Refined Zigzag Theory for Nonlinear Dynamic Response of an Axially Moving Sandwich Nanobeam Embedded on Visco-Pasternak Medium Using MCST

Document Type : Research Paper


1 Faculty of Mechanical Engineering, University of Kashan, Kashan, Islamic Republic of Iran------ Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Islamic Republic of Iran

2 Faculty of Mechanical Engineering, University of Kashan, Kashan, Islamic Republic of Iran


This paper develops the Refined Zigzag Theory (RZT) for nonlinear dynamic response of an axially moving functionally graded (FG) nanobeam integrated with two magnetostrictive face layers based on the modified couple stress theory (MCST). The sandwich nanobeam (SNB) subjected to a temperature difference and both axial and transverse mechanical loads. The material properties of FG core layer depend on the environment temperature and are assumed to vary in thickness direction. The SNB is surrounded by elastic medium which is simulated by visco-Pasternak model. The von-Karman nonlinear strain-displacement relationships are employed to consider the effect of geometric nonlinearities. In order to obtain governing motion equations and boundary conditions the energy method as well as Hamilton’s principle is applied. The differential quadrature method (DQM) is used for space domain and the Newmark-β method is taken into account for time domain response of the axially moving SNB. The detailed parametric study is conducted to investigate the effects of surrounding elastic medium, material length scale parameter, magnetostrictive layers, temperature difference, environment temperature, velocity of the SNB, axial and transverse mechanical loads and volume fraction exponent on the dynamic response of the SNB. Results indicate that the maximum deflection of the system can be controlled by employing negative values of velocity feedback gain values. Also, the system loses its stability when the velocity of SNB is increased.              


[1] Khedir A.A., Aldraihem O.J., 2016, Free vibration of sandwich beams with soft core, Composite Structures 154: 179-189.
[2] Kahya V., 2016, Buckling analysis of laminated composite and sandwich beams by finite element method, Composites Part B: Engineering 91: 126-134.
[3] Jedari Salami S., 2016, Dynamic extended high order sandwich panel theory for transient response of sandwich beams with carbon nanotube reinforced face sheets, Aerospace Science and Technology 56: 56-69.
[4] Simsek M., Al-Shujairi M., 2017, Static, free and forced vibration of functionally graded (FG) sandwich beams excited by two successive moving harmonic loads, Composites Part B: Engineering 108: 18-34.
[5] Kim N., Lee J., 2016, Theory of thin-walled functionally graded sandwich beams with single and double-cell sections, Composite Structures 157: 141-154.
[6] Shi H., Liu W., Fang H., Bai Y., Hui D., 2017, Flexural responses and pseudo-ductile performance of lattice-web reinforced GFRP-wood sandwich beams, Composites Part B: Engineering 108: 364-376.
[7] Arshid E., Khorshidvand A. R., 2018, Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method, Thin-walled Structures 125: 220-233.
[8] Akgoz B., Civalek O., 2014, Mechanical analysis of isolated microtubules based on a higher-order shear deformation theory, Composite Structures 118: 9-18.
[9] Ozutok A., Madenci E., 2017, Static analysis of laminated composite beams based on higher-order shear deformation theory by using mixed-type finite element method, International Journal of Mechanical Sciences 130: 234-243.
[10] Shao D., Hu Sh., Wang Q., Pang F., 2017, Free vibration of refined higher-order shear deformation composite laminated beams with general boundary conditions, Composites Part B: Engineering 108: 75-90.
[11] Zhang Zh., Taheri F., 2003, Dynamic pulsebuckling and postbuckling of composite laminated beam using higher order shear deformation theory, Composites Part B: Engineering 34: 391-398.
[12] Altenbach H., Altenbach J., Kissing W., 2004, Mechanics of Composite Structural Elements, Springer.
[13] Tessler A., Di Sciuva M., Gherlone M., 2007, Refinement of Timoshenko beam theory for composite and sandwich beams using zigzag kinematics, Technical Report NASA/TP-2007-215086.
[14] Tessler A., Di Sciuva M., Gherlone M., 2009, A refined zigzag beam theory for composite and sandwich beams, Journal of Composite Materials 43: 1051-1081.
[15] Nallim L.G., Oller S., Onate E., Flores F.G., 2017, A hierarchical finite element for composite laminated beams using a refined zigzag theory, Composite Structures 163: 168-184.
[16] Di Sciuva M., Gherlone M., Iurlaro L., Tessler A., 2015, A class of higher-order C0 composite and sandwich beam elements based on the Refined Zigzag Theory, Composite Structures 132: 784-803.
[17] Tessler A., Di Sciuva M., Gherlone M., 2010, A consistent refinement of first-order shear deformation theory for laminated composite and sandwich plates using improved zigzag kinematics, Journal of Mechanics of Materials and Structures 5: 341-367.
[18] Gherlone M., Tessler A., Di Sciuva M., 2011, C0 beam elements based on the Refined Zigzag Theory for multilayered composite and sandwich laminates, Composite Structures 93: 2882-2894.
[19] Abdollahian M., Ghorbanpour Arani A., Mosallaie Barzoki AA, Kolahchi R, Loghman A., 2013, Non-local wave propagation in embedded armchair TWBNNTs conveying viscous fluid using DQM, Physica B 418: 1-15.
[20] Bagdatli S.M., 2015, Non-linear vibration of nanobeams with various boundary condition based on nonlocal elasticity theory, Composites Part B: Engineering 80: 43-52.
[21] Aya S.A., Tufekci E., 2017, Modeling and analysis of out-of-plane behavior of curved nanobeams based on nonlocal elasticity, Composites Part B: Engineering 119: 184-195.
[22] Ebrahiminejad S., Marzbanrad J., Boreiry M., Shaghaghi G. R., 2018, On the electro-thermo-mechanical vibration characteristics of elastically restrained functionally graded nanobeams using differential transformation method, Applied Physics A 124: 800.
[23] Amir S., Khani A., Shajari A. R., Dashti P., 2017, Instability analysis of viscoelastic CNTs surrounded by a thermo-elastic foundation, Structural Engineering & Mechanics 63: 171-180.
[24] Amir S., 2016, Orthotropic patterns of visco-Pasternak foundation in nonlocal vibration of orthotropic graphene sheet under thermo-magnetic fields based on new first-order shear deformation theory, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications.
[25] Huang Y., Fu J., Liu A., 2019, Dynamic instability of Euler-Bernoulli nanobeams subject to parametric excitation, Composites Part B: Enginerring 164: 226-234.
[26] Eringen A.C., 1983, On differential equations of nonlocal elasticity theory and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 4703-4710.
[27] Ghorbanpour Arani A., Abdollahian M., Kolahchi R., 2015, Nonlinear vibration of embedded smart composite microtube conveying fluid based on modified couple stress theory, Polymer Composites 36: 1314-1324.
[28] Ke L.L., Wang Y.Sh., 2011, Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory, Composite Structures 93: 342-350.
[29] Ke L.L., Wang Y.Sh., 2011, Flow-induced vibration and instability of embedded double-walled carbon nanotubes based on a modified couple stress theory, Physica E: Low-dimensional Systemns and Nanostructures 43: 1031-1039.
[30] Ke L.L., Wang Y.Sh., Yang J., Kitipornchai S., 2012, Nonlinear free vibration of size-dependent functionally graded microbeams, International Journal of Engineering Science 50: 256-267.
[31] Ghayesh M.H., Farokhi H., Amabili M., 2013, Nonlinear dynamics of a microscale beam based on the modified couple stress theory, Composites Part B: Engineering 50: 318-324.
[32] Ghorbanpour Arani A., Abdollahian M., Kolahchi R., 2015, Nonlinear vibration of a nanobeam elastically bonded with a piezoelectric nanobeam via strain gradient theory, International Journal of Mechanical Sciences 100: 32-40.
[33] Akgoz B., Civalek O., 2011, Strain gradient elasticity theory and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, International Journal of Engineering Science 49: 1268-1280.
[34] Akgoz B., Civalek O., 2015, Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity, Composite Structures 134: 294-301.
[35] Hosseini M., Shishesaz M., Hadi A., 2019, Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness, Thin-Walled Structures 134: 508-523.
[36] Dehghan M., Ebrahimi F., 2018, On wave dispersion characteristics of magneto-electro-elastic nanotubes considering the shell model based on the nonlocal strain gradient elasticity theory, The European Physical Journal Plus 133:466.
[37] Amir S., Khorasani M., BabaAkbar-Zarei H., 2018, Buckling analysis of nanocomposite sandwich plates with piezoelectric face sheets based on flexoelectricity and first order shear deformation theory, Journal of Sandwich Structures & Materials 22: 2186-2209.
[38] Amir S., Bidgoli E. M. R., Arshid E., 2018, Size-dependent vibration analysis of a three-layered porous rectangular nano plate with piezo-electromagnetic face sheets subjected to pre loads based on SSDT, Mechanics of Advanced Materials and Structures 27: 605-619.
[39] Lee J.K., Jeong S., 2016, Flexural and torsional free vibrations of horizontally curved beams on Pasternak foundations, Applied Mathematical Modelling 40: 2242-2256.
[40] Sobamowo M.G., 2017, Nonlinear thermal and flow-induced vibration analysis of fluid-conveying carbon nanotube resting on Winkler and Pasternak foundations, Thermal Science and Engineering Progress 4: 133-149.
[41] Ghorbanpour Arani A., Abdollahian M., Kolahchi R., Rahmati A.H., 2013, Electro-thermo-torsional buckling of an embedded armchair DWBNNT using nonlocal shear deformable shell model, Composites Part B: Engineering 51: 291-299.
[42] Liu J.C., Zhang Y.Q., Fan L.F., 2017, Nonlocal vibration and biaxial buckling of double-viscoelastic-FGM-nanoplate system with viscoelastic Pasternak medium in between, Physics Letters A 381: 1228-1235.
[43] Mohammadi M., Safarabadi M., Rastgoo A., Farajpour A., 2016, Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium in a nonlinear thermal environment, Acta Mechanica 227: 2207-2232.
[44] Jamalpoor A., Bahreman M., Hosseini M., Free transverse vibration analysis of orthotropic multi-viscoelastic microplate system embedded in visco-Pasternak medium via modified strain gradient theory, Journal of Sandwich Structures & Materials 21: 175-210.
[45] Hong C.C., 2010, Transient responses of magnetostrictive plates by using the GDQ method, European Journal of Mechanics A/Solid 29: 1015-1021.
[46] Hong C.C., 2014, Thermal vibration and transient response of magnetostrictive functionally graded material plates, European Journal of Mechanics A/Solids 43: 78-88.
[47] Ghorbanpour Arani A., Abdollahian M., 2017, Transient response of FG higher order nanobeams integrated with magnetostrictive layers using modified couple stress theory, Mechanics of Advanced Materials and Structures 26: 359-371.
[48] Pradhan S.C., 2005, Vibration suppression of FGM shells using embedded magnetostrictive layers, International Journal of Solids and Structures 42: 2465-2488.
[49] Chang J.R., Lin W.J., Huang Ch.J., Choi S.T., 2010, Vibration and stability of an axially moving Rayleigh beam, Applied Mathematical Modelling 34: 1482-1497.
[50] Ghayesh M.H., Amabili M., 2013, Steady-state transverse response of an axially moving beam with time-dependent axial speed, International Journal of Non-Linear Mechanics 49: 40-49.
[51] Lim C.W., Li C., Yu J.L., 2010, Dynamic behavior of axially moving nanobeams based on nonlocal elasticity approach, Acta Mechanica Sinica 26: 755-765.
[52] Rezaee M., Lotfan S., 2015, Non-linear nonlocal vibration and stability analysis of axially moving nanoscale beams with time-dependent velocity, International Journal of Mechanical Sciences 96-97: 36-46.
[53] Ghorbanpour Arani A., Haghparast E., BabaAkbar Zarei H., 2016, Nonlocal vibration of axially moving grapheme sheet resting on orthotropic-Pasternak foundation under longitudinal magnetic field, Physica B: Condensed Matter 495: 35-49.
[54] Ghorbanpour Arani A., Abdollahian M., Jalaei M.H., 2015, Vibration of bioliquid-filled microtubules embedded in cytoplasm including surface effects using modified couple stress theory, Journal of Theoretical Biology 367: 29-38.
[55] Rao S.S., 2007, Vibration of Continuous Systems, John Wiley & Sons.
[56] Dukkipati, R.V., 2010, MATLAB: An Introduction with Applications, New Age International Publishers.