References
- Sahmani S, Aghdam MM. Nonlocal strain gradient shell model for axial buckling and postbuckling analysis of magneto-electro-elastic composite nanoshells. Composites Part B: Engineering. 2018;132:258-74. Available from: https://www.sciencedirect.com/science/article/pii/S1359836817312209.
- Farajpour MR, Shahidi AR, Hadi A, Farajpour A. Influence of initial edge displacement on the nonlinear vibration, electrical and magnetic instabilities of magneto-electro-elastic nanofilms. Mechanics of Advanced Materials and Structures. 2019;26(17):1469-81. Available from: https://doi.org/10.1080/15376494.2018.1432820
- Yakhno VG. An explicit formula for modeling wave propagation in magneto-electro-elastic materials. Journal of Electromagnetic Waves and Applications. 2018;32(7):899-912. Available from: https://doi.org/10.1080/09205071.2017.1410076
- Haghgoo M, Hassanzadeh-Aghdam M-K, Ansari R. Effect of piezoelectric interphase on the effective magneto-electro-elastic properties of three-phase smart composites: A micromechanical study. Mechanics of Advanced Materials and Structures. 2019;26(23):1935-50. Available from: https://doi.org/10.1080/15376494.2018.1455932
- Chen W, Yan Z, Wang L. On mechanics of functionally graded hard-magnetic soft beams. International Journal of Engineering Science. 2020;157:103391. Available from: https://www.sciencedirect.com/science/article/pii/S0020722520301786
- Taati E. On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment. International Journal of Engineering Science. 2018;128:63-78. Available from: https://www.sciencedirect.com/science/article/pii/S0020722518302489
- Dahmane M, Benadouda M, Fellah A, Saimi A, Hassen AA, Bensaid I. Porosities-dependent wave propagation in bi-directional functionally graded cantilever beam with higher-order shear model. Mechanics of Advanced Materials and Structures. Available from: https://doi.org/10.1080/15376494.2023.2253546
- Khaje Khabaz M, Eftekhari SA, Hashemian M, Toghraie D. Optimal vibration control of multi-layer micro-beams actuated by piezoelectric layer based on modified couple stress and surface stress elasticity theories. Physica A: Statistical Mechanics and its Applications. 2020;546:123998. Available from: https://www.sciencedirect.com/science/article/pii/S0378437119322137
- Bhangale RK, Ganesan N. Free vibration of simply supported functionally graded and layered magneto-electro-elastic plates by finite element method. Journal of Sound and Vibration. 2006;294(4):1016-38. Available from: https://www.sciencedirect.com/science/article/pii/S0022460X06000320
- Sladek J, Sladek V, Krahulec S, Chen CS, Young DL. Analyses of Circular Magnetoelectroelastic Plates with Functionally Graded Material Properties. Mechanics of Advanced Materials and Structures. 2015;22(6):479-89. Available from: https://doi.org/10.1080/15376494.2013.807448
- Mahesh V. Porosity effect on the nonlinear deflection of functionally graded magneto-electro-elastic smart shells under combined loading. Mechanics of Advanced Materials and Structures. 2022;29(19):270725. Available from: https://doi.org/10.1080/15376494.2021.1875086
- Zhang GY, Qu YL, Gao XL, Jin F. A transversely isotropic magneto-electro-elastic Timoshenko beam model incorporating microstructure and foundation effects. Mechanics of Materials. 2020;149:103412. Available from: https://www.sciencedirect.com/science/article/pii/S0167663620301137
- Hong J, Wang S, Zhang G, Mi C. On the Bending and Vibration Analysis of Functionally Graded Magneto-Electro-Elastic Timoshenko Microbeams. Crystals. 2021;11(10):1206. Available from: https://www.mdpi.com/2073-4352/11/10/1206
- Hong J, Wang S, Qiu X, Zhang G. Bending and Wave Propagation Analysis of Magneto-Electro-Elastic Functionally Graded Porous Microbeams. Crystals. 2022;12(5):732. Available from: https://doi.org/10.3390/cryst12050732
- Qu YL, Li P, Zhang GY, Jin F, Gao XL. A microstructure-dependent anisotropic magneto-electro-elastic Mindlin plate model based on an extended modified couple stress theory. Acta Mechanica. 2020;231(10):4323-50. Available from: https://doi.org/10.1007/s00707-020-02745-0
- Wang S, Hong J, Gu S, Xiao Z, Zhang G. A size-dependent isogeometric model for magneto-electro-elastic graded curved beams in advanced structures. Composite Structures. 2025;358:118877. Available from: https://doi.org/10.1016/j.compstruct.2025.118877
- Zheng Y-f, Qu D-y, Liu L-c, Chen C-p. Size-dependent nonlinear bending analysis of nonlocal magneto-electro-elastic laminated nanobeams resting on elastic foundation. International Journal of Non-Linear Mechanics. 2023;148:104255. Available from: https://www.sciencedirect.com/science/article/pii/S0020746222002256.
- Alibeigi B, Tadi Beni Y, Mehralian F. On the thermal buckling of magneto-electro-elastic piezoelectric nanobeams. The European Physical Journal Plus. 2018;133(3):133. Available from: https://doi.org/10.1140/epjp/i2018-11954-7
- Ghobadi A, Beni YT, Golestanian H. Nonlinear thermo-electromechanical vibration analysis of size-dependent functionally graded flexoelectric nano-plate exposed magnetic field. Archive of Applied Mechanics. 2020;90(9):2025-70. Available from: https://doi.org/10.1007/s00419-020-01708-0
- Zheng Y-f, Zhou Y, Wang F, Chen C-p. Nonlinear deformation analysis of magneto-electro-elastic nanobeams resting on elastic foundation by using nonlocal modified couple stress theory. European Journal of Mechanics - A/Solids. 2024;103:105158. Available from: https://www.sciencedirect.com/science/article/pii/S0997753823002504
- Habibi B, Beni YT, Mehralian F. Free vibration of magneto-electroelastic nanobeams based on modified couple stress theory in thermal environment. Mechanics of Advanced Materials and Structures. 2019;26(7):601-13. Available from: https://doi.org/10.1080/15376494.2017.1410902
- Alibeigi B, Tadi Beni Y. On the size-dependent magneto/electromechanical buckling of nanobeams. The European Physical Journal Plus. 2018;133(10):398. Available from: https://doi.org/10.1140/epjp/i2018-12208-6
- Zheng Y-f, Liu L-C, Qu D-y, Chen C-p. Nonlinear postbuckling analysis of magneto-electro-thermo-elastic laminated microbeams based on modified couple stress theory. Applied Mathematical Modelling. 2023;118:89-106. Available from: https://www.sciencedirect.com/science/article/pii/S0307904X23000227
- Fattaheian Dehkordi S, Tadi Beni Y. Size-dependent continuum-based model of a truncated flexoelectric/flexomagnetic functionally graded conical nano/microshells. Applied Physics A. 2022;128(4):320. Available from: https://doi.org/10.1007/s00339-022-05386-3
- Lyu Z, Ma M. Nonlinear dynamic modeling of geometrically imperfect magneto-electro-elastic nanobeam made of functionally graded material. Thin-Walled Structures. 2023;191:111004. Available from: https://www.sciencedirect.com/science/article/pii/S0263823123004822
- Wang S, Hong J, Wei D, Zhang G. Bending and wave propagation analysis of axially functionally graded beams based on a reformulated strain gradient elasticity theory. Applied Mathematics and Mechanics. 2023;44(10):1803-20. Available from: https://doi.org/10.1007/s10483-023-3042-6.
- Wang S, Hong J, Yin S, Zhang G. Isogeometric analysis of magneto-electro-elastic functionally graded Mindlin microplates. Thin-Walled Structures. 2024;198:111740. Available from: https://www.sciencedirect.com/science/article/pii/S0263823124001836
- Lam DCC, Yang F, Chong ACM, Wang J, Tong P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids. 2003;51(8):1477-508. Available from: https://www.sciencedirect.com/science/article/pii/S002250960300053X
- McFarland AW, Colton JS. Role of material microstructure in plate stiffness with relevance to microcantilever sensors. Journal of Micromechanics and Microengineering. 2005;15(5):1060. Available from: https://dx.doi.org/10.1088/0960-1317/15/5/024
- Toupin RA. Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis. 1962;11(1):385-414. Available from: https://doi.org/10.1007/BF00253945
- Mindlin RD. Influence of couple-stresses on stress concentrations. Experimental Mechanics. 1963;3(1):1-7. Available from: https://doi.org/10.1007/BF02327219
- Mindlin RD. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis. 1964;16(1):51-78. Available from: https://doi.org/10.1007/BF00248490
- Mindlin RD, Eshel NN. On first strain-gradient theories in linear elasticity. International Journal of Solids and Structures. 1968;4(1):109-24. Available from: https://www.sciencedirect.com/science/article/pii/002076836890036X
- Polizzotto C. A hierarchy of simplified constitutive models within isotropic strain gradient elasticity. European Journal of Mechanics - A/Solids. 2017;61:92-109. Available from: https://www.sciencedirect.com/science/article/pii/S0997753816302145
- B.S. Altan, E.C. Aifantis. On Some Aspects in the Special Theory of Gradient Elasticity. Journal of the Mechanical Behavior of Materials. 1997;8(3):231-82. Available from: https://doi.org/10.1515/JMBM.1997.8.3.231
- Ebrahimi F, Barati MR. Vibration analysis of embedded biaxially loaded magneto-electrically actuated inhomogeneous nanoscale plates. Journal of Vibration and Control. 2018;24(16):3587-607. Available from: https://journals.sagepub.com/doi/abs/10.1177/1077546317708105
- Kiani A, Sheikhkhoshkar M, Jamalpoor A, Khanzadi M. Free vibration problem of embedded magneto-electro-thermo-elastic nanoplate made of functionally graded materials via nonlocal third-order shear deformation theory. Journal of Intelligent Material Systems and Structures. 2018;29(5):741-63. Available from: https://journals.sagepub.com/doi/abs/10.1177/1045389X17721034
- Liu H, Lv Z. Vibration performance evaluation of smart magneto-electro-elastic nanobeam with consideration of nanomaterial uncertainties. Journal of Intelligent Material Systems and Structures. 2019;30(18-19):2932-52. Available from: https://journals.sagepub.com/doi/abs/10.1177/1045389X19873418
- Xiao W-s, Gao Y, Zhu H. Buckling and post-buckling of magneto-electro-thermo-elastic functionally graded porous nanobeams. Microsystem Technologies. 2019;25(6):2451-70. Available from: https://doi.org/10.1007/s00542-018-4145-2
- Lim CW, Zhang G, Reddy JN. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids. 2015;78:298-313. Available from: https://www.sciencedirect.com/science/article/pii/S0022509615000320
- Şimşek M. Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. International Journal of Engineering Science. 2016;105:12-27. Available from: https://www.sciencedirect.com/science/article/pii/S0020722516300520
- Li X, Li L, Hu Y, Ding Z, Deng W. Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Composite Structures. 2017;165:250-65. Available from: https://www.sciencedirect.com/science/article/pii/S0263822316304974
- Nguyen T-K, Vo TP, Thai H-T. Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory. Composites Part B: Engineering. 2013;55:147-57. Available from: https://www.sciencedirect.com/science/article/pii/S1359836813003223
- Li S-R, Batra RC. Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams. Composite Structures. 2013;95:5-9. Available from: https://www.sciencedirect.com/science/article/pii/S0263822312003558
- Nguyen T-K, Nguyen B-D. A new higher-order shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams. Journal of Sandwich Structures & Materials. 2015;17(6):613-31. Available from: https://journals.sagepub.com/doi/abs/10.1177/1099636215589237
- Van Vinh P. Deflections, stresses and free vibration analysis of bi-functionally graded sandwich plates resting on Pasternak’s elastic foundations via a hybrid quasi-3D theory. Mechanics Based Design of Structures and Machines. 2021:1-32. Available from: https://doi.org/10.1080/15397734.2021.1894948
- Adhikari B, Singh BN. Dynamic response of functionally graded plates resting on two-parameter-based elastic foundation model using a quasi-3D theory. Mechanics Based Design of Structures and Machines. 2019;47(4):399-429. Available from: https://doi.org/10.1080/15397734.2018.1555965
- Bensaid I, Saimi A. Dynamic investigation of functionally graded porous beams resting on viscoelastic foundation using generalised differential quadrature method. Australian Journal of Mechanical Engineering. 2022:1-20. Available from: https://doi.org/10.1080/14484846.2021.2017115
- Ahmed S, Abdelhamid H, Ismail B, Ahmed F. An Differential Quadrature Finite Element and the Differential Quadrature Hierarchical Finite Element Methods for the Dynamics Analysis of on Board Shaft. European Journal of Computational Mechanics. 2021;29(4-6):303–44. Available from: https://journals.riverpublishers.com/index.php/EJCM/article/view/5999
- Houalef IE, Bensaid I, Saimi A, Cheikh A. An analysis of vibration and buckling behaviors of nano-composite beams reinforced with agglomerated carbon nanotubes via differential quadrature finite element method. Mechanics of Advanced Materials and Structures. 2023:1-19. Available from: https://doi.org/10.1080/15376494.2023.2185706
- Saimi A, Bensaid I, Houalef IE. Dynamic analysis of a porous microbeam model based on refined beam strain gradient theory via differential quadrature hierarchical finite element method. Advances in Materials Research. 2023;12(2):133-59.
- Saimi A, Bensaid I, Civalek Ö. A study on the crack presence effect on dynamical behaviour of bi-directional compositionally imperfect material graded micro beams. Composite Structures. 2023;316:117032. Available from: https://www.sciencedirect.com/science/article/pii/S0263822323003768
- Saimi A, Bensaid I, Khouani B, Mazari MY, Houalef IE, Cheikh A. A Novel Differential Quadrature Galerkin Method for Dynamic and Stability Behaviour of Bi-directional Functionally Graded Porous Micro Beams. European Journal of Computational Mechanics. 2023;32(04):393-440. Available from: https://journals.riverpublish-ers.com/index.php/EJCM/article/view/23191
- Tadi Beni Y. Size-dependent electromechanical bending, buckling, and free vibration analysis of functionally graded piezoelectric nanobeams. Journal of Intelligent Material Systems and Structures. 2016;27(16):2199-215. Available from: https://journals.sagepub.com/doi/abs/10.1177/1045389X15624798
- Tadi Beni Y. Size-dependent analysis of piezoelectric nanobeams including electro-mechanical coupling. Mechanics Research Communications. 2016;75:67-80. Available from: https://www.sciencedirect.com/science/article/pii/S0093641316300477
- Senthil VG, Vasundara VV, Vijay KV. A review and critique of theories for piezoelectric laminates. Smart Materials and Structures. 2000;9(1):24. Available from: https://dx.doi.org/10.1088/0964-1726/9/1/304
- Wang Q, Quek ST, Sun CT, Liu X. Analysis of piezoelectric coupled circular plate. Smart Materials and Structures. 2001;10(2):229. Available from: https://dx.doi.org/10.1088/0964-1726/10/2/308
- Liu C, Liu B, Zhao L, Xing Y, Ma C, Li H. A differential quadrature hierarchical finite element method and its applications to vibration and bending of Mindlin plates with curvilinear domains. International Journal for Numerical Methods in Engineering. 2017;109(2):174-97. Available from: https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5277
- Li JY. Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. International Journal of Engineering Science. 2000;38(18):1993-2011. Available from: https://www.sciencedirect.com/science/article/pii/S0020722500000148
- Zhang GY, Gao XL. Elastic wave propagation in 3-D periodic composites: Band gaps incorporating microstructure effects. Composite Structures. 2018;204:920-32. Available from: https://www.sciencedirect.com/science/article/pii/S0263822318318270