È presentato un modello di piastra laminata piezoelettrica, dedotto razionalmente dalla teoria tridimensionale della piezoelettricità di Voigt. Si tratta di un'estensione ai laminati piezoelettrici della teoria di laminato elastico detta «alla Reissner-Mindlin». Le equazioni che governano il problema del laminato piezoelettrico sono fornite sia in forma differenziale che in forma variazionale. Il modello proposto è utilizzato nell'analisi di alcuni semplici problemi, allo scopo di saggiarne l'affidabilità. I risultati da esso forniti risultano in soddisfacente accordo con quelli ottenuti tramite l'applicazione della teoria della piezoelettricità di Voigt.
A theory of piezoelectric laminates is rationally derived from the three-dimensional Voigt theory of piezoelectricity. The present theory is a generalization to piezoelectric laminates of the Reissner-Mindlin-type layer-wise theory of elastic laminates. Both a differential formulation and a variational formulation of the piezoelectric laminate problem are presented. The proposed theory is adopted in the analysis of simple problems, in order to verify its effectiveness. The results it provides turn out to be in good agreement with the results supplied by the Voigt theory of piezoelectricity.
@article{RLIN_1997_9_8_2_137_0,
author = {Paolo Bisegna},
title = {Una teoria di piastra laminata piezoelettrica},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
volume = {8},
year = {1997},
pages = {137-165},
zbl = {0894.73129},
mrnumber = {1485326},
language = {it},
url = {http://dml.mathdoc.fr/item/RLIN_1997_9_8_2_137_0}
}
Bisegna, Paolo. Una teoria di piastra laminata piezoelettrica. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 8 (1997) pp. 137-165. http://gdmltest.u-ga.fr/item/RLIN_1997_9_8_2_137_0/
[1] , On the modelling of piezoelectric laminated structures. Mech. Res. Commun., 19, 1992, 541-546. | Zbl 0779.73049
[2] - - , Exact analysis of coupled electroelastic behaviour of a piezoelectric plate under cylindrical bending. Comp. & Struct., 45, 1992, 667-677. | Zbl 0772.73070
[3] - - , Exact solution for static analysis of an intelligent structure under cylindrical bending. Comp. & Struct., 47, 1993, 1031-1042. | Zbl 0780.73066
[4] - , Elastic analysis of laminated composite plates in cylindrical bending due to piezoelectric actuators. Smart Mater. Struct., 3, 1994, 255-265.
[5] - , Static behaviour of piezoelectric laminates with distributed and patched actuators. J. Intell. Mat. Syst. and Struct., 5, 1994, 635-646.
[6] - - , The cylindrical bending vibration of a laminated elastic plate due to piezoelectric actuators. Smart Mater. Struct., 3, 1994, 485-493.
[7] - , Free vibration of piezoelectric laminates in cylindrical bending. Int. J. Solids Structures, 32, 1995, 2945-2960. | Zbl 0869.73039
[8] - , Exact free-vibration analysis of laminated plates with embedded piezoelectric layers. J. Acoust. Soc. Am., 98, 1995, 1547-1557.
[9] - , The vibration of a simply supported rectangular elastic plate due to piezoelectric actuators. Int. J. Solids Structures, 33, 1996, 1597-1618. | Zbl 0900.73388
[10] - , Exact electroelastic analysis of piezoelectric laminae via state space approach. Int. J. Solids Structures, 33, 1996, 977-990. | Zbl 0919.73291
[11] - , An exact three-dimensional solution for simply-supported rectangular piezoelectric plates. J. Appl. Mech., 63, 1996, 628-638. | Zbl 0886.73054
[12] , Electroelastic equations for electroded thin plates subject to large driving voltages. J. Appl. Phys., 74, 1993, 3389-3393.
[13] , High frequency vibrations of piezoelectric crystal plates. Int. J. Solids Structures, 8, 1972, 895-906. | Zbl 0243.73059
[14] , Theory of laminated piezoelectric plates for the design of distributed sensors/actuators. Part I. Governing equations and reciprocal relationships. J. Acoust. Soc. Am., 87, 1990, 1144-1158.
[15] - , An asymptotic theory of thin piezoelectric plates. Q. Jl. Mech. Appl. Math., 43, 1990, 347-362. | MR 1070961 | Zbl 0704.73087
[16] - , Induced strain actuation of isotropic and anisotropic plates. AIAA Jnl., 29, 1991, 944-951.
[17] - - - , Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control-structure interaction. Eur. J. Mech., A/Solids, 11, 1992, 181-213. | MR 1155942
[18] - , Electromechanics and vibrations of piezoelectric shell distributed systems. J. Dynamic Systems, Measurement and Control, 115, 1993, 506-517.
[19] - - - , A refined nonlinear model of composite plates with integrated piezoelectric actuators and sensors. Int. J. Solids Structures, 30, 1993, 1603-1630. | Zbl 0785.73065
[20] - , A laminated plate theory for high frequency piezoelectric thin-film resonators. J. Appl. Phys., 74, 1993, 3028-3046.
[21] - , Theory and applications of cylindrical sandwich shells with piezoelectric sensors and actuators. Smart Mater. Struct., 3, 1994, 391-396.
[22] - , A linear theory of piezoelastic shell vibrations. J. Sound and Vibration, 175, 1994, 77-88. | Zbl 0976.74518
[23] - , A refined hybrid plate theory for composite laminates with piezoelectric laminae. Int. J. Solids Structures, 32, 1995, 2345-2367. | Zbl 0869.73038
[24] , Analisi del comportamento statico di laminati piezoelettrici. In: - (a cura di), Atti del XXIV Convegno Nazionale AIAS (Parma, 27-30/9/1995). Parma 1995, 374-381.
[25] - , A consistent theory of thin piezoelectric plates. J. Intell. Mat. Syst. and Struct., 7, 1996, 372-389.
[26] , The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech., 12, 1945, 69-77. | MR 12579 | Zbl 0063.06470
[27] , Über die Berücksichtigung der Schubverzerrung in ebenen Platten. Ing. Arch., 16, 1947, 72-76. | MR 28766 | Zbl 0030.04301
[28] , Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech., 38, 1951, 31-38. | Zbl 0044.40101
[29] , Fundamentals of piezoelectricity. Oxford University Press, Oxford1990.
[30] - , Electrodynamics of continua. Springer-Verlag, New York 1990. | MR 1031714
[31] , A generalization of two-dimensional theories of laminated plates. Commun. Appl. Numer. Meth., 3, 1987, 173-180. | Zbl 0611.73072
[32] - - , An accurate determination of stresses in thick laminates using a generalized plate theory. Int. J. Numer. Meth. Engng., 29, 1990, 1-14. | Zbl 0724.73238
[33] , An exact derivation of thin plates equations. J. Elasticity, 22, 1989, 121-133. | MR 1040755 | Zbl 0692.73049
[34] , On constrained extrema of functionals. Mat. Sb., 41, 1934, 390-401.
[35] , Optimization by vector space methods. John Wiley & Sons, New York1969. | MR 238472 | Zbl 0176.12701
[36] - , A rational deduction of plate theories from the three-dimensional linear elasticity. Zeit. Angew. Math, und Mech., in press. | Zbl 0885.73023
[37] - , Deduzione di teorie di piastre laminate dalla elasticità tridimensionale. In : (a cura di), Atti del XII Convegno Nazionale Aimeta (Napoli, 3-6/10/1995). Napoli 1995, 257-262.
[38] - , The layer-wise laminate theory rationally deduced from the three-dimensional elasticity. J. Appl. Mech., in press. | Zbl 0901.73049
[39] , Existence theorems in elasticity. In: (ed.), Handbuch der Physik. Springer-Verlag, Berlin 1972, VIa/2. | Zbl 0103.16403
[40] , Linear piezoelectric plate vibrations. Plenum Press, New York1969.
[41] , Variational methods for the study of nonlinear operators. Holden-Day, San Francisco1964. | MR 176364 | Zbl 0122.35501
[42] - , Electrodynamics of continuous media. In: Course of theoretical physics. Pergamon Press, Oxford 1981, vol. 8. | Zbl 0122.45002
[43] , Sur l'équilibre et le mouvement d'une plaque élastique dont l'élasticité n'est pas la même dans tous les sens. Exercises Math., 4, 1829, 1-14.
[44] - , Problèmes aux limites non homogènes et applications. Dunod, Parigi 1968. | Zbl 0165.10801
[45] - , Mixed and hibrid finite element methods. Springer-Verlag, New York 1991. | MR 1115205 | Zbl 0788.73002
[46] , Five modern piezoelectric ceramics. Bulletin of Morgan Matroc Ltd., Vemitron Piezoelectric Division, Bedford, Ohio1983.