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A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment

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Abstract

In this paper, a new size-dependent plate model is developed based on the higher-order nonlocal strain gradient theory. The influences of higher-order deformations in conjunction with the higher- and lower-order nonlocal effects are taken into account. The presence of three different kinds of scale parameters in the formulation results in a theory which is capable of capturing both reduction and increase in the stiffness of structures at nanoscale. The governing differential equations are derived for the buckling of nanoplates resting on a two-parameter elastic foundation using the principle of virtual work. The nanoplate is assumed to be orthotropic with size-dependent material properties. The influence of thermal stress caused by a temperature change is taken into consideration. An exact closed-form solution is obtained for the critical buckling loads of graphene sheets. The higher-order governing differential equation is also solved by the differential quadrature method. The results of the two solution methods are compared with each other. Excellent agreement between the exact and numerical results is observed. For numerical results, three types of graphene sheets with different aspect ratio are considered. The effects of various scale parameters together with the other parameters such as the coefficients of the elastic medium, temperature change and the length of the nanoplate on the buckling behavior of graphene sheets are investigated.

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References

  1. Tsai, C.Y., Lu, S.P., Lin, J.W., Lee, P.T.: High sensitivity plasmonic index sensor using slablike gold nanoring arrays. Appl. Phys. Lett. 98, 153108 (2011)

    Article  Google Scholar 

  2. Bunch, J.S., van der Zande, A.M., Verbridge, S.S., Frank, I.W., Tanenbsum, D.M., Parpia, J.M., Craighead, H.G., McEuen, P.L.: Electromechanical resonators from graphene sheets. Science 315, 490–493 (2007)

    Article  Google Scholar 

  3. Liu, Y., Dong, X., Chen, P.: Biological and chemical sensors based on graphene materials. Chem. Soc. Rev. 41, 2283–2307 (2012)

    Article  Google Scholar 

  4. Zhu, J., Yang, D., Yin, Z., Yan, Q., Zhang, H.: Graphene and graphene-based materials for energy storage applications. Small 10, 3480–3498 (2014)

    Article  Google Scholar 

  5. Istrate, O.M., Paton, K.R., Khan, U., O’Neill, A., Bell, A.P., Coleman, J.N.: Reinforcement in melt-processed polymer-graphene composites at extremely low graphene loading level. Carbon 78, 243–249 (2014)

    Article  Google Scholar 

  6. Kostarelos, K., Novoselov, K.S.: Graphene devices for life. Nat. Nanotechnol. 9, 744–745 (2014)

    Article  Google Scholar 

  7. Legoas, S.B., Coluci, V.R., Braga, S.F., Coura, P.Z., Dantas, S.O., Galvão, D.S.: Molecular-dynamics simulations of carbon nanotubes as gigahertz oscillators. Phys. Rev. Lett. 90, 055504 (2003)

    Article  Google Scholar 

  8. Gao, Y., Hao, P.: Mechanical properties of monolayer graphene under tensile and compressive loading. Phys. E 41, 1561–1566 (2009)

    Article  Google Scholar 

  9. Neek-Amal, M., Peeters, F.M.: Graphene nanoribbons subjected to axial stress. Phys. Rev. B 82, 085432 (2010)

    Article  Google Scholar 

  10. Neek-Amal, M., Peeters, F.M.: Buckled circular monolayer graphene: a graphene nano-bowl. J. Phys. Condens. Matter 23, 045002 (2011)

    Article  Google Scholar 

  11. Xiang, Y., Shen, H.S.: Shear buckling of rippled graphene by molecular dynamics simulation. Mater. Today Commun. 3, 149–155 (2015)

    Article  Google Scholar 

  12. Rahman, R., Foster, J.T.: A molecular dynamics based investigation of thermally vibrating graphene under different boundary conditions. Phys. E 72, 25–47 (2015)

    Article  MathSciNet  Google Scholar 

  13. Shen, L., Shen, H.S., Zhang, C.L.: Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Comput. Mater. Sci. 48, 680–685 (2010)

    Article  Google Scholar 

  14. Toupin, R.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  15. Koiter, W.T.: Couple-stresses in the theory of elasticity: I and II. R. Neth. Acad. Arts Sci. B 67, 17–44 (1964)

    MathSciNet  MATH  Google Scholar 

  16. Mindlin, R.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 414–438 (1965)

    Google Scholar 

  17. Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  18. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)

    Article  Google Scholar 

  19. Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)

    Article  Google Scholar 

  20. Sudak, L.J.: Column buckling of multi-walled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281–7287 (2003)

    Article  Google Scholar 

  21. Duan, W.H., Wang, C.M., Zhang, Y.Y.: Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics. J. Appl. Phys. 101, 024305 (2007)

    Article  Google Scholar 

  22. Civalek, Ö., Akgöz, B.: Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen’s nonlocal elasticity theory. Int. J. Eng. Appl. Sci. 1, 47–56 (2009)

    Google Scholar 

  23. Ansari, R., Rouhi, H., Sahmani, S.: Free vibration analysis of single-and double-walled carbon nanotubes based on nonlocal elastic shell models. J. Vib. Control 20, 670–678 (2014)

    Article  MathSciNet  Google Scholar 

  24. Baghdadi, H., Tounsi, A., Zidour, M., Benzair, A.: Thermal effect on vibration characteristics of armchair and zigzag single-walled carbon nanotubes using nonlocal parabolic beam theory. Fuller. Nanotubes Carbon Nanostruct. 23, 266–272 (2015)

    Article  Google Scholar 

  25. Kiani, K.: Dynamic interactions of doubly orthogonal stocky single-walled carbon nanotubes. Compos. Struct. 125, 144–158 (2015)

    Article  Google Scholar 

  26. Hemmatnezhad, M., Ansari, R.: Finite element formulation for the free vibration analysis of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory. J. Theor. Appl. Phys. 7, 6 (2013)

    Article  Google Scholar 

  27. Danesh, M., Farajpour, A., Mohammadi, M.: Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method. Mech. Res. Commun. 39, 23–27 (2012)

    Article  MATH  Google Scholar 

  28. Aydogdu, M.: Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity. Mech. Res. Commun. 43, 34–40 (2012)

    Article  Google Scholar 

  29. Aydogdu, M., Elishakoff, I.: On the vibration of nanorods restrained by a linear spring in-span. Mech. Res. Commun. 57, 90–96 (2014)

    Article  Google Scholar 

  30. Demir, Ç., Civalek, Ö.: Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models. Appl. Math. Model. 37, 9355–9367 (2013)

    Article  Google Scholar 

  31. Ghorbanpour Arani, A., Kolahchi, R., Khoddami Maraghi, Z.: Nonlinear vibration and instability of embedded double-walled boron nitride nanotubes based on nonlocal cylindrical shell theory. Appl. Math. Model. 37, 7685–7707 (2013)

    Article  MathSciNet  Google Scholar 

  32. Ghorbanpour Arani, A., Abdollahian, M., Kolahchi, R., Rahmati, A.H.: Electro-thermo-torsional buckling of an embedded armchair DWBNNT using nonlocal shear deformable shell model. Compos. Part B Eng. 51, 291–299 (2013)

    Article  Google Scholar 

  33. Mohammadi, M., Farajpour, A., Goodarzi, M., Mohammadi, H.: Temperature effect on vibration analysis of annular graphene sheet embedded on visco-Pasternak foundation. J. Solid Mech. 5, 305–323 (2013)

  34. Narendar, S., Gopalakrishnan, S.: Scale effects on buckling analysis of orthotropic nanoplates based on nonlocal two-variable refined plate theory. Acta Mech. 223, 395–413 (2012)

  35. Murmu, T., Adhikari, S.: Nonlocal vibration of bonded double-nanoplate-systems. Compos. Part B Eng. 42, 1901–1911 (2011)

    Article  Google Scholar 

  36. Bedroud, M., Hosseini-Hashemi, S., Nazemnezhad, R.: Buckling of circular/annular Mindlin nanoplates via nonlocal elasticity. Acta Mech. 224, 2663–2676 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Golmakani, M.E., Rezatalab, J.: Nonuniform biaxial buckling of orthotropic nanoplates embedded in an elastic medium based on nonlocal Mindlin plate theory. Compos. Struct. 119, 238–250 (2015)

    Article  Google Scholar 

  38. Jomehzadeh, E., Saidi, A.R., Jomehzadeh, Z., Bonaccorso, F., Palermo, V., Galiotis, C., Pugno, N.M.: Nonlinear subharmonic oscillation of orthotropic graphene-matrix composite. Comput. Mater. Sci. 99, 164–172 (2015)

    Article  Google Scholar 

  39. Aghababaei, R., Reddy, J.N.: Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J. Sound Vib. 326, 277–289 (2009)

    Article  Google Scholar 

  40. Pradhan, S.C., Murmu, T.: Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory. Phys. E 42, 1293–1301 (2010)

    Article  Google Scholar 

  41. Malekzadeh, P., Setoodeh, A.R., Alibeygi Beni, A.: Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium. Compos. Struct. 93, 2083–2089 (2011)

    Article  Google Scholar 

  42. Farajpour, A., Mohammadi, M., Shahidi, A.R., Mahzoon, M.: Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model. Phys. E 43, 1820–1825 (2011)

    Article  Google Scholar 

  43. Farajpour, A., Shahidi, A.R., Mohammadi, M., Mahzoon, M.: Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics. Compos. Struct. 94, 1605–1615 (2012)

    Article  Google Scholar 

  44. Radebe, I.S., Adali, S.: Buckling and sensitivity analysis of nonlocal orthotropic nanoplates with uncertain material properties. Compos. Part B Eng. 56, 840–846 (2014)

    Article  Google Scholar 

  45. Arash, B., Wang, Q.: A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313 (2012)

    Article  Google Scholar 

  46. Farajpour, A., Dehghany, M., Shahidi, A.R.: Surface and nonlocal effects on the axisymmetric buckling of circular graphene sheets in thermal environment. Compos. Part B Eng. 50, 333–343 (2013)

    Article  Google Scholar 

  47. Lim, C.W., Zhang, G., Reddy, J.N.: A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  49. Farajpour, A., Rastgoo, A., Mohammadi, M.: Surface effects on the mechanical characteristics of microtubule networks in living cells. Mech. Res. Commun. 57, 18–26 (2014)

    Article  Google Scholar 

  50. Akgoz, B., Civalek, O.: A size-dependent shear deformation beam model based on the strain gradient elasticity theory. Int. J. Eng. Sci. 70, 1–14 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  51. Ghayesh, M.H., Amabili, M., Farokhi, H.: Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int. J. Eng. Sci. 63, 52–60 (2013)

    Article  MathSciNet  Google Scholar 

  52. Mohammadi, H., Mahzoon, M.: Thermal effects on postbuckling of nonlinear microbeams based on the modified strain gradient theory. Compos. Struct. 106, 764–776 (2013)

    Article  Google Scholar 

  53. Seyyed Fakhrabadi, M.M., Rastgoo, A., Ahmadian, M.T.: Dynamic behaviours of carbon nanotubes under DC voltage based on strain gradient theory. J. Phys. D Appl. Phys. 46, 405101 (2013)

    Article  Google Scholar 

  54. Akgoz, B., Civalek, O.: Analysis of microtubules based on strain gradient elasticity and modified couple stress theories. Adv. Vib. Eng. 11, 385–400 (2012)

    MATH  Google Scholar 

  55. Nami, M.R., Janghorban, M.: Free vibration analysis of rectangular nanoplates based on two-variable refined plate theory using a new strain gradient elasticity theory. J Braz. Soc. Mech. Sci. Eng. 37, 313–324 (2015)

    Article  Google Scholar 

  56. Sahmani, S., Ansari, R.: On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory. Compos. Struct. 95, 430–442 (2013)

    Article  Google Scholar 

  57. Arani, A.G., Shokravi, M.: Vibration response of visco-elastically coupled double-layered visco-elastic graphene sheet systems subjected to magnetic field via strain gradient theory considering surface stress effects. Proc. Inst. Mech. Eng. Part N J. Nanoeng. Nanosyst. 229, 180–190 (2015)

  58. Malekzadeh, P., Farajpour, A.: Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium. Acta Mech. 223, 2311–2330 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  59. Farajpour, A., Farajpour, A., Arab Solghar, A.R., Shahidi, A.R.: Postbuckling analysis of multi-layered graphene sheets under non-uniform biaxial compression. Phys. E 47, 197–206 (2013)

    Article  Google Scholar 

  60. Shu, C.: Differential quadrature and its application in engineering. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  61. Bert, C.W., Malik, M.: Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49, 1–27 (1996)

    Article  Google Scholar 

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Authors and Affiliations

  1. Young Researchers and Elites Club, North Tehran Branch, Islamic Azad University, Tehran, Iran

    A. Farajpour

  2. School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

    M. R. Haeri Yazdi & A. Rastgoo

  3. Department of Engineering, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran

    M. Mohammadi

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  1. A. Farajpour
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  2. M. R. Haeri Yazdi
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  3. A. Rastgoo
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Farajpour, A., Yazdi, M.R.H., Rastgoo, A. et al. A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment. Acta Mech 227, 1849–1867 (2016). https://doi.org/10.1007/s00707-016-1605-6

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  • DOI: https://doi.org/10.1007/s00707-016-1605-6

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