A New Hyperbolic Two-Unknown Beam Model for Bending and Buckling Analysis of a Nonlocal Strain Gradient Nanobeams

Abstract:

In present paper, a novel two variable shear deformation beam theories are developed and applied to investigate the combined effects of nonlocal stress and strain gradient on the bending and buckling behaviors of nanobeams by using the nonlocal strain gradient theory. The advantage of this theory relies on its two-unknown displacement field as the Euler-Bernoulli beam theory, and it is capable of accurately capturing shear deformation effects, instead of three as in the well-known first shear deformation theory and higher-order shear deformation theory. A shear correction factor is, therefore, not needed. Equations of motion are obtained via Hamilton’s principle. Analytical solutions for the bending and buckling analysis are given for simply supported beams. Efficacy of the proposed model is shown through illustrative examples for bending buckling of nanobeams. The numerical results obtained are compared with those of other higher-order shear deformation beam theory. The results obtained are found to be accurate. Verification studies show that the proposed theory is not only accurate and simple in solving the bending and buckling behaviour of nanobeams, but also comparable with the other shear deformation theories which contain more number of unknowns

• Periodical: Journal of Nano Research (Volume 57)
• Pages: 175-191
• DOI: https://doi.org/10.4028/www.scientific.net/JNanoR.57.175
• Online since: April 2019
• Authors: Wafa Adda Bedia, Mohammed Sid Ahmed Houari, Aicha Bessaim, Abdelmoumen Anis Bousahla, Abdelouahed Tounsi, Tareq Saeed, Mohammed S. Alhodaly
• Keywords: Bending, Buckling, Nanobeams, Nonlocal Strain Gradient, Simple Two-Unknown Theory

References

[1] Ekinci, K.L. and Roukes, M.L. (2005), Nanoelectromechanical systems,, Rev. Sci. Instruments, 76(6), 061101.

[2] Rahmani, O., Refaeinejad, V. and Hosseini, S.A.H. (2017), Assessment of various nonlocal higher order theories for the bending and buckling behavior of functionally graded nanobeams,, Steel Compos. Struct., Int. J., 23(3), 339-350.

DOI: 10.12989/scs.2017.23.3.339

[3] Houari, M.S.A., Bessaim, A,. Bernard, F., Tounsi, A. and Mahmoud, S.R. (2018), Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter,, Steel Compos. Struct., Int. J., 28(1), 13-24.

[4] Eringen, A.C. (1972), Nonlocal polar elastic continua,, Int. J. Eng. Sci., 10(1), 1-16.

[5] Eringen, A.C. (1983), On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves,, J. Appl. Phys., 54(9), 4703-4710.

DOI: 10.1063/1.332803

[6] Mindlin, R.D. (1964), Micro-structure in linear elasticity,, Arch. Rational Mech. Anal., 16(1), 51-78.

DOI: 10.1007/bf00248490

[7] Mindlin, R.D. (1965), Second gradient of strain and surface-tension in linear elasticity,, Int. J. Solids Struct., 1(4), 417-438.

DOI: 10.1016/0020-7683(65)90006-5

[8] Papargyri-Beskou, S., Tsepoura, K.G., Polyzos, D. and Beskos, D.E. (2003), Bending and stability analysis of gradient elastic beams,, Int. J. Solids Struct., 40(2), 385-400.

DOI: 10.1016/s0020-7683(02)00522-x

[9] Yang, F.A.C.M., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), Couple stress based strain gradient theory for elasticity,, Int. J. Solids Struct., 39(10), 2731-2743.

DOI: 10.1016/s0020-7683(02)00152-x

[10] Askes, H. and Aifantis, E.C. (2009), Gradient elasticity and flexural wave dispersion in carbon nanotubes,, Phys. Rev. B, 80(19), 195412.

DOI: 10.1103/physrevb.80.195412

[11] Wang, Q. (2005), Wave propagation in carbon nanotubes via nonlocal continuum mechanics,, Journal of Applied physics, 98(12), 124301.

DOI: 10.1063/1.2141648

[12] Reddy, J. N. (2007), Nonlocal theories for bending, buckling and vibration of beams,, International Journal of Engineering Science, 45(2-8), 288-307.

DOI: 10.1016/j.ijengsci.2007.04.004

[13] Pradhan, S.C. and Murmu, T. (2010), Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever,, Physica E: Low-dimensional Syst. Nanostruct., 42(7), 1944-1949.

DOI: 10.1016/j.physe.2010.03.004

[14] Civalek, Ö., Demir, Ç. and Akgöz, B. (2010), Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model,, Math. Comput. Appl., 15(2), 289-298.

DOI: 10.3390/mca15020289

[15] Tounsi, A., Benguediab, S., Adda, B., Semmah, A., and Zidour, M. (2013), Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes,, Advances in nano research., 1(1), 1-11.

DOI: 10.12989/anr.2013.1.1.001

[16] Benguediab, S., Semmah, A., Chaht, F. L., Mouaz, S. and Tounsi, A. (2014), An investigation on the characteristics of bending, buckling and vibration of nanobeams via nonlocal beam theory,, International Journal of Computational Methods., 11(06), 1350085.

DOI: 10.1142/s0219876213500850

[17] Besseghier, A., Heireche, H., Bousahla, A. A., Tounsi, A., and Benzair, A. (2015), Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix,, Advances in nano research., 3(1), 29-37.

DOI: 10.12989/anr.2015.3.1.029

[18] Bensaid, I. (2017). A refined nonlocal hyperbolic shear deformation beam model for bending and dynamic analysis of nanoscale beams. ADVANCES IN NANO RESEARCH, 5(2), 113-126.

[19] Ehyaei, J., Akbarshahi, A., & Shafiei, N. (2017). Influence of porosity and axial preload on vibration behavior of rotating FG nanobeam. ADVANCES IN NANO RESEARCH, 5(2), 141-169.

[20] Ehyaei, J., & Daman, M. (2017). Free vibration analysis of double walled carbon nanotubes embedded in an elastic medium with initial imperfection. ADVANCES IN NANO RESEARCH, 5(2), 179-192.

[21] Kumar, B. R. (2018). Investigation on mechanical vibration of double-walled carbon nanotubes with inter-tube Van der waals forces. ADVANCES IN NANO RESEARCH, 6(2), 135-145.

[22] Eltaher, M. A., Samir A. Emam, and F. F. Mahmoud. Static and stability analysis of nonlocal functionally graded nanobeams., Composite Structures 96 (2013): 82-88.

DOI: 10.1016/j.compstruct.2012.09.030

[23] Eltaher, M. A., Khater, M. E., &Emam, S. A. (2016). A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Applied Mathematical Modelling, 40(5-6), 4109-4128.

DOI: 10.1016/j.apm.2015.11.026

[24] Thai, Huu-Tai, et al. A review of continuum mechanics models for size-dependent analysis of beams and plates., Composite Structures 177 (2017): 196-219.

DOI: 10.1016/j.compstruct.2017.06.040

[25] Mahmoud, F. F., Eltaher, M. A., Alshorbagy, A. E., & Meletis, E. I. (2012). Static analysis of nanobeams including surface effects by nonlocal finite element. Journal of Mechanical Science and Technology, 26(11), 3555-3563.

DOI: 10.1007/s12206-012-0871-z

[26] Eltaher, M. A., Khairy, A., Sadoun, A. M., & Omar, F. A. (2014a). Static and buckling analysis of functionally graded Timoshenko nanobeams. Applied Mathematics and Computation, 229, 283-295.

DOI: 10.1016/j.amc.2013.12.072

[27] Eltaher, M. A., Hamed, M. A., Sadoun, A. M., & Mansour, A. (2014b). Mechanical analysis of higher order gradient nanobeams. Applied Mathematics and Computation, 229, 260-272.

DOI: 10.1016/j.amc.2013.12.076

[28] Alshorbagy, A. E., Eltaher, M. A., & Mahmoud, F. F. (2013). Static analysis of nanobeams using nonlocal FEM. Journal of Mechanical Science and Technology, 27(7), 2035-2041.

DOI: 10.1007/s12206-013-0212-x

[29] Khater, M. E., Eltaher, M. A., Abdel-Rahman, E., &Yavuz, M. (2014). Surface and thermal load effects on the buckling of curved nanowires. Engineering Science and Technology, an International Journal, 17(4), 279-283.

DOI: 10.1016/j.jestch.2014.07.003

[30] K., Bouafia, A., Kaci, M.S.A., Houri, A.Benzair, and A.Tounsi, A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams, Smart Struct. Syst., 19(2), (2017), 115-126.

DOI: 10.12989/sss.2017.19.2.115

[31] Eltaher, M. A., Kabeel, A. M., Almitani, K. H., &Abdraboh, A. M. (2018a). Static bending and buckling of perforated nonlocal size-dependent nanobeams. Microsystem Technologies, 24 (12), 4881-4893.

DOI: 10.1007/s00542-018-3905-3

[32] Eltaher, M. A., Omar, F. A., Abdalla, W. S., & Gad, E. H. (2018b). Bending and vibrational behaviors of piezoelectric nonlocal nanobeam including surface elasticity. Waves in Random and Complex Media, 1-17.

DOI: 10.1080/17455030.2018.1429693

[33] Eltaher, M. A., Fouda, N., El-midany, T., &Sadoun, A. M. (2018c). Modified porosity model in analysis of functionally graded porous nanobeams. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40(3), 141.

DOI: 10.1007/s40430-018-1065-0

[34] Emam, S., Eltaher, M., Khater, M., &Abdalla, W. (2018). Postbuckling and Free Vibration of Multilayer Imperfect Nanobeams under a Pre-Stress Load. Applied Sciences, 8(11), 2238.

DOI: 10.3390/app8112238

[35] Karami, B., Janghorban, M., Tounsi, A. (2018a), Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles,, Steel and Composite Structures, 27(2), 201-216.

[36] Karami, B., Janghorban, M., Tounsi, A. (2018b), Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory,, Thin-Walled Structures, 129, 251–264.

DOI: 10.1016/j.tws.2018.02.025

[37] Ebrahimi, F., Barati, M. R. and Dabbagh, A. (2016a), A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates,, International Journal of Engineering Science., 107, 169-182.

DOI: 10.1016/j.ijengsci.2016.07.008

[38] Fleck, N. A. and Hutchinson, J. W. (1993), A phenomenological theory for strain gradient effects in plasticity,, Journal of the Mechanics and Physics of Solids., 41(12), 1825-1857.

DOI: 10.1016/0022-5096(93)90072-n

[39] Lam, D. C., Yang, F., Chong, A. C. M., Wang, J. and Tong, P. (2003), Experiments and theory in strain gradient elasticity,, Journal of the Mechanics and Physics of Solids., 51(8), 1477-1508.

DOI: 10.1016/s0022-5096(03)00053-x

[40] Stölken, J. S. and Evans, A. G. (1998), A microbend test method for measuring the plasticity length scale,, Acta Materialia., 46(14), 5109-5115.

DOI: 10.1016/s1359-6454(98)00153-0

[41] Ebrahimi, F., Barati, M. R. and Dabbagh, A. (2016b), A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates,, International Journal of Engineering Science., 107, 169-182.

DOI: 10.1016/j.ijengsci.2016.07.008

[42] Reddy, J. N. (2011), Microstructure-dependent couple stress theories of functionally graded beams,, Journal of the Mechanics and Physics of Solids., 59(11), 2382-2399.

DOI: 10.1016/j.jmps.2011.06.008

[43] Akgöz, B. and Civalek, Ö. (2014), Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium,, International Journal of Engineering Science., 85, 90-104.

DOI: 10.1016/j.ijengsci.2014.08.011

[44] Li, Y. S., Feng, W. J. and Cai, Z. Y. (2014), Bending and free vibration of functionally graded piezoelectric beam based on modified strain gradient theory,, Composite Structures., 115, 41-50.

DOI: 10.1016/j.compstruct.2014.04.005

[45] Mohammadimehr, M., Farahi, M. J. and Alimirzaei, S. (2016), Vibration and wave propagation analysis of twisted micro-beam using strain gradient theory,, Applied Mathematics and Mechanics., 37(10), 1375-1392.

DOI: 10.1007/s10483-016-2138-9

[46] Lim, C. W., Zhang, G. and Reddy, J. N. (2015), A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation,, Journal of the Mechanics and Physics of Solids., 78, 298-313.

DOI: 10.1016/j.jmps.2015.02.001

[47] Li, L., Hu, Y. and Ling, L. (2015), Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory,, Composite Structures., 133, 1079-1092.

DOI: 10.1016/j.compstruct.2015.08.014

[48] Li, L., Li, X. and Hu, Y. (2016), Free vibration analysis of nonlocal strain gradient beams made of functionally graded material,, International Journal of Engineering Science., 102, 77-92.

DOI: 10.1016/j.ijengsci.2016.02.010

[49] Xu, X. J., Wang, X. C., Zheng, M. L. and Ma, Z. (2017), Bending and buckling of nonlocal strain gradient elastic beams,, Composite Structures., 160, 366-377.

DOI: 10.1016/j.compstruct.2016.10.038

[50] Li, X., Li, L., Hu, Y., Ding, Z. and Deng, W. (2017), Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory,, Composite Structures., 165, 250-265.

DOI: 10.1016/j.compstruct.2017.01.032

[51] Sahmani, S., Aghdam, M. M. and Rabczuk, T. (2018), Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory,, Composite Structures., 186, 68-78.

DOI: 10.1016/j.compstruct.2017.11.082

[52] Ghugal, Y. M. and Shimpi, R. P. (2001), A review of refined shear deformation theories for isotropic and anisotropic laminated beams,, Journal of reinforced plastics and composites., 20(3), 255-272.

DOI: 10.1177/073168401772678283

[53] Castellazzi, G., Krysl, P. and Bartoli, I. (2013), A displacement-based finite element formulation for the analysis of laminated composite plates,, Composite Structures., 95, 518-527.

DOI: 10.1016/j.compstruct.2012.08.029

[54] Reddy, J. N. (1984), A simple higher-order theory for laminated composite plates,, Journal of applied mechanics., 51(4), 745-752.

DOI: 10.1115/1.3167719

[55] Soldatos, K. P. (1992), A transverse shear deformation theory for homogeneous monoclinic plates,, Acta Mechanica., 94(3-4), 195-220.

DOI: 10.1007/bf01176650

[56] Hebali, H., Tounsi, A., Houari, M. S. A., Bessaim, A. and Bedia, E. A. A. (2014), New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates,, Journal of Engineering Mechanics., 140(2), 374-383.

DOI: 10.1061/(asce)em.1943-7889.0000665

[57] Houari, M. S. A., Tounsi, A., Bessaim, A. and Mahmoud, S. R. (2016), A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates,, Steel and Composite Structures., 22(2), 257-276.

DOI: 10.12989/scs.2016.22.2.257

[58] Lei, J., He, Y., Zhang, B., Gan, Z. and Zeng, P. (2013), Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory,, International Journal of Engineering Science., 72, 36-52.

DOI: 10.1016/j.ijengsci.2013.06.012

[59] Mohammad-Abadi, M. and Daneshmehr, A. R. (2014), Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions,, International Journal of Engineering Science., 74, 1-14.

DOI: 10.1016/j.ijengsci.2013.08.010

[60] Kolahchi, R. and Bidgoli, A. M. (2016), Size-dependent sinusoidal beam model for dynamic instability of single-walled carbon nanotubes,, Applied Mathematics and Mechanics., 37(2), 265-274.

DOI: 10.1007/s10483-016-2030-8

[61] Refaeinejad, V., Rahmani, O. and Hosseini, S. A. H. (2017), Evaluation of nonlocal higher order shear deformation models for the vibrational analysis of functionally graded nanostructures,, Mechanics of Advanced Materials and Structures., 24(13), 1116-1123.

DOI: 10.1080/15376494.2016.1227496

[62] Ebrahimi, F. and Barati, M. R. (2018), Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magneto-electrical field in thermal environment,, Journal of Vibration and Control., 24(3), 549-564.

DOI: 10.1177/1077546316646239

[63] Arefi, M., Pourjamshidian, M. and Arani, A. G. (2017), Application of nonlocal strain gradient theory and various shear deformation theories to nonlinear vibration analysis of sandwich nano-beam with FG-CNTRCs face-sheets in electro-thermal environment,, Applied Physics A., 123(5), 323.

DOI: 10.1007/s00339-017-0922-5

[64] Lu, L., Guo, X. and Zhao, J. (2017a), A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms,, International Journal of Engineering Science., 119, 265-277.

DOI: 10.1016/j.ijengsci.2017.06.024

[65] Lu, L., Guo, X. and Zhao, J. (2017b), Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory,, International Journal of Engineering Science., 116, 12-24.

DOI: 10.1016/j.ijengsci.2017.03.006

[66] Aifantis, E. C. (1992), On the role of gradients in the localization of deformation and fracture,, International Journal of Engineering Science., 30(10), 1279-1299.

DOI: 10.1016/0020-7225(92)90141-3

[67] Youcef, D.O., Kaci, A., Benzair, A., Bousahla, A.A., Tounsi, A. (2018), Dynamic analysis of nanoscale beams including surface stress effects,, Smart Structures and Systems., 21(1), 65-74.

[68] Younsi, A., Tounsi, A, Zaoui, F.Z., Bousahla, A.A., Mahmoud, S.R. (2018), Novel quasi-3D and 2D shear deformation theories for bending and free vibration analysis of FGM plates,, Geomechanics and Engineering, 14(6), 519-532.

DOI: 10.1016/j.compositesb.2018.09.051

[69] Bourada M, Kaci A, Houari MS, Tounsi A. A new simple shear and normal deformations theory for functionally graded beams. Steel and composite structures. 2015;18(2):409-23.

DOI: 10.12989/scs.2015.18.2.409

[70] Cherif, R.H., Meradjah, M., Zidour, M., Tounsi, A., Belmahi, H., Bensattalah, T. (2018), Vibration analysis of nano beam using differential transform method including thermal effect,, Journal of Nano Research, 54, 1-14.

DOI: 10.4028/www.scientific.net/jnanor.54.1

[71] Bounouara, F., Benrahou, K.H., Belkorissat, I. and Tounsi, A. (2016), A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation,, Steel Compos. Struct., 20(2), 227-249.

DOI: 10.12989/scs.2016.20.2.227

[72] Bakhadda, B., Bachir Bouiadjra, M., Bourada, F., Bousahla, A.A., Tounsi, A., Mahmoud, S.R. (2018), Dynamic and bending analysis of carbon nanotube-reinforced composite plates with elastic foundation,, Wind and Structures, 27(5), 311-324.