Nonlinear analytical buckling analysis of composite cylindrical shells under magneto-electro-mechanical coupling: The role of flexoelectric and flexomagnetic effects
Wei Wang
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorZilong Jiang
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorGaofei Guan
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorJiabin Sun
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Chemical Engineering Ocean and Life Science, Dalian University of Technology, Panjin, P.R. China
Search for more papers by this authorLide Chen
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorRui Li
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorCorresponding Author
Zhenhuan Zhou
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Correspondence
Zhenhuan Zhou, State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian 116024, P.R. China.
Email: [email protected]
Search for more papers by this authorXinsheng Xu
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorWei Wang
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorZilong Jiang
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorGaofei Guan
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorJiabin Sun
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Chemical Engineering Ocean and Life Science, Dalian University of Technology, Panjin, P.R. China
Search for more papers by this authorLide Chen
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorRui Li
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorCorresponding Author
Zhenhuan Zhou
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Correspondence
Zhenhuan Zhou, State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian 116024, P.R. China.
Email: [email protected]
Search for more papers by this authorXinsheng Xu
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, P.R. China
Search for more papers by this authorAbstract
Structures featuring flexoelectric and flexomagnetic effects have great potential for the manufacture of innovative devices, such as data storage, actuators, sensors and energy harvesters. To evaluate the stability of such devices, the buckling behaviors of magneto-electric-elastic (MEE) cylindrical shells with flexoelectric and flexomagnetic effects are investigated. A novel buckling model is developed based on the high-order shear deformation shell theory incorporating with the pre-buckling non-uniform deformation. The critical buckling loads and buckling mode shapes are obtained by the Galerkin's method involving with some newly proposed trial functions. The effects of key parameters including boundary conditions, geometrical parameters, flexoelectric and flexomagnetic coefficients and external magnetoelectric field on the buckling behaviors of MEE cylindrical shells are investigated. The results indicate that the flexoelectric effect enhances the critical buckling stresses, while the flexomagnetic effect weakens them. Therefore, the combination of magnetic and electric fields provides an effective way to adjust the buckling behaviors of MEE cylindrical shells and provide a wider range of control over the critical buckling stresses.
CONFLICT OF INTEREST STATEMENT
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Open Research
DATA AVAILABILITY STATEMENT
Data will be made available on request.
REFERENCES
- 1Lun, Y., Hong, J., Fang, D.: Asymmetric mechanical properties in ferroelectrics driven by flexo-deformation effect. J. Mech. Phys. Solids 164, 104891 (2022)
- 2Liu, L., Chen, W., Zheng, Y.: Flexoresponses of Synthetic antiferromagnetic systems hosting skyrmions. Phys. Rev. Lett. 128(25), 257201 (2022).
- 3Surmenev, R.A., Surmeneva, M.A.: The influence of the flexoelectric effect on materials properties with the emphasis on photovoltaic and related applications: A review. Mater. Today 67, 256–298 (2023)
- 4Wang, B., Gu, Y., Zhang, S., Chen, L.-Q.: Flexoelectricity in solids: Progress, challenges, and perspectives. Prog. Mater Sci. 106, 100570 (2019).
- 5Cheema, S.S., Kwon, D., Shanker, N., dos Reis, R., Hsu, S.-L., Xiao, J., Zhang, H., Wagner, R., Datar, A., McCarter, M.R., Serrao, C.R., Yadav, A.K., Karbasian, G., Hsu, C.-H., Tan, A.J., Wang, L.-C., Thakare, V., Zhang, X., Mehta, A., Karapetrova, E., Chopdekar, R.V., Shafer, P., Arenholz, E., Hu, C., Proksch, R., Ramesh, R., Ciston, J., Salahuddin, S.: Enhanced ferroelectricity in ultrathin films grown directly on silicon. Nature 580(7804), 478–482 (2020)
- 6Narvaez, J., Vasquez-Sancho, F., Catalan, G.: Enhanced flexoelectric-like response in oxide semiconductors. Nature 538(7624), 219–221 (2016)
- 7Spaldin, N.A., Ramesh, R.: Advances in magnetoelectric multiferroics. Nat. Mater. 18(3), 203–212 (2019)
- 8Du, D., Manzo, S., Zhang, C., Saraswat, V., Genser, K.T., Rabe, K.M., Voyles, P.M., Arnold, M.S., Kawasaki, J.K.: Epitaxy, exfoliation, and strain-induced magnetism in rippled Heusler membranes. Nat. Commun. 12(1), 2494 (2021).
- 9Yang, M.-M., Kim, D.J., Alexe, M.: Flexo-photovoltaic effect. Science 360(6391), 904–907 (2018)
- 10Yang, Z., Li, X., Gao, L., Zhang, W., Wang, X., Liu, H., Wang, S., Pan, C., Guo, L.: Ferro-pyro-phototronic effect enhanced self-powered, flexible and ultra-stable photodetectors based on highly crystalized 1D/3D ferroelectric perovskite film. Nano Energy 102, 107743 (2022).
- 11Deng, Q., Liu, L., Sharma, P.: Flexoelectricity in soft materials and biological membranes. J. Mech. Phys. Solids 62, 209–227 (2014)
- 12Hadjesfandiari, A.R.: Size-dependent piezoelectricity. Int. J. Solids Struct. 50(18), 2781–2791 (2013)
- 13Li, A., Zhou, S., Qi, L., Chen, X.: A reformulated flexoelectric theory for isotropic dielectrics. J. Phys. D Appl. Phys. 48(46), 465502 (2015).
- 14Mindlin, R.D.: Polarization gradient in elastic dielectrics. Int. J. Solids Struct. 4(6), 637–642 (1968)
10.1016/0020-7683(68)90079-6 Google Scholar
- 15Sahin, E., Dost, S.: A strain-gradients theory of elastic dielectrics with spatial dispersion. Int. J. Eng. Sci. 26(12), 1231–1245 (1988)
- 16Shen, S., Hu, S.: A theory of flexoelectricity with surface effect for elastic dielectrics. J. Mech. Phys. Solids 58(5), 665–677 (2010)
- 17Ma, W., Cross, L.E.: Large flexoelectric polarization in ceramic lead magnesium niobate. Appl. Phys. Lett. 79(26), 4420–4422 (2001)
- 18Ma, W., Cross, L.E.: Observation of the flexoelectric effect in relaxor Pb(Mg1/3Nb2/3)O3 ceramics. Appl. Phys. Lett. 78(19), 2920–2921 (2001)
- 19Ma, W., Cross, L.E.: Flexoelectric polarization of barium strontium titanate in the paraelectric state. Appl. Phys. Lett. 81(18), 3440–3442 (2002)
- 20Ma, W., Cross, L.E.: Strain-gradient-induced electric polarization in lead zirconate titanate ceramics. Appl. Phys. Lett. 82(19), 3293–3295 (2003)
- 21Ma, W., Cross, L.E.: Flexoelectric effect in ceramic lead zirconate titanate. Appl. Phys. Lett. 86(7), 072905 (2005).
- 22Ma, W., Cross, L.E.: Flexoelectricity of barium titanate. Appl. Phys. Lett. 88(23), 232902 (2006).
- 23Deng, Q., Kammoun, M., Erturk, A., Sharma, P.: Nanoscale flexoelectric energy harvesting. Int. J. Solids Struct. 51(18), 3218–3225 (2014)
- 24Huang, W., Yan, X., Kwon, S.R., Zhang, S., Yuan, F.-G., Jiang, X.: Flexoelectric strain gradient detection using Ba0.64Sr0.36TiO3 for sensing. Appl. Phys. Lett. 101(25), 252903 (2012).
- 25Kwon, S.R., Huang, W.B., Zhang, S.J., Yuan, F.G., Jiang, X.N.: Study on a flexoelectric microphone using barium strontium titanate. J. Micromech. Microeng. 26(4), 045001 (2016).
- 26Wang, L., Liu, S., Feng, X., Zhang, C., Zhu, L., Zhai, J., Qin, Y., Wang, Z.L.: Flexoelectronics of centrosymmetric semiconductors. Nat. Nanotechnol. 15(8), 661–667 (2020)
- 27Zhang, H., Chu, B.: Energy harvesting by exploiting the enhanced flexoelectric-like response of reduced (Na0.5Bi0.5)0.92Ba0.08TiO3 ceramics. J. Eur. Ceram. Soc. 38(6), 2520–2525 (2018)
- 28Zhou, W., Chen, P., Pan, Q., Zhang, X., Chu, B.: Lead-free metamaterials with enormous apparent piezoelectric response. Adv. Mater. 27(41), 6349–6355 (2015)
- 29Tang, Z., Gong, Q., Yi, M.: Flexomagnetism: Progress, challenges, and opportunities. Mater. Sci. Eng. R Rep. 162, 100878 (2025).
- 30Cai, R., Antohe, V.A., Nysten, B., Piraux, L., Jonas, A.M.: Thermally induced flexo-type effects in nanopatterned multiferroic layers. Adv. Funct. Mater. 30(14), 1910371 (2020).
- 31Gu, X., Guo, X.-B., Li, W.-H., Jiang, Y.-P., Liu, Q.-X., Tang, X.-G.: High-entropy materials for application: Electricity, magnetism, and optics. ACS Appl. Mater. Interfaces 16(40), 53372–53392 (2024)
- 32Makushko, P., Kosub, T., Pylypovskyi, O.V., Hedrich, N., Li, J., Pashkin, A., Avdoshenko, S., Hübner, R., Ganss, F., Wolf, D., Lubk, A., Liedke, M.O., Butterling, M., Wagner, A., Wagner, K., Shields, B.J., Lehmann, P., Veremchuk, I., Fassbender, J., Maletinsky, P., Makarov, D.: Flexomagnetism and vertically graded Néel temperature of antiferromagnetic Cr2O3 thin films. Nat. Commun. 13(1), 6745 (2022).
- 33Qiu, G., Li, Z., Zhou, K., Cai, Y.: Flexomagnetic noncollinear state with a plumb line shape spin configuration in edged two-dimensional magnetic CrI3. npj Quantum Mater. 8(1), 15 (2023)
- 34Su, Y., Zong, A., Kogar, A., Lu, D., Hong, S.S., Freelon, B., Rohwer, T., Wang, B.Y., Hwang, H.Y., Gedik, N.: Delamination-assisted ultrafast wrinkle formation in a freestanding film. Nano Lett. 23(23), 10772–10778 (2023)
- 35Chappert, C., Fert, A., Van Dau, F.N.: The emergence of spin electronics in data storage. Nat. Mater. 6(11), 813–823 (2007)
- 36Li, H., Zhan, Q., Liu, Y., Liu, L., Yang, H., Zuo, Z., Shang, T., Wang, B., Li, R.-W.: Stretchable spin valve with stable magnetic field sensitivity by ribbon-patterned periodic wrinkles. ACS Nano 10(4), 4403–4409 (2016)
- 37Kim, Y., Yuk, H., Zhao, R., Chester, S.A., Zhao, X.: Printing ferromagnetic domains for untethered fast-transforming soft materials. Nature 558(7709), 274–279 (2018)
- 38Lenz, J., Edelstein, S.: Magnetic sensors and their applications. IEEE Sensors J. 6(3), 631–649 (2006)
- 39Annapureddy, V., Palneedi, H., Hwang, G.-T., Peddigari, M., Jeong, D.-Y., Yoon, W.-H., Kim, K.-H., Ryu, J.: Magnetic energy harvesting with magnetoelectrics: An emerging technology for self-powered autonomous systems. Sustain. Energy Fuels 1(10), 2039–2052 (2017)
- 40Ke, L.-L., Wang, Y.-S.: Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos. Struct. 93(2), 342–350 (2011)
- 41Ke, L.-L., Yang, J., Kitipornchai, S., Bradford, M.A.: Bending, buckling and vibration of size-dependent functionally graded annular microplates. Compos. Struct. 94(11), 3250–3257 (2012)
- 42Ke, L.-L., Yang, J., Kitipornchai, S., Xiang, Y.: Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials. Mech. Adv. Mater. Struct. 16(6), 488–502 (2009)
- 43Li, Z., Hu, D., Shen, M., Huang, H., Ou, Z.: Thermal upheaval buckling framework of a graphene-reinforced subsea pipeline laid on an arched concave seabed. Eng. Struct. 318, 118750 (2024)
- 44Li, Z., Liu, S., Zhang, Q., Zhang, Z.: Analytical buckling scheme of a functionally graded porous liner reinforced by nanocomposites encased in an egg-shaped pipe. Eng. Struct. 310, 118095 (2024)
- 45Mittelstedt, C.: Buckling and post-buckling of thin-walled composite laminated beams—A review of engineering analysis methods. Appl. Mech. Rev. 72(2), 020802 (2020).
- 46Teng, J.G.: Buckling of thin shells: recent advances and trends. Appl. Mech. Rev. 49(4), 263–274 (1996)
10.1115/1.3101927 Google Scholar
- 47Tovstik, P., Smirnov, A., Steele, C.: Asymptotic methods in the buckling theory of elastic shells. Appl. Mech. Rev. 55(5), B90-B90 (2002)
10.1115/1.1497476 Google Scholar
- 48Zhang, J., Cao, C., Liu, X.: Symplectic method-based analytical solutions of thermal buckling for shear deformable circular functionally graded plates. Int. J. Struct. Stab. Dyn. 24(03), 2450021 (2024)
- 49Zhang, J., Zheng, W.: Elastoplastic buckling of FGM beams in thermal environment. Continuum Mech. Thermodyn. 33(1), 151–161 (2021)
- 50Zhang, Z., Luo, Y., Zhang, H., Kuai, H., Chen, F., Li, Z.: Thermal stability and buckling analyses of novel encased composite cylinder with polyhedral shapes. Compos. Struct. 289, 115460 (2022)
- 51Zhou, Q., Zhang, J.H., Zhao, Y.G.: Nonlinear buckling and postbuckling of circular plates reinforced with graphene platelets using the shooting method. Int. J. Struct. Stab. Dyn. 24(01), 2450001 (2024)
- 52Zhou, Z., Ni, Y., Tong, Z., Zhu, S., Sun, J., Xu, X.: Accurate nonlinear buckling analysis of functionally graded porous graphene platelet reinforced composite cylindrical shells. Int. J. Mech. Sci. 151, 537–550 (2019)
- 53Dastjerdi, S., Malikan, M., Tahani, M., Kadkhodayan, M., Ameli, A.: On three-dimensional dynamics of smart rotating micro-disks. Int. J. Eng. Sci. 211, 104236 (2025).
- 54Malikan, M.: On mechanics of piezocomposite shell structures. Int. J. Eng. Sci. 198, 104056 (2024).
- 55Malikan, M., Dastjerdi, S., Eremeyev, V.A., Kadkhodayan, M.: On a fully three-dimensional bending analysis of very thick smart composite cube-like bulk structures. Compos. Struct. 353, 118733 (2025).
- 56Malikan, M., Dastjerdi, S., Eremeyev, V.A., Sedighi, H.M.: On a 3D material modelling of smart nanocomposite structures. Int. J. Eng. Sci. 193, 103966 (2023).
- 57Liang, X., Hu, S., Shen, S.: Size-dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity. Smart Mater. Struct. 24(10), 105012 (2015).
- 58Zhao, X., Zheng, S., Li, Z.: Bending, free vibration and buckling analyses of AFG flexoelectric nanobeams based on the strain gradient theory. Mech. Adv. Mater. Struct. 29(4), 548–563 (2020)
- 59Ebrahimi, F., Karimiasl, M.: Nonlocal and surface effects on the buckling behavior of flexoelectric sandwich nanobeams. Mech. Adv. Mater. Struct. 25(11), 943–952 (2018)
- 60Ebnali Samani, M.S., Beni, Y.T.: Size dependent thermo-mechanical buckling of the flexoelectric nanobeam. Mater. Res. Express 5(8), 85018 (2018).
10.1088/2053-1591/aad2ca Google Scholar
- 61Barati, M.R., Zenkour, A.M.: Thermal post-buckling analysis of closed circuit flexoelectric nanobeams with surface effects and geometrical imperfection. Mech. Adv. Mater. Struct. 26(17), 1482–1490 (2018)
10.1080/15376494.2018.1432821 Google Scholar
- 62Qu, Y., Jin, F., Yang, J.: Buckling of flexoelectric semiconductor beams. Acta Mech. 232(7), 2623–2633 (2021)
- 63Zhang, G.Y., Guo, Z.W., Qu, Y.L., Gao, X.L., Jin, F.: A new model for thermal buckling of an anisotropic elastic composite beam incorporating piezoelectric, flexoelectric and semiconducting effects. Acta Mech. 233(5), 1719–1738 (2022)
- 64Ebrahimi, F., Barati, M.R.: Static stability analysis of embedded flexoelectric nanoplates considering surface effects. Appl. Phys. A 123(10), 666 (2017).
- 65Amir, S., Khorasani, M., BabaAkbar-Zarei, H.: Buckling analysis of nanocomposite sandwich plates with piezoelectric face sheets based on flexoelectricity and first-order shear deformation theory. J. Sandwich Struct. Mater. 22(7), 2186–2209 (2018)
10.1177/1099636218795385 Google Scholar
- 66Duc, D.H., Thom, D.V., Cong, P.H., Minh, P.V., Nguyen, N.X.: Vibration and static buckling behavior of variable thickness flexoelectric nanoplates. Mech. Based Des. Struct. Mach. 51(12), 7102–7130 (2022)
- 67Zeng, S., Wang, B.L., Wang, K.F.: Static stability analysis of nanoscale piezoelectric shells with flexoelectric effect based on couple stress theory. Microsyst. Technol. 24(7), 2957–2967 (2018)
- 68Wang, W., Qi, Q., Zhang, J., Wang, Z., Sun, J., Zhou, Z., Xu, X.: A size-dependent electro-mechanical buckling analysis of flexoelectric cylindrical nanoshells. Thin Walled Struct. 202, 112118 (2024).
- 69Malikan, M., Eremeyev, V., Żur, K.: Effect of axial porosities on flexomagnetic response of in-plane compressed piezomagnetic nanobeams. Symmetry 12(12), 1935 (2020).
- 70Malikan, M., Wiczenbach, T., Eremeyev, V.A.: Thermal buckling of functionally graded piezomagnetic micro- and nanobeams presenting the flexomagnetic effect. Continuum Mech. Thermodyn. 34(4), 1051–1066 (2021)
- 71Malikan, M., Eremeyev, V.A.: Flexomagneticity in buckled shear deformable hard-magnetic soft structures. Continuum Mech. Thermodyn. 34(1), 1–16 (2021)
- 72Malikan, M., Eremeyev, V.A.: On a flexomagnetic behavior of composite structures. Int. J. Eng. Sci. 175, 103671 (2022).
- 73Zhang, N., Zheng, S., Chen, D.: Size-dependent static bending, free vibration and buckling analysis of curved flexomagnetic nanobeams. Meccanica 57(7), 1505–1518 (2022)
- 74Malikan, M., Uglov, N.S., Eremeyev, V.A.: On instabilities and post-buckling of piezomagnetic and flexomagnetic nanostructures. Int. J. Eng. Sci. 157, 103395 (2020).
- 75Malikan, M., Eremeyev, V.A.: Flexomagnetic response of buckled piezomagnetic composite nanoplates. Compos. Struct. 267, 113932 (2021).
- 76Zhang, N., Zheng, S., Chen, D.: Size-dependent static bending, free vibration and buckling analysis of simply supported flexomagnetic nanoplates. J. Braz. Soc. Mech. Sci. Eng. 44(6), 253 (2022).
- 77Momeni-Khabisi, H., Tahani, M.: Coupled thermal stability analysis of piezomagnetic nano-sensors and nano-actuators considering the flexomagnetic effect. Eur. J. Mech. A/Solids 97, 104773 (2023).
- 78Momeni-Khabisi, H., Tahani, M.: Buckling and post-buckling analysis of double-layer magnetoelectric nano-plate strips considering piezo-flexoelectric and piezo-flexomagnetic effects. Eur. J. Mech. A/Solids 104, 105218 (2024).
- 79Zhang, S., Xu, M., Liu, K., Shen, S.: A flexoelectricity effect-based sensor for direct torque measurement. J. Phys. D Appl. Phys. 48(48), 485502 (2015).
- 80Kashipazha, M., Kheirikhah, M., Meshkinabadi, S.: Analysis of torsional buckling of a cylindrical sandwich shell with a magnetorheological fluid core layer. Smart Mater. Struct. 33(3), 035046 (2024).
- 81Zhang, S., Liu, K., Xu, M., Shen, H., Chen, K., Feng, B., Shen, S.: Investigation of the 2312 flexoelectric coefficient component of polyvinylidene fluoride: Deduction, simulation, and mensuration. Sci. Rep. 7(1), 3134 (2017).
- 82Han, Q., Yan, D., Mu, F., Hornsen, T.: Dynamic analysis of cylindrical shells with flexoelectric actuation. AIAA J. 61(4), 1807–1819 (2023)
- 83Qu, Y.L., Guo, Z.W., Zhang, G.Y., Gao, X.L., Jin, F.: A new model for circular cylindrical Kirchhoff–Love shells incorporating microstructure and flexoelectric effects. J. Appl. Mech. 89(12), 121010 (2022).
- 84Han, J.K., Kim, S., Jang, S., Lim, Y.R., Kim, S.-W., Chang, H., Song, W., Lee, S.S., Lim, J., An, K.-S., Myung, S.: Tunable piezoelectric nanogenerators using flexoelectricity of well-ordered hollow 2D MoS2 shells arrays for energy harvesting. Nano Energy 61, 471–477 (2019)
- 85Scornec, J.Le, Guiffard, B.: Large curvature sensors based on flexoelectric effect in PEDOT: PSS polymer films. ACS mater. Lett. 5(11), 2929–2941 (2023)
10.1021/acsmaterialslett.3c00635 Google Scholar
- 86Thai, T.Q., Zhuang, X., Rabczuk, T.: Curved flexoelectric and piezoelectric micro-beams for nonlinear vibration analysis of energy harvesting. Int. J. Solids Struct. 264, 112096 (2023)
- 87Liu, X., Hu, T., Zhang, Y., Xu, X., Lei, R., Wu, B., Ma, Z., Lv, P., Zhang, Y., Huang, S.-W., Wu, J., Ma, J., Hong, J., Sheng, Z., Jia, C., Kan, E., Nan, C.-W., Zhang, J.: Flexomagnetoelectric effect in Sr2IrO4 thin films. Phys. Rev. Lett. 133(15), 156505 (2024).
- 88Ni, Y., Sun, J., Zhang, J., Tong, Z., Zhou, Z., Xu, X.: Accurate buckling analysis of magneto-electro-elastic cylindrical shells subject to hygro-thermal environments. Appl. Math. Modell. 118, 798–817 (2023)
- 89Ni, Y., Zhu, S., Sun, J., Tong, Z., Zhou, Z., Xu, X.: Analytical buckling solution of magneto-electro-thermo-elastic cylindrical shells under multi-physics fields. Compos. Struct. 239, 112021 (2020).
- 90Li, J.Y., Dunn, M.L.: Micromechanics of magnetoelectroelastic composite materials: Average fields and effective behavior. J. Intell. Mater. Syst. Struct. 9(6), 404–416 (1998)
- 91Reddy, J.N.: Mechanics of laminated composite plates and shells: theory and analysis. CRC Press, Boca Raton, FL (2003)
10.1201/b12409 Google Scholar
- 92Amabili, M.: Nonlinear vibrations and stability of shells and plates. Cambridge University Press, Cambridge (2008)
- 93Brush, D.O., Almroth, B.O., Hutchinson, J.: Buckling of bars, plates, and shells (1975)
- 94SafarPour, H., Ghanbari, B., Ghadiri, M.: Buckling and free vibration analysis of high speed rotating carbon nanotube reinforced cylindrical piezoelectric shell. Appl. Math. Modell. 65, 428–442 (2019)
- 95Mehralian, F., Tadi Beni, Y., Ansari, R.: On the size dependent buckling of anisotropic piezoelectric cylindrical shells under combined axial compression and lateral pressure. Int. J. Mech. Sci. 119, 155–169 (2016)
- 96Abdollahi, A., Peco, C., Millán, D., Arroyo, M., Arias, I.: Computational evaluation of the flexoelectric effect in dielectric solids. J. Appl. Phys. 116(9), 093502 (2014).
- 97Codony, D., Mocci, A., Barceló-Mercader, J., Arias, I.: Mathematical and computational modeling of flexoelectricity. J. Appl. Phys. 130(23), 231102 (2021).
- 98Sun, J., Xu, X., Lim, C.W., Zhou, Z., Xiao, S.: Accurate thermo-electro-mechanical buckling of shear deformable piezoelectric fiber-reinforced composite cylindrical shells. Compos. Struct. 141, 221–231 (2016)
- 99Yamaki, N.: Elastic stability of circular cylindrical shells. Elsevier, North Holland (1984)
- 100Sladek, J., Sladek, V., Xu, M., Deng, Q.: A cantilever beam analysis with flexomagnetic effect. Meccanica 56(9), 2281–2292 (2021)
- 101Gorman, D.J., Evan-Iwanowski, R.M.: An analytical and experimental investigation of the effects of large prebuckling deformations on the buckling of clamped thin-walled circular cylindrical shells subjected to axial loading and internal pressure. Dev. Theor. Appl. Mech. 4, 415–426 (1970)
- 102Weingarten, V.I., Morgan, E.J., Seide, P.: Elastic stability of thin-walled cylindrical and conical shells under axial compression. AIAA J. 3(3), 500–505 (1965)
- 103Tennyson, R.C.: Buckling modes of circular cylindrical shells under axial compression. AIAA J. 7(8), 1481–1487 (1969)