本文探討彈性基礎上磁電彈複材疊層板之邊界元素法分析，透過虎克定律，能夠將原本的邊界積分式擴展成具有彈性支撐的數學形式，並且透過使用二次四邊形元素對疊層板進行離散化，不僅可以在整個疊層板上進行設置，還能針對特定區域進行設置。除此之外，還能夠設置多方向的彈性係數，從而更全面、準確地分析疊層板的力學特性。並且由於應用了類史磋公式，其優點在於只要針對材料係數的矩陣維度擴張，即能將原用於彈性材料，延伸至壓電材料和磁電彈材料，無須重新推導。
為了驗證邊界元素法，透過使用Navier's solution推導之解析解進行比較，包括疊層板和磁電彈性疊層板。同樣能夠設置多方向的彈性係數，但需要指定特定的邊界條件和疊層纖維排向。
將以上運用虎克定律擴展有關彈性支撐之邊界積分式融入本師門研發之程式AEPH (Anisotropic Elastic Plates_Hwu)進行數值分析，本文範例驗證的部分，使用等向性材料、複材疊層板以及磁電彈複材疊層板，並使用商業軟體ANSYS和各個材料之解析解與邊界元素法進行結果比對，藉此來驗證其正確和準確性。

This article presents the boundary element method analysis of magneto-electro-elastic composite laminated plates on an elastic foundation. By utilizing Hooke's law, the original boundary integral equation is expanded into a mathematical form that incorporates elastic support. Additionally, through the discretization of the laminated plate, the method allows for not only the analysis of the entire plate but also the ability to focus on specific regions of interest. Furthermore, the method enables the specification of elastic coefficients in multiple directions, thus facilitating a more comprehensive and accurate analysis of the mechanical properties of the laminated plate. A comparison with Navier's solution, derived using Hooke's law, is conducted, considering specific boundary conditions and fiber orientations.
The expanded boundary integral equation, incorporating elastic support based on Hooke's law, is implemented in our developed program, AEPH, for numerical analysis. In the example verification section, isotropic materials, composite laminated plates, and magneto-electro-elastic composite laminated plates are analyzed. The results are compared with those obtained from the commercial software ANSYS and analytical solutions for each material, serving as a means to validate the correctness and accuracy of the method.

[1] J. T. Katsikadelis and A. E. Armenakas, “Plates on Elastic Foundation by BIE Method”, J. Eng. Mech., 110 (7), (1984), pp. 1086–1105.
[2] G. Bezine, “A New Boundary Element Method for Bending of Plates on Elastic Foundations”, Int. J. Solids Struct., 24 (6), (1988), pp. 557–565.
[3] A. El-Zafrany, S. Fadhil, and K. Al-Hosani, “A New Fundamental Solution for Boundary Element Analysis of this Plates on Winkler Foundation”, Int. J. Numer. Meth. Eng., 38 (1995), pp. 887–903.
[4] R. Buczkowski and W. Torbacki, “Finite Element Modelling of Thick Plates on Two-Parameter Elastic Foundation”, Int. J. Num. Anal. Meth. Geo., 25(2001), pp. 1409–1427.
[5] S. Chucheepsakul and B. Chinnaboon, “Plates on Two-Parameter Elastic Foundations with Nonlinear Boundary Conditions by the Boundary Element Method”, Computers and Structures, 81,(2003), pp. 2739–2748.
[6] S. Chucheepsakul, B. Chinnaboon, An alternative domain/boundary element technique for analyzing plates on two-parameter elastic foundations, Eng. Anal., 26 (2002), pp. 547-555
[7] S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, second ed., McGraw-Hill, New York, 1959.
[8] A.C. Ugural, Stresses in Plates and Shells, second ed., McGraw-Hill, New York, 1999.
[9] E. Reissner, Y. Stavsky, Bending and stretching of certain types of heterogeneous aeolotropic elastic plates, J. Appl. Mech. 28 (1961) 402–408.
[10] J.M. Whitney, A.W. Leissa, Analysis of heterogeneous anisotropic plates, J. Appl. Mech. 36 (1969) 261–266.
[11] C.W. Bert, T.L.C. Chen, Effect of shear deformation on vibration of antisymmetric angle-ply laminated rectangular plates, Int. J. Solids Struct. 14 (1978) 465–473.
[12] J.N. Reddy, W.C. Chao, A comparison of closed-form and finite-element solutions of thick laminated anisotropic rectangular plates, Nucl. Eng. Des. 64 (1981) 153–167.
[13] M. Levinson, D.W. Cooke, Thick rectangular plates-I: the generalized Navier solution, Int. J. Mech. Sci. 25 (1983) 199–205.
[14] S.E. Kim, H.T. Thai, J. Lee, A two variable refined plate theory for laminated composite plates, Compos. Struct. 89 (2009) 197–205.
[15] N. Grover, D.K. Maiti, B.N. Singh, A new inverse hyperbolic shear deformation theory for static and buckling analysis of laminated composite and sandwich plates, Compos. Struct. 95 (2013) 667–675.
[16] K.Y. Lam, T.Y. Ng, Active control of composite plates with integrated piezoelectric sensors and actuators under various dynamic loading conditions, Smart Mater. Struct. 8 (1999) 223–237.
[17] J.N. Reddy, On laminated composite plates with integrated sensors and actuators, Eng. Struct. 21 (1999) 568–593.
[18] Hwu, C. and Hsu, C.-W., “Classical solutions for coupling analysis of unsymmetric magneto-electro-elastic composite laminated thin plates,” Thin-Walled Structures, 2022, 181, 110112.
[19] Akavci SS, Yerli HR. Dogan A. “The first order shear deformation theory for symmetrically laminated composite plates on elastic foundation”. The Arabian Journal for Science and Engineering, 32(2B), 341-8, 2007.
[20] Hwu, C., “Boundary Integral Equations for General Laminated Plates with Coupled Stretching-Bending Deformation,” Proceedings of the Royal Society, Series A, Vol.466, pp.1027-1054, 2010.
[21] Hwu, C., “Boundary Element Formulation for the Coupled Stretching-Bending Analysis of Thin Laminated Plates,” Engineering Analysis with Boundary Elements, Vol. 36, pp. 1027-1039, 2012
[22] Hwu, C., “Green's Function for the Composite Laminates with Bending Extension Coupling,” Composite Structures, Vol. 63, pp. 283-292, 2004
[23] Hwu, C., Anisotropic Elastic Plates, Springer, New York, 2010.
[24] Hwu, C., “Stroh-Like Formalism for the Coupled Stretching-Bending Analysis of Composite Laminates,” International Journal of Solids and Structures, Vol. 40, No.13- 14, pp 3681–3705, 2003.
[25] Eshelby, J.D., Read, W.T. and Shockley, W., “Anisotropic Elasticity with Applications to Dislocation Theory,” Acta Metallurgica, Vol. 1, pp. 251–259, 1953.
[26] Stroh, A.N., “Dislocations and Cracks in Anisotropic Elasticity,” Philosophical Magazine, Vol. 7, pp. 625–646, 1958.
[27] Stroh, A.N., “Steady State Problems in Anisotropic Elasticity,” Journal of Mathematical Physics, Vol. 41, pp. 77–103, 1962.
[28] Ting, T.C.T., Anisotropic Elasticity: Theory and Applications, Oxford Science Publications, New York, 1996.
[29] Hwu, C. and Hsieh, M.C., “Extended Stroh-Like Formalism for the Electro-Elastic Composite Laminates and Its Applications to Hole Problems,” Smart Materials and Structures, Vol. 14, pp. 56-68, 2005.
[30] Jones, R. M., Mechanics of Composite Materials, Scripta, Washington, DC., 1974.
[31] Hwu, C. and Chang, H. W., “Coupled Stretching–Bending Analysis of Laminated Plates with Corners Via Boundary Elements,” Composite Structures, Vol. 120, pp. 300- 314, 2015.
[32] S.R. Soni, Elastic Properties of T300/5208 Bidirectional Symmetric Laminates, Technical Report AFWAL-TR-80-4111, Materials Laboratory, Air Force Wright Aeronautical Laboratories, Air Force Systems Command, Wright-Patterson Air Force Base, Ohio, 1980.
[33] M.L. Dunn, M. Taya, An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities, Proc. R. Soc. A 443 (1993) 265–287.
[34] E. Pan, W. Chen, Static Green’s Functions in Anisotropic Media, Cambridge University Press, Cambridge, 2015.
[35] Hsu CW, Hwu C, Holes/cracks/inclusions in magneto-electro-elastic composite laminates under coupled stretching-bending deformation, submitted for publication.
[36] E. Pan, Exact solution for simply supported and multilayered magneto-electroelastic plates, J. Appl. Mech. 68 (2001) 608–618
[37] Nan CW. Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases. Phys Rev B 1994;50:6082–8.
[38] Benveniste, Y. Magnetoelectric effect in fibrous composites with piezoelectric and piezomagnetic phases 1995 Physical Review B, 51 (22), pp. 16424-16427.
[39] Srinivasan G, Rasmussen ET, Levin BJ, Hayes R. Magnetoelectric effects in bilayers and multilayers of magnetostrictive and piezoelectric perovskite oxides. Phys Rev B 2002;65:134402.
[40] Soh AK, Liu JX. On the constitutive equations of magnetoelectroelastic solids. J Intell Mater Syst Struct 2005;16:597–602.
[41] Hwu C, Hsu CL, Hsu CW, Shiah YC. Fundamental solutions for two dimensional anisotropic thermo-magneto-electro-elasticity. Math Mech Solids 2019;24: 3575–96
[42] Simões Moita JM, Mota Soares CM, Mota Soares CA. Analyses of magneto-electroelastic plates using a higher order finite element model. Compos Struct 2009;91: 421–6.
[43] Alaimo A, Benedetti I, Milazzo A. A finite element formulation for large deflection of multilayered magneto-electro-elastic plates. Compos Struct 2014;107:643–53.
[44] Vinyas M, Kattimani SC. Finite element evaluation of free vibration characteristics of magneto-electro-elastic rectangular plates in hygrothermal environment using higher-order shear deformation theory. Compos Struct 2018;202:1339–52.
[45] Dutta, G., Panda, S.K., Mahapatra, T.R. et al. Electro-Magneto-Elastic Response of Laminated Composite Plate: A Finite Element Approach. Int. J. Appl. Comput. Math 3, 2573–2592 (2017).
[46] Shen, H.-S., Wang, Z.-X. Nonlinear vibration of hybrid laminated plates resting on elastic foundations in thermal environments 2012 Applied Mathematical Modelling, 36 (12), pp. 6275-6290.
[47] Huang, X.-L., Zheng, J.-J. Nonlinear vibration and dynamic response of simply supported shear deformable laminated plates on elastic foundations (2003) Engineering Structures, 25 (8), pp. 1107-1119
[48] Shooshtari, A., Razavi, S. Large amplitude free vibration of symmetrically laminated magneto-electro-elastic rectangular plates on Pasternak type foundation (2015) Mechanics Research Communications, 69, art. no. 2980, pp. 103-113.
[49] Ye, W., Liu, J., Fang, H., Lin, G. Numerical solutions for magneto– electro–elastic laminated plates resting on Winkler foundation or elastic half-space (2020) Computers and Mathematics with Applications, 79 (8), pp. 2388-2410.