Free vibrations of a polar body at elastic range
Authors:
Gülay Altay and M. Cengiz Dökmeci
Journal:
Quart. Appl. Math. 64 (2006), 711-734
MSC (2000):
Primary 74H25, 49R50, 74H45
DOI:
https://doi.org/10.1090/S0033-569X-06-01042-3
Published electronically:
October 31, 2006
MathSciNet review:
2284467
Full-text PDF Free Access
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Abstract: The purpose of this paper is to study certain features of the equations governing the time-harmonic free vibrations of a polar body at 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 the free vibrations as its Euler-Lagrange equations. The governing equations are alternatively expressed in terms of the operators related to the kinetic and potential energies of the body. The basic properties of vibrations are studied and a variational principle in Rayleigh’s quotient is given. As an application, the high-frequency vibrations of an elastic plate are treated.
r1 Cosserat, E. & F. Théorie des corps deformables, Herman et Fils, Paris, 1909.
- 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. MR 0193816
r3 Gauthier, R.D., Experimental investigations on micropolar media, in: O. Brulin, R.K.T. Hsieh (Eds.), Mechanics of micropolar media, World Scientific, Singapore,1982, pp. 395-463.
r4 Chen, Y.; Lee, J.D., Determining material constants in micromorphic theory through phonon dispersion relations, Int. J. Eng. Sci. 41(2003) 871-886.
- 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. MR 664203
- W. Nowacki, Theory of asymmetric elasticity, Pergamon Press, Oxford; PWN—Polish Scientific Publishers, Warsaw, 1986. Translated from the Polish by H. Zorski. MR 894254
- A. Cemal Eringen, Microcontinuum field theories. I. Foundations and solids, Springer-Verlag, New York, 1999. MR 1720520
- G. Capriz, Continua with microstructure, Springer Tracts in Natural Philosophy, vol. 35, Springer-Verlag, New York, 1989. MR 985585
r9 Erofeyev, V.I., Wave processes in solid with microstructure, World Scientific, London, 2003.
r10 Pabst, W., Micropolar materials, Ceramics-Silikaty 49(2005), no.3, 170-180.
- M. Shahinpoor and G. Ahmadi, Uniqueness in elastodynamics of Cosserat and micropolar media, Quart. Appl. Math. 31 (1973/74), 257–261. MR 411315, DOI https://doi.org/10.1090/S0033-569X-1973-0411315-7
r12 Altay, G.; Dökmeci, M. C., Vibrations of 1-D/2-D micropolar elastic continua, ITU and BU, TR 7, November 2001.
r13 Reissner, E., A note on variational principles in elasticity, Int. J. Solids Struct. 1 (1965) 93-95.
r14 Dökmeci, M.C., Dynamic variational principles for discontinuous elastic fields, J. Ship Res. 23 (1979) 115-122.
- 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). MR 1185909
- 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. MR 1455162, DOI https://doi.org/10.1007/BF01262533
- Janusz Dyszlewicz, Micropolar theory of elasticity, Lecture Notes in Applied and Computational Mechanics, vol. 15, Springer-Verlag, Berlin, 2004. MR 2058885
- A. E. H. Love, A treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. Fourth Ed. MR 0010851
- 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. MR 2035527, DOI https://doi.org/10.1007/s00161-003-0141-5
- 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. MR 0421244
- 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. MR 1652204, DOI https://doi.org/10.1007/s000330050124
r22 Altay, G.; Dökmeci, M.C., Fundamental variational equations of discontinuous thermopiezoelectric fields, Int. J. Eng. Sci. 34 (1996) 769-783.
- 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. MR 0118005
r24 Lanczos, C., The variational principles of mechanics, Univ. Toronto Press, Toronto, 1964.
r25 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.
r26 Yang, J.S.; Batra, R.C., Free vibrations of a piezoelectric body, J. Elasticity 34 (1994) 239-254.
r27 Tiersten, H.F., Linear piezoelectric plate vibrations, Plenum Press, New York, 1969.
- J. S. Yang, Variational formulations for the vibration of a piezoelectric body, Quart. Appl. Math. 53 (1995), no. 1, 95–104. MR 1315450, DOI https://doi.org/10.1090/qam/1315450
- J. S. Yang and X. Y. Wu, The vibration of an elastic dielectric with piezoelectromagnetism, Quart. Appl. Math. 53 (1995), no. 4, 753–760. MR 1359509, DOI https://doi.org/10.1090/qam/1359509
r30 Deresiewicz, H.;Bieniek, M.P.; DiMaggio, F.L. (Eds.), The collected papers of Raymond D. Mindlin, vols. I and II, Springer-Verlag, Berlin, 1989.
r31 Altay, G.; Dökmeci, M.C., A polar theory for vibrations of thin elastic shells, Int. J. Solids Struct. 43 (2006) 2578-2601.
r1 Cosserat, E. & F. Théorie des corps deformables, Herman et Fils, Paris, 1909.
r2 C. Truesdell; Noll, W., The non-linear field theories of mechanics, Handbuch der Physik, Springer-Verlag, Berlin, 1965.
r3 Gauthier, R.D., Experimental investigations on micropolar media, in: O. Brulin, R.K.T. Hsieh (Eds.), Mechanics of micropolar media, World Scientific, Singapore,1982, pp. 395-463.
r4 Chen, Y.; Lee, J.D., Determining material constants in micromorphic theory through phonon dispersion relations, Int. J. Eng. Sci. 41(2003) 871-886.
r5 Kunin, I.A., Elastic Media with microstructure, vol. 1: One-dimensional models, vol. 2: Three-dimensional models, Springer-Verlag, New York, 1982-1983. ;
r6 Nowacki, W., Theory of asymmetric elasticity, Pergamon Press, Oxford, 1986.
r7 Eringen, A.C., Microcontinuum field theories, I. Foundations and solids, Springer-Verlag, 1999.
r8 Capriz, G., Continua with microstructure, Springer-Verlag, New York, 1989.
r9 Erofeyev, V.I., Wave processes in solid with microstructure, World Scientific, London, 2003.
r10 Pabst, W., Micropolar materials, Ceramics-Silikaty 49(2005), no.3, 170-180.
r11 Shahinpoor, M.; Ahmadi,G., Uniqueness in elastodynamics of Cosserat and micropolar media, Quart. Appl. Math. 31 (1973), 257-261.
r12 Altay, G.; Dökmeci, M. C., Vibrations of 1-D/2-D micropolar elastic continua, ITU and BU, TR 7, November 2001.
r13 Reissner, E., A note on variational principles in elasticity, Int. J. Solids Struct. 1 (1965) 93-95.
r14 Dökmeci, M.C., Dynamic variational principles for discontinuous elastic fields, J. Ship Res. 23 (1979) 115-122.
r15 Felippa, C.A., Parametrized variational principles for micropolar elasticity, Int. J. Solids Struct. 29 (1992) 2709-2721.
r16 Steinmann, P.; Stein, E., A unifying treatise of variational principles for two types of micropolar continua, Acta Mechanica 121 (1997) 215-232.
r17 Dyszlewicz, J., Micropolar theory of elasticity, Springer-Verlag, New York, 2004.
r18 Love, A.E.H., The mathematical theory of elasticity, Dover Publishers, New York, 1944.
r19 Altay, G.; Dökmeci, M.C., Fundamental equations of certain electromagnetic-acoustic discontinuous fields in variational form, Continuum Mech. Thermodyn.16 (2004) 53-71.
r20 Knops, R.J.; Payne, L.E., Uniqueness Theorems in Linear Elasticity, Springer-Verlag, New York, 1971.
r21 Altay, G.; Dökmeci, M.C., A uniqueness theorem in Biot’s poroelasticity theory, Z. Angew. Math. Phys. (ZAMP) 49 (1998) 838-846.
r22 Altay, G.; Dökmeci, M.C., Fundamental variational equations of discontinuous thermopiezoelectric fields, Int. J. Eng. Sci. 34 (1996) 769-783.
r23 Truesdell, C.; Toupin, R.A., The classical field theories, Handbuch der Physik, Springer-Verlag, New York, 1960.
r24 Lanczos, C., The variational principles of mechanics, Univ. Toronto Press, Toronto, 1964.
r25 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.
r26 Yang, J.S.; Batra, R.C., Free vibrations of a piezoelectric body, J. Elasticity 34 (1994) 239-254.
r27 Tiersten, H.F., Linear piezoelectric plate vibrations, Plenum Press, New York, 1969.
r28 Yang, J.S., Variational formulations for the vibration of a piezoelectric body, Quart. Appl. Math. 53 (1995) 95-104.
r29 Yang, J.S.; Wu, X.Y., The vibration of an elastic dielectric with piezoelectromagnetism, Quart. Appl. Math. 53 (1995), no. 4, 753-760.
r30 Deresiewicz, H.;Bieniek, M.P.; DiMaggio, F.L. (Eds.), The collected papers of Raymond D. Mindlin, vols. I and II, Springer-Verlag, Berlin, 1989.
r31 Altay, G.; Dökmeci, M.C., A polar theory for vibrations of thin elastic shells, Int. J. Solids Struct. 43 (2006) 2578-2601.
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Additional Information
Gülay Altay
Affiliation:
Faculty of Engineering, Boḡaziçi University, Bebek, 34342 Istanbul, Turkey
Email:
askarg@boun.edu.tr
M. Cengiz Dökmeci
Affiliation:
Istanbul Technical University, P.K. 9, Gümüsuyu, 34430 Istanbul, Turkey
Email:
cengiz.dokmeci@itu.edu.tr
Received by editor(s):
February 17, 2006
Published electronically:
October 31, 2006
Additional Notes:
The authors acknowledge the financial support in part by their departments and TUBA, and the second author (M.C.D.) is grateful to Prof. Dr. Oral Büyüköztürk (Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge) for his kind invitation and support to the three-day workshop on engineering materials on June 9-11, 2004, Cambridge, Mass.
Article copyright:
© Copyright 2006
Brown University
