Piotr Jankowski

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The present investigation examines the range of effect of nonlocal parameters on dynamic behavior of a smart beam-like nanostructure modeled as sandwich functionally graded porous nanobeam with piezoelectric layers. Therefore, the study is concentrated on determining length of the structure for which nonlocal effects are observed for vibration of nanobeam under in-plane electro-mechanical forces. The nanobeam-based NEMS device model is obtained based on assumptions of the nonlocal strain gradient theory in conjunction with Reddy higher-order shear deformation theory. The investigation present differences in obtained results for nanostructure’s free vibration based on classical and nonlocal assumptions. To study range of application of nonlocal parameters for different length of simply supported nanobeam, defined eigenvalue problem is solved in view of variation of length to thickness ratio, distribution of material properties, as well as electro-mechanical loads. What is more, the study attempts to determine and calibrate values of size-dependent coefficients based on expected natural frequencies, material properties, and applied loads. The results are completed with extensive discussion on the dependence of nonlocal parameters on nanobeam’s dynamic response, thus may be an important step forward to extend understanding of ultra-small structure’s behavior.


Nanobeam, FGM, Porosity, Piezoelectric effect, Free vibration, Detected nonlocal parameters

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Rieth, M., Schommers, W. (Ed.), 2005, Handbook of Theoretical and Computational Nanotechnology, American Scientific Publishers, USA.

Koochi, A., Abadyan, M., 2020, Nonlinear Differential Equations in Micro/nano Mechanics, Elsevier, Netherlands.

Maurya, D., Pramanick, A., Viehland, D. (Ed.), 2020, Ferroelectric Materials for Energy Harvesting and Storage, Elsevier, Netherlands.

Kamilla, S.K., Ojha, M., 2021, Review on nano-electro-mechanical system devices, Materials Today: Proceedings, in press,

Lin, H., Zaeimbashi, M., Sun, N., Liang, X, Chen, H., Dong, C., Matyushov, A., Wang, X., Guo, Y., Sun, N.-X., 2018, NEMS Magnetoelectric Antennas for Biomedical Application, Proc. 2018 IEEE International Microwave Biomedical Conference (IMBioC).

Ilyas, S., Younis, M.I., 2020, Resonator-based M/NEMS logic devices: Review of recent advances, Sensors and Actuators A: Physical, 302, 111821.

Khazaai, J.J., Qu, H., 2012, Electro-thermal MEMS switch with latching mechanism: design and characterization, IEEE Sensors Journal, 12(9), pp. 2830–2838.

Ghayesh, M.H., Farajpour, A., 2019, A review on the mechanics of functionally graded nanoscale and microscale structures, International Journal of Engineering Science, 137, pp. 8-36.

Daikh, A.A., Houari, M.S.A., Eltaher, M.A., 2021, A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates, Composite Structures, 262, 113347.

Rapaport, D.C., 2004, The art of Molecular Dynamics Simulations, Cambridge University Press, Great Britain.

Farajpour, A., Ghayesh, M.H., Farokhi, H., 2018, A review on the mechanics of nanostructures, International Journal of Engineering Science, 133, pp. 231-263.

Toupin, R.A., 1962, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis, 11, pp. 385-414.

Mindlin, R.D., Tiersten, H.F., 1962, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11, 415-448.

Koiter, W.T., 1964, Couple stresses in the theory of elasticity, I and II. Proceedings Series B, Koninklijke Nederlandse Akademie van Wetenschappen, 67, pp. 17-44.

Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39(10), pp. 2731-2743.

Mindlin, R.D., 1964, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, 16, pp. 51–78.

Mindlin, R.D., 1965, Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures, 1(4), pp. 417–438.

Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P., 2003, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, 51, pp. 1477-1508.

Gurtin, M.E., Murdoch, A.I., 1975, A continuum theory of elastic material surfaces, Archive for Rational Mechanics and Analysis, 57, pp. 291-323.

Kroner, E., 1967, Elasticity theory of materials with long range cohesive forces, International Journal of Solids and Structures, 3, pp. 731-742.

Eringen, A.C., 1972, Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science, 10, pp. 425-435.

Eringen, A.C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science, 10, pp. 1-16.

Eringen, A.C., Edelen, D.G.B., 1972, On nonlocal elasticity, International Journal of Engineering Science, 10, pp. 233-248.

Eringen, A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface wave, Journal of Applied Physics, 54, pp. 4703-4710.

Romano, G., Barretta, R., 2017, Nonlocal elasticity in nanobeams: the stress-driven integral model. International Journal of Engineering Science, 115, pp. 14-27.

Lim, C.W., Zhang, G., Reddy, J.N., 2015, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids, 78, pp. 298-313.

Chandel, V.S., Wang, G., Talha, M., 2020, Advances in modelling and analysis of nano structures: a review, Nanotechnology Reviews, 9(1), pp. 230-258.

Askes, H., Aifantis, E.C., 2009, Gradient elasticity and flexural wave dispersion in carbon nanotubes, Physical Review B, 80, 1995412.

Ouakad, H.M., Younis, M.I., 2010, Nonlinear dynamics of electrically actuated carbon nanotube resonator, Journal of Computational and Nonlinear Dynamics, 5, 011009.

Ke, L.-L., Wang, Y.-S., Yang, J., Kitipornchai, S., 2014, Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory, Acta Mechanica Sinica, 30, pp. 516-525.

Mehralian, F., Beni, Y.T, 2017, Thermo-electro-mechanical buckling analysis of cylindrical nanoshell on the basis of modified couple stress theory, Journal of Mechanical Science and Technology, 31, pp. 1773-1787.

Lu, L., Guo, X., Zhao, J., 2017, A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms, International Journal of Engineering Science, 119, pp. 265-277.

Ouakad, H.M., Sedighi, H.M., Al-Qahtani, H.M., 2020, Forward and backward whirling of a spinning nanotube nano-rotor assuming gyroscopic effects, Advances in Nano Research, 8, pp. 245-254.

Ouakad, H.M., Valipour, A., Żur, K.K., Sedighi, H.M., Reddy, J.N., 2020, On the nonlinear vibration and static deflection problems of actuated hybrid nanotubes based on the stress-driven nonlocal integral elasticity, Mechanics of Materials, 148, 103532,

Sedighi, H.M., Malikan, M., Valipour, A., Żur, K.K., 2020, Nonlocal vibration of carbon/boron-nitridenano-hetero-structure in thermal and magnetic fields by means of nonlinear finite element method, Journal of Computational Design and Engineering, 7(5), pp. 591-602.

Farajpour, A., Żur, K.K., Kim, J., Reddy, J.N., 2021, Nonlinear frequency behaviour of magneto-electromechanical mass nanosensors using vibrating MEE nanoplates with multiple nanoparticles, Composite Structures, 260, 113458.

Żur, K.K., Farajpour, A., Lim, C.W., Jankowski, P., 2021, On the nonlinear dynamics of porous composite nanobeams connected with fullerenes, Composite Structures, 274, 114356.

Hieu, D.V., Hoa N.T., Duy, L.Q., Thoa, N.T.K., 2021, Nonlinear Vibration of an Electrostatically Actuated ‎Functionally Graded Microbeam under Longitudinal Magnetic ‎Field, Journal of Applied and Computational Mechanics, 7, pp. 1537-1549.

Anjum, N., He, J-H., Ain, Q.T., Tian, D., 2021, Li-He’s Modified Homotopy Perturbation Method for Doubly-Clamped ‎Electrically Actuated Microbeams-Based Microelectromechanical System, Facta ‎Universitatis, Series: Mechanical Engineering, 19(4), pp. 601-612. ‎

Abouelregal, A.E., Mohammad-Sedighi, H., Faghidian, S.A., Shirazi A.H., Temperature-Dependent Physical Characteristics of the Rotating Nonlocal Nanobeams Subject to a Varying Heat Source and a Dynamic Load, Facta ‎Universitatis, Series: Mechanical Engineering, 19(4), pp. 633-656.

Goharimanesh, M., Koochi, A., 2021, Nonlinear Oscillationsof CNT Nano-resonator Based on Nonlocal Elasticity: The Energy Balance Method, Reports in Mechanical Engineering, 2, pp. 41-50.

Wang, Q., Varadan, V.K, 2006, Vibration of carbon nanotubes studied using nonlocal continuum mechanics, Smart Materials and Structures, 15(2), pp. 659–666.

Reddy, J.N., 2007, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science, 45, pp. 288-307.

Wang, C.M., Zhang, Y.Y., He, X.Q, 2007, Vibration of nonlocal Timoshenko beams, Nanotechnology, 18(10), 105401.

Aydogdu, M., 2009, Axial vibration of the nanorods with the nonlocal continuum rod model, Physica E: Low-dimensional Systems and Nanostructures, 41, pp. 861-864.

Roque, C.M.C., Ferreira, A.J.M., Reddy, J.N., 2011, Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method, International Journal of Engineering Science, 49, pp. 976-984.

Li, C., Lim, C.W., Yu, J.L., Zeng, Q.C., 2011, Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force, International Journal of Structural Stability and Dynamics, 11(02), pp. 257–271.

Thai, H.T., 2012, A nonlocal beam theory for bending, buckling, and vibration of nanobeams, International Journal of Engineering Science, 52, pp. 56–64.

Thai, H.T., Vo, T.P., 2012, A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams, International Journal of Engineering Science, 54, pp. 58–66.

Ke, L.-L., Wang, Y.-S., Wang, Z.-D., 2012, Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory, Composite Structures, 94, pp. 2038–2047.

Eltaher, M.A., Emam, S.A., Mahmoud, F.F., 2012, Free vibration analysis of functionally graded size-dependent nanobeams, Applied Mathematics and Computation, 218(14), pp. 7406–7420.

Berrabah, H.M., Tounsi, A., Semmah, A., Adda Bedia, E.A., 2013, Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams, Structural Engineering and Mechanics, 48(3), pp. 351–365.

Şimşek, M., 2014, Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory, Composites Part B: Engineering, 56, pp. 621-628.

Rahmani, O., Pedram, O., 2014, Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, International Journal of Engineering Science, 77, pp. 55-70.

Nejad, M.Z., Hadi, A., 2016, Non-local analysis of free vibration of bi-directional functionally graded Euler-Bernoulli nano-beams, International Journal of Engineering Science, 105, pp. 1–11.

Beni, Y.T., 2016, A Nonlinear Electro-Mechanical Analysis of Nanobeams Based on the Size-Dependent Piezoelectricity Theory, Journal of Mechanics, 33, pp. 289–301.

Arefi, M., Zenkour, A.M., 2017, Size-dependent vibration and bending analyses of the piezomagnetic three-layer nanobeams, Applied Physics A, 123, 202.

Lu, L., Guo, X., Zhao, J., 2017, Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory, International Journal of Engineering Science, 116, pp. 12–24.

Liu, H., Liu, H., Yang, J., 2018, Vibration of FG magneto-electro-viscoelastic porous nanobeams on visco-Pasternak foundation, Composites Part B: Engineering, 155, pp. 244–256.

Thai, S., Thai, H.T., Vo, T.P., Patel, V.I., 2018, A simple shear deformation theory for nonlocal beams, Composite Structures, 183, pp. 262–270.

Karami, B., Janghorban, M., 2019, A new size-dependent shear deformation theory for free vibration analysis of functionally graded/anisotropic nanobeams, Thin-Walled Structures, 143, 106227.

Liu, H., Lv, Z., Wu, H., 2019, Nonlinear free vibration of geometrically imperfect functionally graded sandwich nanobeams based on nonlocal strain gradient theory, Composite Structures, 214, pp. 47–61.

Jankowski, P., Żur, K.K., Kim, J., Reddy, J.N., 2020, On the bifurcation buckling and vibration of porous nanobeams, Composite Structures, 250, 112632.

Jankowski, P., Żur, K.K., Farajpour, A., 2022, Analytical and meshless DQM approaches to free vibration analysis of symmetric FGM porous nanobeams with piezoelectric effect, Engineering Analysis with Boundary Elements, 136, pp. 266-289.

Nasr, M.E., Abouelregal, A.E., Soleiman, A., Khalil, K.M., 2021, Thermoelastic Vibrations of Nonlocal Nanobeams Resting on a Pasternak Foundation via DPL Model, Journal of Applied and Computational Mechanics, 7, pp. 34-44.

Abdelrahman, A.A., Esen, I., Özarpa, C., Eltaher, M.A., 2021, Dynamics of perforated nanobeams subject to moving mass using the nonlocal strain gradient theory, Applied Mathematical Modelling, 96, pp. 215–235.

Wang, Q., 2002, On buckling of column structures with a pair of piezoelectric layers, Engineering Structures, 24, pp. 199-205.

Reddy, J.N., 2007, Energy principles and variational methods in applied mechanics, John Wiley & Sons, USA.

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

Ghavanloo, E., Fazelzadeh S.A., 2016, Evaluation of nonlocal parameter for single-walled carbon nanotubes with arbitrary chirality, Meccanica, 51, pp. 41-54.

Zhang, Z., Wang, C.M., Challamel, N., 2014, Eringen’s length scale coefficient for buckling of nonlocal rectangular plates from microstructured beam-grid model, International Journal of Solids and Structures, 51, pp. 4307-4315.

Mehralian, F., Beni, Y.T., Zeverdejani, M.K., 2017, Calibration of nonlocal strain gradient shell model for buckling analysis of nanotubes using molecular dynamics simulations, Physica B: Condensed Matter, 521, pp. 102-111.



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