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Journal articles on the topic 'Plates theory'

Author: Grafiati

Published: 5 June 2025

Last updated: 24 June 2025

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1

Zheng, Yu Fang, Tao Chen, Feng Wang, and Chang Ping Chen. "Nonlinear Responses of Rectangular Magnetoelectroelastic Plates with Transverse Shear Deformation." Key Engineering Materials 689 (April 2016): 103–7. http://dx.doi.org/10.4028/www.scientific.net/kem.689.103.

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With employing the transverse shear deformation theory and von Karman plate theory, the nonlinear static behavior of a simply supported rectangular magnetoelectroelastic plates is investigated. According to the Maxwell’s equations, when applying the magnetoelectric load on the plate’s surfaces and neglecting the in-plane electric and magnetic fields in thin plates, the electric and magnetic potentials varying along the thickness direction of the magnetoelectroelastic plates are determined. The nonlinear differential equations for magnetoelectroelastic plates are established based on the Hamilton’s principle. The Galerkin procedure furnishes an infinite system of differential equations into algebraic equations. In the numerical calculations, the effects of the nonlinearity and span-thickness ratio on the nonlinear load-deflection curves and electric/magnetic potentials for magnetoelectroelastic plates are discussed.

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2

Zheng-hua, Zhong, and Luo Jian-hui. "Theory and refined theory of elasticity for transversely isotropic plates and a new theory for tnick plates." Applied Mathematics and Mechanics 9, no. 4 (1988): 375–89. http://dx.doi.org/10.1007/bf02456118.

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3

Ugrimov, S. V. "Generalized theory of multilayer plates." International Journal of Solids and Structures 39, no. 4 (2002): 819–39. http://dx.doi.org/10.1016/s0020-7683(01)00253-0.

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4

Eringen, A. Cemal. "Theory of electromagnetic elastic plates." International Journal of Engineering Science 27, no. 4 (1989): 363–75. http://dx.doi.org/10.1016/0020-7225(89)90128-6.

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5

Ugrimov, S. V., and A. N. Shupikov. "Layered orthotropic plates. Generalized theory." Composite Structures 129 (October 2015): 224–35. http://dx.doi.org/10.1016/j.compstruct.2015.04.004.

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6

Lim, Teik Cheng. "Auxetic Plates on Auxetic Foundation." Advanced Materials Research 974 (June 2014): 398–401. http://dx.doi.org/10.4028/www.scientific.net/amr.974.398.

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Auxetic solids are materials that exhibit negative Poisson’s ratio. This paper evaluates the maximum stresses in point-loaded (a) auxetic plates on conventional elastic foundation, (b) conventional plates on auxetic elastic foundation, and (c) auxetic plates on auxetic elastic foundation vis-à-vis conventional plates on conventional elastic foundation. Using thick plate theory for infinite plates on elastic foundation, it was found that in most cases the auxetic plates and auxetic foundation play the primary and secondary roles, respectively, in reducing the plate’s maximum stresses. It is herein suggested that, in addition to materials selection technique and other design considerations, the use of auxetic plates and/or auxetic foundation be introduced for reducing stresses in plates on elastic foundations.

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7

Trien, Trang Tan, Le Thanh Phong, and Pham Tan Hung. "Isogeometric free vibration of the porous metal foam plates resting on an elastic foundation using a quasi-3D refined theory." Journal of Science and Technology in Civil Engineering (JSTCE) - HUCE 19, no. 1 (2025): 119–30. https://doi.org/10.31814/stce.huce2025-19(1)-10.

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This study investigates the free vibration behavior of porous metal foam plates using the Quasi-3D refined plate theory. We consider three types of pores across the plate thickness: uniform, symmetric, and asymmetric distributions. Besides, the metal foam plate is reinforced by a Winkler-Pasternak foundation. By employing the variational principle and Quasi-3D refined theory, we derive the weak form for free vibration analysis. The Quasi-3D theory is essential for analyzing plates, as it accurately captures transverse shear and normal deformations, which are vital for understanding the behavior of thick and moderately thick plates. Unlike simpler models, it provides a detailed representation of stress and strain distributions across the plate’s thickness, enabling precise modeling of complex structural behaviors. The natural frequency of the porous metal foam plates is determined by solving the explicit governing equation using the isogeometric approach. Additionally, we examine how the porous coefficient, porous distribution, and geometry impact the vibrational frequency of the porous metal foam plate.

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8

Kasimov, A. T. "INVESTIGATION OF LAYERED ORTHOTROPIC STRUCTURES BASED ON ONE MODIFIED REFINED BENDING THEORY." Eurasian Physical Technical Journal 18, no. 4 (38) (2021): 37–44. http://dx.doi.org/10.31489/2021no4/37-44.

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In the article, constructions made of orthotropic multilayer composite material, in particular, layered orthotropic plates are considered. Numerical modeling and analysis of the stress-strain state for the plates are carried out on the basis of one version for the refined theory of layered plates. The bending problems for plates of medium thickness and thin multilayer plates of symmetric and asymmetric structures are investigated. All studies are conducted taking into account the properties of orthotropy and multilayering for the composite material from which the plates are made. The general algorithm for the numerical calculation of the stress-strain state for layered plates with orthotropic layers is developed on the basis of the finite difference method. This algorithm is implemented on a PC by a software package.

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9

Javed, Saira. "A Numerical Solution of Symmetric Angle Ply Plates Using Higher-Order Shear Deformation Theory." Symmetry 15, no. 3 (2023): 767. http://dx.doi.org/10.3390/sym15030767.

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This research aims to provide the numerical analysis solution of symmetric angle ply plates using higher-order shear deformation theory (HSDT). The vibration of symmetric angle ply composite plates is analyzed using differential equations consisting of supplanting and turning functions. These supplanting and turning functions are numerically approximated through spline approximation. The obtained global eigenvalue problem is solved numerically to find the eigenfrequency parameter and a related eigenvector of spline coefficients. The plates of different constituent components are used to study the parametric effects of the plate’s aspect ratio, side-to-thickness ratio, assembling sequence, number of composite layers, and alignment of each layer on the frequency of the plate. The obtained results are validated by existing literature.

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10

Voyiadjis, George Z., and Mohammed H. Baluch. "Refined Theory for Thick Composite Plates." Journal of Engineering Mechanics 114, no. 4 (1988): 671–87. http://dx.doi.org/10.1061/(asce)0733-9399(1988)114:4(671).

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11

Düring, Gustavo, Christophe Josserand, and Sergio Rica. "Wave turbulence theory of elastic plates." Physica D: Nonlinear Phenomena 347 (May 2017): 42–73. http://dx.doi.org/10.1016/j.physd.2017.01.002.

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12

Volokh, K. Yu. "On the classical theory of plates." Journal of Applied Mathematics and Mechanics 58, no. 6 (1994): 1101–10. http://dx.doi.org/10.1016/0021-8928(94)90129-5.

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13

Barrett, K. E., and S. Ellis. "An exact theory of elastic plates." International Journal of Solids and Structures 24, no. 9 (1988): 859–80. http://dx.doi.org/10.1016/0020-7683(88)90038-8.

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14

Berdichevsky, Victor L. "An asymptotic theory of sandwich plates." International Journal of Engineering Science 48, no. 3 (2010): 383–404. http://dx.doi.org/10.1016/j.ijengsci.2009.09.001.

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15

Busse, Anke, and Martin Schanz. "A consistent theory for poroelastic plates." PAMM 5, no. 1 (2005): 381–82. http://dx.doi.org/10.1002/pamm.200510167.

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16

Erbay, H. A., and E. S. Şuhubi. "An asymptotic theory of thin hyperelastic plates—I. General theory." International Journal of Engineering Science 29, no. 4 (1991): 447–66. http://dx.doi.org/10.1016/0020-7225(91)90087-j.

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17

Soltani, HM, and M. Kharazi. "Investigation of the incremental and deformation theories of plasticity on the elastoplastic postbuckling of plates." Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications 234, no. 7 (2020): 1017–31. http://dx.doi.org/10.1177/1464420720922866.

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This article investigates the elastoplastic response of buckling and postbuckling behavior of plates under uniaxial and biaxial end-shortening considering incremental theory and deformation theory of plasticity. According to elastoplastic buckling and postbuckling behavior of plates, the finite element code considering geometrically and material nonlinearities is developed based on incremental theory and deformation theory of plasticity. The results show that boundary conditions, loading ratios, and aspect ratios of a plate have a significant effect on the discrepancy between incremental theory and deformation theory. Moreover, differences in estimating the buckling point using incremental theory and deformation theory are less than 10%, while in a number of plates at the last loading steps, postbuckling paths determined by incremental theory and deformation theory are diverted from each other. Also the difference between these two theories in the postbuckling region is more noticeable by increasing the thickness of plates.

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18

Özdemir, Y. I., and Y. Ayvaz. "Parametric Earthquake Analysis of Thick Plates Using Mindlin’s Theory." Shock and Vibration 17, no. 6 (2010): 771–85. http://dx.doi.org/10.1155/2010/687549.

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The purpose of this paper is to study parametric earthquake analysis of thick plates using Mindlin's theory, to determine the effects of the thickness/span ratio, the aspect ratio and the boundary conditions on the linear responses of thick plates subjected to earthquake excitations and to present the frequency parameters and the mode shapes of the same plates. In the analysis, finite element method is used for spatial integration and the Newmark-βmethod is used for time integration. A computer program using finite element method is coded in C++ to analyze the plates clamped or simply supported along all four edges. In the analysis, 8-noded finite element is used. Graphs and tables are presented that should help engineers in the design of thick plates subjected to earthquake excitations. It is concluded that, in general, the changes in the thickness/span ratio are more effective on the maximum responses considered in this study than the changes in the aspect ratio. It is also concluded that the effects of the change in the thickness/span ratio on the frequency parameter of the thick plates are always larger than those of the change in the aspect ratio.

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19

Nguyen, Hoang Nam, Tran Thi Hong, Pham Van Vinh, Nguyen Dinh Quang, and Do Van Thom. "A Refined Simple First-Order Shear Deformation Theory for Static Bending and Free Vibration Analysis of Advanced Composite Plates." Materials 12, no. 15 (2019): 2385. http://dx.doi.org/10.3390/ma12152385.

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A refined simple first-order shear deformation theory is developed to investigate the static bending and free vibration of advanced composite plates such as functionally graded plates. By introducing the new distribution shape function, the transverse shear strain and shear stress have a parabolic distribution across the thickness of the plates, and they equal zero at the surfaces of the plates. Hence, the new refined theory needs no shear correction factor. The Navier solution is applied to investigate the static bending and free vibration of simply supported advanced composite plates. The proposed theory shows an improvement in calculating the deflections and frequencies of advanced composite plates. The formulation and transformation of the present theory are as simple as the simple first-order shear deformation. The comparisons of deflection, axial stresses, transverse shear stresses, and frequencies of the plates obtained by the proposed theory with published results of different theories are carried out to show the efficiency and accuracy of the new theory. In addition, some discussions on the influence of various parameters such as the power-law index, the slenderness ratio, and the aspect ratio are carried out, which are useful for the design and testing of advanced composite structures.

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20

Hayrapetyan, G. S., and S. H. Sargsyan. "Theory of micropolar orthotropic elastic thin plates." Mechanics - Proceedings of National Academy of Sciences of Armenia 65, no. 3 (2012): 22–33. http://dx.doi.org/10.33018/65.3.3.

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21

Vijayakumar, K. "New Look at Kirchoff's Theory of Plates." AIAA Journal 47, no. 4 (2009): 1045–46. http://dx.doi.org/10.2514/1.38471.

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22

Spencer, A. J. M., P. Watson, and T. G. Rogers. "Exact Theory of Heterogeneous Anisotropic Elastic Plates." Materials Science Forum 123-125 (January 1993): 235–44. http://dx.doi.org/10.4028/www.scientific.net/msf.123-125.235.

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23

Bîrsan, Mircea. "A BENDING THEORY OF POROUS THERMOELASTIC PLATES." Journal of Thermal Stresses 26, no. 1 (2003): 67–90. http://dx.doi.org/10.1080/713855760.

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24

Dimitrienko, Yu I., and I. D. Dimitrienko. "Asymptotic theory for vibrations of composite plates." Applied Mathematical Sciences 10 (2016): 2993–3002. http://dx.doi.org/10.12988/ams.2016.68231.

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25

Bogomolny, E., and E. Hugues. "Semiclassical theory of flexural vibrations of plates." Physical Review E 57, no. 5 (1998): 5404–24. http://dx.doi.org/10.1103/physreve.57.5404.

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26

Shimpi, Rameshchandra P. "Zeroth-Order Shear Deformation Theory for Plates." AIAA Journal 37, no. 4 (1999): 524–26. http://dx.doi.org/10.2514/2.750.

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27

Paroni, Roberto. "Theory of Linearly Elastic Residually Stressed Plates." Mathematics and Mechanics of Solids 11, no. 2 (2005): 137–59. http://dx.doi.org/10.1177/1081286504036221.

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28

Ciarlet, Philippe G., and Cristinel Mardare. "The intrinsic theory of linearly elastic plates." Mathematics and Mechanics of Solids 24, no. 4 (2018): 1182–203. http://dx.doi.org/10.1177/1081286518776047.

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In an intrinsic approach to a problem in elasticity, the only unknown is a tensor field representing an appropriate ‘measure of strain’, instead of the displacement vector field in the classical approach. The objective of this paper is to study the displacement traction problem in the special case where the elastic body is a linearly elastic plate of constant thickness, clamped over a portion of its lateral face. In this respect, we first explicitly compute the intrinsic three-dimensional boundary condition of place in terms of the Cartesian components of the linearized strain tensor field, thus avoiding the recourse to covariant components in curvilinear coordinates and providing an interesting example of actual computation of an intrinsic boundary condition of place in three-dimensional elasticity. Second, we perform a rigorous asymptotic analysis of the three-dimensional equations as the thickness of the plate, considered as a parameter, approaches zero. As a result, we identify the intrinsic two-dimensional equations of a linearly elastic plate modelled by the Kirchhoff–Love theory, with the linearized change of metric and change of curvature tensor fields of the middle surface of the plate as the new unknowns, instead of the displacement field of the middle surface in the classical approach.

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29

Karama, M., K. S. Afaq, and S. Mistou. "A new theory for laminated composite plates." Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications 223, no. 2 (2009): 53–62. http://dx.doi.org/10.1243/14644207jmda189.

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30

Haddad, Madjid, Yves Gourinat, and Miguel Charlotte. "Equivalence Theory Applied to Anisotropic Thin Plates." Engineering 03, no. 07 (2011): 669–79. http://dx.doi.org/10.4236/eng.2011.37080.

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31

Baluch, M. H., and A. K. Muhammad. "GENERALIZED THEORY FOR BENDING OF THICK PLATES." Transactions of the Canadian Society for Mechanical Engineering 14, no. 4 (1990): 113–23. http://dx.doi.org/10.1139/tcsme-1990-0015.

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32

Reissner, Eric. "Reflections on the Theory of Elastic Plates." Applied Mechanics Reviews 38, no. 11 (1985): 1453–64. http://dx.doi.org/10.1115/1.3143699.

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We depart from a three-dimensional statement of the problem of small bending of elastic plates, for a survey of approximate two-dimensional theories, beginning with Kirchhoff’s fourth-order formulation. After discussing various variational statements of the three-dimensional problem, we describe the development of two-dimensional sixth-order theories by Bolle´, Hencky, Mindlin, and Reissner which take account of the effect of transverse shear deformation. Additionally, we report on an early analysis by Le´vy, on a direct two-dimensional formulation of sixth-order theory, on constitutive coupling of bending and stretching of laminated plates, on higher than sixth-order theories, and on an asymptotic analysis of sixth-order theory which leads to a fourth-order interior solution contribution with first-order transverse shear deformation effects included, as well as to a sequentially determined second-order edge zone solution contribution.

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33

Hodges, Dewey H., Ali R. Atilgan, and D. A. Danielson. "A Geometrically Nonlinear Theory of Elastic Plates." Journal of Applied Mechanics 60, no. 1 (1993): 109–16. http://dx.doi.org/10.1115/1.2900732.

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A set of kinematical and intrinsic equilibrium equations are derived for plates undergoing large deflection and rotation but with small strain. The large rotation is handled by means of the general finite rotation of a frame in which the material points that are originally along a normal line in the undeformedplate undergo only small displacements. The unit vector fixed in this frame, which coincides with the normal when the plate is undeformed, is not in general normal to the deformed plate average surface because of transverse shear. The arbitrarily large displacement and rotation of this frame, which vary over the surface of the plate, are termed global deformation; the small relative displacement is termed warping. It is shown that rotation of the frame about the normal is not zero and that it can be expressed in terms of other global deformation variables. Exact intrinsic virtual strain-displacement relations are derived; based on a reduced two-dimensional strain energy function from which the warping has been systematically eliminated, a set of intrinsic equilibrium equations follow. It is shown that only five equilibrium equations can be derived in this manner, because the component of virtual rotation about the normal is not independent. These equilibrium equations contain terms which cannot be obtained without the use of a finite rotation vector which contains three nonzero components. These extra terms correspond to the difference of in-plane shear stress resultants in other theories; this difference is a reactive quantity in the present theory.

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34

Voyiadjis, George Z., and Shahram Sarkani. "Engineering Large Deflection Theory for Thick Plates." Journal of Engineering Mechanics 115, no. 5 (1989): 935–51. http://dx.doi.org/10.1061/(asce)0733-9399(1989)115:5(935).

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35

MAUGIN, G. A., and D. ATTOU. "AN ASYMPTOTIC THEORY OF THIN PIEZOELECTRIC PLATES." Quarterly Journal of Mechanics and Applied Mathematics 43, no. 3 (1990): 347–62. http://dx.doi.org/10.1093/qjmam/43.3.347.

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36

Erbay, H. A. "An asymptotic theory of thin micropolar plates." International Journal of Engineering Science 38, no. 13 (2000): 1497–516. http://dx.doi.org/10.1016/s0020-7225(99)00118-4.

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37

Efrati, E., E. Sharon, and R. Kupferman. "Elastic theory of unconstrained non-Euclidean plates." Journal of the Mechanics and Physics of Solids 57, no. 4 (2009): 762–75. http://dx.doi.org/10.1016/j.jmps.2008.12.004.

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38

De Cicco, S., and D. Ieşan. "A Theory of Chiral Cosserat Elastic Plates." Journal of Elasticity 111, no. 2 (2012): 245–63. http://dx.doi.org/10.1007/s10659-012-9400-7.

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39

Atoyan,, A. A., and S. H. Sargsyan,. "Dynamic Theory of Micropolar Elastic Thin Plates." Journal of the Mechanical Behavior of Materials 18, no. 2 (2007): 81–88. http://dx.doi.org/10.1515/jmbm.2007.18.2.81.

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40

Bisegna, Paolo, and Franco Maceri. "A Consistent Theory of Thin Piezoelectric Plates." Journal of Intelligent Material Systems and Structures 7, no. 4 (1996): 372–89. http://dx.doi.org/10.1177/1045389x9600700402.

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41

Shimpi, Rameshchandra P. "Zeroth-order shear deformation theory for plates." AIAA Journal 37 (January 1999): 524–26. http://dx.doi.org/10.2514/3.14205.

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42

Gao, Yang, and Bao-sheng Zhao. "The Refined Theory of Thermoelastic Rectangular Plates." Journal of Thermal Stresses 30, no. 5 (2007): 505–20. http://dx.doi.org/10.1080/01495730701212773.

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43

Askes, Harm, and Alexandra R. Wallace. "Dynamic Bergan–Wang theory for thick plates." Mathematics and Mechanics of Complex Systems 10, no. 2 (2022): 191–204. http://dx.doi.org/10.2140/memocs.2022.10.191.

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44

Touratier, Maurice. "A refined theory for thick composite plates." Mechanics Research Communications 15, no. 4 (1988): 229–36. http://dx.doi.org/10.1016/0093-6413(88)90016-x.

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45

Altenbach, H., and V. A. Eremeyev. "On the linear theory of micropolar plates." ZAMM 89, no. 4 (2009): 242–56. http://dx.doi.org/10.1002/zamm.200800207.

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46

Deepak, S. A., Rajesh A. Shetty, K. Sudheer Kini, and G. L. Dushyanthkumar. "Buckling analysis of thick plates using a single variable simple plate theory." Journal of Mines, Metals and Fuels 69, no. 12A (2022): 67. http://dx.doi.org/10.18311/jmmf/2021/30097.

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Buckling analysis of thick plates has been carried out herein by using a single variable simple plate theory. Theory used herein is a third order shear deformation plate theory which uses a single displacement function for the complete formulation of plates. Plate formulation is governed by only one governing differential equation. Governing equation of the theory has close resemblance to that of Classical Plate Theory. Thus, plate problems can be solved in the similar lines as in case of classical plate theory. Plate theory used herein does not require a shear correction coefficient. To check the efficacy of the theory buckling analysis of simply supported thick rectangular plates is carried out. Critical buckling loads for simply supported plates are evaluated and the results obtained are compared to other shear deformation plate theories. Buckling load results are found to be in good agreement with other plate theory results.

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47

Dong, S. B., and C. K. Chun. "Shear Constitutive Relations for Laminated Anisotropic Shells and Plates: Part I—Methodology." Journal of Applied Mechanics 59, no. 2 (1992): 372–79. http://dx.doi.org/10.1115/1.2899530.

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Shear constitutive relations of a first-order shear deformation theory for laminated anisotropic shells and plates are formulated following Mindlin’s procedure for homogeneous isotropic plates. Because thickness-shear motions for laminated anisotropic thickness profiles may not be polarized in planes normal to the reference surface, the concept of generalized principal shear planes is needed. These planes are established by least-squares minimization of the out-of-plane motions of infinitely long thickness-shear waves based on an elasticity analysis of the profile. Typical shear rigidities for a variety of laminated composite and sandwich profiles are given. In a companion paper, the efficacy of this form of shear constitutive relations in predicting the response of a class of laminated composite and sandwich cylindrical shells is demonstrated.

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48

Rakočević, Marina, and Nikolay Vatin. "Bending of Laminated Composite Plates." Applied Mechanics and Materials 725-726 (January 2015): 667–73. http://dx.doi.org/10.4028/www.scientific.net/amm.725-726.667.

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In this work, there are presented overviews of theoretical and numerical models for defining the stress-strain state in the cross section of moderately thick and thick composite plates in case of bending. Layered composite plates are constructed by combination of layers of various materials and geometrical characteristics, wherein each of them has got a bearing capacity in previously defined directions. By applying Equivalent Single-layer Laminate Theory (or ESL theory) these layers' problems cannot be solved successfully. That is the reason to apply contemporary theories of plates, in literature known as Layerwise theories. At the end of this work, there are given numeric examples of applying Partial Layerwise Theory.

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49

Krysko, A. V., J. Awrejcewicz, K. S. Bodyagina, and V. A. Krysko. "Mathematical modeling of planar physically nonlinear inhomogeneous plates with rectangular cuts in the three-dimensional formulation." Acta Mechanica 232, no. 12 (2021): 4933–50. http://dx.doi.org/10.1007/s00707-021-03096-0.

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AbstractMathematical models of planar physically nonlinear inhomogeneous plates with rectangular cuts are constructed based on the three-dimensional (3D) theory of elasticity, the Mises plasticity criterion, and Birger’s method of variable parameters. The theory is developed for arbitrary deformation diagrams, boundary conditions, transverse loads, and material inhomogeneities. Additionally, inhomogeneities in the form of holes of any size and shape are considered. The finite element method is employed to solve the problem, and the convergence of this method is examined. Finally, based on numerical experiments, the influence of various inhomogeneities in the plates on their stress–strain states under the action of static mechanical loads is presented and discussed. Results show that these imbalances existing with the plate’s structure lead to increased plastic deformation.

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50

Wankhade, Rajan L., and Kamal M. Bajoria. "Vibration Analysis of Piezolaminated Plates for Sensing and Actuating Applications Under Dynamic Excitation." International Journal of Structural Stability and Dynamics 19, no. 10 (2019): 1950121. http://dx.doi.org/10.1142/s0219455419501219.

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The vibration characteristics of piezolaminated plates under coupled electromechanical loading are investigated using the finite element method. Higher order shear deformation theory is adopted to incorporate the effect of shear in the formulation. In the finite element formulation, an isoperimetric eight-nodded rectangular element is employed with linear through-the-thickness electric potential distribution. In the parametric study, the influences of geometry and boundary conditions on the vibration characteristics of piezolaminated plates are evaluated. Numerical results are presented for the frequencies of simply supported, clamped plates fitted with piezoelectric patches at the top and bottom surfaces of the laminate. For this, a model simulating the effect of the electromechanical loading with control potential is developed. Studies are also performed for piezolaminated plates with various thickness-to-span ratios, along with convergence test. The effect of the provision of piezoelectric layers to improve the vibration response of plates is investigated. This analysis reveals that the electro-mechanical coupling can strengthen the plate’s resistance to vibration.

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