Abstract
A new model for thermal buckling of an anisotropic elastic composite beam consisting of a dielectric core and two thin semiconductor surface layers is developed. The field equations and boundary conditions for the beam are obtained by using the piezoelectricity, flexoelectricity, strain gradient elasticity and semiconductor theories and the kinematic relations for a Timoshenko beam. The current model includes piezoelectric, flexoelectric and semiconducting effects simultaneously, unlike existing models. A variational formulation based on the principle of minimum potential energy is employed for the dielectric core, where the contribution of the two thin semiconductor surface layers is incorporated through the work done by the free charge density. Two simplified models for piezoelectric and flexoelectric composite beams incorporating the semiconducting effect are obtained as two special cases of the new model for the dielectric composite beam. Thermal buckling of a simply supported composite beam with a piezoelectric or flexoelectric core and two thin semiconductor surface layers is analytically studied by directly applying the two simplified models, leading to the determination of the critical buckling temperature and concentration perturbation of free carriers in the composite beam. Numerical results show that the presence of the piezoelectric or flexoelectric effect results in an increased critical buckling temperature, while the inclusion of the semiconducting effect leads to a reduced value in both cases. In addition, it is seen that the redistributions of free carriers in the piezoelectric composite beam are uniform, whereas those in the flexoelectric composite beam are non-uniform along the beam thickness direction.
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Abbas, I.A., Kumar, R., Rani, L.: Thermoelastic interaction in a thermally conducting cubic crystal subjected to ramp-type heating. Appl. Math. Comput. 254, 360–369 (2015)
Abdollahi, A., Peco, C., Millán, D., Arroyo, M., Arias, I.: Computational evaluation of the flexoelectric effect in dielectric solids. J. Appl. Phys. 116, 093502 (2014)
Ai, L., Gao, X.-L.: Micromechanical modeling of 3-D printable interpenetrating phase composites with tailorable effective elastic properties including negative Poisson’s ratio. J. Micromech. Mol. Phys. 2, 1750015 (2017)
Auld, B.A.: Acoustic Fields and Waves in Solids. Wiley, New York (1973)
Bell, A.J., Comyn, T.P., Stevenson, T.J.: Expanding the application space for piezoelectric materials. APL Mater. 9, 010901 (2021)
Chen, W., Liang, X., Shen, S.: Forced vibration of piezoelectric and flexoelectric Euler-Bernoulli beams by dynamic Green’s functions. Acta Mech. 232, 449–460 (2021)
Chen, W.-R., Chen, C.-S., Chang, H.: Thermal buckling analysis of functionally graded Euler-Bernoulli beams with temperature-dependent properties. J. Appl. Comput. Mech. 6, 457–470 (2020)
Cheng, R., Zhang, C., Chen, W., Yang, J.: Piezotronic effects in the extension of a composite fiber of piezoelectric dielectrics and nonpiezoelectric semiconductors. J. Appl. Phys. 124, 064506 (2018)
Chu, L., Li, Y., Dui, G.: Size-dependent electromechanical coupling in functionally graded flexoelectric nanocylinders. Acta Mech. 230, 3071–3086 (2019)
Deng, Q., Lv, S., Li, Z., Tan, K., Liang, X., Shen, S.: The impact of flexoelectricity on materials, devices, and physics. J. Appl. Phys. 128, 080902 (2020)
Ebrahimi, F., Barati, M.R.: Dynamic modeling of a thermo-piezo-electrically actuated nanosize beam subjected to a magnetic field. Appl. Phys. A 122, 451 (2016)
Elahi, H., Munir, K., Eugeni, M., Abrar, M., Khan, A., Arshad, A., Gaudenzi, P.: A review on applications of piezoelectric materials in aerospace industry. Integr. Ferroelectr. 211, 25–44 (2020)
Fu, Y., Wang, J., Mao, Y.: Nonlinear analysis of buckling, free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment. Appl. Math. Model. 36, 4324–4340 (2012)
Gao, X.-L.: A new Timoshenko beam model incorporating microstructure and surface energy effects. Acta Mech. 226, 457–474 (2015)
Gao, X.-L., Mall, S.: Variational solution for a cracked mosaic model of woven fabric composites. Int. J. Solids Struct. 38(5), 855–874 (2001)
Gao, X.-L., Park, S.K.: Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int. J. Solids Struct. 44, 7486–7499 (2007)
Guin, L., Jabbour, M., Triantafyllidis, N.: The p-n junction under nonuniform strains: general theory and application to photovoltaics. J. Mech. Phys. Solids 110, 54–79 (2018)
Kim, B., Hopcroft, M.A., Candler, R.N., Jha, C.M., Agarwal, M., Melamud, R., Chandorkar, S.A., Yama, G., Kenny, T.W.: Temperature dependence of quality factor in MEMS resonators. J. Microelectromech. Syst. 17, 755–766 (2008)
Lei, J., He, Y., Guo, S., Li, Z., Liu, D.: Thermal buckling and vibration of functionally graded sinusoidal microbeams incorporating nonlinear temperature distribution using DQM. J. Therm. Stresses 40(6), 665–689 (2017)
Liang, X., Hu, S., Shen, S.: Effects of surface and flexoelectricity on a piezoelectric nanobeam. Smart. Mater. Struct. 23, 035020 (2014)
Ma, H.M., Gao, X.-L., Reddy, J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008)
Ma, W., Cross, L.E.: Flexoelectricity of barium titanate. Appl. Phys. Lett. 88, 232902 (2006)
Majdoub, M.S., Sharma, P., Cagin, T.: Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Phys. Rev. B 77, 125424 (2008)
Mao, S., Purohit, P.K.: Insights into flexoelectric solids from strain-gradient elasticity. J. Appl. Mech. 81, 081004 (2014)
Maranganti, R., Sharma, N.D., Sharma, P.: Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions. Phys. Rev. B 74, 014110 (2006)
Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)
Mindlin, R.D.: Polarization gradient in elastic dielectrics. Int. J. Solids Struct. 4, 637–642 (1968)
Mindlin, R.D.: High frequency vibrations of piezoelectric crystal plates. Int. J. Solids Struct. 8, 895–906 (1972)
Mindlin, R.D.: Equations of high frequency vibrations of thermopiezoelectric crystal plates. Int. J. Solids Struct. 10, 625–637 (1974)
Ootao, Y., Tanigawa, Y.: Transient analysis of multilayered magneto-electro-thermoelastic strip due to nonuniform heat supply. Compos. Struct. 68, 471–480 (2005)
Papargyri-Beskou, S., Tsepoura, K.G., Polyzos, D., Beskos, D.E.: Bending and stability analysis of gradient elastic beams. Int. J. Solids Struct. 40, 385–400 (2003)
Polizzotto, C.: A hierarchy of simplified constitutive models within isotropic strain gradient elasticity. Eur. J. Mech. A Solid 61, 92–109 (2017)
Quan, T.Q., Dinh Duc, N.: Nonlinear thermal stability of eccentrically stiffened FGM double curved shallow shells. J. Therm. Stresses 40(2), 211–236 (2017)
Qu, Y., Jin, F., Yang, J.: Effects of mechanical fields on mobile charges in a composite beam of flexoelectric dielectrics and semiconductors. J. Appl. Phys. 127, 194502 (2020)
Qu, Y.L., Li, P., Zhang, G.Y., Jin, F., Gao, X.-L.: A microstructure-dependent anisotropic magneto-electro-elastic Mindlin plate model based on an extended modified couple stress theory. Acta Mech. 231(10), 4323–4350 (2020)
Qu, Y., Jin, F., Yang, J.: Temperature effects on mobile charges in thermopiezoelectric semiconductor plates. Int. J. Appl. Mech. 13(3), 2150037 (2021)
Qu, Y., Jin, F., Yang, J.: Magnetically induced charge redistribution in the bending of a composite beam with flexoelectric semiconductor and piezomagnetic dielectric layers. J. Appl. Phys. 129, 064503 (2021)
Qu, Y., Jin, F., Yang, J.: Buckling of flexoelectric semiconductor beams. Acta Mech. 232, 2623–2633 (2021)
Qu, Y., Jin, F., Yang, J.: Torsion of a flexoelectric semiconductor rod with a rectangular cross section. Arch. Appl. Mech. 91, 2027–2038 (2021)
Qu, Y.L., Zhang, G.Y., Fan, Y.M., Jin, F.: A non-classical theory of elastic dielectrics incorporating couple stress and quadrupole effects: part I—reconsideration of curvature-based flexoelectricity theory. Math. Mech. Solids 26(11), 1647–1659 (2021)
Qu, Y.L., Zhang, G.Y., Gao, X.-L., Jin, F.: A new model for thermally induced redistributions of free carriers in centrosymmetric flexoelectric semiconductor beams. Manuscript under review (2022)
Ray, M.C.: Analysis of smart nanobeams integrated with a flexoelectric nano actuator layer. Smart Mater. Struct. 25(5), 055011 (2016)
Ren, C., Wang, K.F., Wang, B.L.: Adjusting the electromechanical coupling behaviors of piezoelectric semiconductor nanowires via strain gradient and flexoelectric effects. J. Appl. Phys. 128, 215701 (2020)
Sadd, M.H.: Elasticity: Theory, Applications, and Numerics, 3rd edn. Academic Press, Oxford (2014)
Samani, M.S.E., Beni, Y.T.: Size dependent thermo-mechanical buckling of the flexoelectric nanobeam. Mater. Res. Express 5, 085018 (2018)
Shen, S., Hu, S.: A theory of flexoelectricity with surface effect for elastic dielectrics. J. Mech. Phys. Solids 58(5), 665–677 (2010)
Shivashankar, P., Gopalakrishnan, S.: Review on the use of piezoelectric materials for active vibration, noise, and flow control. Smart Mater. Struct. 29, 053001 (2020)
Shu, L., Liang, R., Rao, Z., Fei, L., Ke, S., Wang, Y.: Flexoelectric materials and their related applications: a focused review. J. Adv. Ceram. 8, 153–173 (2019)
Shu, L., Wei, X., Pang, T., Yao, X., Wang, C.: Symmetry of flexoelectric coefficients in crystalline medium. J. Appl. Phys. 110, 104106 (2011)
Sze, S.M., Ng, K.K.: Physics of Semiconductor Devices, 3rd edn. Wiley, New York (2006)
Tagantsev, A.K., Meunier, V., Sharma, P.: Novel electromechanical phenomena at the nanoscale: phenomenological theory and atomistic modeling. MRS Bull. 34(9), 643–647 (2009)
Tian, X., Xu, M., Deng, Q., Sladek, J., Sladek, V., Repka, M., Li, Q.: Size-dependent direct and converse flexoelectricity around a micro-hole. Acta Mech. 231, 4851–4865 (2020)
Tiersten, H.F.: Linear Piezoelectric Plate Vibrations. Plenum, New York (1969)
Vineyard, E., Gao, X.-L.: Topology and shape optimization of 2-D and 3-D micro-architectured thermoelastic metamaterials using a parametric level set method. CMES-Comput. Model. Eng. Sci. 127, 819–854 (2021)
Wang, B., Gu, Y., Zhang, S., Chen, L.Q.: Flexoelectricity in solids: progress, challenges, and perspectives. Prog. Mater. Sci. 106, 100570 (2019)
Wang, G.-F., Yu, S.-W., Feng, X.-Q.: A piezoelectric constitutive theory with rotation gradient effects. Eur. J. Mech. A Solid. 23(3), 455–466 (2004)
Wang, 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)
Yan, Z., Jiang, L.Y.: Flexoelectric effect on the electroelastic responses of bending piezoelectric nanobeams. J. Appl. Phys. 113, 194102 (2013)
Yang, J.: Analysis of Piezoelectric Semiconductor Structures. Springer, Switzerland (2020)
Yang, J.: Mechanics of Piezoelectric Structures, 2nd edn. World Scientific, Singapore (2020)
Yang, J., Zhou, H.: Amplification of acoustic waves in piezoelectric semiconductor plates. Int. J. Solids Struct. 42, 3171–3183 (2005)
Zhang, G.Y., Gao, X.-L., Zheng, C.Y., Mi, C.W.: A non-classical Bernoulli-Euler beam model based on a simplified micromorphic elasticity theory. Mech. Mater. 161, 103967 (2021)
Zhang, G.Y., He, Z.Z., Gao, X.-L., Zhou, H.W.: Band gaps in a periodic electro-elastic composite beam structure incorporating microstructure and flexoelectric effects. Arch. Appl. Mech. (2022). https://doi.org/10.1007/s00419-021-02088-9
Zhang, G.Y., Qu, Y.L., Gao, X.-L., Jin, F.: A transversely isotropic magneto-electro-elastic Timoshenko beam model incorporating microstructure and foundation effects. Mech. Mater. 149, 103412 (2020)
Zhang, R., Liang, X., Shen, S.: A Timoshenko dielectric beam model with flexoelectric effect. Meccanica 51(5), 1181–1188 (2016)
Zhao, M., Liu, X., Fan, C., Lu, C., Wang, B.: Theoretical analysis on the extension of a piezoelectric semi-conductor nanowire: effects of flexoelectricity and strain gradient. J. Appl. Phys. 127(8), 085707 (2020)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 12002086 (G. Y. Zhang); Grant No. 12072253 (F. Jin)) and Zhishan Youth Scholar Program of SEU (G. Y. Zhang). The authors also would like to thank Prof. Shaofan Li and two anonymous reviewers for their encouragement and helpful comments on an earlier version of the paper.
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Zhang, G.Y., Guo, Z.W., Qu, Y.L. et al. A new model for thermal buckling of an anisotropic elastic composite beam incorporating piezoelectric, flexoelectric and semiconducting effects. Acta Mech 233, 1719–1738 (2022). https://doi.org/10.1007/s00707-022-03186-7
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DOI: https://doi.org/10.1007/s00707-022-03186-7