Considering Bending and Vibration of Homogeneous Nanobeam Coated by a FG Layer

Document Type: Research Paper

Authors

1 Department of Mechanical Engineering, Ilam University, Ilam 69315-516, Iran

2 Department of Mechanical Engineering, Kermanshah University of Technology, Kermanshah, Iran

10.22034/jsm.2019.1870709.1457

Abstract

In this research static deflection and free vibration of homogeneous nanobeams coated by a functionally graded (FG) layer is investigated according to the nonlocal elasticity theory. A higher order beam theory is used that does not need the shear correction factor. The equations of motion (equilibrium equations) are extracted by using Hamilton’s principle. The material properties are considered to vary in the thickness direction of FG coated layer. This nonlocal nanobeam model incorporates the length scale parameter (nonlocal parameter) that can capture the small scale effects. In the numerical results section, the effects of different parameters, especially the ratio of thickness of FG layer to the total thickness of the beam are considered and discussed. The results reveal that the frequency is maximum for a special value of material power index. Also, increasing the ratio of thickness of FG layer to the total thickness of the beam increases the static deflection and decreases the natural frequencies. These results help with the understanding such coated structures and designing them carefully. The results also show that the new nonlocal FG nanobeam model produces larger vibration and smaller deflection than homogeneous nonlocal nanobeam.

Keywords

[1] Iijima S., 1991, Helical Microtubules of Graphitic Carbon, Nature 354: 56-58.
[2] Zhang Y.Y., Wang C. M., Duan W.H., Xiang Y., Zong Z., 2009, Assessment of continuum mechanics models in predicting buckling strains of singlewalled carbon nanotubes, Nanotechnology 20: 395707.
[3] Nix W.D., GAO H., 1998, Indentation size effects in crystalline materials: A law for strain gradient plasticity, Journal of the Mechanics and Physics of Solids 46: 411-425.
[4] Hadjesfandiari A.R., Dargush G.F., 2011, Couple stress theory for solids, International Journal of Solids and Structures 48: 2496-2510.
[5] Asghari M., Kahrobaiyan M.H., Ahmadian M.T., 2010, A nonlinear Timoshenko beam formulation based on the modified couple stress theory, International Journal of Engineering Science 48: 1749-1761.
[6] Ma H.M., GAO X.L., Reddy J.N., 2008, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, Journal of Mechanics Physics and Solids 56: 3379-3391.
[7] Reddy J., 2011, Microstructure-dependent couple stress theories of functionally graded beams, Journal of the Mechanics and Physics of Solids 59(11): 2382-2399.
[8] Eringen A.C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science 10: 1-16.
[9] Eringen A.C., Edelen D.G.B., 1972, On nonlocal elasticity, International Journal of Engineering Science 10: 233-248.
[10] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 4703.
[11] Thai H-T., 2012, A nonlocal beam theory for bending, buckling, and vibration of nanobeams, International Journal of Engineering Science 52: 56-64.
[12] Eringen A.C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science 10: 1-16.
[13] Eringen A.C., Edelen D.G.B., 1972, On non-local elasticity, International Journal of Engineering Science 10: 230.
[14] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 4703-4710.
[15] Eringen A.C., 2002, Nonlocal Continuum Field Theories, Springer, New York.
[16] Peddieson J., Buchanan G.R., McNitt R.P., 2003, Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science 41: 305-312.
[17] Sudak L.J., 2003, Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics, Journal of Applied Physics 94: 7281.
[18] Ece M.C., Aydogdu M., 2007, Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes, Acta Mechanica 190: 185-195.
[19] Lu P., Lee H.P., Lu C., Zhang P.Q., 2007, Application of nonlocal beam models for carbon nanotubes, International Journal Solids and Structures 44: 5289-5300.
[20] Aydogdu M., 2009, Axial vibration of the nanorods with the nonlocal continuum rod model, Physica E 41: 861-864.
[21] Hu Y., Liew K.M., Wang Q., He X.Q., Yakobson B.I., 2008, Nonlocal shell model for elastic wave propagation in single-and double-walled carbon nanotubes, Journal of the Mechanics and Physics of Solids 56: 3475-3485.
[22] Sakhaee-Pour A., 2009, Elastic buckling of single-layer grapheme sheet, Computational Materials Science 45: 266-270.
[23] Aydogdu M., 2009, Ageneral nonlocal beam theory: Its application to nanobeam bending, buckling and vibration Physica E 41:1651-1655.
[24] Simsek M., Yurtcu H., 2012, Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory, Composite Structures 97: 378-386.
[25] Byrd L.W., Birman V., 2007, Modeling and analysis of functionally graded materials and structures, Applied Mechanics Reviews 60: 195-216.
[26] 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: 55-70.
[27] Ebrahimi F., Rastgoo A., Atai A., 2009, A theoretical analysis of smart moderately thick shear deformable annular functionally graded plate, European Journal of Mechanics-A/Solids 28(5): 962-973.
[28] Ebrahimi F., Naei M.H., Rastgoo A., 2009, Geometrically nonlinear vibration analysis of piezoelectrically actuated FGM plate with an initial large deformation, Journal of Mechanical Science and Technology 23(8): 2107-2124.
[29] Ghadiri M., 2017, On size-dependent thermal buckling and free vibration of circular FG Microplates in thermal environments, Microsystem Technologies 23: 4989-5001.
[30] Shafiei N., Ghadiri M., Mahinzare M., 2017, Flapwise bending vibration analysis of rotary tapered functionally graded nanobeam in thermal environment, Mechanics of Advanced Materials and Structures 26: 139-155.
[31] Reddy J.N., Chin C.D., 1998, Thermomechanical analysis of functionally graded cylinders and plates, Journal of Thermal Stresses 21: 593-626.
[32] Praveen G.N., Reddy J.N., 1998, Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates, International Journal Solids and Structures 35(33): 4457-4476.
[33] Pisano A.A., Sofi A., Fuschi P., 2009, Nonlocal integral elasticity: 2D finite element based solutions, International Journal of Solids and Structures 46: 3836-3849.
[34] Pisano A.A., Sofi A., Fuschi P., 2009, Finite element solutions for nonhomogeneous nonlocal elastic problems, Mechanics Research Communications 36: 755-761.
[35] Eltaher M.A., Samir A.E., Mahmoud F.F., 2013, Static and stability analysis of nonlocal functionally graded nanobeams, Composite Structures 96: 82-88.
[36] Song M., Yang J., Kitipornchai S., 2018, Bending and buckling analyses of functionally graded polymer composite plates reinforced with graphene nanoplatelets, Composites Part B: Engineering 134: 106-113.
[37] Wu H., Kitipornchai S., Yang J., 2017, Thermal buckling and postbuckling of functionally graded graphene nanocomposite plates, Materials& Design 132: 430-441.
[38] Khanchehgardan A., Rezazadeh G., Shabani R., 2013, Effect of mass diffusion on the damping ratio in a functionally graded micro-beam, Composite Structures 106: 15-29.
[39] Zhen W., Wanji C., 2006, A higher-order theory and refined three-node triangular element for functionally graded plates, European Journal of Mechanics-A/Solids 25(3): 447-463.
[40] Reddy J., 2000, Analysis of functionally graded plates, International Journal of Numerical Methods Engineering 47(1-3): 663-684.
[41] Ke L-L., Wang Y-S., Yang J., Kitipornchai S., 2012, Nonlinear free vibration of size-dependent functionally graded microbeams, International Journal of Engineering Science 50(1): 256-267.
[42] Jia X., Ke L., Feng C., Yang J., Kitipornchai S., 2015, Size effect on the free vibration of geometrically nonlinear functionally graded micro-beams under electrical actuation and temperature change, Composite Structures 133: 1137-1148.
[43] Li L., Hu Y., 2017, Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects, International Journal of Mechanical Sciences 120: 159-170.
[44] Chen D., Yang J., Kitipornchai S., 2017, Nonlinear vibration and postbuckling of functionally graded graphene reinforced porous nanocomposite beams, Composites Science and Technology 142: 235-245.
[45] Arbind A., Reddy J., 2013, Nonlinear analysis of functionally graded microstructure-dependent beams, Composite Structures 98: 272-281.
[46] Reddy J.N., 2011, Microstructure-dependent couple stress theories of functionally graded beams, Journal of the Mechanics and Physics of Solids 59: 2382-2399.
[47] Yang J., Wu H., Kitipornchai S., 2017, Buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams, Composite Structures 161: 111-118.
[48] Wu H., Kitipornchai S., Yang J., 2017, Thermal buckling and postbuckling of functionally graded graphene nanocomposite plates, Materials& Design 132: 430-441.
[49] Simsek M., 2015, Size dependent nonlinear free vibration of an axially functionally graded (AFG) microbeam using he’s variational method,. Composite Structures 131: 207-214.
[50] Ke L-L., Yang J., Kitipornchai S., Wang Y-S., 2014, Axisymmetric postbuckling analysis of size-dependent functionally graded annular microplates using the physical neutral plane, International Journal of Mechanical Sciences 81: 66-81.
[51] Ke L-L., Yang J., Kitipornchai S., Bradford M.A., 2012, Bending, buckling and vibration of size-dependent functionally graded annular microplates, Composite Structures 94(11): 3250-3257.
[52] Kitipornchai S., Ke L., Yang J., Xiang Y., 2009, Nonlinear vibration of edge cracked functionally graded timoshenko beams, Journal of Sound and Vibration 324(3): 962-982.
[53] Farokhi H., Ghayesh M.H., Gholipour A., 2017, Dynamics of functionally graded micro-cantilevers, International Journal of Mechanical Sciences 115: 117-130.
[54] Shafiei N., Kazemi M., 2017, Nonlinear buckling of functionally graded nano-/micro-scaled porous beams, Composite Structures 178: 483-492.
[55] Lee J.W., Lee J.Y., 2017, Free vibration analysis of functionally graded bernoulli-euler beams using an exact transfer matrix expression, International Journal of Mechanical Sciences 122: 1-17.
[56] Tianzhi Y., Ye T., Qian L., Xiao-Dong Y., 2018, Nonlinear bending, buckling and vibration of bi directional functionally graded nanobeams, Composite Structures 204: 313-319.
[57] Leissa A.W., 1986, Conditions for laminated plates to remain flat under inplane loading, Composite Structures 6(4): 262-270.
[58] Leissa A.W., 1987, A review of laminated composite plate buckling, Applied Mechanics Reviews 40(5): 575-591.
[59] Qatu M.S., Leissa A.W., 1993, Buckling or transverse deflections of unsymmetrically laminated plates subjected to in-plane loads, AIAA Journal 31(1): 189-194.
[60] Shen H.S., 2004, Bending, buckling and vibration of functionally graded plates and shells, Advances in Mechanics 34(1): 53-60.
[61] Aydogdu M., 2008, Conditions for functionally graded plates to remain flat under inplane loads by classical plate theory, Composite Structures 82: 155-157.
[62] Shen H.S., 2002, Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments, International Journal of Mechanical Sciences 44: 561-584.
[63] Ma L.S., Wang T.J., 2003, Axisymmetric post-buckling of a functionally graded circular plate subjected to uniformly distributed radial compression, Materials Science Forum 423-424:719-724.
[64] Ma L.S., Wang T.J., 2003, Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings, International Journal Solids and Structures 40(13-14): 3311-3330.
[65] Ma L.S., Lee D.W., 2011, A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading, Composite Structures 93: 831-842.
[66] Wang Q., Wang C.M., 2007, The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes, Nanotechnology 18: 075702.