Size-Dependent Analysis of Orthotropic Mindlin Nanoplate on Orthotropic Visco-Pasternak Substrate with Consideration of Structural Damping

Document Type : Research Paper


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

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

3 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran



This paper discusses static and dynamic response of nanoplate resting on an orthotropic visco-Pasternak foundation based on Eringen’s nonlocal theory. Graphene sheet modeled as nanoplate which is assumed to be orthotropic and viscoelastic. By considering the Mindlin plate theory and viscoelastic Kelvin-Voigt model, equations of motion are derived using Hamilton’s principle which are then solved analytically by means of Fourier series -Laplace transform method. The parametric study is thoroughly accomplished, concentrating on the influences of size effect, elastic foundation type, structural damping, orthotropy directions and damping coefficient of the foundation, modulus ratio, length to thickness ratio and aspect ratio. Results depict that the structural and foundation damping coefficients are effective parameters on the dynamic response, particularly for large damping coefficients, where response of nanoplate is damped rapidly.  


[1] Lee C., Wei X.D., Kysar J.W., Hone J., 2008, Measurement of the elastic properties and intrinsic strength of monolayer graphene, Science 321: 385-388.
[2] Robinson J.T., Zalalutdinov M., Baldwin J.W., Snow E.S., Wei Z., Sheehan P., Houston B.H., 2008, Wafer-scale reduced graphene oxide films for nanomechanical devices, Nano Letters 8: 3441-3445.
[3] Pisana S., Braganca P.M., Marinero E.E., Gurney B.A., 2010, Graphene magnetic field sensors, IEEE Transactions on Magnetics 46: 1910-1913.
[4] Kuilla T., Bhadra S., Yao D., Kim N.H., Bose S., Lee J.H., 2010, Recent advances in graphene based polymer composites, Progress in Polymer Science 35: 1350-1375.
[5] Ryzhii M., Satou A., Ryzhii V., Otsuji T., 2008, High-frequency properties of a graphene nanoribbon field-effect transistor, Journal of Applied Physics 104: 114505.
[6] Murmu T., Adhikari S., 2013, Nonlocal mass nanosensors based on vibrating monolayer graphene sheets, Sensors and Actuators B: Chemical 188: 1319-1327.
[7] Kuila T., Bose S., Khanra P., Mishra A.K., Kim N.H., Lee J.H., 2011, Recent advances in graphene-based biosensors, Biosensors and Bioelectronics 26: 4637-4648.
[8] Sun X., Liu Z., Welsher K., Robinson J.T., Goodwin A., Zaric S., Dai H., 2008, Nano-graphene oxide for cellular imaging and drug delivery, Nano Research 1: 203-212.
[9] 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: 1477-1508.
[10] Akgöz, B., Civalek Ö., 2012, Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory, Materials and Design 42: 164-171.
[11] 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.
[12] Eringen A.C., 2002, Nonlocal Continuum Field Theories, New York, Springer.
[13] Pradhan S.C., Murmu T., 2009, Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics, Computational Materials Science 47: 268-274.
[14] Hosseini Hashemi Sh., Tourki Samaei A., 2011, Buckling analysis of micro/nanoscale plates via nonlocal elasticity theory, Physica E 43: 1400-1404.
[15] Shen H.S., 2011, Nonlocal plate model for nonlinear analysis of thin films on elastic foundations in thermal environments, Composite Structures 93: 1143-1152.
[16] Ansari R., Rouhi H., 2012, Explicit analytical expressions for the critical buckling stresses in a monolayer graphene based on nonlocal elasticity, Solid State Communications 152: 56-59.
[17] Mohammadi M., Ghayour M., Farajpour A., 2013, Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Composites: Part B 45: 32-42.
[18] Zenkour A.M., Sobhy M., 2013, Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler–Pasternak elastic substrate medium, Physica E 53: 251-259.
[19] Ghorbanpour Arani A., Maboudi M.J., Kolahchi R., 2014, Nonlinear vibration analysis of viscoelastically coupled DLAGS-system, European Journal of Mechanics A/Solids 45: 185-197.
[20] Kananipour H., 2014, Static analysis of nanoplates based on the nonlocal Kirchhoff and Mindlin plate theories using DQM, Latin American Journal of Solids and Structures 11: 1709-1720.
[21] Mohammadi M., Farajpour A., Goodarzi M., Shehni nezhad pour H., 2014, Numerical study of the effect of shear in-plane load on the vibration analysis of graphene sheet embedded in an elastic medium, Computational Materials Science 82: 510-520.
[22] Golmakani M.E., Rezatalab J., 2014, Nonlinear bending analysis of orthotropic nanoscale plates in an elastic matrix based on nonlocal continuum mechanics, Composite Structures 111: 85-97.
[23] Liu C.C., Chen Z.B., 2014, Dynamic analysis of finite periodic nanoplate structures with various boundaries, Physica E 60: 139-146.
[24] Ghorbanpour Arani A., Jalaei M.H., 2015, Nonlocal dynamic response of embedded single-layered graphene sheet via analytical approach, Journal of Engineering Mathematics 92: 129-144.
[25] Pouresmaeeli S., Ghavanloo E., Fazelzadeh S.A., 2013, Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium, Composite Structures 96: 405-410.
[26] Karličić D., Kozić P., Pavlović R., 2014, Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium, Composite Structures 115: 89-99.
[27] Wang Y., Li F.M., Wang Y.Z., 2015, Nonlinear vibration of double layered viscoelastic nanoplates based on the nonlocal theory, Physica E 67: 65-76.
[28] Hosseini Hashemi Sh., Mehrabani H., Ahmadi-Savadkoohi A., 2015, Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium, Composites Part B 78: 377-383.
[29] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Boca Raton.
[30] Ghorbanpour Arani A., Shiravand A., Rahi M., Kolahchi R., 2012, Nonlocal vibration of coupled DLGS systems embedded on visco-Pasternak foundation, Physica B 407: 4123-4131.
[31] Kutlu A., Uğurlu B., Omurtag M.H., Ergin A., 2012, Dynamic response on Mindlin plates resting on arbitrarily orthotropic Pasternak foundation and partially in contact with fluid, Ocean Engineering 42: 112-125.
[32] Kiani Y., Sadighi M., Eslami M.R., 2013, Dynamic analysis and active control of smart doubly curved FGM panels, Composite Structures 102: 205-216.
[33] Krylov V.I., Skoblya N.S., 1977, A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation, Moscow, Mir Publishers.
[34] Mohammadi M., Farajpour A., Goodarzi M., Heydarshenas., 2013, Levy type solution for nonlocal thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium, Journal of Solid Mechanics 5: 116-132.
[35] Thai H-T., Kim S-E., 2012, Analytical solution of a two variable refined plate theory for bending analysis of orthotropic Levy-type plates, International Journal of Mechanical Sciences 54: 269-276.
[36] Reddy J.N., 2000, Analysis of functionally graded plates, International Journal for Numerical Methods in Engineering 47: 663-684.