Buckling Analysis of Orthotropic Nanoscale Plates Resting on Elastic Foundations

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This work presents the buckling investigation of embedded orthotropic nanoplates by using a new hyperbolic plate theory and nonlocal small-scale effects. The main advantage of this theory is that, in addition to including the shear deformation effect, the displacement field is modeled with only three unknowns and three governing equation as the case of the classical plate theory (CPT) and which is even less than the first order shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT). A shear correction factor is, therefore, not required. Nonlocal differential constitutive relations of Eringen is employed to investigate effects of small scale on buckling of the rectangular nanoplate. The elastic foundation is modeled as two-parameter Pasternak foundation. The equations of motion of the nonlocal theories are derived and solved via Navier's procedure for all edges simply supported boundary conditions. The proposed theory is compared with other plate theories. Analytical solutions for buckling loads are obtained for single-layered graphene sheets with isotropic and orthotropic properties. The results presented in this study may provide useful guidance for design of orthotropic graphene based nanodevices that make use of the buckling properties of orthotropic nanoplates. Verification studies show that the proposed theory is not only accurate and simple in solving the buckling nanoplates, but also comparable with the other higher-order shear deformation theories which contain more number of unknowns. Keywords: Buckling; orthotropic nanoplates; a simple 3-unknown theory; nonlocal elasticity theory; Pasternak’s foundations. * Corresponding author; Email-tou_abdel@yahoo.com

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42-56

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B. Kadari et al., "Buckling Analysis of Orthotropic Nanoscale Plates Resting on Elastic Foundations", Journal of Nano Research, Vol. 55, pp. 42-56, 2018

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November 2018

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* - Corresponding Author

[1] K.L. Ekinci, M.L. Roukes, Nanoelectromechanical systems, Review of Scientific Instruments., 76 (2005) 061101.

DOI: https://doi.org/10.1063/1.1927327

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

DOI: https://doi.org/10.12989/scs.2017.23.3.339

[3] H. Bellifa, K.H. Benrahou, A.A. Bousahla, A. Tounsi, S.R. Mahmoud, A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams, Structural Engineering and Mechanics, 62 (2017) 695 - 702.

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

DOI: https://doi.org/10.12989/sss.2017.19.2.115

[5] B. Karami, M. Janghorban, A. Tounsi, 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 (2018) 251–264.

DOI: https://doi.org/10.1016/j.tws.2018.02.025

[6] D.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., 51 (2003) 1477-1508.

DOI: https://doi.org/10.1016/s0022-5096(03)00053-x

[7] A.W. McFarland, M.A. Poggi, M.J. Doyle, L.A. Bottomley, J.S. Colton, Influence of surface stress on the resonance behavior of microcantilevers, Applied Physics Letters., 87 (2005) 053505.

DOI: https://doi.org/10.1063/1.2006212

[8] A.C. Eringen, Nonlocal polar elastic continua, International Journal of Engineering Science., 10 (1972) 1-16.

[9] A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics., 54 (1983) 4703-4710.

DOI: https://doi.org/10.1063/1.332803

[10] R. Aghababaei, J.N. Reddy, Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates, Journal of Sound and Vibration., 326 (2009) 277-289.

DOI: https://doi.org/10.1016/j.jsv.2009.04.044

[11] R. Kolahchi, R., A.M.M. Bidgoli, M.M. Heydari, Size-dependent bending analysis of FGM nano-sinusoidal plates resting on orthotropic elastic medium, Structural Engineering and Mechanics., 55 (2015) 1001-1014.

DOI: https://doi.org/10.12989/sem.2015.55.5.1001

[12] S.C. Pradhan, J.K. Phadikar, Nonlocal elasticity theory for vibration of nanoplates, Journal of Sound and Vibration., 325 (2009) 206-223.

DOI: https://doi.org/10.1016/j.jsv.2009.03.007

[13] T. Murmu, S.C. Pradhan,Vibration analysis of nanoplates under uniaxial prestressed conditions via nonlocal elasticity, J.Appl.Phys.,106 (2009) 104301.

DOI: https://doi.org/10.1063/1.3233914

[14] A. Zemri, M.S.A. Houari, A.A. Bousahla, A. Tounsi, A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory, Structural Engineering and Mechanics, 54 (2015) 693-710.

DOI: https://doi.org/10.12989/sem.2015.54.4.693

[15] G.L. She, Y.R. Ren, F.G. Yuan, W.S. Xiao, On vibrations of porous nanotubes, International Journal of Engineering Science, 125 (2018) 23-35.

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

DOI: https://doi.org/10.12989/scs.2016.20.2.227

[17] S.D. Akbas, Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium, Smart Structures and Systems., 18 (2016) 1125-1143.

DOI: https://doi.org/10.12989/sss.2016.18.6.1125

[18] A. Besseghier, M.S.A. Houari, A. Tounsi, A., S.R. Mahmoud, Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory, Smart Structures and Systems., 19 (2017) 601-614.

[19] A. Mouffoki, E.A. Adda Bedia, M.S.A. Houari, A. Tounsi, S.R. Mahmoud, Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory, Smart Structures Systems, 20 (2017).

[20] S.C. Pradhan, Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory, Physics Letters A, 373 (2009) 4182-4188.

DOI: https://doi.org/10.1016/j.physleta.2009.09.021

[21] H. Babaei, A.R. Shahidi, Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method, Arch. Appl. Mech., 81 (2010) 1051-1062.

DOI: https://doi.org/10.1007/s00419-010-0469-9

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

DOI: https://doi.org/10.1007/s10483-016-2030-8

[23] H. Khetir, M. Bachir Bouiadjra, M.S.A. Houari, A. Tounsi, S.R. Mahmoud, A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates, Struct. Eng. Mech., 64 (2017) 391-402.

[24] M. Yazid, H. Heireche, A. Tounsi, A.A. Bousahla, M.S.A. Houari, A novel nonlocal refined plate theory for stability response of orthotropic single-layer graphene sheet resting on elastic medium, Smart Structures and Systems., 21 (2018) 15-25.

[25] A. Bouadi, A.A. Bousahla, M.S.A. Houari, H. Heireche, A. Tounsi, A new nonlocal HSDT for analysis of stability of single layer graphene sheet, Advances in Nano Research, 6 (2018) 147-162.

[26] H.T. Thai, T.P. Vo, T.K. Nguyen, S.E. Kim, A review of continuum mechanics models for size-dependent analysis of beams and plates, Composite Structures, 177 (2017) 196-219.

DOI: https://doi.org/10.1016/j.compstruct.2017.06.040

[27] P. Lu, P.Q. Zhang, H.P. Lee, C.M. Wang, J.N. Reddy, Non-local elastic plate theories, Proc R Soc A, 463 (2007) 3225-3240.

[28] W.H. Duan, C.M. Wang, Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory, Nanotechnology, 18 (2007) 385704.

DOI: https://doi.org/10.1088/0957-4484/18/38/385704

[29] T. Aksencer, M. Aydogdu, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E, 43 (2011) 954–959.

DOI: https://doi.org/10.1016/j.physe.2010.11.024

[30] S. Pouresmaeeli, S.A. Fazelzadeh, E. Ghavanloo, Exact solution for nonlocal vibration of double-orthotropic nanoplates embedded in elastic medium, Composites Part B Engineering, 43 (2012) 3384–3390.

DOI: https://doi.org/10.1016/j.compositesb.2012.01.046

[31] M. Sari, W.G. Al-Kouz, Vibration analysis of non-uniform orthotropic Kirchhoff plates resting on elastic foundation based on nonlocal elasticity theory, 114 (2016) 1-11.

DOI: https://doi.org/10.1016/j.ijmecsci.2016.05.008

[32] S.C. Pradhan, J.K. Phadikar, Nonlocal elasticity theory for vibration of nanoplates, Journal of Sound and Vibration, 325 (2009) 206-223.

DOI: https://doi.org/10.1016/j.jsv.2009.03.007

[33] R. Ansari, S. Sahmani, Prediction of biaxial buckling behavior of single-layered graphene sheets based on nonlocal plate models and molecular dynamics simulations, Applied Mathematical Modelling, 37 (2013) 7338-7351.

DOI: https://doi.org/10.1016/j.apm.2013.03.004

[34] S. Narendar, Buckling analysis of micro-nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects, Compos. Struct., 93 (2011) 3093–3103.

DOI: https://doi.org/10.1016/j.compstruct.2011.06.028

[35] N. Satish, S. Narendar, S. Gopalakrishnan, Thermal vibration analysis of orthotropic nanoplates based on nonlocal continuum mechanics, Physica E Low-dimensional Systems and Nanostructures, 44 (2012) 1950–(1962).

DOI: https://doi.org/10.1016/j.physe.2012.05.024

[36] T.J.P. Kumar, S. Narendar, S. Gopalakrishnan, Thermal vibration analysis of monolayer graphene embedded in elastic medium based on nonlocal continuum mechanics, Composite Structures, 100 (2013) 332–342.

DOI: https://doi.org/10.1016/j.compstruct.2012.12.039

[37] T. Murmu, S.C. Pradhan, Vibration analysis of nanoplates under uniaxial prestressed conditions via nonlocal elasticity, J. Appl. Phys., 106 (2009) 104301.

DOI: https://doi.org/10.1063/1.3233914

[38] M. Sobhy, Generalized two-variable plate theory for multi-layered graphene sheets with arbitrary boundary conditions, 225 (2014) 2521–2538.

DOI: https://doi.org/10.1007/s00707-014-1093-5

[39] A. Bessaim, M.S.A. Houari, F. Bernard, A. Tounsi, A nonlocal quasi-3D trigonometric plate model for free vibration behaviour of micro/nanoscale plates, Structural Engineering and Mechanics, 56 (2015) 223-240.

DOI: https://doi.org/10.12989/sem.2015.56.2.223

[40] M. Sobhy, A.F. Radwan, A new quasi 3d nonlocal plate theory for vibration and buckling of FGM nanoplates, International Journal of Applied Mechanics, 9 (2017) 1750008.

DOI: https://doi.org/10.1142/s1758825117500089

[41] Y.M. Ghugal, R.P. Shimpi, A review of refined shear deformation theories of isotropic and anisotropic laminated plates, Journal of Reinforced Plastics and Composites., 21 (2002) 775-813.

DOI: https://doi.org/10.1177/073168402128988481

[42] P. Malekzadeh, M. Monajjemzadeh, Dynamic response of functionally graded plates in thermal environment under moving load, Composites Part B Engineering, 45 (2013) 1521-1533.

DOI: https://doi.org/10.1016/j.compositesb.2012.09.022

[43] H. Bellifa, K.H. Benrahou, L. Hadji, M.S.A. Houari, A. Tounsi, Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position, J Braz. Soc. Mech. Sci. Eng., 38 (2016).

DOI: https://doi.org/10.1007/s40430-015-0354-0

[44] B. Bouderba, M.S.A. Houari, A. Tounsi, S.R. Mahmoud, Thermal stability of functionally graded sandwich plates using a simple shear deformation theory, Struct. Eng. Mech., 58 (2016) 397-422.

DOI: https://doi.org/10.12989/sem.2016.58.3.397

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

[46] A. Ahmed, Post buckling analysis of sandwich beams with functionally graded faces using a consistent higher order theory, Int. J. Civil, Struct. Envir., 4 (2014) 59-64.

[47] V.R. Kar, S.K. Panda, Nonlinear flexural vibration of shear deformable functionally graded spherical shell panel, Steel and Composite Structures., 18 (2015) 693-709.

DOI: https://doi.org/10.12989/scs.2015.18.3.693

[48] A. Behravan Rad, Thermo-elastic analysis of functionally graded circular plates resting on a gradient hybrid foundation, Applied Mathematics and Computation, 256 (2015) 276-298.

DOI: https://doi.org/10.1016/j.amc.2015.01.026

[49] S.S. Akavci, Mechanical behavior of functionally graded sandwich plates on elastic foundation, Composites Part B, 96 (2016) 136 – 152.

DOI: https://doi.org/10.1016/j.compositesb.2016.04.035

[50] K. Mehar, S. K. Panda, Free vibration and bending behaviour of CNT reinforced composite plate using different shear deformation theory, In IOP Conference Series: Materials Science and Engineering, 115 (2016) 012014.

DOI: https://doi.org/10.1088/1757-899x/115/1/012014

[51] V.R Kar, S.K. Panda, T.R. Mahapatra, Thermal buckling behaviour of shear deformable functionally graded single/doubly curved shell panel with TD and TID properties, Advances in Materials Research, 5 (2016) 205-221.

DOI: https://doi.org/10.12989/amr.2016.5.4.205

[52] S.M. Aldousari, Bending analysis of different material distributions of functionally graded beam, Appl. Phys. A, 123 (2017) 296.

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

DOI: https://doi.org/10.12989/scs.2016.22.2.257

[54] H. Bellifa, A. Bakora, A. Tounsi, A.A. Bousahla, S.R. Mahmoud, An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates, Steel Compos. Struct., 25 (2017) 257-270.

DOI: https://doi.org/10.12989/scs.2016.22.3.473

[55] A.A. Bousahla, S. Benyoucef, A. Tounsi, S.R. Mahmoud, On thermal stability of plates with functionally graded coefficient of thermal expansion, Struct. Eng. Mech., 60 (2016) 313-335.

DOI: https://doi.org/10.12989/sem.2016.60.2.313

[56] A. Chikh, A. Tounsi, H. Hebali, S.R. Mahmoud, Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT, Smart Structures Systems, 19 (2017) 289-297.

DOI: https://doi.org/10.12989/sss.2017.19.3.289

[57] F. El-Haina, A. Bakora, A.A. Bousahla, A. Tounsi, S.R. Mahmoud, A simple analytical approach for thermal buckling of thick functionally graded sandwich plates, Struct. Eng. Mech., 63 (2017) 585-595.

[58] A. Menasria, A. Bouhadra, A. Tounsi, A.A. Bousahla, S.R. Mahmoud, A new and simple HSDT for thermal stability analysis of FG sandwich plates, Steel. Compos. Struct., 25 (2017) 157-175.

[59] Y. Beldjelili, A. Tounsi, S.R. Mahmoud, Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory, Smart Structures and Systems, 18 (2016) 755-786.

DOI: https://doi.org/10.12989/sss.2016.18.4.755

[60] Z. Belabed, A.A. Bousahla, M.S.A. Houari, A. Tounsi, S.R. Mahmoud, A new 3-unknown hyperbolic shear deformation theory for vibration of functionally graded sandwich plate, Earthquakes and Structures, 14 (2018) 103-115.

[61] H. Hachemi, A. Kaci, M.S.A. Houari, A. Bourada, A. Tounsi, S.R. Mahmoud, A new simple three-unknown shear deformation theory for bending analysis of FG plates resting on elastic foundations, Steel and Composite Structures, 25 (2017) 717-726.

DOI: https://doi.org/10.12989/scs.2016.22.2.257

[62] A. Kaci, M.S.A. Houari, A.A. Bousahla, A. Tounsi, S.R. Mahmoud, Post-buckling analysis of shear-deformable composite beams using a novel simple two-unknown beam theory, Structural Engineering and Mechanics, 65 (2018) 621-631.

[63] F. Klouche, L. Darcherif, M. Sekkal, A. Tounsi, S.R. Mahmoud, An original single variable shear deformation theory for buckling analysis of thick isotropic plates, Structural Engineering and Mechanics, 63 (2017) 439-446.

[64] M. Zidi, M.S.A. Houari, A. Tounsi, A. Bessaim, S.R. Mahmoud, A novel simple two-unknown hyperbolic shear deformation theory for functionally graded beams, Struct. Eng. Mech., 64 (2017) 145-153.

DOI: https://doi.org/10.12989/scs.2016.22.2.257

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

[66] P. Nali, E. Carrera, S. Lecca, Assessments of refined theories for buckling analysis of laminated plates, Composite Structures, 93 (2011) 456-464.

DOI: https://doi.org/10.1016/j.compstruct.2010.08.035

[67] E.J. Brunelle, S.R. Robertson, Vibrations of an initially stressed thick plate, Journal of Sound and Vibration., 45 (1976) 405-416.

DOI: https://doi.org/10.1016/0022-460x(76)90395-3

[68] H. Matsunaga, Free vibration and stability of thick elastic plates subjected to in-plane forces. Int. J. Solids Struct., 31 (1994) 3113-3124.

DOI: https://doi.org/10.1016/0020-7683(94)90044-2

[69] H. Akhavan, H.R. Damavandi Taher, S. Hosseini-Hashemi, Sh. Vahabi, Exact solutions for rectangular Mindlin plates under in-plane loads resting on Pasternak elastic foundation. Part II: Frequency analysis, Computational Materials Science, 44 (2009).

DOI: https://doi.org/10.1016/j.commatsci.2008.07.001

[70] M. Sobhy, Natural frequency and buckling of orthotropic nanoplates resting on two-parameter elastic foundations with various boundary conditions, Journal of Mechanics, 30 (2014) 443-453.

DOI: https://doi.org/10.1017/jmech.2014.46