903. Free vibration studies of functionally graded magneto-electro-elastic plates/shells by using solid-shell elements

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1 903 Free vibration studies of functionally graded magneto-electro-elastic plates/shells by using solid-shell elements Zheng Shijie Fen Yan 2 Wang Hongtao 3 2 State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and Astronautics Nanjing 2006 China 3 School of Mechanical and Electrical Engineering Nanjing University of Aeronautics and Astronautics Nanjing 2006 China sjzheng@nuaaeducn 3 xwwang@nuaaeducn 3 meehtwang@nuaaeducn (Received 23 September 202; accepted 4 December 202) Abstract In this article free vibration studies on functionally graded magneto-electro-elastic plates and cylindrical shells have been carried out by means of finite element method he functionally graded material is assumed to be exponential in the thickness direction he present finite element is formulated on the basis of assumed natural strain enhanced assumed strain method and using displacement components electric potential and magnetic potentials as nodal degrees of freedom his element can be used as solid element and can also be applied to model thin curved shell structures Numerical studies include the influence of the different exponential factor magnetic and piezoelectric effect on the natural frequencies Obtained numerical results are in good agreement with the semi-analytical finite element solutions available in the literature Keywords: functionally graded materials magneto-electro-elastic free vibration finite element plates/shells Introduction Smart or intelligent materials such as the piezoelectric and piezomagnetic ones have attracted considerable interest recently due to their ability of converting energy from one type to the other (among magnetic electric and mechanical energies) Magneto-electro-elastic materials simultaneously possess piezoelectric piezomagnetic and magneto-electro-elastic effects which is two orders higher than that of the individual constituent materials With application in ultrasonic imaging devices sensors actuators transducers and many other emerging components there is a strong need for theories or techniques that can predict the coupled response of these smart materials as well as structure composed of them Studies on static and dynamic behavior on simply supported plates as well as cylindrical shells have been dealt in literature [-9] In recent years functionally graded (FG) materials a novel microscopically inhomogeneous composites usually made from a mixture of metals and ceramics with continuously varying volume fractions of two materials along the thickness direction have gained big attention in engineering community But the studies on non-homogeneous magneto-electro-elastic structure are relatively scarce Cao Shi and Jin [0] employed the power series technique for solving the propagation behavior of Lamb waves in the FG piezoelectric-piezomagnetic material plate with material parameters varying continuously along the thickness direction Wu and sai [] used the method of perturbation to present a three-dimensional (3D) free vibration analysis of simply supported doubly-curved FG magneto-electro-elastic shells with closed-circuit surface conditions Pan and Han [2] presented an exact solution for FG and layered magneto-electro-elastic plates by pseudo-stroh formalism Huang Ding and Chen [3] derived the analytical and semi-analytical solutions for anisotropic FG magneto-electro-elastic beams 787 VIBROENGINEERING JOURNAL OF VIBROENGINEERING DECEMBER 202 VOLUME 4 ISSUE 4 ISSN

2 903 FREE VIBRAION SUDIES OF FUNCIONALLY GRADED MAGNEO-ELECRO-ELASIC PLAES/SHELLS BY USING SOLID-SHELL ELEMENS ZHENG SHIJIE FEN YAN WANG HONGAO subjected to an arbitrary load which can be expanded in terms of sinusoidal series heir analysis is applicable to beams with various boundary conditions at the two ends Free vibration analysis of FGM and layered magneto-electro-elastic plates has been carried out by using series solution in conjunction with finite element approach [4] From the literature survey it is found that only several studies have been reported on magneto-electro-elastic structures analyzed by semi-analytical finite element analysis o the author s knowledge there is no finite element formulation without combining with series solution available for vibration studies on FG magneto-electro-elastic plates and shells Hence in present study a solid-shell element formulation is presented for the free vibration analysis of FG magneto-electro-elastic plates and shells by using enhanced assumed strain and assumed natural strain methods For the study magneto-electro-elastic material made of a homogenous mixture of piezoelectric barium titanate (BaiO 3 ) as the embedded material and piezomagnetic cobalt ferrite (CoFe 2 O 4 ) as the matrix material is considered 2 Constitutive equations 2 Governing equation For anisotropic and linearly magneto-electro-elastic solid the coupled constitutive equations for a general three-dimensional solid is as follows []: σ = C S e E q H j jk k kj k kj k D = e S + ε E + m H j jk k jk k jk k B = q S + m E + µ H j jk k jk k jk k () where σ D and B j indicate the stress electric displacement and magnetic induction S k E k j j and H k are strain electric field and magnetic field respectively C ε and µ jk are the elastic dielectric and magnetic permeability coefficients respectively e q and m jk are the piezoelectric piezomagnetic and magneto-electro-elastic material coefficients respectively A completely coupled magneto-electro-elastic material matrix assuming a hexagonal crystal class for above constitutive equations is given by Buchanan [7] he present study considers FG material composed of piezoelectric and magnetostrictive material he grading is accounted across the thickness of the shell his has been achieved by grading the volume fraction distribution of either piezoelectric or magnetostrictive material governed by a simple power-law exponent In the present case a simple power-law-type definition for the volume fraction of BaiO 3 across the thickness direction of the FGM plate is defined as: jk kj jk kj 2z+ h VB = 2h n (2) where h - thickness of the plate z - thickness coordinates (0 z h) and n - power-law index he bottom surface of the plate (0 z h) is CoFe 2 O 4 -rich whereas the top surface (0 z h) of the plate is BaiO 3 -rich and the sum of the total volume fractions of the constituent materials BaiO 3 (B) and CoFe 2 O 4 (F) should be one While in the case of an FGM cylindrical shell the volume fraction of CoFe 2 O 4 (F) across the radial direction of the shell is assumed as: 788 VIBROENGINEERING JOURNAL OF VIBROENGINEERING DECEMBER 202 VOLUME 4 ISSUE 4 ISSN

3 903 FREE VIBRAION SUDIES OF FUNCIONALLY GRADED MAGNEO-ELECRO-ELASIC PLAES/SHELLS BY USING SOLID-SHELL ELEMENS ZHENG SHIJIE FEN YAN WANG HONGAO V F r z i = ro ri n r (3) where r z represents radius at any point along the radial direction of the shell r i is the inner radius r o is the outer radius of the shell Based on the above definition it follows that the inner surface of the cylindrical shell will be piezo-rich On the basis of the volume fraction definition and law of mixtures the effective material property definition follows: P ( ς ) = PV ( ς ) (4) eff 3 j j 3 j= where P j and V j are material properties and volume fraction of the constituent material j comprising the functionally graded material P eff is general notation for material property Making use of equation (2)-(4) the effective elastic piezoelectric piezomagnetic dielectric and magnetic permeability definitions can be derived 3 Finite element formulation In this section the eight-node magneto-electro-elastic solid shell element is described he element has four nodes at each of its top (ζ = ) and bottom faces (ζ = ) he mapping between the global Cartesian co-ordinates (X Y Z) and the natural coordinates (ξ η ζ) is: 4 X ( ξ η ζ ) = N ( ξ η)( ζ X + ζ X ) = N ( ξ η) X + ζ N ( ξ η) X i= 4 i= + i i 4+ i 0 e n e U ( ξ η ζ ) = N ( ξ η)( ζ U + ζ U ) = N ( ξ η) U + ζ N ( ξ η) U + i i 4+ i 0 e n e (5) where N i s are the two-dimensional Lagrangian interpolation functions [ N N N N 2 ] [ N 2 3 N4 3 N 3 N4 3] { X Y Z X 8 Y8 Z8} { U V W U V W } N = I I I I N = I I I I X U n e e = = (6) (7) Here the strain-displacement relation of the element will be presented With reference to the interpolations of X and U the strain with respect to (ξ η ζ) is computed ie: S = X U S = X U S = X U ξ ξ ξ η η η ζ ζ ζ S = X U + X U S = X U + X U S = X U + X U ξη ξ η η ξ ζξ ζ ξ ξ ζ ζη ζ η η ζ (8) It has been rather standard practice to use an assumed natural strain (ANS) method for resolving the shear locking and trapezoidal locking Shear locking is due to the excess number of transverse shear strains sampled in the process of integrating the element stiffness matrix 789 VIBROENGINEERING JOURNAL OF VIBROENGINEERING DECEMBER 202 VOLUME 4 ISSUE 4 ISSN

4 903 FREE VIBRAION SUDIES OF FUNCIONALLY GRADED MAGNEO-ELECRO-ELASIC PLAES/SHELLS BY USING SOLID-SHELL ELEMENS ZHENG SHIJIE FEN YAN WANG HONGAO he ANS method is an effective method of resolving shear locking By using ANS method the number of independent shear strains in the system level can be reduced he natural transverse shear strains are modified to be [5 6]: Sɶ η + η = S + S 2 2 ξ ξ Sɶ + = S + S 2 2 ξ= 0 η= ξ= 0 η=+ ζξ ζξ ζξ ξ= η= 0 ξ=+ η= 0 ζξ ζη ζη (9) Similar to shear locking the excessive number of sampled thickness strains lead to trapezoidal locking rapezoidal locking occurs when the common shell element is used to model curved shells It can also be reduced in the system level by sampling the strain along the element edges ie [5 6]: Sɶ N Sɶ N Sɶ N Sɶ N Sɶ (0) ξ= η= ξ=+ η= ξ=+ η=+ ξ= η=+ ζ = ζ + 2 ζ + 3 ζ + 4 ζ he conventional eight-node solid elements have a significant deficiency when the element thickness is small compared with the element span the excessive shear strain energy stored in the thickness direction will lead to a much higher stiffness coefficient than those in planar directions o overcome the thickness locking of solid shell elements a linear extension EAS over the thickness is used to enhance the strain in the thickness direction [6]: S = S + ζ S ζ EAS ζ { λ λ λ λ } EAS [ EAS Sζ = ξ η ξη] λ= Bζ λ J λ= ζ () where J is the value of the determinant of the Jacobian As the material properties are often defined in a local orthogonal frame (x y z) it is necessary to obtain the local physical strains from the ones with respect to (ξ η ζ) It will be assumed that z-axis is perpendicular to the mid-surface of the shell Hence the relation between the nature coordinate infinitesimal strains and the local physical strains is: 790 { S S Sɶ EAS ξ η ζ ζ Sz Sɶ Sɶ ζη ζξ Sξη} u e S = + = B U (2) where is the strain transformation matrix [6] For eight-node magneto-electro-elastic solid shell element with one electric degree and magnetic degree per node the electrical potential field and the magnetic field are adopted as follows: E = B φ H = B ψ (3) φ e N φ ξ N ψ ξ Bφ = φ Nφ η Bψ = ψ Nψ η N N ψ φ ζ ψ ζ e VIBROENGINEERING JOURNAL OF VIBROENGINEERING DECEMBER 202 VOLUME 4 ISSUE 4 ISSN (4)

5 903 FREE VIBRAION SUDIES OF FUNCIONALLY GRADED MAGNEO-ELECRO-ELASIC PLAES/SHELLS BY USING SOLID-SHELL ELEMENS ZHENG SHIJIE FEN YAN WANG HONGAO [ N N ] [ N N ] N = N = (5) φ 8 ψ 8 where N i s are the three-dimensional Lagrangian interpolation functions φ ψ is the transformation matrix he thermodynamic potential for a 3-D magneto-electro-elastic solid is given by: G = G( S E H ) (6) where S E and H are independent variables which represent strain electric field and magnetic field respectively By using Eq () in Eq (6) the variation expression for magneto-electro-elastic solid can be obtained as follows: G = S CS E εe H µh SeE SqH EmH (7) where C ε µ represent elastic dielectric and magnetic permeability coefficients and e q m indicate the piezoelectric piezo-magnetic and magneto-electric coefficients respectively By minimizing Eq (7) for nodal variables of shape functions for strain-displacement electric field-electric potential and magnetic field-magnetic potential the finite element equations for magneto-electro-elastic solid can be obtained A formulation for such coupled field variables can be written as [2]: 2 ( ω ) K M u+ K φ+ K ψ = 0 uu uφ uψ K u K φ K ψ = 0 uφ φφ φψ K u K φ K ψ = 0 uψ φψ ψψ (8) Various stiffness matrices are defined as follows: K = B CB dv K = B eb dv uu v u u uφ v u K = B qb dv K = B mb dv uψ v u φ φψ v φ ψ K = B εb dv K = B µ B dv φφ v φ φ ψψ v ψ ψ φ (9) B u B φ and B ψ denotes the strain-displacement electric field-electric potential and magnetic field-magnetic potential relations respectively Mass matrix for the structure is: ( ζ ) ρ( ζ ) M = N + N N + N (20) 0 n 0 n dv In equation (8) by using a condensation technique ϕ and ψ are eliminated to obtain an equivalent system stiffness matrix K he equation of motion for the system can be written as: eq MUɺɺ + K U = 0 (2) eq VIBROENGINEERING JOURNAL OF VIBROENGINEERING DECEMBER 202 VOLUME 4 ISSUE 4 ISSN

6 903 FREE VIBRAION SUDIES OF FUNCIONALLY GRADED MAGNEO-ELECRO-ELASIC PLAES/SHELLS BY USING SOLID-SHELL ELEMENS ZHENG SHIJIE FEN YAN WANG HONGAO where: K = K + K K K + K K K (22) eq uu uφ ΙΙ Ι uψ ΙV ΙΙΙ he component matrices for Eq (9) are: K = K K K K Ι ΙΙ ΙΙΙ ΙV uφ φψ ψψ uψ K = K K K K φφ φψ ψψ φψ K = K K K K uψ φψ φφ uφ K = K K K K ψψ φψ φφ φψ (23) he eigenvectors that correspond to the distribution of ϕ and ψ are shown below: φ= K K u II I ψ = K K u (24) IV III o study the magneto-electric effect on the system frequencies the equivalent stiffness matrix K is derived by neglecting the coupling between the piezoelectric BaiO 3 and eq _ reduced piezomagnetic CoFe 2 O 4 materials he magneto-electric material coefficient (m) is zero for single phase BaiO 3 and CoFe 2 O 4 From equation (8) by putting K φψ = 0 the reduced finite element equations are as follows: 2 ( ω ) K M u+ K φ+ K ψ = 0 uu uφ uψ K u K φ= 0 uφ K u K ψ = 0 uψ φφ ψψ (25) he reduced stiffness matrix K is shown below: eq _ reduced K K K K K K K K (26) eq _ reduced= uu+ uφ φφ uφ+ uψ ψψ uψ o study the piezoelectric effect due to piezoelectric BaiO 3 on the frequency of the magneto-electro-elastic structures the stiffness matrix K is derived by putting magnetic potential to zero as follows: eq _ φφ = uu uφ φφ uφ eq _ φφ K K K K K (27) o study the magnetic effect of piezomagnetic CoFe 2 O 4 on the frequency of the system K is used as the stiffness matrix and is expressed as: eq _ ψψ K K K K K (28) eq _ ψψ = uu + uψ ψψ uψ 4 Results and discussion In this section we present some numerical results by using the finite element formulation proposed in this paper Plate and shell problems are considered for validating the presented finite element method Material constants are given in several papers and Aboudi [7] is the source for the material parameters used here and hence not repeated here he density of BaiO 3 and CoFe 2 O 4 is assumed to be the same 5730 kg/m 3 fuu feq feq _ reduce f ψψ and f φφ 792 VIBROENGINEERING JOURNAL OF VIBROENGINEERING DECEMBER 202 VOLUME 4 ISSUE 4 ISSN

7 903 FREE VIBRAION SUDIES OF FUNCIONALLY GRADED MAGNEO-ELECRO-ELASIC PLAES/SHELLS BY USING SOLID-SHELL ELEMENS ZHENG SHIJIE FEN YAN WANG HONGAO are system frequencies which are computed by using K K K _ K and uu eq eq reduced eq _ φφ K eq _ ψψ 4 Simply supported FGM plate Frequency analysis of simply supported FG magneto-electro-elastic plate for different power-law index as n = and 0000 has been carried out he dimensions of the FGM plate are L = L = m h = 03 m x y ables -4 show the results of frequency behavior of FGM plate by using present element and Bhangale s element [8] Here power-law index is n = 0 representing an isotropic plate with properties corresponding to that of homogeneous BaiO 3 (piezoelectric) plate and n = 0000 corresponds to isotropic plate made up of magneto-electro-elastic material (CoFe 2 O 4 ) Comparing with the results given by Bhangale a good correlation is observed when applying the presented finite element able Normalized natural frequency for power-law index n = 0 of FGM plate No f uu f eq f eq_reduced f ψψ f φφ 879(980) 900(209) 900(209) 879(980) 900(209) (23003) 23020(23003) 23020(23003) 23020(23003) 23020(23003) (26503) 25225(2804) 25225(2804) 24824(26503) 25225(2804) (2804) 27077(2853) 27077(2853) 26330(2804) 27077(2853) (37772) 36373(39397) 36373(39397) 3578(37772) 36373(39397) able 2 Normalized natural frequency for power-law index n = of FGM plate No f uu f eq f eq_reduced f ψψ f φφ 7383(8946) 7566(9897) 7566(9897) 7369(890) 7577(9938) (23394) 2077(23394) 2077(23394) 2074(23394) 2077(23394) (26934) 2268(27800) 2267(27800) 22465(26925) 22640(27807) (2870) 24659(2870) 24658(2870) 2430(2870) 24673(2870) (38237) 34684(38905) 34679(38905) 33824(3823) 34699(38905) able 3 Normalized natural frequency for power-law index n = 5 of FGM plate No f uu f eq f eq_reduced f ψψ f φφ 6545(73467) 663(74004) 663(74004) 6532(72695) 6626(74769) (220267) 20093(220267) 20093(220267) 20092(220267) 20093(220267) (253548) 2533(254305) 2533(254305) 2479(253399) 2549(254447) (260829) 233(260829) 233(260829) 23002(260829) 2336(260829) 5 325(357426) 32352(357426) 3235(357426) 3200(357426) 3240(357426) able 4 Normalized natural frequency for power-law index n = 000 of FGM plate No f uu f eq f eq_reduced f ψψ f φφ 5032(5477) 4989(5403) 4989(5403) 4989(5403) 5032(5477) (9747) 9763(9747) 9763(9747) 9763(9747) 9763(9747) 3 272(22607) 253(22594) 253(22594) 253(22594) 272(22607) (23372) 22436(23372) 22436(23372) 22436(23372) 2246(23372) (3866) 28996(3866) 28996(3866) 28996(3866) 29033(3866) Note: Data format: results of present element / Bhangale s element [8] VIBROENGINEERING JOURNAL OF VIBROENGINEERING DECEMBER 202 VOLUME 4 ISSUE 4 ISSN

8 903 FREE VIBRAION SUDIES OF FUNCIONALLY GRADED MAGNEO-ELECRO-ELASIC PLAES/SHELLS BY USING SOLID-SHELL ELEMENS ZHENG SHIJIE FEN YAN WANG HONGAO From ables 4 it is determined that as the value of power-law index increases the natural frequency decreases as it is approaching toward the homogeneous magneto-electro-elastic material which corresponds to n = 0000 his is due to the fact that piezoelectric effect has a tendency to increase the stiffness of the plate by induced electric field while magnetostrictive material has a tendency to decrease the stiffness of the system by inducing the magnetic field It is observed from ables and 4 that f eq and f eq _ reduce classes of eigenvalues are the same; this is due to the fact that n = 0 represents the plate fully made up of BaiO 3 only and n = 0000 corresponds to CoFe 2 O 4 (magnetostrictive) where magneto-electro coupling coefficients are zero In contrast for n = 0 and 50 similar behavior is not observed for f and f classes of vibration in ables 2-3 Further it is observed that as power-law index eq _ reduce n increases the influence of the magnetic effect is more pronounced when compared to piezoelectric effect as the material approaches to homogeneous magnetostrictive his is evidenced by looking at ables Simply supported FG cylindrical shells Studies have been carried out for FGM (BaiO 3 CoFe 2 O 4 ) cylindrical shells of radius r and thickness h In present analysis it is assumed that the composition is varied from the inner to the outer surfaces ie the inner surface of the shell is metal piezoelectric-rich whereas the outer surface is magnetostrictive-rich In addition material properties are graded throughout the thickness direction according to the volume fraction power-law distribution Fig provides the first axial mode frequency results associated with first 0 circumferential modes for a simply supported boundary condition obtained for different power law indexes for different r/h ratios he influence of the power law index is mainly to change the magnitude of the first axial mode frequency As the power law index increases the frequencies increase as well his fact can be understood easily as the elastic properties of magnetostrictive are higher compared frequencies associated with 0 circumferential harmonics for FGM cylindrical shells for different r/h ratios and power law index n to piezoelectric counterpart he power law index does not have a great influence to alter the associated circumferential mode number to the lowest of the first axial mode frequency However the change in the r/h ratios of the FGM shell does shift the mode number of the lowest frequency It is observed from Fig as r/h ratios increases the lowest frequency of the shells occurs at higher circumferential mode his behavior is similar to conventional shells It is also detected that the frequency does not vary much with r/h ratios for lower circumferential modes his is also to be expected as the membrane effect is more predominant at the lowest circumferential mode As expected at higher circumferential mode with the increase of r/h ratios the shell frequencies diminish and this is due to the fact that the bending stiffness reduces as r/h ratios increases In addition it is also observed that as the r/h ratio increases the lowest circumferential mode is shifted 5 Conclusions eq In this paper the frequency behavior of FG magneto-electro-elastic structures is studied for plates and shells having different volume fractions of BaiO 3 he analysis is carried out using the proposed finite element formulation he study established the following: (a) he frequency characteristics of magneto-electro-elastic structure are similar to those of homogeneous isotropic structures (b) In general the piezoelectric effect has the tendency of stiffening the plate and hence increases the natural frequency of a structure In contrast pure magnetic effect has an inverse influence on the system frequency and reduces the structural natural frequency marginally 794 VIBROENGINEERING JOURNAL OF VIBROENGINEERING DECEMBER 202 VOLUME 4 ISSUE 4 ISSN

9 903 FREE VIBRAION SUDIES OF FUNCIONALLY GRADED MAGNEO-ELECRO-ELASIC PLAES/SHELLS BY USING SOLID-SHELL ELEMENS ZHENG SHIJIE FEN YAN WANG HONGAO Fig Variations of natural frequencies of the first axial mode (c) For FG magneto-electro-elastic structure the volume fraction of BaiO 3 in BaiO 3 -CoFe 2 O 4 composite has considerable effect on the frequency of the system As power-law index increases the frequency decreases as the constituent of material reaches homogeneous magnetostrictive material for FGM plate model; the frequencies increases as exponential factor increases (d) While in the case of FGM shell model II the influence of the power law index is mainly to change the magnitude of the first axial mode frequency he power law index does not have a significant influence to alter the associated circumferential mode number to the lowest of the first axial mode frequency However the change in the thickness of the FGM shell does shift the mode number of the lowest frequency Acknowledgments his research is supported by a grant from the Chinese Natural Science Foundation (No ) and Jiangsu Colleges and Universities Superior Discipline Construction Project References [] Pan E Exact solution for simply supported and multilayered magneto-electro-elastic plates J Appl Mech Vol 68 Issue p [2] Pan E Heyliger P R Free vibrations of simply supported and multilayered magneto-electro-elastic plates Journal of Sound and Vibration Vol 252 Issue p [3] Wang Yun Xu Rongqiao Ding Haojiang et al hree-dimensional exact solutions for free vibrations of simply supported magneto-electro-elastic cylindrical panels International Journal of Engineering Science Vol 48 Issue p [4] Zhong Xian-Ci Lee Kang Yong Dielectric crack problem for a magnetoelectroelastic strip with functionally graded properties Arch Appl Mech Vol 82 Issue p VIBROENGINEERING JOURNAL OF VIBROENGINEERING DECEMBER 202 VOLUME 4 ISSUE 4 ISSN

10 903 FREE VIBRAION SUDIES OF FUNCIONALLY GRADED MAGNEO-ELECRO-ELASIC PLAES/SHELLS BY USING SOLID-SHELL ELEMENS ZHENG SHIJIE FEN YAN WANG HONGAO [5] Alberto Milazzo An equivalent single-layer model for magnetoelectroelastic multilayered plate dynamics Composite Structures Vol 94 Issue p [6] Milazzo A Orlando C An equivalent single-layer approach for free vibration analysis of smart laminated thick composite plates Smart Mater Struct Vol 2 Issue p (8 pages) [7] Buchanan G R Layered versus multiphase magneto-electro-elastic composites Composites Part B Engineering Vol 35 Issue p [8] Wang Jianguo Qu Lei Qian Feng State vector approach of free-vibration analysis of magnetoelectro-elastic hybrid laminated plates Composite Structures Vol 92 Issue p [9] Gülay Altay M Cengiz Dömeci On the fundamental equations of electromagnetoelastic media in variational form with an application to shell laminae equations International Journal of Solids and Structures Vol 47 Issue p [0] Cao Xiaoshan Shi Junping Jin Feng Lamb wave propagation in the functionally graded piezoelectric-piezomagnetic material plate Acta Mech Vol 223 Issue p [] Wu Chih-Ping sai Yi-Hwa Dynamic responses of functionally graded magneto-electro-elastic shells with closed-circuit surface conditions using the method of multiple scales European Journal of Mechanics A/Solids Vol 29 Issue p 66 8 [2] Pan E Han F Exact solution for functionally graded and layered magneto-electro-elastic plates International Journal of Engineering Science Vol 43 Issue p [3] Huang D J Ding H J Chen W Q Static analysis of anisotropic functionally graded magnetoelectro-elastic beams subjected to arbitrary loading European Journal of Mechanics A/Solids Vol 29 Issue p [4] 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 Vol 288 Issue p [5] Sze K Y Yao L Q Modeling smart structures with segmented piezoelectric sensors and actuators Journal of Sound and Vibration Vol 235 Issue p [6] Zheng S J Finite element simulation of wireless structural vibration control with photostrictive actuators Science China echnological Sciences Vol 55 Issue p [7] Jacob Aboudi Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multiphase composites Smart Mater Struct Vol 0 Issue p [8] Bhangale R K Ganesan N Free vibration of simply supported functionally graded and layered magneto-electro-elastic plates by finite element method Journal of Sound and Vibration Vol 294 Issue p VIBROENGINEERING JOURNAL OF VIBROENGINEERING DECEMBER 202 VOLUME 4 ISSUE 4 ISSN