Abstract
Free vibration and buckling analyses of the magneto-electro-elastic functionally graded (MEE FG) microplates in thermal environment are investigated for the first time. The MEE FG microplate is composed of two phases: piezoelectric (barium titanate) and piezomagnetic (cobalt ferrite) materials, which are distributed across the thickness direction based on the power law model. To satisfy Maxwell’s equation in the quasi-static approximation, the electric and magnetic fields are assumed a combination of trigonometric and linear functions across the plate thickness. To capture small effects on microstructures, the modified strain gradient theory (MSGT), including three length scale parameters combined with the generalized higher-order shear deformation theory (HSDT), is presented. The equilibrium equations for free vibration and buckling analyses of MEE FG microplates are derived by using Hamilton’s principle. Through those equations, the natural frequency and critical buckling load of MEE FG microplates are computed by using isogeometric analysis (IGA). Based on the Non-uniform rational B-splines (NURBs) basic functions, which achieve any desired degree of continuity of basis functions, the IGA easily satisfy the MSGT model’s higher-order derivatives. The advantage and accuracy of the proposed model are demonstrated through comparisons between the present results and those provided in the literature. The effect of the electric voltage, magnetic potential, power index, geometrical parameter and length scale parameters on the dimensionless frequencies and critical buckling loads of the MEE FG microplates is fully reported. The article’s results can be considered as benchmark solutions for the vibration and buckling of MEE FG microplates and they are helpful for manufacturing sensors, actuators, stability control, etc.
Similar content being viewed by others
Data Availability Statement
The authors can confirm that all relevant data are included in this article.
References
Abazid, M.A.: The nonlocal strain gradient theory for hygrothermo-electromagnetic effects on buckling, vibration and wave propagation in piezoelectromagnetic nanoplates. Int. J. Appl. Mech. 11, 1950067 (2019)
Ansari, R., Shojaei, M.F., Mohammadi, V., Gholami, R., Rouhi, H.: Size-dependent thermal buckling and postbuckling of functionally graded annular microplates based on the modified strain gradient theory. J. Therm. Stresses 37, 174–201 (2014)
Arefi, M., Zenkour, A.M.: Size-dependent free vibration and dynamic analyses of piezo-electro-magnetic sandwich nanoplates resting on viscoelastic foundation. Physica B 521, 188–197 (2017)
Ashoori, A., Mahmoodi, M.: A nonlinear thick plate formulation based on the modified strain gradient theory. Mech. Adv. Mater. Struct. 25, 813–819 (2018)
Badri, T.M., Al-Kayiem, H.H.: Analytical solution for simply supported and multilayered magneto-thermo-electro-elastic plates. Asian J. Sci. Res. 6, 236–244 (2012)
Chen, W., Lee, K.Y., Ding, H.: On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates. J. Sound Vib. 279, 237–251 (2005)
Chunjiang Guo, Z.Z., Yue, W.,Yawei, W., Guangtong, M., Jiangtao, S., Zhuoyan, Z., Zhiyong, H., Zhijian, J., Yue, Z.: Synergic realization of electrical insulation and mechanical strength in liquid nitrogen for high-temperature superconducting tapes with ultra-thin acrylic resin coating. Superconductor Sci. Technol. 35, 075014 (2022).
Ebrahimi, F., Barati, M.R.: Vibration analysis of embedded biaxially loaded magneto-electrically actuated inhomogeneous nanoscale plates. J. Vib. Control 24, 3587–3607 (2018)
Ebrahimi, F., Jafari, A.: Buckling behavior of smart MEE-FG porous plate with various boundary conditions based on refined theory. Adv. Mater. Res. 5, 279 (2016)
Ebrahimi, F., Farazmandnia, N., Kokaba, M.R., Mahesh, V.: Vibration analysis of porous magneto-electro-elastically actuated carbon nanotube-reinforced composite sandwich plate based on a refined plate theory. Eng. Comput. 37, 921–936 (2021)
Farzam, A., Hassani, B.: Size-dependent analysis of FG microplates with temperature-dependent material properties using modified strain gradient theory and isogeometric approach. Compos. B Eng. 161, 150–168 (2019)
Garcia-Sanchez, F., Rojas-Diaz, R., Sáez, A., Zhang, C.: Fracture of magnetoelectroelastic composite materials using boundary element method (BEM). Theoret. Appl. Fract. Mech. 47, 192–204 (2007)
Gholami, R., Ansari, R.: A unified nonlocal nonlinear higher-order shear deformable plate model for postbuckling analysis of piezoelectric-piezomagnetic rectangular nanoplates with various edge supports. Compos. Struct. 166, 202–218 (2017)
Gholami, R., Ansari, R., Gholami, Y.: Size-dependent bending, buckling and vibration of higher-order shear deformable magneto-electro-thermo-elastic rectangular nanoplates. Materials Research Express. 4, 065702 (2017)
Gu M, Cai X, Fu Q, Li H, Wang X, Mao B. Numerical analysis of passive piles under surcharge load in extensively deep soft soil. Buildings (2022).
Huang, D., Ding, H., Chen, W.: Analytical solution for functionally graded magneto-electro-elastic plane beams. Int. J. Eng. Sci. 45, 467–485 (2007)
Hughes, T.J., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)
Hung P, Phung-Van P, Thai CH. A refined isogeometric plate analysis of porous metal foam microplates using modified strain gradient theory. Composite Struct., 115467 (2022).
Jamalpoor, A., Hosseini, M.: Biaxial buckling analysis of double-orthotropic microplate-systems including in-plane magnetic field based on strain gradient theory. Compos. B Eng. 75, 53–64 (2015)
Jamalpoor, A., Ahmadi-Savadkoohi, A., Hosseini, M., Hosseini-Hashemi, S.: Free vibration and biaxial buckling analysis of double magneto-electro-elastic nanoplate-systems coupled by a visco- Pasternak medium via nonlocal elasticity theory. Eur. J. Mech. A. Solids 63, 84–98 (2017)
Jamalpoor, A., Bahreman, M., Hosseini, M.: Free transverse vibration analysis of orthotropic multi-viscoelastic microplate system embedded in visco-Pasternak medium via modified strain gradient theory. J. Sandwich Struct. Mater. 21, 175–210 (2019)
Jun Zhang, X.W., Long, Z., Gangxiu, L., Devashibhai, T.A., Ivan da, S., Franz, D., Dmitry, K., Jhuma, S., Hari S.N., Lei, D., Jianfa, Z., Zeng, D., Runze, Y., Xi, S., Richeng, Y., Hui, Z., Jimin, Z., Youwen, L., Zhiwei, H., Hong-Ji, L., Ting-Shan, C., Chien-Te, C., Wei, W., Changqing, J.: A Ferrotoroidic Candidate with Well-Separated Spin Chains. Adv. Mater. 34:2106728 (2022).
Kattimani, S., Ray, M.: Control of geometrically nonlinear vibrations of functionally graded magneto-electro-elastic plates. Int. J. Mech. Sci. 99, 154–167 (2015)
Ke, L.-L., Wang, Y.-S., Yang, J., Kitipornchai, S.: Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory. Acta. Mech. Sin. 30, 516–525 (2014)
Lam, D.C., Yang, F., Chong, A., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)
Lu, C., Zhu, R., Yu, F., Jiang, X., Liu, Z., Dong, L., et al.: Gear rotational speed sensor based on FeCoSiB/Pb(Zr, Ti)O3 magnetoelectric composite. Measurement 168, 108409 (2021)
Malikan, M., Nguyen, V.B.: Buckling analysis of piezo-magnetoelectric nanoplates in hygrothermal environment based on a novel one variable plate theory combining with higher-order nonlocal strain gradient theory. Physica E 102, 8–28 (2018)
Malikan, M., Nguyen, V.B., Tornabene, F.: Electromagnetic forced vibrations of composite nanoplates using nonlocal strain gradient theory. Mater. Res. Express. 5, 075031 (2018)
Mindlin, R.D.: Microstructure in linear elasticity. Columbia Univ, New York (1963).
Mohammadi, H., Mahzoon, M.: Investigating thermal effects in nonlinear buckling analysis of micro beams using modified strain gradient theory. Iran. J. Sci. Technol. Trans. Mech. Eng. 38, 303 (2014)
Moita, J.M.S., Soares, C.M.M., Soares, C.A.M.: Analyses of magneto-electro-elastic plates using a higher order finite element model. Compos. Struct. 91, 421–426 (2009)
Ni, Y., Zhu, S., Sun, J., Tong, Z., Zhou, Z., Xu, X., et al.: An accurate model for free vibration of porous magneto-electro-thermo-elastic functionally graded cylindrical shells subjected to multi-field coupled loadings. J. Intell. Mater. Syst. Struct. 32, 2006–2023 (2021)
Pan, E., Han, F.: Exact solution for functionally graded and layered magneto-electro-elastic plates. Int. J. Eng. Sci. 43, 321–339 (2005)
Quan Zhang, Z.L., Xiaomeng, J., Yan, P., Chuan, Z., Zhongjie, L.: Experimental investigation on performance improvement of cantilever piezoelectric energy harvesters via escapement mechanism from extremely Low-Frequency excitations. Sustain. Energy Technol. Assess. 53, 2213–1388 (2022).
Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51, 745–752 (1984)
Sahmani, S., Bahrami, M., Ansari, R.: Nonlinear free vibration analysis of functionally graded third-order shear deformable microbeams based on the modified strain gradient elasticity theory. Compos. Struct. 110, 219–230 (2014)
Shooshtari, A., Razavi, S.: Large amplitude free vibration of symmetrically laminated magneto-electro-elastic rectangular plates on Pasternak type foundation. Mech. Res. Commun. 69, 103–113 (2015)
Singh, S.S., Nair, D.K., Rajagopal, A., Pal, P., Pandey, A.K.: Dynamic analysis of microbeams based on modified strain gradient theory using differential quadrature method. Euro. J. Comput. Mech. 27, 187–203 (2018)
Sladek, J., Sladek, V., Krahulec, S., Chen, C., Young, D.: Analyses of circular magnetoelectroelastic plates with functionally graded material properties. Mech. Adv. Mater. Struct. 22, 479–489 (2015)
Thai, S., Thai, H.-T., Vo, T.P., Patel, V.I.: Size-dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis. Comput. Struct. 190, 219–241 (2017)
Thai, C.H., Ferreira, A.J.M., Rabczuk, T., Nguyen-Xuan, H.: Size-dependent analysis of FG-CNTRC microplates based on modified strain gradient elasticity theory. Eur. J. Mech. a. Solids 72, 521–538 (2018a)
Thai, C.H., Ferreira, A.J.M., Nguyen-Xuan, H.: Isogeometric analysis of size-dependent isotropic and sandwich functionally graded microplates based on modified strain gradient elasticity theory. Compos Struct. 192, 274–288 (2018b)
Thai, C.H., Ferreira, A., Phung-Van, P.: Size dependent free vibration analysis of multilayer functionally graded GPLRC microplates based on modified strain gradient theory. Compos. B Eng. 169, 174–188 (2019)
Thai, C.H., Ferreira, A., Phung-Van, P.: Free vibration analysis of functionally graded anisotropic microplates using modified strain gradient theory. Eng. Anal. Boundary Elem. 117, 284–298 (2020)
Vinyas, M., Harursampath, D.: Computational evaluation of electro-magnetic circuits’ effect on the coupled response of multifunctional magneto-electro-elastic composites plates exposed to hygrothermal fields. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 235, 2832–2850 (2021)
Wang, Y.Q., Zhao, H.L., Ye, C., Zu, J.W.: A porous microbeam model for bending and vibration analysis based on the sinusoidal beam theory and modified strain gradient theory. Int. J. Appl. Mech. 10, 1850059 (2018)
Wu, C.-P., Lu, Y.-C.: A modified Pagano method for the 3D dynamic responses of functionally graded magneto-electro-elastic plates. Compos. Struct. 90, 363–372 (2009)
Wu, C.-P., Chen, S.-J., Chiu, K.-H.: Three-dimensional static behavior of functionally graded magneto-electro-elastic plates using the modified Pagano method. Mech. Res. Commun. 37, 54–60 (2010)
Yue Wu, Y.Z., Xiaocang, H., Guangyu, J., Jiangtao, S., Pan, L., Mukarram, Z.K., Hannu, H., Jiamin, Z., Zhijian, J., Yutaka, Y.: Ultra-fast growth of cuprate superconducting films: Dual-phase liquid assisted epitaxy and strong flux pinning. Mater. Today Phys. 18:100400 (2021).
Zanoosi, A.A.P.: Size-dependent thermo-mechanical free vibration analysis of functionally graded porous microbeams based on modified strain gradient theory. J. Braz. Soc. Mech. Sci. Eng. 42, 1–18 (2020)
Zhang, H., Wu, Z.: Analysis of electromagnetic properties of new graphene partial discharge sensor electrode plate material. Sensors. 22, 2550 (2022)
Zhang, B., He, Y., Liu, D., Shen, L., Lei, J.: An efficient size-dependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation. Appl. Math. Model. 39, 3814–3845 (2015)
Zhang, H., Ouyang, Z., Li, L., Ma, W., Liu, Y., Chen, F., et al.: Numerical study on welding residual stress distribution of corrugated steel webs. Metals (2022).
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
Conflict of interest
The authors state that there is no potential conflict of interest in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hung, P.T., Phung-Van, P. & Thai, C.H. Small scale thermal analysis of piezoelectric–piezomagnetic FG microplates using modified strain gradient theory. Int J Mech Mater Des 19, 739–761 (2023). https://doi.org/10.1007/s10999-023-09651-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10999-023-09651-y