Variational principles and vibrations of a functionally graded plate

Computers and Structures

Volumn 83, Issue 15-16, 2005, Pages 1340-1354

  Altay, Gülay ^a   Dökmeci, M Cengiz ^b  

^a BOGAZICI UNIVERSITY   (Turkey)

^b ISTANBUL TECHNICAL UNIVERSITY   (Turkey)

Author keywords

Functionally graded invariant plate equations; High frequency vibrations; Piezoelectromagnetic plates; Piezoelectromagnetism; Unified variational principles; Uniqueness of solutions

Indexed keywords

BOUNDARY VALUE PROBLEMS; CONVERGENCE OF NUMERICAL METHODS; ELECTRIC FIELDS; ELECTROMAGNETISM; ERROR ANALYSIS; ESTIMATION; MAGNETIC FIELDS; MATHEMATICAL TRANSFORMATIONS; PIEZOELECTRICITY; PLATES (STRUCTURAL COMPONENTS);

FUNCTIONALLY GRADED INVARIANT PLATE EQUATIONS; HIGH-FREQUENCY VIBRATIONS; PEIZOELECTROMAGNETISM; PIEZOELECTROMAGNETIC PLATES; UNIFIED VARIATIONAL PRINCIPLES; UNIQUENESS OF SOLUTIONS;

VIBRATIONS (MECHANICAL);

EID: 17044405117     PISSN: 00457949     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.compstruc.2004.09.017     Document Type: Article

References (84)

1. M.V. Gandhi, and B.S. Thompson Smart materials and structures 1992 Chapman & Hall London
2. H.T. Banks, R.C. Smith, and Y. Wang Smart material structures: modeling, estimation and control 1996 John Wiley New York
3. Tzou HS, Guran A. Structronic systems: smart structures, devices and systems, vols. 1-2. London: World Scientific; 1997-1998
4. Z.L. Wang, and Z.C. Kang Functional and smart materials 1998 Plenum Press New York
5. H. Janocha Adaptronics and smart structures 1999 Springer New York
6. S. Suresh, and A. Mortensen Fundamentals of functionally graded materials 1998 IOM Communications Ltd. London
7. D. Haojiang, and C. Weiqiu Three dimensional problems of piezoelasticity 2001 Nova Sci. Publ. New York
8. K. Ichikawa Functionally graded materials in the 21st century 2001 Kluwer New York
9. G.A. Altay, and M.C. Dökmeci Some comments on the higher order theories of piezoelectric, piezothermoelastic and thermopiezoelectric rods and shells Int J Solids Struct 40 2003 4699 4706
10. Gol'denveizer AL. The principles of reducing three-dimensional problems of elasticity to two-dimensional problems of the theory of plates and shells. In: Görtler H, editor. Proc. 11th Int Cong Appl Mech. New York: Springer; 1964. p. 306-11
11. Kil'chevskiy NA. Fundamentals of the analytical mechanics of shells. NASA TT F-292; 1965
12. Dökmeci MC. A dynamic analysis of piezoelectric strained elements. U.S.A.R.D.S.G., December 1992
13. V.V. Pikul Theory of shells: state of the art and prospects of development Mech Solids 35 2 2000 130 142
14. A.L. Gol'denveizer, and A.l. Lur'e On the mathematical theory of equilibrium of elastic shells Appl Math Mech 11 5 1947 565 592 [english ed.]
15. Mindlin RD. Lecture notes on the theory of plates and shells. New York: Columbia University; 1968. See also, In: Deresiewicz H, Bieniek MP, DiMaggio FL, editors. The collected papers of Raymond D. Mindlin, vols. 1-2. New York: Springer; 1989
16. Mindlin RD. Lecture notes on the theory of plates and shells. New York: Columbia University; 1968. See also, In: Deresiewicz H, Bieniek MP, DiMaggio FL, editors. The collected papers of Raymond D. Mindlin, vols. 1-2. New York: Springer; 1989
17. G.A. Altay, and M.C. Dökmeci Coupled thermoelastic shell equations with second sound for high-frequency vibrations of temperature-dependent materials Int J Solids Struct 38 2001 2737 2768
18. H.L. Zhang On variational principles of a piezoelectric body Acta Acoust 1 1985
19. G.A. Altay, and M.C. Dökmeci Fundamental variational equations of discontinuous thermopiezoelectric fields Int J Eng Sci 34 1996 769 782
20. A.A. Cannarozzi, and F. Ubertini Some hybrid variational methods for linear electroelasticity problems Int J Solids Struct 38 2001 2573 2596
21. J.-H. He Coupled variational principles of piezoelectricity Int J Eng Sci 39 2001 323 341
22. G.A. Altay, and M.C. Dökmeci High-frequency equations for non-linear vibrations of thermopiezoelectric shells Int J Eng Sci 40 2002 957 989
23. T.R. Tauchert, and F. Ashida Control of transient response in intelligent piezothermoelastic structures J Therm Stresses 26 2003 559 582
24. R.A. Toupin The elastic dielectric J Rat Mech Anal 5 1956 849 915
25. Mindlin RD. A variational principle for the equations of piezoelectromagnetism in a compound medium. In: Kompleksnii analiz i ego Prilozhenia, in Vekua 70th birthday volume. Moscow: Nauka; 1978. p. 397-400. See also, in [13], p. 1093-6
26. K. Morishita, and N. Kumagai Systematic derivation of variational expressions for electromagnetic and/or acoustic waves IEEE Trans MTT 26 1978 684 689
27. P.C.Y. Lee A variational principle for the equations of piezoelectromagnetism in elastic dielectric crystals J Appl Phys 69 11 1991 4770 4773
28. M.C. Dökmeci, and G.A. Aşkar The variational piezoelectric equations in elastic dielectric crystals J Acoust Soc Am 1995 3007
29. E.P. Hadjigeorgiou, V.K. Kalpakides, and C.V. Massalas A general theory for elastic dielectrics 11. The variational approach Int J Non-linear Mech 34 1999 967 980
30. M. Sunar, A.Z. AL-Garni, M.H. Ali, and R. Kahraman Finite element modeling of thermopiezomagnetic smart structures AIAA J 40 9 2002 1846 1851
31. J.S. Yang A generalized variational principle for piezoelectromagnetism in an elastic medium Arch Mech 43 6 1991 795 798
32. G.A. Altay, and M.C. Dökmeci Fundamental equations of certain electromagnetic-acoustic discontinuous fields in variational form Continuum Mech Thermodyn 16 1-2 2003 53 71
33. P.C.Y. Lee, and J.-D. Yu Governing equations for a piezoelectric plate with graded properties across the thickness IEEE Trans UFFC 45 1 1998 236 250
34. J.N. Reddy, and Z.Q. Cheng Three dimensional solutions of smart functionally graded plates ASME J Appl Mech 68 2001 234 241
35. C. Zhang, Y.K. Cheung, S. Di, and N. Zhang The exact solution of coupled thermoelectroelastic behavior of piezoelectric laminates Comput Struct 80 2002 1201 1212
36. R. Garcia Lage, C.M. Mota Soares, C.A. Mota Soares, and J.N. Reddy Layerwise partial mixed finite element analysis of magneto-electro-elastic plates Comput Struct 82 2004 1293 1301
37. W.Q. Chen, and H.J. Ding On free vibration of a functionally graded piezoelectric rectangular plate Acta Mech 153 2002 207 216
38. Z. Zhong, and E.T. Shang Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate Int J Solids Struct 40 2003 5335 5352
39. D.A. Saravanos, and P.R. Heyliger Mechanics and computational models for laminated piezoelectric beams, plates, and shells Appl Mech Rev 52 10 1999 305 320
40. J. Wang, and J. Yang Higher-order theories of piezoelectric plates and applications Appl Mech Rev 53 4 2000 87 99
41. H.F. Tiersten Linear piezoelectric plate vibrations 1969 Plenum Press New York
42. Mikhailov GK, Parton VZ. Applied mechanics: soviet reviews, vol. 2: electromagnetoelasticity. New York: Hamisphere publishing; 1990
43. M.C. Dökmeci Dynamic applications of piezoelectric crystals, Parts I-III: fundamentals, theoretical studies, experimental studies Shock Vib Digest 15 3-5 1983 9 20 15-26, 11-22
44. M.C. Dökmeci Recent advances in vibrations of piezoelectric crystals Int J Eng Sci 18 1980 431 448
45. M.C. Dökmeci Recent progress in the dynamic applications of piezoelectric crystals Shock Vib Digest 20 2 1985 3 20
46. J. Tani, T. Toshiyuki, and J. Qiu Intelligent material systems: applications of functional materials Appl Mech Rev 51 8 1998 5050 5521
47. M. Sunar, and S.S. Rao Recent advances in sensing and control of flexible structures via piezoelectric materials technology Appl Mech Rev 52 1 1999 1 16
48. R.D. Mindlin Electromagnetic radiation from a vibrating quartz plate Int J Solids Struct 9 1973 697 702 See also, in [13], p. 975-80
49. P.C.Y. Lee Electromagnetic radiation from an AT-cut quartz plate under lateral-field excitation J Appl Phys 65 4 1989 1395 1399
50. P.C.Y. Lee, Y.-G. Kim, and J.H. Prevost Electromagnetic radiation from doubly rotated piezoelectric crystal plates vibrating at thickness frequencies J Appl Phys 67 11 1990 6633 6642
51. A.C. Eringen Theory of electromagnetic elastic plates Int J Eng Sci 27 4 1989 363 375
52. Y. Shindo Magneto-mechanical behavior of first wall materials Fusion Eng Des 16 1991 317 322
53. E. Pan, and P.R. Heyliger Free vibrations of simply supported and multilayered magneto-electro-elastic plates J Sound Vib 252 3 2002 429 442
54. A. Ballato Modeling piezoelectric and piezomagnetic devices and structures via equivalent networks IEEE Trans UFFC 48 5 2001 1189 1240
55. J.L. Synge, and A. Schild Tensor calculus 1949 Univ.Toronto Press Toronto
56. Deresiewicz H, Bieniek MP, DiMagio FL, editors. The collected papers of Raymond D. Mindlin, vols 1-2. New York: Springer; 1989
57. Hellwege K-H, Hellwege AM, editors. Landolt-Börnstein, numerical data and functional relationships in science and technology, new series, group III, vol 2. New York: Springer; 1979
58. IEEE Standard on piezoelectricity 176-1978. IEEE Trans SU 1984;31(2):1-55
59. GOST (All-union state standard) 12370-72: piezoceramic materials. Testing methods. Moscow: Izd standartov; 1973
60. A.J. Moulson, and J.M. Herbert Electroceramics (materials, properties, applications) 1990 Chapman & Hall London
61. W.R. Hamilton On a general method in dynamics Philos Trans R Soc London 124 1834 247 308
62. W.R. Hamilton Second essay on a general method in dynamics Philos Trans R Soc London 125 1835 95 144
63. Kirchhoff G. Vorlesungen über mathematische physik: Mechanik Leipzig; 1876
64. M.C. Dökmeci Certain integral and differential types of variational principles in nonlinear piezoelectricity IEEE Trans UFFC 35 6 1988 775 787
65. K.O. Friedrichs Ein verfahren der variationsrechnung das Minimum eines Integrals das Maximum eines anderen Ausdruckes darzustellen Math Phys (Ges Wiss Göttingen; Nachrichten) 1 1929 13 20
66. De Veubeke BF. Dual principles of elastodynamics-finite element applications. In: Lectures on finite element methods in continuum mechanics. Huntsville: Univ. Alabama Press; 1973. p. 357-77
67. M.C. Dökmeci Laminae theory for motions of micropolar materials Bull Tech Univ Istanbul 47 4 1994 19 77
68. H.F. Tiersten Natural boundary and initial conditions from a modification of Hamilton's principle J Math Phys 9 1968 1445 1451
69. G. Chen Unconstrained variational statements for initial and boundary-value problems via the principle of total virtual action Int J Eng Sci 28 1990 875 887
70. G.A. Altay, and M.C. Dökmeci Some variational principles for linear coupled thermoelasticity Int J Solids Struct 33 1996 3937 3948
71. J. Lambros, A. Narayanaswamy, M.H. Santare, and G. Anlas Manufacture and testing of a functionally graded material J Eng Mater Technol 121 1999 488 493
72. T. Reiter, G.J. Dvorak, and V. Tvergaard Micromechanical models for graded composite materials J Mech Phys Solids 435 1997 1281 1302
73. C.C.M. Wu, M. Khan, and W. Moy Piezoelectric ceramics with functional gradients: a new application in material design J Am Ceram Soc 79 1996 809 812
74. A.J. Markworth, K.S. Ramesh, and W.P. Parks Jr. Review: modelling studies applied to functionally graded materials J Mater Sci 30 1995 2183 2193
75. Fichera G. Existence theorems in elasticity. In: Encyclopedia of physics, vol VIa/2. New York: Springer; 1972. p. 347-424
76. G.A. Altay, and M.C. Dökmeci A uniqueness theorem in Biot's poroelasticity Z angew Math Phys 49 1998 838 846
77. D.S. Jones Methods in electromagnetic wave propagation 1995 Oxford University Press Oxford
78. G. Duvaut, and P.L. Lions Inequations in physics and mechanics 1979 Springer New York
79. J.E. Santos, and D. Sheen On the existence and uniqueness of solutions to Maxwell's equations in bounded domains with application to magnetotellurics Math Models Methods Appl Sci 10 2000 615 628
80. R.D. Mindlin Equations of high frequency vibrations of thermopiezoelectric crystal plates Int J Solids Struct 10 1974 625 637
81. R.J. Knops, and L.E. Payne Uniqueness theorems in linear elasticity 1972 Springer New York
82. R. Leis Initial boundary value problems in mathematical physics 1986 John Wiley New York
83. R.D. Mindlin High frequency vibrations of crystal plates Q Appl Math 1961 51 61
84. R.D. Mindlin High frequency of piezoelectric crystal plates Int J Solids Struct 8 1972 895 906