×

On the fundamental equations of electromagnetoelastic media in variational form with an application to shell/laminae equations. (English) Zbl 1183.74066

Summary: The fundamental equations of elasticity with extensions to electromagnetic effects are expressed in differential form for a regular region of materials, and the uniqueness of solutions is examined. Alternatively, the fundamental equations are stated as the Euler-Lagrange equations of a unified variational principle, which operates on all the field variables. The variational principle is deduced from a general principle of physics by modifying it through an involutory transformation. Then, a system of two-dimensional shear deformation equations is derived in differential and fully variational forms for the high frequency waves and vibrations of a functionally graded shell. Also, a theorem is given, which states the conditions sufficient for the uniqueness in solutions of the shell equations. On the basis of a discrete layer modeling, the governing equations are obtained for the motions of a curved laminae made of any numbers of functionally graded distinct layers, whenever the displacements and the electric and magnetic potentials of a layer are taken to vary linearly across its thickness. The resulting equations in differential and fully variational, invariant forms account for various types of waves and coupled vibrations of one and two dimensional structural elements as well. The invariant form makes it possible to express the equations in a particular coordinate system most suitable to the geometry of shell (plate) or laminae. The results are shown to be compatible with and to recover some of earlier equations of plane and curved elements for special material, geometry and/or effects.

MSC:

74F15 Electromagnetic effects in solid mechanics
74E30 Composite and mixture properties
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aboudi, J.: Micromechanical analyses of smart composite materials, Progress in smart materials and structures, 291-361 (2007)
[2] Alhazza, K. A.; Alhazza, A. A.: A review of the vibrations of plates and shells, The shock and vibration digest 36, No. 5, 377-395 (2004)
[3] Alshits, I.; Darinskii, A. N.; Lothe, J.: On the existence of surface waves in half-anisotropic elastic media with piezoelectric and piezomagnetic properties, Wave motion 16, 265-283 (1992) · Zbl 0764.73022 · doi:10.1016/0165-2125(92)90033-X
[4] Altay, G.; Dökmeci, M. C.: Fundamental variational equations of discontinuous thermopiezoelectric fields, International journal of engineering science 34, 769-782 (1996) · Zbl 0899.73452 · doi:10.1016/0020-7225(95)00133-6
[5] Altay, G.; Dökmeci, M. C.: A uniqueness theorem in Biot’s poroelasticity theory, Journal of applied mathematics and physics (ZAMP) 49, 838-846 (1998) · Zbl 0911.73003 · doi:10.1007/PL00001489
[6] Altay, G.; Dökmeci, M. C.: Thermo-viscoelastic analysis of high-frequency motions of thin plates, Acta mechanica 143, 91-111 (2000) · Zbl 1001.74053 · doi:10.1007/BF01250020
[7] Altay, G. A.; Dökmeci, M. C.: Coupled thermoelastic shell equations with second sound for high-frequency vibrations of temperature dependent materials, International journal of solids and structures 38, 2737-2768 (2001) · Zbl 1049.74678 · doi:10.1016/S0020-7683(00)00179-7
[8] Altay, G. A.; Dökmeci, M. C.: High-frequency equations for non-linear vibrations of thermopiezoelectric shells, International journal of engineering science 40, 957-989 (2002)
[9] Altay, G. A.; Dökmeci, M. C.: A non-linear rod theory for high-frequency vibrations of thermopiezoelectric materials, International journal of non-linear mechanics 37, 225-243 (2002) · Zbl 1116.74371 · doi:10.1016/S0020-7462(00)00108-6
[10] Altay, G.; Dökmeci, M. C.: Some comments on the higher order theories of piezoelectric, piezothermoelastic and thermopiezoelectric rods and shells, International journal of solids and structures 40, 4699-4706 (2003) · Zbl 1054.74568 · doi:10.1016/S0020-7683(03)00185-9
[11] Altay, G. A.; Dökmeci, M. C.: Variational and differential equations for dynamics of a coated piezolaminated composite bar, Mechanics of electromagnetic material systems and structures, 209-217 (2003)
[12] Altay, G.; Dökmeci, M. C.: Fundamental equations of certain electromagnetic-acoustic discontinuous fields in variational form, Continuum mechanics and thermodynamics 16, 53-71 (2004) · Zbl 1100.74548 · doi:10.1007/s00161-003-0141-5
[13] Altay, G.; Dökmeci, M. C.: Variational principles and vibrations of a functionally graded plate, Computers and structures 83, 1340-1354 (2005)
[14] Altay, G.; Dökmeci, M. C.: A polar theory for vibrations of thin elastic shells, International journal of solids and structures 43, 2578-2601 (2006) · Zbl 1120.74522 · doi:10.1016/j.ijsolstr.2005.06.027
[15] Altay, G.; Dökmeci, M. C.: Variational principles for piezoelectric, thermopiezoelectric, and hygrothermopiezoelectric continua revisited, Mechanics of advanced materials and structures 14, No. 7, 549-562 (2007)
[16] Altay, G.; Dökmeci, M. C.: Certain hygrothermopiezoelectric multi-field vaariational principles for smart elastic laminae, Mechanics of advanced materials and structures 14, No. 7, 549-562 (2008)
[17] Altay, G., Dökmeci, M.C., 2009. Piezomagnetic/Electromagnetoelastic Structural Elements; Rods, Shells and Laminated Elements. ITU & BU-TR. January 2009.
[18] Ambartsumian, S.A., 1964. Theory of Anisotropic Shells. NASA TTF-118.
[19] Annigeri, A. R.; Ganesan, N.; Swarnamani, S.: Free vibrations of simply supported layered and multiphase magneto-electro-elastic cylindrical shells, Smart materials and structures 15, 459-467 (2006)
[20] Annigeri, A. R.; Ganesan, N.; Swarnamani, S.: Free vibration behavior of multiphase and layered magneto- electro-elastic beam, Journal of sound and vibration 299, 44-63 (2007)
[21] Aouadi, M.: On the coupled theory of thermo-magnetoelectroelasticity, Quarterly journal of mechanics and applied mathematics 60, No. 4, 443-456 (2007) · Zbl 1169.74017 · doi:10.1093/qjmam/hbm016
[22] Avellaneda, M.; Harsche, G.: Magnetoelectric effect in piezoelectric/magnetostrictive multilayer (2-2) composites, Journal of intelligent material systems and structures 5, 501-513 (1994)
[23] Bardzokas, D. I.; Filshtinsky, M. L.; Filshtinsky, L. A.: Mathematical methods in electro- magneto-elasticity, (2007) · Zbl 1147.74001
[24] Basset, A. B.: On the extension and flexure of cylindrical and spherical thin elastic shells, Philosophical transactions of the royal society of London A 181, 433-480 (1890) · JFM 22.1006.01 · doi:10.1098/rsta.1890.0007
[25] Benveniste, Y.: Magnetoelectric effect in fibrous composite with piezoelectric and piezomagnetic phases, Physical review B 51, 16424-16427 (1995)
[26] Berdichevskii, V. L.; Khan’chau, L.: High-frequency vibrations of shells, Soviet physics doklady 27, No. 11, 988-990 (1982) · Zbl 0537.73047
[27] Berger, N.: Estimates for stress derivatives and error in interior equations for shells of variable thickness with applied forces, SIAM journal of applied mathematics 24, 97-120 (1973) · Zbl 0253.73072 · doi:10.1137/0124011
[28] Berlincourt, D. A.; Curran, D. R.; Jaffe, H.: Piezoelectric and piezomagnetic materials and their function in transducers, Physical acoustics, 169-270 (1964)
[29] Bhangale, R. K.; Ganesan, N.: Free vibration studies of simply supported non-homogeneous functionally graded magneto-electro-elastic finite cylindrical shells, Journal of sound and vibration 288, 412-422 (2005)
[30] Bhangale, R. K.; Ganesan, N.: Free vibration of simply supported functionally graded and layered magneto- electro-elastic plates, Journal of sound and vibration 294, 1016-1038 (2006) · Zbl 1120.74597
[31] Birman, V.; Byrd, L. W.: Modeling and analysis of functionally graded materials and structures, Applied mechanics reviews 60, No. 9, 195-215 (2007)
[32] Buchanan, G. R.: Free vibration of an infinite magneto-electro-elastic cylinder, Journal of sound and vibration 268, 413-426 (2003)
[33] Buchanan, G. R.: Layered versus multiphase magneto-electro-elastic composites, Composites part B: engineering 35, 413-420 (2004)
[34] Byrne, R.,1944. Theory of small deformations of a thin shell. Seminar Reports in Mathematics. University of California (Los Angeles) Publications in Mathematics (N.S.), vol. 2, pp. 103 – 152. · Zbl 0060.42602
[35] Carrera, E.: Theories and finite elements for multilayered, anisotropic, composite plates and shells, Archives of computational methods in engineering 9, No. 2, 87-140 (2002) · Zbl 1062.74048 · doi:10.1007/BF02736649
[36] Carrera, E.: Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking, Archives of computational methods in engineering 10, No. 3, 215-296 (2003) · Zbl 1140.74549 · doi:10.1007/BF02736224
[37] Carrera, E.: A historical review of zig-zag theories for multi layered plates and shells, Applied mechanics reviews 56, 290-309 (2003)
[38] Cauchy, A.L., 1829. Sur l’équilibre et le mouvement d’une plaque élastique don’t l’élasticité n’est pas la méme dans tous les sens. Exercices Mathématique, vol. 4, pp. 1 – 14.
[39] Chen, Z. R.; Yu, S. W.; Meng, Lu.M.; Ye, L.: Effective properties of layered magneto-electro-elastic composites, Composite structures 57, No. 1, 177-182 (2002)
[40] Chen, W. Q.; Lee, K. Y.; Ding, H. J.: On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates, Journal of sound and vibration 279, 237-251 (2005)
[41] Chen, J.; Chen, H.; Pan, E.; Heyliger, P. R.: Modal analysis of magneto-electro-elastic plates using the state vector approach, Journal of sound and vibration 304, 722-734 (2007)
[42] Chen, J. Y.; Pan, E.; Chen, H. L.: Wave propagation in magneto-electro-elastic multilayered plates, International journal of solids and structures 44, 1073-1085 (2007) · Zbl 1178.74090 · doi:10.1016/j.ijsolstr.2006.06.003
[43] Chopra, I.: Review of state of art of smart structures and integrated systems, AIAA journal 40, 2145-2187 (2002)
[44] Christoffel, E. B.: Untersuchungen über die mit dem fortbestehen linearer partieller. Differentialgleichungen vertraglichen unstetigkeiten, Annals of mathematics 8, 81-113 (1877) · JFM 09.0663.01
[45] Ciarlet, P. G.: Mathematical elasticity, Theory of shells 3 (2000) · Zbl 0953.74004
[46] Cicala, P.: Systematic approach to linear shell theory, (1965)
[47] Daga, A.; Ganesan, N.; Shankar, K.: Transient response of magneto-electro-elastic simply supported cylinder using finite element, Journal of mechanics of materials and structures 3, No. 2, 374-388 (2008)
[48] Daga, A.; Ganesan, N.; Shankar, K.: Comparative studies of the transient response for PECP, MSCP, barium titanate, magneto-electro-elastic finite cylindrical shell under constant internal pressure using finite element method, Finite elements in analysis and design 44, 89-104 (2008)
[49] Dai, H. L.; Fu, Y. M.; Liu, T. X.: Electromagnetoelastic dynamic response of transversely isotropic piezoelectric hollow spheres in a uniform magnetic field, ASME – journal of applied mechanics 74, No. 1, 65-73 (2007) · Zbl 1111.74380 · doi:10.1115/1.2178361
[50] De Lacheisserie, E. T.: Magnetostriction – theory and applications of magnetoelasticity, (1993)
[51] Delia, C. N.; Shu, D. W.: Vibration of delaminated composite laminates: a review, Applied mechanics reviews 60, 1-20 (2007)
[52] , The collected papers of raymond D. Mindlin & II, 989-1001 (1989)
[53] De Veubeke, B. Fraeijs: Dual principles of elastodynamics finite element applications, Lectures on finite element methods in continuum mechanics (1973) · Zbl 0281.73049
[54] Dikmen, M., 1982. Theory of Thin Elastic Shells. Pitman, London · Zbl 0478.73043
[55] Dökmeci, M. C.: A general theory of elastic beams, International journal of solids and structures 8, 1205-1222 (1972) · Zbl 0247.73065
[56] Dökmeci, M. C.: Variational principles in piezoelectricity, Lettere al nuovo cimento 7, 449-454 (1973)
[57] Dökmeci, M. C.: On the higher order theories of piezoelectric crystal surfaces, Journal of mathematical physics 15, No. 12, 2248-2252 (1974)
[58] Dökmeci, M. C.: A theory of high frequency vibrations of piezoelectric crystal bars, International journal of solids and structures 10, 401-409 (1974) · Zbl 0279.73036 · doi:10.1016/0020-7683(74)90109-7
[59] Dökmeci, M. C.: An isothermal theory of anisotropic rods, Journal of engineering mathematics 9, 311-322 (1975) · Zbl 0315.73025
[60] Dökmeci, M. C.: Theory of vibrations of coated, thermopiezoelectric laminae, Journal of mathematical physics 19, No. 1, 109-126 (1978)
[61] Dökmeci, M. C.: Recent advances: vibrations of piezoelectric crystals, International journal of engineering science 18, 431-448 (1980) · Zbl 0439.73098
[62] Dökmeci, M. C.: Certain integral and differential types of variational principles in non-linear piezoelectricity, IEEE transactions on ultrasonics, ferroelectrics, and frequency control 37, No. 5, 775-787 (1988)
[63] Dökmeci, M.C., 1992. A Dynamic Analysis of Piezoelectric Strained Elements. T.R.U.S.A.R.D.S.G., June 1992.
[64] Dökmeci, M. C.: Laminae theory for motions of micropolar materials, Bulletin of the technical university of Istanbul 47, No. 4, 19-77 (1994) · Zbl 0945.74522
[65] Duvaut, G.; Lions, P. L.: Inequations in physics and mechanics, (1979)
[66] Ekstein, H.: High frequency vibrations of thin crystal plates, Physical review 68, 11-23 (1945) · Zbl 0063.01235
[67] El-Karamany, A. S.; Ezzat, M. A.: Uniqueness and reciprocal theorems in linear micropolar electro-magnetic themoelasticity with two relaxation times, Mechanics of time-dependent materials 13, 93-115 (2009)
[68] Fichera, G.: Existence theorems in elasticity, Encyclopedia of physics /2, 347-424 (1972)
[69] Fiebig, M.: Revival of the magnetoelectric effect, Journal of physics D 38, 123-152 (2005)
[70] , Proceedings of symposium on magnetoelectric interaction phenomena in crystals (1975)
[71] Friedrichs, K. O.: Ein verfahren der variationsrechnung das minimum eines integrals als das maximum eines anderen ausdruckes darzustellen, Mathematik und physik (Ges. Wiss. göttingen, nachrichten) 1, 13-20 (1929) · JFM 55.0294.01
[72] Lage, R. Garcia; Soares, C. M. Mota; Soares, C. A. Mota; Reddy, J. N.: Layerwise partial mixed finite element analysis of magneto-electro-elastic plates, Computers and structures 82, 1293-1301 (2004)
[73] Ghugal, Y. M.; Shimpi, R. P.: A review of refined shear deformation theories for isotropic and anisotropic laminated beams, Journal of reinforced plastics and composites 20, No. 3, 255-272 (2001)
[74] Ghugal, Y. M.; Shimpi, R. P.: A review of refined shear deformation theories for isotropic and anisotropic laminated plates, Journal of reinforced plastics and composites 21, No. 9, 755-813 (2002)
[75] Gol’denveizer, A. L.: On the applicability of general theorems of the theory of elasticity to thin shells (in russian), PMM (Soviet applied mathematics and mechanics) 8, 3-14 (1944) · Zbl 0060.42302
[76] Gol’denveizer, A. L.: On approximate methods for analyzing thin elastic shells and plates, Mechanics of solids 32, No. 3, 115-126 (1997)
[77] Green, A. E.; Naghdi, P. M.: On uniqueness in the linear theory of elastic shells and plates, Journal de mécanique 10, 251-261 (1971) · Zbl 0214.52001
[78] Green, A. E.; Naghdi, P. M.: On electromagnetic effects in the theory of shells and plates, Philosophical transactions of royal society of London A 309, 559-610 (1983) · Zbl 0536.73091 · doi:10.1098/rsta.1983.0058
[79] Gurtin, M. E.: The linear theory of elasticity, Encyclopedia of physics /2, 1-295 (1972)
[80] Hamilton, W. R.: On a general method in dynamics, Philosophical transactions of royal society of London 125, 95-144 (1834)
[81] Hamilton, W. R.: Second essay on a general method in dynamics, Philosophical transactions of royal society of London 125, 247-308 (1835)
[82] Hashin, Z.: Analysis of composite materials – a survey, ASME journal of applied mechanics 50, 481-505 (1983) · Zbl 0542.73092 · doi:10.1115/1.3167081
[83] He, J-H.: Coupled variational principles of piezoelectricity, International journal of engineering science 39, 323-341 (2001) · Zbl 1210.74175 · doi:10.1016/S0020-7225(00)00035-5
[84] He, J. -H.: Variational theory for linear magneto-electro-elasticity, International journal of nonlinear sciences and numerical simulation 2, No. 4, 309-316 (2001) · Zbl 1083.74526
[85] Heyliger, P. R.; Pan, E.: Static fields in magnetoelectroelastic laminates, AIAA journal 42, 1435-1443 (2004)
[86] Heyliger, P. R.; Ramirez, F.; Pan, E.: Two dimensional static fields in magnetoelectroelastic laminates, Journal of intelligent material systems and structures 15, 689-709 (2004)
[87] Hou, P. F.; Leung, A. Y. T.: The transient responses of magneto-electro-elastic hollow cylinders, Smart materials and structures 13, 762-776 (2004)
[88] Hou, P. F.; Ding, H. J.; Leung, A. Y. T.: The transient responses of a special non-homogeneous magneto-electro-elastic hollow cylinder for axisymmetric plane strain problem, Journal of sound and vibration 291, 19-47 (2006)
[89] Hu, H-C.: On some variational principles in the theory of elasticity and the theory of plasticity, Acta physica sinica 10, No. 3, 259-290 (1954)
[90] Hutter, K.; Van De Ven, A. A. F.; Ursescu, A.: Electromagnetic field matter interactions in thermoelastic solids and viscous fluids, (2006)
[91] , Functionally graded materials in the 21st century (2001)
[92] Jiang, A.; Ding, H.: Analytical solutions to magneto-electro-elastic beams, Structural engineering and mechanics 18, 195-209 (2004)
[93] Jiangong, Y.; Qiujuan, M.; Shan, S.: Wave propagation in non-homogeneous magneto-electro-elastic hollow cylinders, Ultrasonics 48, 664-677 (2008)
[94] John, F.: Estimates for the derivatives of the stresses in a thin shell and interior shell equations, Communcations in pure and applied mathematics 18, 235-267 (1965)
[95] Kaplunov, J. D.; Kossovich, L. Y.; Nolde, E. V.: Dynamics of thin walled elastic bodies, (1998) · Zbl 0927.74001
[96] Karlash, V. L.: Resonant electromechanical vibrations of piezoelectric shells of revolution (Review), International applied mechanics 44, No. 4, 361-387 (2008) · Zbl 1177.74158
[97] Kellogg, O. D.: Foundations of potential theory, (1946) · Zbl 0053.07301
[98] Kienzler, R.; Altenbach, H.; Ott, L.: Theories of plates and shells: critical review and new aplications, (2004) · Zbl 1045.74003
[99] Kil’chevskiy, N.A., 1965. Fundamentals of the Analytical Mechanics of Shells. NASA TT F-292, Washington, DC.
[100] Kiral, E.; Eringen, A. C.: Constitutive equations of nonlinear electromagnetic-elastic crystals, (1990)
[101] Kirchhoff, G.: Über das gleichgewicht und die bewegung einer elastischen scheibe, Crelles journal 40, 51-88 (1850) · ERAM 040.1086cj · doi:10.1515/crll.1850.40.51
[102] Kirchhoff, G.: Vorlesungen über matematische physik: mechanik, (1876) · JFM 08.0542.01
[103] Knops, R. J.; Payne, L. E.: Uniqueness theorems in linear elasticity, (1972) · Zbl 0224.73016
[104] Koiter, W. T.; Simmonds, J. G.: Foundations of shell theory, Theoretical and applied mechanics, Proceedings of the thirteenth international congress of theoretical and applied mechanics, 150-176 (1973) · Zbl 0286.73068
[105] Kudryavtsev, B. A.; Parton, V. Z.; Senik, N. A.: Electromagnetoelasticity, (1990)
[106] Lanczos, C.: The variational principles of mechanics, (1986) · Zbl 1198.70001
[107] Landau, L. D.; Lifshitz, E.: Electrodynamics of continuous media, (1960) · Zbl 0122.45002
[108] Langley, R. S.; Bardell, N. S.: A review of current analysis capabilities applicable to the high frequency vibration prediction of aerospace structures, Aeronautical journal 102, 287-297 (1998)
[109] Le, K. C.: Vibrations of shells and rods, (1999) · Zbl 0933.74003
[110] Lee, P. C. Y.: A variational principle for the equations of piezoelectromagnetism in elastic dielectric crystals, Journal of applied physics 69, No. 11, 4770-4773 (1991)
[111] Lee, C-Y.; Hodges, D. H.: Dynamic variational-asymptotic procedure for laminated composite shells – part I: Low-frequency vibration analysis, ASME – journal of applied mechanics 76, No. 1 (2009)
[112] Lee, C-Y.; Hodges, D. H.: Dynamic variational-asymptotic procedure for laminated composite shells – part II: High-frequency vibration analysis, ASME – journal of applied mechanics 76, No. 1 (2009)
[113] Lee, J.; Boyd, J.; Lagoudas, D.: Effective properties of three-phase electro-magneto-elastic composites, International journal of engineering science 43, No. 10, 790-825 (2005) · Zbl 1211.74182 · doi:10.1016/j.ijengsci.2005.01.004
[114] Leibniz, G. W.: Demonstrationes novae de resistentia solidorum, Acta erud., leibnizens math. Schriften 6, 106-112 (1684)
[115] Leo, D. J.: Engineering analysis of smart material systems, (2007)
[116] Li, J. Y.: Uniqueness and reciprocity theorems for linear thermo-electro-magneto-elasticity, Quarterly journal of mechanics and applied mathematics 56, No. 1, 35-43 (2003) · Zbl 1043.74015 · doi:10.1093/qjmam/56.1.35
[117] Li, J. Y.; Dunn, M. L.: Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior, Journal of intelligent material systems and structures 9, 404-416 (1998)
[118] Libai, A.; Simmonds, J. G.: The nonlinear theory of elastic shells, (1998) · Zbl 0918.73003
[119] Librescu, L.: Elastostatics and kinetics of heterogeneous shell type structures, (1975) · Zbl 0335.73027
[120] Luo, E.; Zhu, H.; Yuan, L.: Unconventional Hamilton-type variational principles for electromagnetic elastodynamics, Science in China: series G physics mechanics and astronomy 49, No. 1, 119-128 (2006) · Zbl 1148.74301 · doi:10.1007/s11433-005-0209-2
[121] Lur’e, S. A.; Shumova, N. P.: Kinematic models of refined theories concerning composite beams, plates, and shells, Mechanics of composite materials 32, No. 5, 422-430 (1996)
[122] Markworth, A. J.; Ramesh, K. S.; Parks, Jr. W. P.: Review: modeling studies applied to functionally graded materials, Journal of material science 30, 2183-2193 (1995)
[123] Mason, W. P.: Crystal physics of interaction processes, (1966)
[124] Miloh, T.: Hamilton’s principle, Lagrange’s method, and ship motion theory, Journal of ship research 28, No. 4, 229-237 (1984)
[125] Milton, G. W.: The theory of composites, (2002) · Zbl 0993.74002
[126] Mindlin, R.D., 1955. An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. US Army Signals Corps Engineering Laboratories, Fort Monmouth, New Jersey. · Zbl 0066.42203
[127] Mindlin, R. D.: High frequency vibrations of crystal plates, Quarterly of applied mathematics 19, 51-61 (1961) · Zbl 0109.17707
[128] Mindlin, R. D.: Lecture notes on the theory of beams and plates, (1967)
[129] Mindlin, R. D.: High frequency vibrations of piezoelectric crystal plates, International journal of solids and structures 8, 895-906 (1972) · Zbl 0243.73059 · doi:10.1016/0020-7683(72)90004-2
[130] Mindlin, R. D.: Equations of high frequency vibrations of thermopiezoelectric crystal plates, International journal of solids and structures, 10 (1974) · Zbl 0282.73068 · doi:10.1016/0020-7683(74)90047-X
[131] Miyamoto, Y.; Kaysser, W. A.; Rabin, B. H.; Kawasaki, A.; Ford, R. G.: Functionally graded materials: design processing and applications, (1999)
[132] Naghdi, P. M.: Foundations of elastic shell theory, Progress in solid mechanics, 1-90 (1963)
[133] Naghdi, P. M.: The theory of shells and plates, Mechanics of solids /2, 425-640 (1972)
[134] Naghdi, P. M.; Trapp, J. A.: A uniqueness theorem in the theory of Cosserat surface, Journal of elasticity 2, 9-20 (1972)
[135] Nan, C. -W.: Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases, Physical review B 50, No. 9, 6082-6088 (1994)
[136] Pan, E.: Exact solution for simply supported and multilayered magnet-electro-elastic plates, ASME – journal of applied mechanics 68, 608-618 (2001) · Zbl 1110.74612 · doi:10.1115/1.1380385
[137] Pan, E.; Han, F.: Exact solution for functionally graded and layered magneto-electro- elastic plates, International journal of engineering science 43, 321-339 (2005)
[138] Pan, E.; Heyliger, P. R.: Free vibrations of simply supported and multilayered magneto-electro-elastic plates, Journal of sound and vibration 252, No. 3, 429-442 (2002)
[139] Pao, Y. H.: Electromagnetic forces in deformable continua, Mechanics today, 209-305 (1978) · Zbl 0379.73100
[140] Pao, Y. H.; Yeh, C. S.: A linear theory for soft ferromagnetics elastic solids, International journal of engineering science 11, 415-436 (1973) · Zbl 0252.73070
[141] Parton, V. Z.; Kudryavtsev, B. A.: Electromagnetoelasticity, piezoelectrics and electrically conductive solids, (1988)
[142] Pikul, V. V.: Theory of shells: state of the art and prospects of development, Mechanics of solids 35, No. 2, 130-142 (2000)
[143] Poisson, S. D.: Mémoire sur l’équilibre et le movement des corps élastiques, Mémoire de l’academie sciences 8, 357-570 (1829)
[144] Priimenko, V.; Vishnevskii, M.: An initial boundary-value problem for model electromagnetoelasticity system, Journal of differential equations 235, 31-55 (2007) · Zbl 1117.35078 · doi:10.1016/j.jde.2006.12.016
[145] Qatu, M. S.: Recent research advances in the dynamic behavior of shells 1989 – 2000. Part 1: laminated composite shells, Applied mechanics reviews 55, 325-349 (2002)
[146] Qatu, M. S.: Recent research advances in the dynamic behavior of shells 1989 – 2000. Part 2: homogeneous shells, Applied mechanics reviews 55, 415-434 (2002)
[147] Qatu, M. S.: Vibration of laminated shells and plates, (2004) · Zbl 1098.74596
[148] Qing, G. -H.; Qui, J. -J.; Liu, Y-H.: Mixed H-R mixed variational principle for magneto-electroelastic bodies and state-vector equation, Applied mathematics and mechanics 26, No. 6, 722-728 (2005) · Zbl 1144.74325 · doi:10.1007/BF02465422
[149] Rado, G. T.; Folen, V. J.: Observation of the magnetically induced magnetoelectric effect and evidence to antiferromagnetic domains, Physics review letters 7, 310-311 (1961)
[150] Ramirez, F.; Heyliger, P. R.; Pan, E.: Discrete layer solution to free vibrations of functionally graded magneto-electro-elastic plates, Mechanics of advanced materials and structures 13, 249-266 (2006)
[151] Reddy, J. N.: Mechanics of laminated composite plates and shells, (2004) · Zbl 1075.74001
[152] Reddy, J. N.; Robbins, D. H.: Theories and computational models for composite laminates, Applied mechanics reviews 47, No. 6, 147-169 (1994)
[153] , Progress in smart materials and structures (2007)
[154] Rubin, M. B.: Cosserat theories: shells, rods and points, (2000) · Zbl 0984.74003
[155] Ryu, J.; Priya, S.; Uchino, K.; Kim, H-E.: Magnetoelectric effect in composites of magnetostrictive and piezoelectric materials, Journal of electroceramics 8, 107-119 (2002)
[156] Santos, J. E.; Sheen, D.: On the existence and uniqueness of solutions to Maxwell’s equations in bounded domains with application to magnetotellurics, Mathematical models and methods in applied sciences 10, 615-628 (2000) · Zbl 1034.35138 · doi:10.1142/S0218202500000331
[157] Saravanos, D. A.; Heyliger, P. R.: Mechanics and computational models for laminated piezoelectric beams, plates, and shells, Applied mechanics reviews 52, 305-320 (1999)
[158] Schwartz, M.: Encyclopedia of smart materials, (2002)
[159] Shen, H-S.: Functionally graded materials: nonlinear analysis of plates and shells, (2009)
[160] Soh, A. K.; Liu, J. X.: On the constitutive equations of magnetoelectroelastic solids, Journal of intelligent material systems and structures 16, 597-602 (2005)
[161] Strang, G.: Science and engineering, (1975) · Zbl 0334.65087
[162] Suresh, S.; Mortensen, A.: Fundamentals of functionally graded materials, (1998)
[163] Tan, P.; Tong, L.: Modeling for the electro-magneto-elastic properties of piezoelectric- magnetic fiber reinforced composites, Composites part A 33, 631-645 (2002)
[164] Tang, T.; Yu, W.: Variational asymptotic homogenization of heterogeneous electromagnetoelastic materials, International journal of engineering science 46, 741-757 (2008) · Zbl 1213.74140 · doi:10.1016/j.ijengsci.2008.03.002
[165] Tani, J.; Tagaki, T.; Qui, J.: Intelligent material systems: application of functional materials, Applied mechanics reviews 51, No. 8, 505-521 (1998)
[166] Tiersten, H. F.: Linear piezoelectric plate vibrations, (1969)
[167] Tiersten, H. F.; Mindlin, R. D.: Forced vibrations of piezoelectric crystal plates, Quarterly of applied mathematics 20, No. 2, 107-119 (1962) · Zbl 0114.17505
[168] Tsai, Y. H.; Wu, C. P.: Dynamic responses of functionally graded magneto-electro-elastic shells with open-circuit surface conditions, International journal of engineering science 46, No. 9, 843-857 (2008) · Zbl 1213.74220 · doi:10.1016/j.ijengsci.2008.03.005
[169] Tzou, H. S.; Lee, H. -J.; Arnold, S. M.: Smart materials, precision sensors/actuators, smart structures, and structronic systems, Mechanics of advanced materials and structures 11, 367-393 (2004)
[170] Van Suchtelen, J.: Product properties: a new application of composite materials, Philips research reports 27, No. 1, 28-37 (1972)
[171] Vasiliev, V. V.; Lurie, S. A.: On refined theories of beams, plates, and shells, Journal of composite materials 26, No. 4 (1992)
[172] Vekovishcheva, I. A.: Variational principles in the theory of electroelasticity, Soviet applied mechanics 7, 1049-1052 (1974)
[173] Venkataraman, G.; Feldkamp, L. A.; Sahni, V. C.: Dynamics of perfect crystals, (1975)
[174] Villaggio, P.: Mathematical models for elastic structures, (1997) · Zbl 0978.74002
[175] Wang, Z. L.; Kang, Z. C.: Functional and smart materials, (1998)
[176] Wang, B. L.; Mai, Y. -M.: Fracture of piezoelectromagnetic materials, Mechanics research communications 31, 65-73 (2004) · Zbl 1045.74586 · doi:10.1016/j.mechrescom.2003.08.002
[177] Wang, B. -L.; Mai, Y. -W.: Self-consistent analysis of coupled magnetoelectroelastic fracture-theoretical investigation and finite element verification, Computer methods in applied mechanics and engineering 196, 2044-2054 (2007) · Zbl 1173.74330 · doi:10.1016/j.cma.2006.11.006
[178] Wang, X. M.; Shen, Y. P.: Some fundamental theory of electro-magneto-thermo-elastic material, Journal of applied mechanics 12, No. 2, 28-39 (1995)
[179] Wang, J.; Yang, J.: Higher order theories of piezoelectric plates and applications, Applied mechanics reviews 53, No. 4, 87-99 (2000)
[180] Wang, J.; Chen, L.; Fang, S.: State vector approach to analysis of multilayered magneto-electro-elastic plates, International journal of solids and structures 40, 1669-1680 (2003) · Zbl 1033.74029 · doi:10.1016/S0020-7683(03)00027-1
[181] Washizu, K., 1955. On the variational principles of elasticity and plasticity. Aeroelastic and Structural Research Lab., M.I.T., Tech. Rep. N0.25-18, Cambridge, Massachusetts.
[182] Weinitschke, H. J.: On uniqueness of axisymmetric deformations of elastic plates and shells, SIAM journal of mathematical analysis 19, No. 3, 580-592 (1988) · Zbl 0654.73031 · doi:10.1137/0519041
[183] Wu, C. P.; Tsai, Y. H.: Static behavior of functionally graded magneto-electro-elastic shells under electric displacement and magnetic flux, International journal of engineering science 45, 744-769 (2007)
[184] Wu, C. C. M.; Khan, M.; Moy, W.: Piezoelectric ceramics with functional gradients: a new application in material design, Journal of American ceramic society 79, 809-812 (1996)
[185] Wu, C. P.; Chiu, K. H.; Wang, Y. M.: A mesh-free DRK-based collocation method for the coupled analysis of functionally graded magneto-electro-elastic shells and plates, CMES – computer modeling in engineering and sciences 35, No. 3, 181-214 (2008) · Zbl 1153.74381
[186] Wu, C. P.; Chiu, K. H.; Wang, Y. M.: A review of the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells, CMC – computers materials and continua 8, No. 2, 93-132 (2008)
[187] Yao, W.: Generalized variational principles of three-dimensional problems in magneto-electroelastic bodies, Chinese journal of computational mechanics 20, No. 4, 487-489 (2003)
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.