Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

doi:10.1007/s10483-020-2679-7

Abstract

Abstract: Based on the nonlocal theory and Mindlin plate theory, the governing equations (i.e., a system of partial differential equations (PDEs) for bending problem) of magnetoelectroelastic (MEE) nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle. The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions (MPS) to solve the governing equations numerically. It is confirmed that for the present bending model, the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points. The effects of different boundary conditions, applied loads, and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method. Some important conclusions are drawn, which should be helpful for the design and applications of electromagnetic nanoplate structures.

Key words: magnetoelectroelastic (MEE) nanoplate bending, nonlocal theory, Mindlin plate theory, method of particular solution (MPS), polynomial basis function

2010 MSC Number:

Wenjie FENG, Zhen YAN, Ji LIN, C. Z. ZHANG. Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory. Applied Mathematics and Mechanics (English Edition), 2020, 41(12): 1769-1786.

TrendMD

References

[1] LI, Y. S. Buckling analysis of magnetoelectroelastic plate resting on Pasternak elastic foundation. Mechanics Research Communications, 56, 104-114(2014)
[2] ZHOU, Y. H., WANG, X. Z., and ZHENG X. J. Magnetoelastic bending and stability of soft ferromagnetic rectangular plates. Applied Mathematics and Mechanics (English Edition), 19(7), 669-676(1998) https://doi.org/10.1007/BF02452375
[3] LIU, M. F. An exact deformation analysis for the magneto-electro-elastic fiber-reinforced thin plate. Applied Mathematical Modelling, 35(5), 2443-2461(2011)
[4] WANG, Y., XU, R. Q., and DING, H. J. Axisymmetric bending of functionally graded circular magneto-electro-elastic plates. European Journal of Mechanics A:Solids, 30(6), 999-1011(2011)
[5] YANG, Y. and LI, X. F. Bending and free vibration of a circular magnetoelectroelastic plate with surface effects. International Journal of Mechanical Sciences, 157, 858-871(2019)
[6] AREFI, M. and ZENKOUR, A. M. Size-dependent electro-magneto-elastic bending analyses of the shear-deformable axisymmetric functionally graded circular nanoplates. European Physical Journal Plus, 132(10), 423(2017)
[7] KARIMI, M. and FARAJPOUR, M. R. Bending and buckling analyses of BiTiO₃-CoFe₂O₄ nanoplates based on nonlocal strain gradient and modified couple stress hypotheses:rate of surface layers variations. Applied Physics A-Materials Science & Processing, 125(8), 530(2019)
[8] ERINGEN, A. C. and EDELEN, D. G. B. On nonlocal elasticity. International Journal of Engineering Science, 10(3), 233-248(1972)
[9] ERINGEN, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703-4710(1983)
[10] LI, Y. S., CAI, Z. Y., and SHI, S. Y. Buckling and free vibration of magnetoelectroelastic nanoplate based on nonlocal theory. Composite Structures, 111, 522-529(2014)
[11] PAN, E. and WAKSMANSKI, N. Deformation of a layered magnetoelectroelastic simply-supported plate with nonlocal effect, an analytical three-dimensional solution. Smart Materials and Structures, 25(9), 095013(2016)
[12] GUO, J. H., SUN, T. Y., and PAN, E. Three-dimensional buckling of embedded multilayered magnetoelectroelastic nanoplates/graphene sheets with nonlocal effect. Journal of Intelligent Material Systems and Structures, 30(18-19), 2870-2893(2019)
[13] LI, Y. S., MA, P., and WANG, W. Bending, buckling, and free vibration of magnetoelectroelastic nanobeam based on nonlocal theory. Journal of Intelligent Material Systems and Structures, 27(9), 1139-1149(2016)
[14] VANMAELE, C., VANDEPITTE, D., and DESMET, W. An efficient wave based prediction technique for plate bending vibrations. Computer Methods in Applied Mechanics and Engineering, 196(33-34), 3178-3189(2007)
[15] VANMAELE, C., VANDEPITTE, D., and DESMET, W. An efficient wave based prediction technique for dynamic plate bending problems with corner stress singularities. Computer Methods in Applied Mechanics and Engineering, 198(30-32), 2227-2245(2009)
[16] YAN, F., FENG, X. T., and ZHOU, H. Dual reciprocity hybrid radial boundary node method for the analysis of kirchhoff plates. Applied Mathematical Modelling, 35(12), 5691-5706(2011)
[17] TAN, F. and ZHANG, Y. The regular hybrid boundary node method in the bending analysis of thin plate structures subjected to a concentrated load. European Journal of Mechanics A:Solids, 38, 79-89(2013)
[18] MANOLIS, G. D., RANGELOV, T. V., and SHAW, R. P. The non-homogeneous biharmonic plate equation:fundamental solutions. International Journal of Solids and Structures, 40(21), 5753-5767(2003)
[19] HEYLIGER, P. R. and KIENHOLZ, J. The mechanics of pyramids. International Journal of Solids and Structures, 43, 2693-2709(2006)
[20] LIN, J., CHEN, C. S., WANG, F., and DANGAL, T. Method of particular solutions using polynomial basis functions for the simulation of plate bending vibration problems. Applied Mathematical Modelling, 49, 452-469(2017)
[21] MULESHKOV, A. S., GOLBERG, M. A., and CHEN, C. S. Particular solutions of Helmholtz-type operators using higher order polyharmonic splines. Computational Mechanics, 23, 411-419(1999)
[22] CHENG, H. D. Particular solutions of Laplacian, Helmholtz-type, and polyharmonic operators involving higher order radial basis functions. Engineering Analysis with Boundary Elements, 24(7-8), 531-538(2000)
[23] MULESHKOV, A. S. and GOLBERG, M. A. Particular solutions of the multi-Helmholtz-type equation, Engineering Analysis with Boundary Elements. 31(7), 624-630(2007)
[24] CHEN, C. S., FAN, C. M., and WEN, P. H. The method of particular solutions for solving certain partial differential equations. Numerical Methods for Partial Differential Equations, 28(2), 506-522(2012)
[25] DANGAL, T., CHEN, C. S., and LIN, J. Polynomial particular solutions for solving elliptic partial differential equations. Computers & Mathematics with Applications, 73(1), 60-70(2017)
[26] CHEN, C. S., LEE, S., and HUANG, C. S. Derivation of particular solutions using Chebyshev polynomial based functions. International Journal of Computational Methods, 4(1), 15-32(2007)
[27] GHIMIRE, B. K., TIAN, H. Y., and LAMICHHANE, A. R. Numerical solutions of elliptic partial differential equations using Chebyshev polynomials. Computers & Mathematics with Applications, 72(4), 1042-1054(2016)
[28] LIU, C. S. A highly accurate MCTM for inverse Cauchy problems of Laplace equation in arbitrary plane domains. Computer Modeling in Engineering & Sciences, 35(2), 91-111(2008)
[29] LIU, C. S. A multiple-scale Trefftz method for an incomplete Cauchy problem of biharmonic equation. Engineering Analysis with Boundary Elements, 37(11), 1445-1456(2013)
[30] LI, X. L. and DONG, H. Y. Error analysis of the meshless finite point method. Applied Mathematics and Computation, 382, 125326(2020)
[31] QU, W. Z., FAN, C. M., and LI, X. L. Analysis of an augmented moving least squares approximation and the associated localized method of fundamental solutions. Computers & Mathematics with Applications, 80, 13-30(2020)
[32] PARK, S. and SUN, C. T. Fracture criteria for piezoelectric ceramics. Journal of the American Ceramic Society, 78(6), 1475-1480(1995)
[33] FANG, D. N., ZHANG, Z. K., SOH, A. K., and LEE, K. L. Fracture criteria of piezoelectric ceramics with defects. Mechanics of Materials, 36, 917-928(2004)
[34] HASHEMI, S. H. and ARSANJANI, M. Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates. International Journal of Solids and Structures, 42(3), 819-853(2005)
[35] REDDY, J. N. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45(2-8), 288-307(2007)
[36] ESMAEILZADEH, M., KADKHODAYAN, M., MOHAMMADI, S., and TURVEY, G. J. Nonlinear dynamic analysis of moving bilayer plates resting on elastic foundations. Applied Mathematics and Mechanics (English Edition), 41(3), 439-458(2020) https://doi.org/10.1007/s10483-020-2587-8
[37] WANG, B. L. and HAN, J. C. Multiple cracking of magnetoelectroelastic materials in coupling thermo-electro-magneto-mechanical loading environments. Computational Materials Science, 39(2), 291-304(2007)
[38] LI, X., ZHU, J., and ZHANG, S. A meshless method based on boundary integral equations and radial basis functions for biharmonic-type problems. Applied Mathematical Modelling, 35(2), 737-751(2011)
[39] GUO, J. H., SUN, T. Y., and PAN, E. Three-dimensional nonlocal buckling of composite nanoplates with coated one-dimensional quasicrystal in an elastic medium. International Journal of Solids and Structures, 185, 272-280(2020)