A concise review of nano-plates

Document Type: Review Paper


1 Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

2 Department of Mechanical Engineering, University of Hormozgan, Bandar Abbas, Iran


Recent works done by nano-engineers and nano-sciences about mechanical behavior of nano-plates including bending, buckling and vibration response were reviewed. The authors used non-classical elasticity theories to explain these behaviors of plate structures. Some of them employed Hamilton’s principle along with stain gradient theory, nonlocal theory, surface theory and couple stress theory to derive the governing equation of nanostructures. Also, the authors have used various plate theories such as classical plate theory (CPT), first-order shear deformation theory (FSDT) and higher-order shear deformation plate theory (HSDT) to explain the linear and nonlinear behavior of nano-plates. Few researchers utilized molecular dynamics or experimental tests to explain size-dependent behavior of nano-plates. Investigated nano-plates were made of homogeneous and functionally graded materials (FGM) and were under mechanical and/or thermal loads. The effect of the magnetic field was considered, in other few researches. Governing equations solved using numerical methods such as differential quadrature method (DQM). The results of recent researches were presented and discussed.


[1]         F. Khademolhosseini, R. K. N. D. Rajapakse, and A. Nojeh, “Torsional buckling of carbon nanotubes based on nonlocal elasticity shell models,” Comput. Mater. Sci., vol. 48, no. 4, pp. 736–742, 2010.

[2]         A. Hadi, M. Z. Nejad, A. Rastgoo, and M. Hosseini, “Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory,” Steel Compos. Struct., vol. 26, no. 6, pp. 663–672, Mar. 2018.

[3]         M. Z. Nejad, A. Hadi, and A. Farajpour, “Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials,” Struct. Eng. Mech., vol. 63, no. 2, pp. 161–169, Jul. 2017.

[4]         M. M. Adeli, A. Hadi, M. Hosseini, and H. H. Gorgani, “Torsional vibration of nano-cone based on nonlocal strain gradient elasticity theory,” Eur. Phys. J. Plus, vol. 132, no. 9, Sep. 2017.

[5]         M. Hosseini, H. H. Gorgani, M. Shishesaz, and A. Hadi, “Size-dependent stress analysis of single-wall carbon nanotube based on strain gradient theory,” Int. J. Appl. Mech., vol. 9, no. 6, Sep. 2017.

[6]         Z. Mazarei, M. Z. Nejad, and A. Hadi, “Thermo-Elasto-Plastic Analysis of Thick-Walled Spherical Pressure Vessels Made of Functionally Graded Materials,” Int. J. Appl. Mech., vol. 8, no. 4, Jun. 2016.

[7]         M. R. Farajpour, A. R. Shahidi, A. Hadi, and A. Farajpour, “Influence of initial edge displacement on the nonlinear vibration, electrical and magnetic instabilities of magneto-electro-elastic nanofilms,” Mech. Adv. Mater. Struct., vol. 26, no. 17, pp. 1469–1481, 2019.

[8]         M. Hosseini, M. Shishesaz, K. N. Tahan, and A. Hadi, “Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials,” Int. J. Eng. Sci., vol. 109, pp. 29–53, Dec. 2016.

[9]         A. Hadi, M. Z. Nejad, and M. Hosseini, “Vibrations of three-dimensionally graded nanobeams,” Int. J. Eng. Sci., vol. 128, pp. 12–23, Jul. 2018.

[10]       A. Daneshmehr, A. Rajabpoor, and A. Hadi, “Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories,” Int. J. Eng. Sci., vol. 95, pp. 23–35, Jul. 2015.

[11]       M. Z. Nejad, A. Hadi, and A. Rastgoo, “Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory,” Int. J. Eng. Sci., vol. 103, pp. 1–10, Jun. 2016.

[12]       M. Z. Nejad, N. Alamzadeh, and A. Hadi, “Thermoelastoplastic analysis of FGM rotating thick cylindrical pressure vessels in linear elastic-fully plastic condition,” Compos. Part B Eng., vol. 154, pp. 410–422, Dec. 2018.

[13]       A. Soleimani, K. Dastani, A. Hadi, and M. H. Naei, “Effect of out-of-plane defects on the postbuckling behavior of graphene sheets based on nonlocal elasticity theory,” Steel Compos. Struct., vol. 30, no. 6, pp. 517–534, 2019.

[14]       M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, and A. Rastgoo, “Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads,” Eur. J. Mech. A/Solids, vol. 77, Sep. 2019.

[15]       E. Zarezadeh, V. Hosseini, and A. Hadi, “Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory,” Mech. Based Des. Struct. Mach., pp. 1–16, Jul. 2019.

[16]       M. Hosseini, M. Shishesaz, and A. Hadi, “Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness,” Thin-Walled Struct., vol. 134, pp. 508–523, Jan. 2019.

[17]       M. Z. Nejad, A. Hadi, A. Omidvari, and A. Rastgoo, “Bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams using integral form of Eringen’s non-local elasticity theory,” Struct. Eng. Mech., vol. 67, no. 4, pp. 417–425, Aug. 2018.

[18]       M. Shishesaz, M. Hosseini, K. Naderan Tahan, and A. Hadi, “Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory,” Acta Mech., vol. 228, no. 12, pp. 4141–4168, Dec. 2017.

[19]       M. Gharibi, M. Zamani Nejad, and A. Hadi, “Elastic analysis of functionally graded rotating thick cylindrical pressure vessels with exponentially-varying properties using power series method of Frobenius,” J. Comput. Appl. Mech., vol. 48, no. 1, pp. 89–98, 2017.

[20]       D. Karličić, M. Cajić, S. Adhikari, P. Kozić, and T. Murmu, “Vibrating nonlocal multi-nanoplate system under inplane magnetic field,” Eur. J. Mech. A/Solids, vol. 64, pp. 29–45, 2017.

[21]       J. Fernández-Sáez, A. Morassi, L. Rubio, and R. Zaera, “Transverse free vibration of resonant nanoplate mass sensors: Identification of an attached point mass,” Int. J. Mech. Sci., vol. 150, pp. 217–225, 2019.

[22]       A. G. Arani, Z. K. Maraghi, and H. K. Arani, “Orthotropic patterns of Pasternak foundation in smart vibration analysis of magnetostrictive nanoplate,” Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci., vol. 230, no. 4, pp. 559–572, 2016.

[23]       M. S. Atanasov, D. Karličić, and P. Kozić, “Forced transverse vibrations of an elastically connected nonlocal orthotropic double-nanoplate system subjected to an in-plane magnetic field,” Acta Mech., vol. 228, no. 6, pp. 2165–2185, 2017.

[24]       H. Bakhshi Khaniki and S. Hosseini-Hashemi, “Dynamic response of biaxially loaded double-layer viscoelastic orthotropic nanoplate system under a moving nanoparticle,” Int. J. Eng. Sci., vol. 115, pp. 51–72, 2017.

[25]       N. Despotovic, “Stability and vibration of a nanoplate under body force using nonlocal elasticity theory,” Acta Mech., vol. 229, no. 1, pp. 273–284, 2018.

[26]       M. Ghadiri, N. Shafiei, and H. Alavi, “Thermo-mechanical vibration of orthotropic cantilever and propped cantilever nanoplate using generalized differential quadrature method,” Mech. Adv. Mater. Struct., vol. 24, no. 8, pp. 636–646, 2017.

[27]       A. Ghorbanpour Arani, R. Kolahchi, and H. Gharbi Afshar, “Dynamic analysis of embedded PVDF nanoplate subjected to a moving nanoparticle on an arbitrary elliptical path,” J. Brazilian Soc. Mech. Sci. Eng., vol. 37, no. 3, pp. 973–986, 2015.

[28]       A. Ghorbanpour-Arani, F. Kolahdouzan, and M. Abdollahian, “Nonlocal buckling of embedded magnetoelectroelastic sandwich nanoplate using refined zigzag theory,” Appl. Math. Mech. (English Ed., vol. 39, no. 4, pp. 529–546, 2018.

[29]       M. Hosseini and A. Jamalpoor, “Analytical Solution for Thermomechanical Vibration of Double-Viscoelastic Nanoplate-Systems Made of Functionally Graded Materials,” J. Therm. Stress., vol. 38, no. 12, pp. 1428–1456, 2015.

[30]       M. Hosseini, A. Jamalpoor, and M. Bahreman, “Small-scale effects on the free vibrational behavior of embedded viscoelastic double-nanoplate-systems under thermal environment,” Acta Astronaut., vol. 129, pp. 400–409, 2016.

[31]       S. Hosseini Hashemi, H. Mehrabani, and A. Ahmadi-Savadkoohi, “Forced vibration of nanoplate on viscoelastic substrate with consideration of structural damping: An analytical solution,” Compos. Struct., vol. 133, pp. 8–15, 2015.

[32]       C. Li, J. J. Liu, M. Cheng, and X. L. Fan, “Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces,” Compos. Part B Eng., vol. 116, pp. 153–169, 2017.

[33]       J. C. Liu, Y. Q. Zhang, and L. F. Fan, “Nonlocal vibration and biaxial buckling of double-viscoelastic-FGM-nanoplate system with viscoelastic Pasternak medium in between,” Phys. Lett. Sect. A Gen. At. Solid State Phys., vol. 381, no. 14, pp. 1228–1235, 2017.

[34]       M. Malikan, “Electro-mechanical shear buckling of piezoelectric nanoplate using modified couple stress theory based on simplified first order shear deformation theory,” Appl. Math. Model., vol. 48, pp. 196–207, 2017.

[35]       M. Malikan, “Analytical predictions for the buckling of a nanoplate subjected to non-uniform compression based on the fourvariable plate theory,” J. Appl. Comput. Mech., vol. 3, no. 3, pp. 218–228, 2017.

[36]       K. Mehar, T. R. Mahapatra, S. K. Panda, P. V. Katariya, and U. K. Tompe, “Finite-element solution to nonlocal elasticity and scale effect on frequency behavior of shear deformable nanoplate structure,” J. Eng. Mech., vol. 144, no. 9, pp. 1–8, 2018.

[37]       A. Moradi, A. Yaghootian, M. Jalalvand, and A. Ghanbarzadeh, “Magneto-Thermo mechanical vibration analysis of FG nanoplate embedded on Visco Pasternak foundation,” J. Comput. Appl. Mech., vol. 49, no. 2, pp. 395–407, 2018.

[38]       P. Ponnusamy and A. Amuthalakshmi, “Dispersion analysis of thermo magnetic effect on double layered nanoplate embedded in an elastic medium,” J. Comput. Theor. Nanosci., vol. 12, no. 8, pp. 1729–1736, 2015.

[39]       E. A. Shahrbabaki, “On three-dimensional nonlocal elasticity: Free vibration of rectangular nanoplate,” Eur. J. Mech. A/Solids, vol. 71, pp. 122–133, 2018.

[40]       Y. Wang, F. Li, Y. Wang, and X. Jing, “Nonlinear responses and stability analysis of viscoelastic nanoplate resting on elastic matrix under 3:1 internal resonances,” Int. J. Mech. Sci., vol. 128–129, pp. 94–104, 2017.

[41]       A. M. Zenkour and M. Sobhy, “Nonlocal piezo-hygrothermal analysis for vibration characteristics of a piezoelectric Kelvin–Voigt viscoelastic nanoplate embedded in a viscoelastic medium,” Acta Mech., vol. 229, no. 1, pp. 3–19, 2018.

[42]       Z. Zhou, D. Rong, C. Yang, and X. Xu, “Rigorous vibration analysis of double-layered orthotropic nanoplate system,” Int. J. Mech. Sci., vol. 123, pp. 84–93, 2017.

[43]       M. R. Barati and H. Shahverdi, “Dynamic modeling and vibration analysis of double-layered multi-phase porous nanocrystalline silicon nanoplate systems,” Eur. J. Mech. A/Solids, vol. 66, pp. 256–268, 2017.

[44]       M. R. Barati, “Magneto-hygro-thermal vibration behavior of elastically coupled nanoplate systems incorporating nonlocal and strain gradient effects,” J. Brazilian Soc. Mech. Sci. Eng., vol. 39, no. 11, pp. 4335–4352, 2017.

[45]       M. R. Barati and H. Shahverdi, “Hygro-thermal vibration analysis of graded double-refined-nanoplate systems using hybrid nonlocal stress-strain gradient theory,” Compos. Struct., vol. 176, pp. 982–995, 2017.

[46]       M. R. Barati and H. Shahverdi, “Frequency analysis of nanoporous mass sensors based on a vibrating heterogeneous nanoplate and nonlocal strain gradient theory,” Microsyst. Technol., vol. 24, no. 3, pp. 1479–1494, 2018.

[47]       F. Ebrahimi, A. Dabbagh, and M. Reza Barati, “Wave propagation analysis of a size-dependent magneto-electro-elastic heterogeneous nanoplate,” Eur. Phys. J. Plus, vol. 131, no. 12, 2016.

[48]       F. Ebrahimi and A. Dabbagh, “Wave dispersion characteristics of orthotropic double-nanoplate-system subjected to a longitudinal magnetic field,” Microsyst. Technol., vol. 24, no. 7, pp. 2929–2939, 2018.

[49]       F. Ebrahimi and M. R. Barati, “Dynamic modeling of embedded nanoplate systems incorporating flexoelectricity and surface effects,” Microsyst. Technol., vol. 25, no. 1, pp. 175–187, 2019.

[50]       M. H. Jalaei and H. T. Thai, “Dynamic stability of viscoelastic porous FG nanoplate under longitudinal magnetic field via a nonlocal strain gradient quasi-3D theory,” Compos. Part B Eng., vol. 175, no. June, p. 107164, 2019.

[51]       B. Karami, D. Shahsavari, and L. Li, “Temperature-dependent flexural wave propagation in nanoplate-type porous heterogenous material subjected to in-plane magnetic field,” J. Therm. Stress., vol. 41, no. 4, pp. 483–499, 2018.

[52]       B. Karami and S. Karami, “Buckling analysis of nanoplate-type temperature-dependent heterogeneous materials,” Adv. nano Res., vol. 7, Jan. 2019.

[53]       Y. Li, L. Yang, Y. Gao, and E. Pan, “Cylindrical bending analysis of a layered two-dimensional piezoelectric quasicrystal nanoplate,” J. Intell. Mater. Syst. Struct., vol. 29, no. 12, pp. 2660–2676, 2018.

[54]       M. Mohammadia and A. Rastgoo, “Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core,” Struct. Eng. Mech., vol. 69, Jan. 2019.

[55]       C. Xu, D. Rong, Z. Tong, Z. Zhou, J. Hu, and X. Xu, “Coupled effect of in-plane magnetic field and size effect on vibration properties of the completely free double-layered nanoplate system,” Phys. E Low-Dimensional Syst. Nanostructures, vol. 108, pp. 215–225, 2019.

[56]       X. Zhang and L. Zhou, “Melnikov’s method for chaos of the nanoplate postulating nonlinear foundation,” Appl. Math. Model., vol. 61, no. 16, pp. 744–749, 2018.

[57]       A. Ghorbanpour Arani and M. H. Zamani, “Nonlocal Free Vibration Analysis of FG-Porous Shear and Normal Deformable Sandwich Nanoplate with Piezoelectric Face Sheets Resting on Silica Aerogel Foundation,” Arab. J. Sci. Eng., vol. 43, no. 9, pp. 4675–4688, 2018.

[58]       A. Ghorbanpour Arani and M. H. Zamani, “Investigation of electric field effect on size-dependent bending analysis of functionally graded porous shear and normal deformable sandwich nanoplate on silica Aerogel foundation,” J. Sandw. Struct. Mater., vol. 21, no. 8, pp. 2700–2734, 2019.

[59]       M. Arefi and A. M. Zenkour, “Thermo-electro-mechanical bending behavior of sandwich nanoplate integrated with piezoelectric face-sheets based on trigonometric plate theory,” Compos. Struct., vol. 162, pp. 108–122, 2017.

[60]       M. Arefi and A. M. Zenkour, “Nonlocal electro-thermo-mechanical analysis of a sandwich nanoplate containing a Kelvin–Voigt viscoelastic nanoplate and two piezoelectric layers,” Acta Mech., vol. 228, no. 2, pp. 475–493, 2017.

[61]       M. Arefi, M. H. Zamani, and M. Kiani, “Size-dependent free vibration analysis of three-layered exponentially graded nanoplate with piezomagnetic face-sheets resting on Pasternak’s foundation,” J. Intell. Mater. Syst. Struct., vol. 29, no. 5, pp. 774–786, 2018.

[62]       M. Arefi, M. Kiani, and T. Rabczuk, “Application of nonlocal strain gradient theory to size dependent bending analysis of a sandwich porous nanoplate integrated with piezomagnetic face-sheets,” Compos. Part B Eng., vol. 168, no. December 2018, pp. 320–333, 2019.

[63]       M. Arefi and A. M. Zenkour, “Effect of thermo-magneto-electro-mechanical fields on the bending behaviors of a three-layered nanoplate based on sinusoidal shear-deformation plate theory,” J. Sandw. Struct. Mater., vol. 21, no. 2, pp. 639–669, 2019.

[64]       A. Jamalpoor and A. Kiani, “Vibration analysis of bonded double-FGM viscoelastic nanoplate systems based on a modified strain gradient theory incorporating surface effects,” Appl. Phys. A Mater. Sci. Process., vol. 123, no. 3, p. 0, 2017.

[65]       I. Mechab, B. Mechab, S. Benaissa, B. Serier, and B. B. Bouiadjra, “Free vibration analysis of FGM nanoplate with porosities resting on Winkler Pasternak elastic foundations based on two-variable refined plate theories,” J. Brazilian Soc. Mech. Sci. Eng., vol. 38, no. 8, pp. 2193–2211, 2016.

[66]       B. Mechab, I. Mechab, S. Benaissa, M. Ameri, and B. Serier, “Probabilistic analysis of effect of the porosities in functionally graded material nanoplate resting on Winkler-Pasternak elastic foundations,” Appl. Math. Model., vol. 40, no. 2, pp. 738–749, 2016.

[67]       M. Hosseini, A. Jamalpoor, and A. Fath, “Surface effect on the biaxial buckling and free vibration of FGM nanoplate embedded in visco-Pasternak standard linear solid-type of foundation,” Meccanica, vol. 52, no. 6, pp. 1381–1396, 2017.

[68]       M. Hosseini, M. Mofidi, A. Jamalpoor, and M. Safi Jahanshahi, “Nanoscale mass nanosensor based on the vibration analysis of embedded magneto-electro-elastic nanoplate made of FGMs via nonlocal Mindlin plate theory,” Microsyst. Technol., vol. 8, no. 2016, 2017.

[69]       E. Jafari, M. Fakoor, and E. Karvand, “Hygrothermal free vibration of multiple magneto-electro-elastic nanoplate system via higher-order nonlocal strain gradient theory,” Appl. Phys. A, vol. 7, 2019.

[70]       A. Jamalpoor, A. Ahmadi-savadkoohi, M. Hosseini, and S. Hosseini-hashemi, “Free vibration and biaxial buckling analysis of double magneto-electro-elastic nanoplate-systems coupled by a visco- Pasternak medium via nonlocal elasticity theory,” Eur. J. Mech. / A Solids, vol. 63, pp. 84–98, 2017.

[71]       E. Khanmirza, A. Jamalpoor, and A. Kiani, “Nano-scale mass sensor based on the vibration analysis of a magneto-electro-elastic nanoplate resting on a visco-Pasternak substrate,” Eur. Phys. J. Plus, 2017.

[72]       A. Kiani, M. Sheikhkhoshkar, A. Jamalpoor, and M. Khanzadi, “Free vibration problem of embedded nanoplate made of functionally graded materials via nonlocal third-order shear deformation theory,” J. Intell. Mater. Syst. Struct., 2017.

[73]       J. Liu, H. Liu, J. Yang, and X. Feng, “Transient response of a circular nanoplate subjected to low velocity impact,” Int. J. Appl. Mech., 2017.

[74]       H. Liu, J. Liu, J. Yang, and X. Feng, “Low velocity impact of a nanoparticle on a rectangular nanoplate: A theoretical study,” Int. J. Mech. Sci., 2016.

[75]       R. Ansari, M. F. Shojaei, V. Mohammadi, R. Gholami, and M. A. Darabi, “A nonlinear shear deformable nanoplate model including surface effects for large amplitude vibrations of rectangular nanoplates with various boundary conditions,” Int. J. Appl. Mech., vol. 7, no. 5, 2015.

[76]       F. Ebrahimi and S. H. S. Hosseini, “Double nanoplate-based NEMS under hydrostatic and electrostatic actuations,” Eur. Phys. J. Plus, 2016.

[77]       F. Ebrahimi and S. H. S. Hosseini, “Effect of temperature on pull-in voltage and nonlinear vibration behavior of nanoplate-based NEMS under hydrostatic and electrostatic actuations,” Acta Mech. Solida Sin., 2017.

[78]       M. Hosseini, A. Bahreman, and M. Jamalpoo, “Thermomechanical vibration analysis of FGM viscoelastic multi ‑ nanoplate system incorporating the surface effects via nonlocal elasticity theory,” Microsyst. Technol., 2016.

[79]       M. Lin, S. Lee, and C. Chen, “Dynamic characteristic analysis of an electrostatically-actuated circular nanoplate subject to surface effects,” Appl. Math. Model., 2018.

[80]       M. L. H. Lai and C. Chen, “Analysis of nonlocal nonlinear behavior of graphene sheet circular nanoplate actuators subject to uniform hydrostatic pressure,” Microsyst. Technol., 2017.

[81]       S. A. Mirkalantari, M. Hashemian, S. A. Eftekhari, and D. Toghraie, “Pull-in instability analysis of rectangular nanoplate based on strain gradient theory considering surface stress effects,” Phys. B Phys. Condens. Matter, 2017.

[82]       W. D. Yang, F. P. Yang, and X. Wang, “Dynamic instability and bifurcation of electrically actuated circular nanoplate considering surface behavior and small scale effect,” Int. J. Mech. Sci., 2017.

[83]       F. Abbasi and A. Ghassemi, “Static bending behaviors of piezoelectric nanoplate considering thermal and mechanical loadings based on the surface elasticity and two variable refined plate theories,” Microsyst. Technol., 2016.

[84]       M. Fathi and A. Ghassemi, “The effects of surface stress and nonlocal small scale on the uniaxial and biaxial buckling of the rectangular piezoelectric nanoplate based on the two variable-refined plate theory,” J. Brazilian Soc. Mech. Sci. Eng., 2017.

[85]       L. Jamali and A. Ghassemi, “Fundamental frequency analysis of rectangular piezoelectric nanoplate under in-plane forces based on surface layer , non-local elasticity , and two variable re fi ned plate hypotheses,” vol. 13, pp. 47–53, 2018.

[86]       M. Karimi, “Surface layer and nonlocal parameter effects on the in-phase and out-of-phase natural frequencies of a double-layer piezoelectric nanoplate under thermo-electro-mechanical loadings,” Microsyst. Technol., 2017.

[87]       M. Karimi and A. R. Shahidi, “A general comparison the surface layer degree on the out-of- phase and in-phase vibration behavior of a skew double-layer magneto – electro – thermo-elastic nanoplate,” Appl. Phys. A, 2019.

[88]       M. Hosseini, M. Mahinzare, and M. Ghadiri, “Magnetic field effect on vibration of a rotary smart size-dependent two-dimensional porous functionally graded nanoplate,” J. Intell. Mater. Syst. Struct. Struct., 2018.

[89]       M. Mahinzare, H. Ranjbarpur, and M. Ghadiri, “Free vibration analysis of a rotary smart two directional functionally graded piezoelectric material in axial symmetry circular nanoplate,” Mech. Syst. Signal Process., vol. 100, pp. 188–207, 2018.

[90]       M. Jamali, T. Shojaee, and B. Mohammadi, “Uniaxial Buckling Analysis Comparison of Nanoplate and Nanocomposite Plate with Central Square Cut out Using Domain Decomposition Method,” vol. 2, no. 4, pp. 230–242, 2016.

[91]       S. Ghahnavieh, S. Hosseini-Hashemi, K. Rajabi, and S. Ghahnavieh, “A higher-order nonlocal strain gradient mass sensor based on vibrating heterogeneous magneto-electro-elastic nanoplate via third-order shear deformation theory,” Eur. Phys. J. Plus, vol. 133, no. 12, pp. 1–21, 2018.

[92]       A. O. Bochkarev, “Compressive buckling of a rectangular nanoplate,” AIP Conf. Proc., vol. 1959, 2018.

[93]       A. O. Bochkarev and M. A. Grekov, “Influence of Surface Stresses on the Nanoplate Stiffness and Stability in the Kirsch Problem,” Phys. Mesomech., vol. 22, no. 3, pp. 209–223, 2019.

[94]       Y. S. Li and E. Pan, “Bending of a sinusoidal piezoelectric nanoplate with surface effect,” Compos. Struct., vol. 136, pp. 45–55, 2016.

[95]       Y. Guo, T. Ma, J. Wang, B. Huang, and H. S. Kim, “Bending stress analysis of a piezoelectric nanoplate with flexoelectricity under inhomogeneous electric fields,” AIP Adv., vol. 9, no. 5, 2019.

[96]       X. Wang, R. Zhang, and L. Jiang, “A Study of the Flexoelectric Effect on the Electroelastic Fields of a Cantilevered Piezoelectric Nanoplate,” Int. J. Appl. Mech., vol. 9, no. 4, pp. 1–25, 2017.

[97]       S. Sahmani and A. M. Fattahi, “Development an efficient calibrated nonlocal plate model for nonlinear axial instability of zirconia nanosheets using molecular dynamics simulation,” J. Mol. Graph. Model., vol. 75, pp. 20–31, 2017.

[98]       D. T. Ho, S. D. Park, S. Y. Kwon, K. Park, and S. Y. Kim, “Negative Poisson’s ratios in metal nanoplates,” Nat. Commun., vol. 5, pp. 1–8, 2014.

[99]       H. I. Virus, A. N. Disorders, C. Report, M. C. Author, and A. C. Author, “Dynamic characteristics of viscoelastic nanoplates under moving load embedded within visco-Pasternak substrate and hygrothermal environment,” Mater. Res. Express, pp. 1–46, 2012.

[100]     C. C. Liu and Z. B. Chen, “Dynamic analysis of finite periodic nanoplate structures with various boundaries,” Phys. E Low-Dimensional Syst. Nanostructures, vol. 60, pp. 139–146, 2014.

[101]     M. Jamshidian, A. Dehghani, M. S. Talaei, and T. Rabczuk, “Size dependent surface energy of nanoplates: Molecular dynamics and nanoscale continuum theory correlations,” Phys. Lett. Sect. A Gen. At. Solid State Phys., vol. 382, no. 2–3, pp. 61–65, 2018.

[102]     S. K. Jalali, N. M. Pugno, and E. Jomehzadeh, Influence of out-of-plane defects on vibration analysis of graphene sheets: Molecular and continuum approaches, vol. 91. Elsevier Ltd, 2016.

[103]     S. Kamal Jalali, M. Hassan Naei, and N. M. Pugno, “Graphene-based resonant sensors for detection of ultra-fine nanoparticles: Molecular dynamics and nonlocal elasticity investigations,” Nano, vol. 10, no. 2, pp. 1–18, 2015.

[104]     M. Mohammadimehr, M. Emdadi, H. Afshari, and B. Rousta Navi, “Bending, buckling and vibration analyses of MSGT microcomposite circular-annular sandwich plate under hydro-thermo-magneto-mechanical loadings using DQM,” Int. J. Smart Nano Mater., vol. 9, no. 4, pp. 233–260, 2018.