Pull-in behaviors of micro-beam made of bidirectional functionally grade materials based on modified couple stress theory
Document Type : Research Paper
Authors
1 Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran
2 Department of physics, center of basic science khatam ol-anbia (PBU) university, Tehran, Iran
Abstract
In this paper, pull-in behavior of cantilever micro/nano beams made of bi directional functionally graded materials (BDFGM) with small scale effects under electrostatic force is investigated. Couple stress theory is employed to study the influence of small-scale on pull-in behavior. The material properties except Poisson’s ratio obey the arbitrary function in the thickness and length direction. The approximate analytical solutions for the pull-in voltage and pull-in displacement of the microbeams are derived using the Galerkin method. Comparison between the results of present work with other article for pull-in behavior of a microbeams made of isotropic material reveals the accuracy of this study. Numerical results explored the effects of material length scale parameter, inhomogeneity constant, gap distance and dimensionless thickness.
Keywords
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[6] A. Hadi, A. Rastgoo, N. Haghighipour, A. Bolhassani, Numerical modelling of a spheroid living cell membrane under hydrostatic pressure, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, No. 8, pp. 083501, 2018.
[7] A. C. Eringen, 2002, Nonlocal continuum field theories, Springer Science & Business Media,
[8] A. C. Eringen, Nonlocal polar elastic continua, International Journal of Engineering Science, Vol. 10, No. 1, pp. 1-16, 1972/01/01/, 1972.
[9] A. C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of applied physics, Vol. 54, No. 9, pp. 4703-4710, 1983.
[10] A. C. Eringen, Theory of micromorphic materials with memory, International Journal of Engineering Science, Vol. 10, No. 7, pp. 623-641, 1972/07/01/, 1972.
[11] D. C. C. Lam, F. Yang, A. C. M. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, Vol. 51, No. 8, pp. 1477-1508, 8//, 2003.
[12] R. A. Toupin, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis, Vol. 11, No. 1, pp. 385-414, January 01, 1962.
[13] R. D. Mindlin, H. F. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, Vol. 11, No. 1, pp. 415-448, January 01, 1962.
[14] W. Koiter, Couple stresses in the theory of elasticity, I & II, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Vol. 67, pp. 17-44, 1964.
[15] F. Yang, A. C. M. Chong, D. C. C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, Vol. 39, No. 10, pp. 2731-2743, 2002/05/01/, 2002.
[16] S. Kong, Size effect on pull-in behavior of electrostatically actuated microbeams based on a modified couple stress theory, Applied Mathematical Modelling, Vol. 37, No. 12, pp. 7481-7488, 2013/07/01/, 2013.
[17] H. M. Ma, X. L. Gao, J. N. Reddy, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids, Vol. 56, No. 12, pp. 3379-3391, 2008/12/01/, 2008.
[18] S. Park, X. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, Vol. 16, No. 11, pp. 2355, 2006.
[19] W. Xia, L. Wang, L. Yin, Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration, International Journal of Engineering Science, Vol. 48, No. 12, pp. 2044-2053, 2010/12/01/, 2010.
[20] M. Asghari, M. H. Kahrobaiyan, M. T. Ahmadian, A nonlinear Timoshenko beam formulation based on the modified couple stress theory, International Journal of Engineering Science, Vol. 48, No. 12, pp. 1749-1761, 2010/12/01/, 2010.
[21] M. Rahaeifard, M. H. Kahrobaiyan, M. Asghari, M. T. Ahmadian, Static pull-in analysis of microcantilevers based on the modified couple stress theory, Sensors and Actuators A: Physical, Vol. 171, No. 2, pp. 370-374, 2011/11/01/, 2011.
[22] L.-L. Ke, Y.-S. Wang, J. Yang, S. Kitipornchai, Nonlinear free vibration of size-dependent functionally graded microbeams, International Journal of Engineering Science, Vol. 50, No. 1, pp. 256-267, 2012.
[23] E. Jomehzadeh, H. R. Noori, A. R. Saidi, The size-dependent vibration analysis of micro-plates based on a modified couple stress theory, Physica E: Low-dimensional Systems and Nanostructures, Vol. 43, No. 4, pp. 877-883, 2011/02/01/, 2011.
[24] L. Yin, Q. Qian, L. Wang, W. Xia, Vibration analysis of microscale plates based on modified couple stress theory, Acta Mechanica Solida Sinica, Vol. 23, No. 5, pp. 386-393, 2010.
[25] M. Asghari, Geometrically nonlinear micro-plate formulation based on the modified couple stress theory, International Journal of Engineering Science, Vol. 51, No. Supplement C, pp. 292-309, 2012/02/01/, 2012.
[26] M. Farajpour, A. Shahidi, A. Hadi, A. Farajpour, Influence of initial edge displacement on the nonlinear vibration, electrical and magnetic instabilities of magneto-electro-elastic nanofilms, Mechanics of Advanced Materials and Structures, Vol. 26, No. 17, pp. 1469-1481, 2019.
[27] M. Mohammad-Abadi, A. R. Daneshmehr, Modified couple stress theory applied to dynamic analysis of composite laminated beams by considering different beam theories, International Journal of Engineering Science, Vol. 87, No. Supplement C, pp. 83-102, 2015/02/01/, 2015.
[28] M. Z. Nejad, A. Hadi, A. Rastgoo, Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory, International Journal of Engineering Science, Vol. 103, pp. 1-10, 2016.
[29] M. Z. Nejad, A. Hadi, Non-local analysis of free vibration of bi-directional functionally graded Euler–Bernoulli nano-beams, International Journal of Engineering Science, Vol. 105, pp. 1-11, 2016/08/01/, 2016.
[30] M. Z. Nejad, A. Hadi, Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler–Bernoulli nano-beams, International Journal of Engineering Science, Vol. 106, pp. 1-9, 2016/09/01/, 2016.
[31] A. Daneshmehr, A. Rajabpoor, A. Hadi, Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories, International Journal of Engineering Science, Vol. 95, pp. 23-35, 2015.
[32] A. Hadi, M. Z. Nejad, M. Hosseini, Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, Vol. 128, pp. 12-23, 2018.
[33] M. Z. Nejad, A. Hadi, A. Farajpour, Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials, Structural Engineering and Mechanics, Vol. 63, No. 2, pp. 161-169, 2017.
[34] M. Hosseini, M. Shishesaz, K. N. Tahan, A. Hadi, Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials, International Journal of Engineering Science, Vol. 109, pp. 29-53, 2016.
[35] Z. Mazarei, M. Z. Nejad, A. Hadi, Thermo-elasto-plastic analysis of thick-walled spherical pressure vessels made of functionally graded materials, International Journal of Applied Mechanics, Vol. 8, No. 04, pp. 1650054, 2016.
[36] A. Hadi, M. Z. Nejad, A. Rastgoo, M. Hosseini, Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory, Steel and Composite Structures, Vol. 26, No. 6, pp. 663-672, 2018.
[37] M. Gharibi, M. Zamani Nejad, A. Hadi, Elastic analysis of functionally graded rotating thick cylindrical pressure vessels with exponentially-varying properties using power series method of Frobenius, Journal of Computational Applied Mechanics, Vol. 48, No. 1, pp. 89-98, 2017.
[38] E. Zarezadeh, V. Hosseini, A. Hadi, Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory, Mechanics Based Design of Structures and Machines, Vol. 48, No. 4, pp. 480-495, 2020.
[39] M. Shishesaz, M. Hosseini, K. N. Tahan, A. Hadi, Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, Acta Mechanica, Vol. 228, No. 12, pp. 4141-4168, 2017.
[40] M. Zamani Nejad, M. Jabbari, A. Hadi, A review of functionally graded thick cylindrical and conical shells, Journal of Computational Applied Mechanics, Vol. 48, No. 2, pp. 357-370, 2017.
[41] M. Zamani Nejad, A. Rastgoo, A. Hadi, Effect of exponentially-varying properties on displacements and stresses in pressurized functionally graded thick spherical shells with using iterative technique, Journal of Solid Mechanics, Vol. 6, No. 4, pp. 366-377, 2014.
[42] M. Hosseini, M. Shishesaz, A. Hadi, Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness, Thin-Walled Structures, Vol. 134, pp. 508-523, 2019.
[43] M. Z. Nejad, A. Hadi, A. Omidvari, A. Rastgoo, Bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams using integral form of Eringen's non-local elasticity theory, Structural Engineering and Mechanics, Vol. 67, No. 4, pp. 417-425, 2018.
[44] M. Z. Nejad, N. Alamzadeh, A. Hadi, Thermoelastoplastic analysis of FGM rotating thick cylindrical pressure vessels in linear elastic-fully plastic condition, Composites Part B: Engineering, Vol. 154, pp. 410-422, 2018.
[45] M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, A. Rastgoo, Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads, European Journal of Mechanics-A/Solids, Vol. 77, pp. 103793, 2019.
[46] A. Hadi, A. Rastgoo, A. Daneshmehr, F. Ehsani, Stress and strain analysis of functionally graded rectangular plate with exponentially varying properties, Indian Journal of Materials Science, Vol. 2013, 2013.
[47] A. Barati, A. Hadi, M. Z. Nejad, R. Noroozi, On vibration of bi-directional functionally graded nanobeams under magnetic field, Mechanics Based Design of Structures and Machines, pp. 1-18, 2020.
[48] R. Noroozi, A. Barati, A. Kazemi, S. Norouzi, A. Hadi, Torsional vibration analysis of bi-directional FG nano-cone with arbitrary cross-section based on nonlocal strain gradient elasticity, Advances in nano research, Vol. 8, No. 1, pp. 13-24, 2020.
[49] A. Barati, M. M. Adeli, A. Hadi, Static torsion of bi-directional functionally graded microtube based on the couple stress theory under magnetic field, International Journal of Applied Mechanics, Vol. 12, No. 02, pp. 2050021, 2020.
[50] M. Najafzadeh, M. M. Adeli, E. Zarezadeh, A. Hadi, Torsional vibration of the porous nanotube with an arbitrary cross-section based on couple stress theory under magnetic field, Mechanics Based Design of Structures and Machines, pp. 1-15, 2020.
[51] K. Dehshahri, M. Z. Nejad, S. Ziaee, A. Niknejad, A. Hadi, Free vibrations analysis of arbitrary three-dimensionally FGM nanoplates, Advances in nano research, Vol. 8, No. 2, pp. 115-134, 2020.
[52] M. M. Khoram, M. Hosseini, A. Hadi, M. Shishehsaz, Bending Analysis of Bi-Directional FGM Timoshenko Nano-Beam subjected to mechanical and magnetic forces and resting on Winkler-Pasternak foundation, International Journal of Applied Mechanics, 2020.
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Volume 51, Issue 2
December 2020Pages 472-481
- Receive Date: 12 November 2020
- Revise Date: 21 November 2020
- Accept Date: 21 November 2020