VIBRATION AND STABILITY OF A NONLINEAR NONLOCAL STRAIN-GRADIENT FG BEAM ON A VISCO-PASTERNAK FOUNDATION

Nikola Nešić, Milan Cajić, Danilo Karličić, Mihailo Lazarević, Sondipon Adhikari

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Abstract


This study investigates the stability of periodic solutions of a nonlinear nonlocal strain gradient functionally graded Euler–Bernoulli beam model resting on a visco-Pasternak foundation and subjected to external harmonic excitation. The nonlinearity of the beam arises from the von Kármán strain-displacement relation. Nonlocal stress gradient theory combined with the strain gradient theory is used to describe the stress-strain relation. Variations of material properties across the thickness direction are defined by the power-law model. The governing differential equation of motion is derived by using Hamilton's principle and discretized by the Galerkin approximation. The methodology for obtaining the steady-state amplitude-frequency responses via the incremental harmonic balance method and continuation technique is presented. The obtained periodic solutions are verified against the numerical integration method and stability analysis is performed by utilizing the Floquet theory.

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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.

Mahamood, R.M., Akinlabi, E.T., Shukla, M., Pityana, S.L., 2012, Functionally graded material: an overview, Proceedings of the World Congress on Engineering, Vol. 3. London, UK: International Association of Engineers (IAENG).

Peddieson, J., Buchanan, G.R., McNitt, R.P., 2003, Application of nonlocal continuum models to nanotechnology, International journal of engineering science, 41(3-5), pp. 305-312.

Li, X., Bhushan, B., Takashima, K., Baek, C.W., Kim, Y.K., 2003, Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques, Ultramicroscopy, 97(1-4), pp. 481-494.

Najar, F., Nayfeh, A.H., Abdel-Rahman, E.M., Choura, S., El-Borgi, S., 2010, Dynamics and global stability of beam-based electrostatic microactuators, Journal of Vibration and Control, 16(5), pp. 721-748.

Zavracky, P.M., McGruer, N.E., Morrison, R.H., Potter, D., 1999, Microswitches and microrelays with a view toward microwave applications, International Journal of RF and Microwave Computer‐Aided Engineering, 9(4), pp. 338-347.

Moser, Y., Gijs, M.A., 2007, Miniaturized flexible temperature sensor, Journal of Microelectromechanical Systems, 16(6), pp. 1349-1354.

Ceballes, S., Abdelkefi, A., 2021, Uncertainty analysis and stochastic characterization of carbon nanotube-based mass sensor with multiple deposited nanoparticles, Sensors and Actuators A: Physical, 332(2), 113182.

He, X.Q., Kitipornchai, S., Liew, K.M., 2005, Resonance analysis of multi-layered graphene sheets used as nanoscale resonators, Nanotechnology, 16(10), 2086.

Wang, Z.L., Wu, W., 2012, Nanotechnology‐enabled energy harvesting for self‐powered micro‐/nanosystems, Angewandte Chemie International Edition, 51(47), pp. 11700-11721.

Khan, A.A., Khan, I.U., Noman, M., Khan, U.A. and Bilal, M., 2022, Performance Evaluation of Flexible Pavement Using Carbon Nanotubes and Plastic Waste as Admixtures, Tehnički vjesnik, 29(1), pp. 9-14.

Moory-Shirbani, M., Sedighi, H.M., Ouakad, H.M., Najar, F., 2018, Experimental and mathematical analysis of a piezoelectrically actuated multilayered imperfect microbeam subjected to applied electric potential, Composite Structures, 184, pp. 950-960.

Lam, D.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(8), pp. 1477-1508.

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.

El-Borgi, S., Fernandes, R., Reddy, J.N., 2015, Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation, International Journal of Non-Linear Mechanics, 77, pp. 348-363.

Fan, Y., Xiang, Y., Shen, H.S., 2020, Nonlinear dynamics of temperature-dependent FG-GRC laminated beams resting on visco-Pasternak foundations, International Journal of Structural Stability and Dynamics, 20(01), 2050012.

Reddy, J., 2007, Nonlocal theories for bending, buckling and vibration of beams, International journal of engineering science, 45(2-8), pp. 288-307.

Pradhan, S.C., Phadikar, J.K., 2009, Nonlocal elasticity theory for vibration of nanoplates, Journal of Sound and Vibration, 325(1-2), pp. 206-223.

Limkatanyu, S., Sae-Long, W., Rungamornrat, J., Buachart, C., Sukontasukkul, P., Keawsawasvong, S. and Chindaprasirt, P., 2022. Bending, buckling and free vibration analyses of nanobeam-substrate medium systems, Facta Universitatis, Series: Mechanical Engineering, 20(3), pp. 561-587.

Yang, F.A.C.M., 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.

Wang, G.F., Feng, X.Q., 2007, Effects of surface elasticity and residual surface tension on the natural frequency of microbeams, Applied physics letters, 90(23), 231904.

Lim, C.W., Zhang, G., Reddy, J., 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.

Garg, A., Chalak, H.D., Zenkour, A.M., Belarbi, M.O., Houari, M.S.A., 2022, A review of available theories and methodologies for the analysis of nano isotropic, nano functionally graded, and CNT reinforced nanocomposite structures, Archives of Computational Methods in Engineering, 29, pp. 2237–2270.

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.

Fallah, A., Aghdam, M.M., 2011, Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation, European Journal of Mechanics-A/Solids, 30(4), pp. 571-583.

Şimşek, M., 2016, Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach, International Journal of Engineering Science, 105, pp. 12-27.

Nešić, N., Cajić, M., Karličić, D., Obradović, A., Simonović, J., 2022, Nonlinear vibration of a nonlocal functionally graded beam on fractional visco-Pasternak foundation, Nonlinear Dynamics, 107(3), pp. 2003-2026.

Barretta, R., de Sciarra, F.M., 2018, Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams, International Journal of Engineering Science, 130, pp. 187-198.

Apuzzo, A., Barretta, R., Faghidian, S.A., Luciano, R., De Sciarra, F.M., 2019, Nonlocal strain gradient exact solutions for functionally graded inflected nano-beams, Composites Part B: Engineering, 164, pp. 667-674.

Barretta, R., Faghidian, S.A., Luciano, R., Medaglia, C.M., Penna, R., 2018, Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress-driven nonlocal models, Composites Part B: Engineering, 154, pp. 20-32.

Pinnola, F.P., Faghidian, S.A., Barretta, R., de Sciarra, F.M., 2020, Variationally consistent dynamics of nonlocal gradient elastic beams, International Journal of Engineering Science, 149, 103220.

Sahmani, S., Safaei, B., 2020, Influence of homogenization models on size-dependent nonlinear bending and postbuckling of bi-directional functionally graded micro/nano-beams, Applied Mathematical Modelling, 82, pp. 336-358.

Li, L., Hu, Y., 2016, Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Science, 107, pp. 77-97.

Liu, H., Wu, H., Lyu, Z., 2020, Nonlinear resonance of FG multilayer beam-type nanocomposites: effects of graphene nanoplatelet-reinforcement and geometric imperfection, Aerospace Science and Technology, 98, 105702.

Penna, R., Feo, L., Fortunato, A., Luciano, R., 2021, Nonlinear free vibrations analysis of geometrically imperfect FG nano-beams based on stress-driven nonlocal elasticity with initial pretension force, Composite Structures, 255, 112856.

Penna, R., Feo, L., 2020, Nonlinear dynamic behavior of porous and imperfect Bernoulli-Euler functionally graded nanobeams resting on Winkler elastic foundation, Technologies, 8(4), 56.

Vaccaro, M.S., Barretta, R., Marotti de Sciarra, F., Reddy, J.N., 2022, Nonlocal integral elasticity for third-order small-scale beams, Acta Mechanica, 233(6). pp. 2393–2403.

Ansari, R., Faraji Oskouie, M., Nesarhosseini, S., Rouhi, H., 2022, Vibrations of piezoelectric nanobeams considering flexoelectricity influence: a numerical approach based on strain-driven nonlocal differential/integral models, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 44(2), 57.

Romano, G., Barretta, R., 2017, Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams, Composites Part B: Engineering, 114, pp. 184-188.

Faraji Oskouie, M., Ansari, R., Rouhi, H., 2018, Bending of Euler–Bernoulli nanobeams based on the strain-driven and stress-driven nonlocal integral models: a numerical approach, Acta Mechanica Sinica, 34(5), pp. 871-882.

Faraji Oskouie, M., Ansari, R., Rouhi, H., 2018, A numerical study on the buckling and vibration of nanobeams based on the strain and stress-driven nonlocal integral models, International Journal of Computational Materials Science and Engineering, 7(03), 1850016.

Vaccaro, M.S., Pinnola, F.P., Marotti de Sciarra, F., Barretta, R., 2021, Elastostatics of Bernoulli–Euler beams resting on displacement-driven nonlocal foundation, Nanomaterials, 11(3), 573.

Patnaik, S., Sidhardh, S., Semperlotti, F., 2022, Displacement-driven approach to nonlocal elasticity, European Journal of Mechanics-A/Solids, 92, 104434.

Pinnola, F.P., Vaccaro, M.S., Barretta, R., Marotti de Sciarra, F., Ruta, G., 2023, Elasticity problems of beams on reaction-driven nonlocal foundation, Archive of Applied Mechanics, 93, pp. 41–71.

Gao, Y., Xiao, W., Zhu, H., 2019, Nonlinear vibration of functionally graded nano-tubes using nonlocal strain gradient theory and a two-steps perturbation method, Structural Engineering and Mechanics, 69(2), pp. 205-219.

Gao, Y., Xiao, W.S., Zhu, H., 2020, Snap-buckling of functionally graded multilayer graphene platelet-reinforced composite curved nanobeams with geometrical imperfections, European Journal of Mechanics-A/Solids, 82, 103993.

Gao, Y., Xiao, W.S., Zhu, H., 2019, Nonlinear vibration analysis of different types of functionally graded beams using nonlocal strain gradient theory and a two-step perturbation method, The European Physical Journal Plus, 134(1), 23.

Wang, J., Shen, H., 2019, Nonlinear vibrations of axially moving simply supported viscoelastic nanobeams based on nonlocal strain gradient theory. Journal of Physics: Condensed Matter, 31(48), 485403.

Jafarsadeghi-Pournaki, I., Azizi, S., Zamanzadeh, M., Madinei, H., Shabani, R., Rezazadeh, G., 2020, Size-dependent dynamics of a FG nanobeam near nonlinear resonances induced by heat, Applied Mathematical Modelling, 86, pp. 349-367.

Li, L., Hu, Y., Li, X., 2016, Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory, International Journal of Mechanical Sciences, 115-116, pp. 135-144.

Li, X., Li, L., Hu, Y., Ding, Z., Deng, W., 2017. Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Composite Structures, 165, pp. 250-265.

Bhattiprolu, U., Bajaj, A.K., Davies, P., 2016, Periodic response predictions of beams on nonlinear and viscoelastic unilateral foundations using incremental harmonic balance method, International Journal of Solids and Structures, 99, pp. 28-39.

Jalaei, M.H., Arani, A.G., Nguyen-Xuan, H., 2019, Investigation of thermal and magnetic field effects on the dynamic instability of FG Timoshenko nanobeam employing nonlocal strain gradient theory, International Journal of Mechanical Sciences, 161-162, 105043.

Nayfeh, A.H., Lacarbonara, W., 1997, On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities, Nonlinear Dynamics, 13(3), pp. 203-220.

Hashemian, M., Falsafioon, M., Pirmoradian, M., Toghraie, D., 2020, Nonlocal dynamic stability analysis of a Timoshenko nanobeam subjected to a sequence of moving nanoparticles considering surface effects, Mechanics of Materials, 148, 103452.

Trabelssi, M., El-Borgi, S., Friswell, M.I., 2020, A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method, Archive of Applied Mechanics, 90(10), pp. 2133-2156.

Nešić, N., Cajić, M., Karličić, D., Janevski, G., 2021, Nonlinear superharmonic resonance analysis of a nonlocal beam on a fractional visco-Pasternak foundation, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 235(20), pp. 4594-4611.

Engelnkemper, S., Gurevich, S.V., Uecker, H., Wetzel, D., Thiele, U., 2019, Continuation for thin film hydrodynamics and related scalar problems, Computational modelling of bifurcations and instabilities in fluid dynamics, Computational Methods in Applied Sciences, 50, pp. 459-501

Wang, S., Hua, L., Yang, C., Han, X.. Su, Z., 2019, Applications of incremental harmonic balance method combined with equivalent piecewise linearization on vibrations of nonlinear stiffness systems, Journal of Sound and Vibration, 441, pp. 111-125.

Shen, J.H., Lin, K.C., Chen, S.H., Sze, K.Y., 2008, Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method, Nonlinear Dynamics, 52(4), pp. 403-414.

Huang, J.L., Su, R.K.L., Lee, Y.Y., Chen, S.H., 2011, Nonlinear vibration of a curved beam under uniform base harmonic excitation with quadratic and cubic nonlinearities, Journal of Sound and Vibration, 330(21), pp. 5151-5164.

Hsu, C.S., 1974, On approximating a general linear periodic system, Journal of Mathematical Analysis and Applications, 45(1), pp. 234-251.

Togun, N., Bağdatlı, S.M., 2016, Nonlinear vibration of a nanobeam on a Pasternak elastic foundation based on non-local Euler-Bernoulli beam theory, Mathematical and Computational Applications, 21(1), 3.

Eyebe, G.J., Betchewe, G., Mohamadou, A., Kofane, T.C., 2018, Nonlinear vibration of a nonlocal nanobeam resting on fractional-order viscoelastic Pasternak foundations, Fractal and Fractional, 2(3), 21.

Yokoyama, T., 1987, Vibrations and transient responses of Timoshenko beams resting on elastic foundations, Ingenieur-Archiv, 57(2), pp. 81-90.

Mustapha, K.B., Zhong, Z.W., 2010, Free transverse vibration of an axially loaded non-prismatic single-walled carbon nanotube embedded in a two-parameter elastic medium, Computational Materials Science, 50(2), pp. 742-751.

Ansari, R., Gholami, R., Hosseini, K., Sahmani, S., 2011, A sixth-order compact finite difference method for vibrational analysis of nanobeams embedded in an elastic medium based on nonlocal beam theory, Mathematical and Computer Modelling, 54(11-12), pp. 2577-2586.

Zhao, X., Wang, C.F., Zhu, W.D., Li, Y.H., Wan, X.S., 2021, Coupled thermoelastic nonlocal forced vibration of an axially moving micro/nano-beam, International Journal of Mechanical Sciences, 206, 106600.

Brennan, M.J., Kovacic, I., Carrella, A., Waters, T.P., 2008, On the jump-up and jump-down frequencies of the Duffing oscillator, Journal of Sound and Vibration, 318(4-5), pp. 1250-1261.


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