vibration of an elastic dielectric with piezoelectromagnetism, Quart. Appl. Math. 53 (1995), no. 4, 753 – 760. · Zbl 0840.73054 · doi:10.1090/qam/1359509
[30] Deresiewicz, H.;Bieniek, M.P.; DiMaggio, F.L. , The collected papers of Raymond D. Mindlin, vols. I and II, Springer-Verlag, Berlin, 1989.
[31] Altay, G.; Dökmeci, M.C., A polar theory for vibrations of thin elastic shells, Int. J. Solids Struct. 43 (2006) 2578-2601. · Zbl 1120.74522
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.