Abstract
Based on the nonlocal elasticity theory, the vibration behavior of circular double-layered graphene sheets (DLGSs) resting on the Winkler- and Pasternak-type elastic foundations in a thermal environment is investigated. The governing equation is derived on the basis of Eringen’s nonlocal elasticity and the classical plate theory (CLPT). The initial thermal loading is assumed to be due to a uniform temperature rise throughout the thickness direction. Using the generalized differential quadrature (GDQ) method and periodic differential operators in radial and circumferential directions, respectively, the governing equation is discretized. DLGSs with clamped and simply-supported boundary conditions are studied and the influence of van der Waals (vdW) interaction forces is taken into account. In the numerical results, the effects of various parameters such as elastic medium coefficients, radius-to-thickness ratio, thermal loading and nonlocal parameter are examined on both in-phase and anti-phase natural frequencies. The results show that the thermal load and elastic foundation respectively decreases and increases the fundamental frequencies of DLGSs.
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Li, X., Bhushan, B., Takashima, K., et al.: Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques. Ultramicroscopy 97, 481–494 (2003)
Belytschko, T., Xiao, S.P., Schatz, G.C., et al.: Atomistic simulations of nanotube fracture. Phys. Rev. B 65, 235430 (2002)
Natsuki, T., Matsuyama, N., Shi, J.X., et al.: Vibration analysis of nanomechanical mass sensor using carbon nanotubes under axial tensile loads. Appl. Phys. A 116, 1001–1007 (2014)
Natsuki, T., Shi, J.X., Ni, Q.Q.: Vibration analysis of circular double-layered graphene sheets. J. Appl. Phys. 111, 044310 (2012)
Wang, J., He, X., Kitipornchai, S., et al.: Geometrical nonlinear free vibration of multi-layered graphene sheets. J. Phys. D Appl. Phys. 44, 135401 (2011)
Yang, F.A.C.M., Chong, A.C.M., Lam, D.C.C., et al.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)
Park, S.K., Gao, X.L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355 (2006)
Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)
Ansari, R., Gholami, R., Shojaei, M.F., et al.: Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory. Compos. Struct. 100, 385–397 (2013)
Gurtin, M.E., Weissmüller, J., Larche, F.: A general theory of curved deformable interfaces in solids at equilibrium. Philos. Mag. A 78, 1093–1109 (1998)
Dingreville, R., Qu, J., Cherkaoui, M.: Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. J. Mech. Phys. Solids 53, 1827–1854 (2005)
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)
Asemi, S.R., Farajpour, A.: Decoupling the nonlocal elasticity equations for thermo-mechanical vibration of circular graphene sheets including surface effects. Phys. E Low Dimens. Syst. Nanostruct. 60, 80–90 (2014)
Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Rahmani, O., Jandaghian, A.A.: Buckling analysis of functionally graded nanobeams based on a nonlocal third-order shear deformation theory. Appl. Phys. A 119, 1019–1032 (2015)
Moosavi, H., Mohammadi, M., Farajpour, A., et al.: Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory. Phys. E Low Dimens. Syst. Nanostruct. 44, 135–140 (2011)
Mohammadi, M., Farajpour, A., Moradi, A., et al.: Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment. Compos. Part B Eng. 56, 629–637 (2014)
Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)
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, 24305–24305 (2007)
Ansari, R., Rouhi, H.: Analytical treatment of the free vibration of single-walled carbon nanotubes based on the nonlocal Flugge shell theory. J. Eng. Mater. Technol. 134, 011008 (2012)
Ansari, R., Rouhi, H., Sahmani, S.: Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics. Int. J. Mech. Sci. 53, 786–792 (2011)
Aydogdu, M.: Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics. Int. J. Eng. Sci. 56, 17–28 (2012)
Gibson, R.F., Ayorinde, E.O., Wen, Y.F.: Vibrations of carbon nanotubes and their composites: a review. Compos. Sci. Technol. 67, 1–28 (2007)
Pradhan, S.C., Phadikar, J.K.: Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models. Phys. Lett. A 373, 1062–1069 (2009)
Mohammadi, M., Moradi, A., Ghayour, M., et al.: Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium. Lat. Am. J. Solids Struct. 11, 437–458 (2014)
Shen, L.E., 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)
Ansari, R., Rajabiehfard, R., Arash, B.: Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets. Comput. Mater. Sci. 49, 831–838 (2010)
Ansari, R., Sahmani, S., Arash, B.: Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys. Lett. A 375, 53–62 (2010)
Shen, H.S., Shen, L., Zhang, C.L.: Nonlocal plate model for nonlinear bending of single-layer graphene sheets subjected to transverse loads in thermal environments. Appl. Phys. A 103, 103–112 (2011)
Pradhan, S.C., Murmu, T.: Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics. Comput. Mater. Sci. 47, 268–274 (2009)
Pradhan, S.C., Phadikar, J.K.: Scale effect and buckling analysis of multilayered graphene sheets based on nonlocal continuum mechanics. J. Comput. Theor. Nanosci. 7, 1948–1954 (2010)
Farajpour, A., Mohammadi, M., Shahidi, A.R., et al.: Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model. Phys. E Low Dimens. Syst. Nanostruct. 43, 1820–1825 (2011)
Mohammadi, M., Goodarzi, M., Ghayour, M., et al.: Influence of in-plane pre-load on the vibration frequency of circular graphene sheet via nonlocal continuum theory. Compos. Part B Eng. 51, 121–129 (2013)
Mohammadi, M., Farajpour, A., Goodarzi, M., et al.: Temperature effect on vibration analysis of annular graphene sheet embedded on visco-pasternak foundation. J. Solid Mech. 5, 305–323 (2013)
Arash, B., Wang, Q.: Vibration of single-and double-layered graphene sheets. J. Nanotechnol. Eng. Med. 2, 011012 (2011)
Pradhan, S.C., Kumar, A.: Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method. Compos. Struct. 93, 774–779 (2011)
Jomehzadeh, E., Saidi, A.R.: A study on large amplitude vibration of multilayered graphene sheets. Comput. Mater. Sci. 50, 1043–1051 (2011)
Jomehzadeh, E., Saidi, A.R., Pugno, N.M.: Large amplitude vibration of a bilayer graphene embedded in a nonlinear polymer matrix. Phys. E Low Dimens. Syst. Nanostruct. 44, 1973–1982 (2012)
Babaei, H., Shahidi, A.R.: Vibration of quadrilateral embedded multilayered graphene sheets based on nonlocal continuum models using the Galerkin method. Acta Mech. Sin. 27, 967–976 (2011)
Murmu, T., McCarthy, M.A., Adhikari, S.: In-plane magnetic field affected transverse vibration of embedded single-layer graphene sheets using equivalent nonlocal elasticity approach. Compos. Struct. 96, 57–63 (2013)
Mohammadi, M., Ghayour, M., Farajpour, A.: Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model. Compos. Part B Eng. 45, 32–42 (2013)
Mohammadi, M., Farajpour, A., Goodarzi, M., et al.: Thermo-mechanical vibration analysis of annular and circular graphene sheet embedded in an elastic medium. Lat. Am. J. Solids Struct. 11, 659–682 (2014)
Mohammadi, M., Farajpour, A., Goodarzi, M.: Numerical study of the effect of shear in-plane load on the vibration analysis of graphene sheet embedded in an elastic medium. Comput. Mater. Sci. 82, 510–520 (2014)
Asemi, S.R., Farajpour, A., Borghei, M., et al.: Thermal effects on the stability of circular graphene sheets via nonlocal continuum mechanics. Lat. Am. J. Solids Struct. 11, 704–724 (2014)
Shen, H.S., Xu, Y.M., Zhang, C.L.: Prediction of nonlinear vibration of bilayer graphene sheets in thermal environments via molecular dynamics simulations and nonlocal elasticity. Comput. Methods Appl. Mech. Eng. 267, 458–470 (2013)
Shi, J.X., Ni, Q.Q., Lei, X.W., et al.: Nonlocal vibration analysis of nanomechanical systems resonators using circular double-layer graphene sheets. Appl. Phys. A 115, 213–219 (2014)
Sarrami-Foroushani, S., Azhari, M.: Nonlocal vibration and buckling analysis of single and multi-layered graphene sheets using finite strip method including van der Waals effects. Phys. E Low Dimens. Syst. Nanostruct. 57, 83–95 (2014)
Shu, C.: Differential Quadrature and its Application in Engineering. Springer, London (2000)
Ansari, R., Mohammadi, V., Shojaei, M.F., et al.: Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory. Eur. J. Mech. A Solids 45, 143–152 (2014)
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Ansari, R., Torabi, J. Nonlocal vibration analysis of circular double-layered graphene sheets resting on an elastic foundation subjected to thermal loading. Acta Mech. Sin. 32, 841–853 (2016). https://doi.org/10.1007/s10409-016-0574-2
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DOI: https://doi.org/10.1007/s10409-016-0574-2