Effect of Different Kernel Normalization Procedures on Vibration Behaviour of Calibrated Nonlocal Integral Continuum Model of Nanobeams

Document Type : Research Paper


Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran


Although the nonlocal integral (NI) model circumvents the inconsistencies associated with the differential model, it is shown in the present study that the way its nonlocal kernel function is normalized noticeably affects the dynamic response of nanobeams. To this aim, a two-phase nonlocal integral nanobeam model with different boundary conditions and kernel functions is considered and its natural frequencies are obtained using the Rayleigh-Ritz method. Also, the kernel function is normalized via two procedures to see the influence of each one on the vibration characteristics of nanobeam. From the results it is found that kernel normalization has a significant effect on vibration response of nanobeam and therefore must be taken into account. Further, it is found that the results from each normalized model are noticeably different from the other. Furthermore, by comparing the results of continuum NI models with those from atomistic models, it is revealed that for certain normalization schemes a calibrated nonlocal parameter cannot be found due to twofold hardening-softening behavior. Moreover, the effect of kernel type, boundary conditions and mode number is thoroughly studied. The results from current study can shed light on the way of choosing or developing more reliable equivalent continuum NI models for nanostructures.


Main Subjects

[1]          R. K. Joshi, H. Gomez, F. Alvi, A. Kumar, Graphene films and ribbons for sensing of O2, and 100 ppm of CO and NO2 in practical conditions, The Journal of Physical Chemistry C, Vol. 114, No. 14, pp. 6610-6613, 2010.
[2]          Q. Liang, J. Dong, Superconducting switch made of graphene–nanoribbon junctions, Nanotechnology, Vol. 19, No. 35, pp. 355706, 2008.
[3]          M. Liu, Y.-E. Miao, C. Zhang, W. W. Tjiu, Z. Yang, H. Peng, T. Liu, Hierarchical composites of polyaniline–graphene nanoribbons–carbon nanotubes as electrode materials in all-solid-state supercapacitors, Nanoscale, Vol. 5, No. 16, pp. 7312-7320, 2013.
[4]          M. A. Rafiee, W. Lu, A. V. Thomas, A. Zandiatashbar, J. Rafiee, J. M. Tour, N. A. Koratkar, Graphene nanoribbon composites, ACS nano, Vol. 4, No. 12, pp. 7415-7420, 2010.
[5]          R. J. Young, I. A. Kinloch, L. Gong, K. S. Novoselov, The mechanics of graphene nanocomposites: a review, Composites Science and Technology, Vol. 72, No. 12, pp. 1459-1476, 2012.
[6]          H. Kim, A. A. Abdala, C. W. Macosko, Graphene/polymer nanocomposites, Macromolecules, Vol. 43, No. 16, pp. 6515-6530, 2010.
[7]          J. L. Johnson, A. Behnam, S. Pearton, A. Ural, Hydrogen sensing using Pd‐functionalized multi‐layer graphene nanoribbon networks, Advanced materials, Vol. 22, No. 43, pp. 4877-4880, 2010.
[8]          J. W. Kang, S. Lee, Molecular dynamics study on the bending rigidity of graphene nanoribbons, Computational Materials Science, Vol. 74, pp. 107-113, 2013.
[9]          R. Faccio, P. A. Denis, H. Pardo, C. Goyenola, A. W. Mombrú, Mechanical properties of graphene nanoribbons, Journal of Physics: Condensed Matter, Vol. 21, No. 28, pp. 285304, 2009.
[10]        A. C. Eringen, J. Wegner, Nonlocal continuum field theories, Appl. Mech. Rev., Vol. 56, No. 2, pp. B20-B22, 2003.
[11]        A. C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of applied physics, Vol. 54, No. 9, pp. 4703-4710, 1983.
[12]        J. Reddy, Nonlocal theories for bending, buckling and vibration of beams, International journal of engineering science, Vol. 45, No. 2-8, pp. 288-307, 2007.
[13]        A. I. Aria, M. Friswell, A nonlocal finite element model for buckling and vibration of functionally graded nanobeams, Composites Part B: Engineering, Vol. 166, pp. 233-246, 2019.
[14]        Ö. Civalek, Ç. Demir, B. Akgöz, Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model, Mathematical and Computational Applications, Vol. 15, No. 2, pp. 289-298, 2010.
[15]        M. Aydogdu, A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E: Low-dimensional Systems and Nanostructures, Vol. 41, No. 9, pp. 1651-1655, 2009.
[16]        T. Aksencer, M. Aydogdu, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E: Low-dimensional Systems and Nanostructures, Vol. 43, No. 4, pp. 954-959, 2011.
[17]        W. Duan, C. M. Wang, Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory, Nanotechnology, Vol. 18, No. 38, pp. 385704, 2007.
[18]        T. Murmu, S. Pradhan, Small-scale effect on the free in-plane vibration of nanoplates by nonlocal continuum model, Physica E: Low-dimensional Systems and Nanostructures, Vol. 41, No. 8, pp. 1628-1633, 2009.
[19]        T. Murmu, J. Sienz, S. Adhikari, C. Arnold, Nonlocal buckling of double-nanoplate-systems under biaxial compression, Composites Part B: Engineering, Vol. 44, No. 1, pp. 84-94, 2013.
[20]        R. Ansari, S. Sahmani, B. Arash, Nonlocal plate model for free vibrations of single-layered graphene sheets, Physics Letters A, Vol. 375, No. 1, pp. 53-62, 2010.
[21]        R. Li, G. A. Kardomateas, Vibration characteristics of multiwalled carbon nanotubes embedded in elastic media by a nonlocal elastic shell model, 2007.
[22]        F. Khademolhosseini, R. Rajapakse, A. Nojeh, Torsional buckling of carbon nanotubes based on nonlocal elasticity shell models, Computational materials science, Vol. 48, No. 4, pp. 736-742, 2010.
[23]        J. Peddieson, G. R. Buchanan, R. P. McNitt, Application of nonlocal continuum models to nanotechnology, International journal of engineering science, Vol. 41, No. 3-5, pp. 305-312, 2003.
[24]        H. Ersoy, K. Mercan, Ö. Civalek, Frequencies of FGM shells and annular plates by the methods of discrete singular convolution and differential quadrature methods, Composite Structures, Vol. 183, pp. 7-20, 2018.
[25]        Ö. Civalek, Ç. Demir, Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory, Applied Mathematical Modelling, Vol. 35, No. 5, pp. 2053-2067, 2011.
[26]        S. Shaw, P. Murthy, S. Pradhan, The effect of body acceleration on two dimensional flow of Casson fluid through an artery with asymmetric stenosis, The Open Conservation Biology Journal, Vol. 2, No. 1, 2010.
[27]        J. Fernández-Sáez, R. Zaera, J. Loya, J. Reddy, Bending of Euler–Bernoulli beams using Eringen’s integral formulation: a paradox resolved, International Journal of Engineering Science, Vol. 99, pp. 107-116, 2016.
[28]        C. Li, L. Yao, W. Chen, S. Li, Comments on nonlocal effects in nano-cantilever beams, International Journal of Engineering Science, Vol. 87, pp. 47-57, 2015.
[29]        M. Tuna, M. Kirca, Exact solution of Eringen's nonlocal integral model for bending of Euler–Bernoulli and Timoshenko beams, International Journal of Engineering Science, Vol. 105, pp. 80-92, 2016.
[30]        M. Tuna, M. Kirca, Exact solution of Eringen's nonlocal integral model for vibration and buckling of Euler–Bernoulli beam, International Journal of Engineering Science, Vol. 107, pp. 54-67, 2016.
[31]        Y. Wang, X. Zhu, H. Dai, Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model, Aip Advances, Vol. 6, No. 8, pp. 085114, 2016.
[32]        J. Fernández-Sáez, R. Zaera, Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory, International Journal of Engineering Science, Vol. 119, pp. 232-248, 2017.
[33]        X. Zhu, Y. Wang, H.-H. Dai, Buckling analysis of Euler–Bernoulli beams using Eringen’s two-phase nonlocal model, International Journal of Engineering Science, Vol. 116, pp. 130-140, 2017.
[34]        K. Eptaimeros, C. C. Koutsoumaris, G. Tsamasphyros, Nonlocal integral approach to the dynamical response of nanobeams, International Journal of Mechanical Sciences, Vol. 115, pp. 68-80, 2016.
[35]        K. Eptaimeros, C. C. Koutsoumaris, I. Dernikas, T. Zisis, Dynamical response of an embedded nanobeam by using nonlocal integral stress models, Composites Part B: Engineering, Vol. 150, pp. 255-268, 2018.
[36]        A. Anjomshoa, B. Hassani, On the importance of proper kernel normalization procedure in nonlocal integral continuum modeling of nanobeams, ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 101, No. 10, pp. e202000126, 2021.
[37]        L. Shen, H.-S. Shen, C.-L. Zhang, Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Computational Materials Science, Vol. 48, No. 3, pp. 680-685, 2010.
[38]        A. Shakouri, T. Ng, R. Lin, A study of the scale effects on the flexural vibration of graphene sheets using REBO potential based atomistic structural and nonlocal couple stress thin plate models, Physica E: Low-dimensional Systems and Nanostructures, Vol. 50, pp. 22-28, 2013.
[39]        S. B. Altan, Uniqueness of initial-boundary value problems in nonlocal elasticity, International journal of solids and structures, Vol. 25, No. 11, pp. 1271-1278, 1989.
[40]        G. Borino, B. Failla, F. Parrinello, A symmetric nonlocal damage theory, International Journal of Solids and Structures, Vol. 40, No. 13-14, pp. 3621-3645, 2003.
[41]        Z. P. Bažant, M. Jirásek, Nonlocal integral formulations of plasticity and damage: survey of progress, Journal of engineering mechanics, Vol. 128, No. 11, pp. 1119-1149, 2002.
[42]        S. Chakraverty, L. Behera, Free vibration of non-uniform nanobeams using Rayleigh–Ritz method, Physica E: Low-dimensional Systems and Nanostructures, Vol. 67, pp. 38-46, 2015.
[43]        Q. Wang, C. Wang, The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes, Nanotechnology, Vol. 18, No. 7, pp. 075702, 2007.
[44]        A. Shakouri, T. Ng, R. Lin, A new REBO potential based atomistic structural model for graphene sheets, Nanotechnology, Vol. 22, No. 29, pp. 295711, 2011.
Volume 53, Issue 4
December 2022
Pages 639-656
  • Receive Date: 16 October 2022
  • Revise Date: 15 November 2022
  • Accept Date: 16 November 2022