Vibration Analysis of Magneto-Electro-Elastic Timoshenko Micro Beam Using Surface Stress Effect and Modified Strain Gradient Theory under Moving Nano-Particle

Document Type: Research Paper


Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran


In this article, the free vibration analysis of magneto-electro-elastic (MEE) Timoshenko micro beam model based on surface stress effect and modified strain gradient theory (MSGT) under moving nano-particle is presented. The governing equations of motion using Hamilton’s principle are derived and these equations are solved using differential quadrature method (DQM). The effects of dimensionless electric potential, dimensionless magnetic parameter, material length scale parameter, external electric voltage, external magnetic parameter, slenderness ratio, temperature change, surface stress effect, two parameters of elastic foundation on the dimensionless natural frequency are investigated. It is shown that the effect of electric potential and magnetic parameter simultaneously increases the dimensionless natural frequency. On the other hands, with considering two parameters, the stiffness of MEE Timoshenko micro beam model increases. It can be seen that the dimensionless natural frequency of micro structure increases by MSGT more than modified couple stress theory (MCST) and classical theory (CT). It is found that by increasing the mass of nano-particle, the dimensionless natural frequency of system decreases. The results of this study can be employed to design and manufacture micro-devices to prevent resonance phenomenon or as a sensor to control the dynamic stability of micro structures.


[1] Sun K.H., Kim Y.Y., 2010, Layout design optimization for magneto-electro-elastic laminate composites for maximized energy conversion under mechanical loading, Smart Materials and Structures 19: 055008.
[2] Wang B.L., Niraula O.P., 2007, Transient thermal fracture analysis of transversely isotropic magneto-electro-elastic materials, Journal of Thermal Stresses 30: 297-317.
[3] Priya S., Islam R., Dong S., Viehland D., 2007, Recent advancements in magneto-electric particulate and laminate composites, Journal of Electroceramics 19: 149-166.
[4] Zhai J., Xing Z., Dong S., Li J., Viehland D., 2008, Magnetoelectric laminate composites: an overview, Journal of the American Ceramic Society 91: 351-358.
[5] Nan C.W., Bichurin M., Dong S., Viehland D., Srinivasan G., 2008, Multiferroic magnetoelectric composites: historical perspective, status, and future directions, Journal of Applied Physics 103: 031101.
[6] Bhangale R.K., Ganesan N., 2006, Free vibration of functionally graded non-homogeneous magneto-electro-elastic cylindrical shell, International Journal for Computational Methods in Engineering Science and Mechanics 7: 191-200.
[7] Lang Z., Xuewu L., 2013, Buckling and vibration analysis of functionally graded magneto-electro-thermo-elastic circular cylindrical shells, Applied Mathematical Modelling 37: 2279-2292.
[8] Razavi S., Shooshtari A., 2015, Nonlinear free vibration of magneto-electro-elastic rectangular plates, Composite Structures 119: 377-384.
[9] 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: 516-525.
[10] Li Y.S., Cai Z.Y., Shi S.Y., 2014, Buckling and free vibration of magneto-electro-elastic nanoplate based on nonlocal theory, Composite Structures 111: 522-529.
[11] Shooshtari A., Razavi S., 2015, Linear and nonlinear free vibration of a multilayered magneto-electro-elastic doubly-curved shell on elastic foundation, Composite Part B 78: 95-108.
[12] Shooshtari A., Razavi S., 2015, Large amplitude free vibration of symmetrically laminated magneto-electro-elastic rectangular plates on Pasternak type foundation, Mechanics Research Communications 69: 103-113.
[13] Mohammadimehr M., Rostami R., Arefi M., 2016, Electro-elastic analysis of a sandwich thick plate considering FG core and composite piezoelectric layers on Pasternak foundation using TSDT, Steel and Composite Structures 20: 513-543.
[14] Ansari R., Gholami R., Rouhi H., 2015, Size-dependent nonlinear forced vibration analysis of magneto-electro-thermo-elastic Timoshenko Nano beams based upon the nonlocal elasticity theory, Composite Structures 126: 216-226.
[15] Xin L., Hu Z., 2015, Free vibration of layered magneto-electro-elastic beams by SS-DSC approach, Composite Structures 125: 96-103.
[16] Xin L., Hu Z., 2015, Free vibration of simply supported and multilayered magneto-electro-elastic plates, Composite Structures 121: 344-350.
[17] Mohammadimehr M., Monajemi A.A., Moradi M., 2015, Vibration analysis of viscoelastic tapered micro-rod based on strain gradient theory resting on visco-Pasternak foundation using DQM, Journal of Mechanical Science and Technology 29 (6): 2297-2305.
[18] Rahmati A.H., Mohammadimehr M., 2014, Vibration analysis of non-uniform and non-homogeneous boron nitride nanorods embedded in an elastic medium under combined loadings using DQM, Physica B: Condensed Matter 440: 88-98.
[19] Ke L.L., Wang Y.S., 2014, Free vibration of size-dependent magneto-electro-elastic Nano beams based on the nonlocal theory, Phisyca E 63: 52-61.
[20] Wang Y., Xu R., Ding H., 2011, Axisymmetric bending of functionally graded circular magneto-electro-elastic plates, European Journal of Mechanics-A/Solid 30: 999-1011.
[21] Rao M.N., Schmidt R., Schröder K.U., 2015, Geometrically nonlinear static FE-simulation of multilayered magneto-electro-elastic, Composite Structures 127: 120-131.
[22] Mohammadimehr M., Rousta Navi B., Ghorbanpour Arani A., 2015, Free vibration of viscoelastic double-bonded polymeric nanocomposite plates reinforced by FG-SWCNTs using MSGT, sinusoidal shear deformation theory and meshless method, Composite Structures 131: 654-671.
[23] Mohammadimehr M., Rousta Navi B., Ghorbanpour Arani A., 2016, Modified strain gradient Reddy rectangular plate model for biaxial buckling and bending analysis of double-coupled piezoelectric polymeric nanocomposite reinforced by FG-SWNT, Composite Part B: Engineering 87: 132-148.
[24] Kattimani S.C., Ray M.C., 2015, Control of geometrically nonlinear vibrations of functionally graded magneto-electro-elastic plates, International Journal of Mechanical Sciences 99: 154-167.
[25] Liu Y., Han Q., Li C., Liu X., Wu B., 2015, Guided wave propagation and mode differentiation in the layered magneto-electro-elastic hollow cylinder, Composite Structures 132: 558-566.
[26] Sedighi H. M., Farjam N., 2016, A modified model for dynamic instability of CNT based actuators by considering rippling deformation, tip-charge concentration and Casimir attraction, Microsystem Technologies 23: 2175-2191.
[27] Zare J., 2015, Pull-in behavior analysis of vibrating functionally graded micro-cantilevers under suddenly DC voltage, Journal of Applied and Computational Mechanics 1(1): 17-25.
[28] Sedighi H. M., 2014, The influence of small scale on the pull-in behavior of nonlocal nano bridges considering surface effect, Casimir and van der Waals attractions, International Journal of Applied Mechanics 6(3): 1450030.
[29] Fleck N. A., Hutchinson J. W., 1993, Phenomenological theory for strain gradient effects in plasticity, Journal of the Mechanics and Physics of Solids 41(12): 1825-1857.
[30] Fleck N. A., Hutchinson J. W., 1997, Strain gradient plasticity, Advances in Applied Mechanics 33: 296-358.
[31] Fleck N. A., Hutchinson J. W., 2001, A reformulation of strain gradient plasticity, Journal of the Mechanics and Physics of Solids 49(10): 2245- 2271.
[32] Lam D.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 Solids 51: 1477-1508.
[33] Akgöz B., Civalek Ö., 2013, A size-dependent shear deformation beam model based on the strain gradient elasticity theory, International Journal of Engineering Science 70: 1-14.
[34] Mohammadimehr M., Salemi M., Rousta Navi B., 2016, Bending, buckling, and free vibration analysis of MSGT microcomposite Reddy plate reinforced by FG-SWCNTs with temperature- dependent material properties under hydro-thermo-mechanical loadings using DQM, Composite Structures 138: 361-380.
[35] Gurtin M., Ian Murdoch A., 1975, A continuum theory of elastic material surfaces, Archive for Rational Mechanics and Analysis 57: 291-323.
[36] Gurtin M., Ian Murdoch A., 1987, Surface stress in solids, International Journal of Solids and Structures 14: 431-440.
[37] Mohammadimehr M., Rousta Navi B., Ghorbanpour Arani A., 2015, Surface stress effect on the nonlocal biaxial buckling and bending analysis of polymeric piezoelectric Nano plate reinforced by CNT using Eshelby-Mori-Tanaka approach, Journal of Solid Mechanics 7( 2): 173-190.
[38] Karimi M., Shokrani M. H., Shahidi A. R., 2015, Size-dependent free vibration analysis of rectangular nanoplates with the consideration of surface effects using finite difference method, Journal of Applied and Computational Mechanics 1(3): 122-133.
[39] Ghorbanpour Arani A., Kolahchi R., Mosayebi M., Jamali M., 2016, Pulsating fluid induced dynamic instability of visco-double-walled carbon nano-tubes based on sinusoidal strain gradient theory using DQM and Bolotin method, International Journal of Mechanics and Materials in Design 12(1): 17-38.
[40] Ansari R., Gholami R., Sahmani S., 2011, Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory, Composite Structures 94 : 221-228.
[41] Şimşek M., Kocatürk T., Akbaş Ş.D., 2013, Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory, Composite Structures 95: 740-747.
[42] Li Y.S., Feng W.J., Cai Z.Y., 2014, Bending and free vibration of functionally graded piezoelectric beam based on modified strain gradient theory, Composite Structures 115: 41-50.
[43] Ghorbanpour Arani A., Abdollahian M., Kolahchi R., 2015, Nonlinear vibration of a Nano beam elastically bonded with a piezoelectric Nano beam via strain gradient theory, International Journal of Mechanical Sciences 100: 32-40.
[44] Ansari R., Mohammadi V., Faghih Shojaei M., Gholami R., Rouhi H., 2013, Nonlinear vibration analysis of Timoshenko Nano beams based on surface stress elasticity theory, European Journal of Mechanics-A/Solid 45 :143-152.
[45] Ke L.L., Wang Y.S., Wang Z.D., 2012, Nonlinear vibration of the piezoelectric Nano beams based on the nonlocal theory, Composite Structures 94: 2038-2047.
[46] Şimşek M., 2011, Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle, Computational Materials Science 50: 2112-2123.
[47] Ghorbanpour Arani A., Mortazavi S.A., Kolahchi R., Ghorbanpour Arani A.H., 2015, Vibration response of an elastically connected double-Smart Nano beam-system based nano-electro-mechanical sensor, Journal of Solid Mechanics 7: 121-130.
[48] Ghorbanpour Arani A., Atabakhshian V., Loghman A., Shajari A.R., Amir S., 2012, Nonlinear vibration of embedded SWBNNTs based on nonlocal Timoshenko beam theory using DQ method, Phisyca B 407: 2549-2555.
[49] Murmu T., Pradhan S.C., 2009, Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM, Phisyca E 41: 1232-1239.
[50] Civalek O., 2006, Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation, Journal of Sound and Vibrations 294: 966-980.
[51] Akgoz B., Civalek O., 2011, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, International Journal of Engineering Science 49: 1268-1280.
[52] Zhang B., He Y., Liu D., Gan Z., Shen L., 2014, Non - classical Timoshenko beam element based on the strain gradient elasticity theory, Finite Element in Analysis and Design 79: 22-39.
[53] Ansari R., Gholami R., Darabi M.A., 2012, A non-linear Timoshenko beam formulation based on strain gradient theory, Journal of Mechanics of Materials and Structures 7: 195-211.
[54] Ghorbanpour Arani A., Kolahchi R., Zarei M.Sh., 2015, Visco-surface-nonlocal piezo-elasticity effects on nonlinear dynamic stability of graphene sheets integrated with ZnO sensors and actuators using refined zigzag theory, Composite Structures 132: 506-526.