Nonlinear post-buckling modeling of a magneto-electro-elastic cylindrical shell with flexomagnetic and flexoelectric effects

doi:10.1007/s10483-025-3317-8

Abstract

Abstract:

When the size of a magneto-electro-elastic cylindrical shell (MEECS) is reduced to micro-/nano-scale, the size-dependent flexomagnetic effect (FME) and flexoelectric effect (FEE) significantly influence their multi-physical coupling behaviors. To investigate these effects on the post-buckling behaviors of an MEECS, a nonlinear post-buckling model is developed based on the higher-order shear deformation theory (HSDT) and magneto-electro-elastic (MEE) constitutive relations with the FME and FEE. The equilibrium path and the corresponding shell deformations are obtained with a set of newly developed generalized displacement functions within the framework of the Galerkin approach. These displacement functions are established based on the trigonometric series expansions, which accurately satisfy the clamped boundary conditions (BCs). The effects of geometry, flexomagnetic/flexoelectric coefficients, and external electromagnetic fields on the post-buckling behaviors of an MEECS with the FME and FEE are analyzed. Numerical results indicate that the FME decreases the upper critical load of an MEECS, whereas the FEE exhibits an opposite effect by increasing it.

Key words: flexomagnetic effect (FME), flexoelectric effect (FEE), post-buckling, cylindrical shell, equilibrium path

2010 MSC Number:

O343.9

Wei WANG, Gaofei GUAN, Yongqi LI, Jiabin SUN, Zhenhuan ZHOU, Xinsheng XU. Nonlinear post-buckling modeling of a magneto-electro-elastic cylindrical shell with flexomagnetic and flexoelectric effects. Applied Mathematics and Mechanics (English Edition), 2025, 46(11): 2135-2154.

TrendMD

Figures/Tables 16

Fig.?1

Fig.?2

Fig.?3

Fig.?4

Fig.?5

Fig.?6

Fig.?7

Fig.?8

Fig.?9

Fig.?10

Fig.?11

Fig.?12

Fig.?13

Fig.?14

Fig.?15

Fig.?16

References 58

[1]	QU, Y. L., JIN, F., and ZHANG, G. Y. Mechanically induced electric and magnetic fields in the bending and symmetric-shear deformations of a microstructure-dependent FG-MEE composite beam. Composite Structures, 278, 114554 (2021)
[2]	ARUNKUMAR, M. P., BHAGAT, V., SWETHA, S., GENG, Q., PITCHAIMANI, J., and LI, Y. M. An exact solution for vibro-acoustic response of MEE composite plate. Thin-Walled Structures, 179, 109598 (2022)
[3]	ZHANG, S. Q., ZHAO, Y. F., WANG, X., CHEN, M., and SCHMIDT, R. Static and dynamic analysis of functionally graded magneto-electro-elastic plates and shells. Composite Structures, 281, 114950 (2022)
[4]	LI, L. Z., NIE, L., and REN, Y. R. On multiple impacts of functional gradient magneto-electro-elastic plates with convex and concave configurations and prediction based on PSO-BPNN. Aerospace Science and Technology, 151, 109277 (2024)
[5]	HU, F. W. and LIU, Y. T. A magneto-elastica reinforced elastomer makes soft robotic grippers. Sensors and Actuators A: Physical, 379, 115977 (2024)
[6]	MARTINS, P., BRITO-PEREIRA, R., RIBEIRO, S., LANCEROS-MENDEZ, S., and RIBEIRO, C. Magnetoelectrics for biomedical applications: 130 years later, bridging materials, energy, and life. Nano Energy, 126, 109569 (2024)
[7]	ZHAO, X., ZHOU, Y. H., XU, J., CHEN, G. R., FANG, Y. S., TAT, T., XIAO, X., SONG, Y., LI, S., and CHEN, J. Soft fibers with magnetoelasticity for wearable electronics. Nature Communications, 12(1), 6755 (2021)
[8]	LING, Y. C., YU, X., YUAN, S. J., HE, A. P., HAN, Z. D., DU, J., FAN, Q., YAN, S. C., and XU, Q. Y. Flexomagnetic effect enhanced ferromagnetism and magnetoelectrochemistry in freestanding high-entropy alloy films. ACS Nano, 17(17), 17299–17307 (2023)
[9]	TANG, Z. M., GONG, Q. H., and YI, M. Flexomagnetism: progress, challenges, and opportunities. Materials Science and Engineering: R: Reports, 162, 100878 (2025)
[10]	JIN, S. Z., LIU, Y. J., DENG, Z. Y., WANG, T. J., XU, S. Q., CHEN, Y. C., JIANG, X. G., LIANG, C. B., HONG, J. W., CHEONG, S. W., and WANG, X. Y. Strain gradient induced skyrmion in a van der Waals magnet by wrinkling. Advanced Materials, 37(29), e2501935 (2025)
[11]	LADUCA, Z., SAMANTA, T., HAGOPIAN, N., JUNG, T., SU, K., GENSER, K., RABE, K. M., VOYLES, P. M., ARNOLD, M. S., and KAWASAKI, J. K. Cold seeded epitaxy and flexomagnetism in GdAuGe membranes exfoliated from graphene/Ge(111). Nano Letters, 24(33), 10284–10289 (2024)
[12]	SU, Y. F., ZONG, A., KOGAR, A., LU, D., HONG, S. S., FREELON, B., ROHWER, T., WANG, B. Y., HWANG, H. Y., and GEDIK, N. Delamination-assisted ultrafast wrinkle formation in a freestanding film. Nano Letters, 23(23), 10772–10778 (2023)
[13]	LIANG, X., HU, S. L., and SHEN, S. P. Size-dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity. Smart Materials and Structures, 24(10), 105012 (2015)
[14]	ANSARI, R., NESARHOSSEINI, S., FARAJI OSKOUIE, M., and ROUHI, H. Size-dependent buckling analysis of piezoelectric nanobeams resting on elastic foundation considering flexoelectricity effect using the stress-driven nonlocal model. The European Physical Journal Plus, 136(8), 876 (2021)
[15]	EBNALI-SAMANI, M. S. and BENI, Y. T. Size dependent thermo-mechanical buckling of the flexoelectric nanobeam. Materials Research Express, 5(8), 085018 (2018)
[16]	ZHANG, G. Y., GUO, Z. W., QU, Y. L., GAO, X. L., and JIN, F. A new model for thermal buckling of an anisotropic elastic composite beam incorporating piezoelectric, flexoelectric and semiconducting effects. Acta Mechanica, 233(5), 1719–1738 (2022)
[17]	XIONG, Z. Z., ZHUANG, J. Y., and GU, B. D. Piezoelectric thermal elastic buckling analysis on nanostructures via nonlocal elastic theory. Archive of Applied Mechanics, 95(5), 117 (2025)
[18]	BARATI, M. R. and ZENKOUR, A. M. Thermal post-buckling analysis of closed circuit flexoelectric nanobeams with surface effects and geometrical imperfection. Mechanics of Advanced Materials and Structures, 26(17), 1482–1490 (2019)
[19]	ZHAO, X., ZHENG, S. J., and LI, Z. J. Bending, free vibration and buckling analyses of AFG flexoelectric nanobeams based on the strain gradient theory. Mechanics of Advanced Materials and Structures, 29(4), 548–563 (2022)
[20]	ABDELRAHMAN, A. A., ABDEL-MOTTALEB, H. E., ALJABRI, A., MAHMOUD, E. R. I., and ELTAHER, M. A. Modeling of size dependent buckling behavior of piezoelectric sandwich perforated nanobeams rested on elastic foundation with flexoelectricity. Mechanics Based Design of Structures and Machines, 53(1), 374–400 (2025)
[21]	EBRAHIMI, F. and KARIMIASL, M. Nonlocal and surface effects on the buckling behavior of flexoelectric sandwich nanobeams. Mechanics of Advanced Materials and Structures, 25(11), 943–952 (2018)
[22]	QU, Y. L., JIN, F., and YANG, J. S. Buckling of flexoelectric semiconductor beams. Acta Mechanica, 232(7), 2623–2633 (2021)
[23]	LIANG, X., YANG, W. J., HU, S. L., and SHEN, S. P. Buckling and vibration of flexoelectric nanofilms subjected to mechanical loads. Journal of Physics D: Applied Physics, 49(11), 115307 (2016)
[24]	DUC, D. H., THOM, D. V., CONG, P. H., MINH, P. V., and NGUYEN, N. X. Vibration and static buckling behavior of variable thickness flexoelectric nanoplates. Mechanics Based Design of Structures and Machines, 51(12), 7102–7130 (2023)
[25]	QU, Y. L., JIN, F., and YANG, J. S. Bending of a flexoelectric semiconductor plate. Acta Mechanica Solida Sinica, 35(3), 434–445 (2022)
[26]	THI, T. H. N., TRAN, V. K., TU, P. H., and THAO, P. H. Dynamic instability analysis of piezoelectric fluid-infiltrated porous metal foam nanosheet considering surface and flexoelectricity effects in hygro-thermal environment. International Journal of Mechanics and Materials in Design, 21(2), 261–296 (2025)
[27]	LE, H. H., TRAN, V. K., HOANG, N. T., and HUONG, N. N. M. The impacts of variable nonlocal, length-scale factors and surface energy on hygro-thermo-mechanical vibration and buckling behaviors of viscoelastic FGP nanosheet on viscoelastic medium. Acta Mechanica Sinica, 41(5), 124135 (2025)
[28]	ZENG, S., WANG, B. L., and WANG, K. F. Static stability analysis of nanoscale piezoelectric shells with flexoelectric effect based on couple stress theory. Microsystem Technologies, 24(7), 2957–2967 (2018)
[29]	WANG, W., QI, Q. S., ZHANG, J. L., WANG, Z. K., SUN, J. B., ZHOU, Z. H., and XU, X. S. A size-dependent electro-mechanical buckling analysis of flexoelectric cylindrical nanoshells. Thin-Walled Structures, 202, 112118 (2024)
[30]	WANG, W., YIN, H. L., ZHANG, J. L., SUN, J. B., ZHOU, Z. H., and XU, X. S. Nonlinear stability characteristics of piezoelectric cylindrical shells with flexoelectric effects. Acta Mechanica Sinica, 41(9), 424412 (2024)
[31]	WANG, W., YIN, H. L., YU, Q. Y., WANG, Z. K., SUN, J. B., ZHOU, Z. H., and XU, X. S. Torsion stability analysis of functionally graded piezoelectric cylindrical shell with flexoelectric effect. Thin-Walled Structures, 208, 112820 (2025)
[32]	ZHANG, N., ZHENG, S. J., and CHEN, D. J. Size-dependent static bending, free vibration and buckling analysis of curved flexomagnetic nanobeams. Meccanica, 57(7), 1505–1518 (2022)
[33]	MALIKAN, M., WICZENBACH, T., and EREMEYEV, V. A. Thermal buckling of functionally graded piezomagnetic micro- and nanobeams presenting the flexomagnetic effect. Continuum Mechanics and Thermodynamics, 34(4), 1051–1066 (2022)
[34]	MALIKAN, M., WICZENBACH, T., and EREMEYEV, V. A. On thermal stability of piezo-flexomagnetic microbeams considering different temperature distributions. Continuum Mechanics and Thermodynamics, 33(4), 1281–1297 (2021)
[35]	JENA, S. K., CHAKRAVERTY, S., and MALIKAN, M. Stability analysis of nanobeams in hygrothermal environment based on a nonlocal strain gradient Timoshenko beam model under nonlinear thermal field. Journal of Computational Design and Engineering, 7(6), 685–699 (2020)
[36]	MALIKAN, M. and EREMEYEV, V. A. Flexomagneticity in buckled shear deformable hard-magnetic soft structures. Continuum Mechanics and Thermodynamics, 34(1), 1–16 (2022)
[37]	MALIKAN, M., UGLOV, N. S., and EREMEYEV, V. A. On instabilities and post-buckling of piezomagnetic and flexomagnetic nanostructures. International Journal of Engineering Science, 157, 103395 (2020)
[38]	ZHANG, N., ZHENG, S. J., and CHEN, D. J. Size-dependent static bending, free vibration and buckling analysis of simply supported flexomagnetic nanoplates. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 44(6), 253 (2022)
[39]	MOMENI-KHABISI, H. and TAHANI, M. Coupled thermal stability analysis of piezomagnetic nano-sensors and nano-actuators considering the flexomagnetic effect. European Journal of Mechanics-A/Solids, 97, 104773 (2023)
[40]	MOMENI-KHABISI, H. and TAHANI, M. Buckling and post-buckling analysis of double-layer magnetoelectric nano-plate strips considering piezo-flexoelectric and piezo-flexomagnetic effects. European Journal of Mechanics-A/Solids, 104, 105218 (2024)
[41]	MALIKAN, M. and EREMEYEV, V. A. On a flexomagnetic behavior of composite structures. International Journal of Engineering Science, 175, 103671 (2022)
[42]	OVEISSI, S., GHASSEMI, A., SALEHI, M., EFTEKHARI, S. A., and ZIAEI-RAD, S. Hydro-hygro-thermo-magneto-electro elastic wave propagation of axially moving nano-cylindrical shells conveying various magnetic-nano-fluids resting on the electromagnetic-visco-Pasternak medium. Thin-Walled Structures, 173, 108926 (2022)
[43]	WU, P., HU, C., and QIN, Q. H. Time-dependent behavior of layered magneto-electro-elastic cylindrical shell with viscoelastic interlayer. Composite Structures, 200, 874–885 (2018)
[44]	YAMAKI, N., OTOMO, K., and MATSUDA, K. Experiments on the postbuckling behavior of circular cylindrical shells under compression. Experimental Mechanics, 15(1), 23–28 (1975)
[45]	YAMAKI, N. and KODAMA, S. Postbuckling behavior of circular cylindrical shells under compression. International Journal of Non-Linear Mechanics, 11(2), 99–111 (1976)
[46]	REDDY, J. N. A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics, 51(4), 745–752 (1984)
[47]	JIA, J. F., YIN, H. L., YU, Q. Y., SUN, J. B., XU, X. S., and ZHOU, Z. H. New analytical solutions for free vibration of embedded magneto-electro-elastic cylindrical shells with step-wise thickness variations. Applied Mathematics and Mechanics (English Edition), 46(3), 447–466 (2025) https://doi.org/10.1007/s10483-025-3228-7
[48]	WANG, X. T., LIU, J., HU, B., ZHANG, B., and SHEN, H. M. Wave propagation responses of porous bi-directional functionally graded magneto-electro-elastic nanoshells via nonlocal strain gradient theory. Applied Mathematics and Mechanics (English Edition), 44(10), 1821–1840 (2023) https://doi.org/10.1007/s10483-023-3043-7
[49]	FU, G. Y., ZHANG, Z. J., DONG, C. M., SUN, Y. F, WANG, J. J., and ZHENG, H. Y. On the magneto-mechanical response of piezomagnetic microbeam with size effects. Thin-Walled Structures, 191, 111040 (2023)
[50]	WANG, T. Q., ZHU, F., LI, P., XU, Z. L., MA, T. F., KUZNETSOVA, I., and QIAN, Z. H. Analysis of the electromechanical coupling characteristics of piezoelectric semiconductor PN junction shell structures. Applied Mathematics and Mechanics (English Edition), 46(6), 1167–1186 (2025) https://doi.org/10.1007/s10483-025-3259-6
[51]	DU, D. X., YANG, J., SUN, W., MA, H. W., and XU, K. P. The semi-analytical modeling and vibration reduction analysis of the cylindrical shell with piezoelectric shunt damping patches. Applied Mathematics and Mechanics (English Edition), 44(10), 1675–1700 (2023) https://doi.org/10.1007/s10483-023-3034-7
[52]	LIU, Y. F., WANG, J., HU, J. X., QIN, Z. Y., and CHU, F. L. Multiple internal resonances of rotating composite cylindrical shells under varying temperature fields. Applied Mathematics and Mechanics (English Edition), 43(10), 1543–1554 (2022) https://doi.org/10.1007/s10483-022-2904-9
[53]	NI, Y. W., SUN, J. B., ZHANG, J. L., TONG, Z. Z., ZHOU, Z. H., and XU, X. S. Accurate buckling analysis of magneto-electro-elastic cylindrical shells subject to hygro-thermal environments. Applied Mathematical Modelling, 118, 798–817 (2023)
[54]	ZHU, S. B., SUN, J. B., TONG, Z. Z., LI, Q. D., ZHOU, Z. H., and XU, X. S. Post-buckling analysis of magneto-electro-elastic composite cylindrical shells subjected to multi-field coupled loadings. Composite Structures, 270, 114061 (2021)
[55]	YAMAKI, N. Elastic Stability of Circular Cylindrical Shells, North-Holland, New York (1984)
[56]	KOBAYASHI, T., MIHARA, Y., and FUJII, F. Path-tracing analysis for post-buckling process of elastic cylindrical shells under axial compression. Thin-Walled Structures, 61, 180–187 (2012)
[57]	LU, L., LEANZA, S., LIU, Y., and ZHAO, R. R. Buckling and post-buckling of cylindrical shells under combined torsional and axial loads. European Journal of Mechanics-A/Solids, 112, 105653 (2025)
[58]	LAN, M. D., YANG, W. J., LIANG, X., HU, S. L., and SHEN, S. P. Vibration modes of flexoelectric circular plate. Acta Mechanica Sinica, 38(12), 422063 (2022)