Size-Dependent Higher Order Thermo-Mechanical Vibration Analysis of Two Directional Functionally Graded Material Nanobeam
Subject Areas : Engineering
M
Mahinzare
1
(School of Engineering, Tehran University, Tehran, Iran)
S
Amanpanah
2
(Faculty of Engineering, Department of Mechanics, Imam Khomeini International University, Qazvin, Iran)
M
Ghadiri
3
(Faculty of Engineering, Department of Mechanics, Imam Khomeini International University, Qazvin, Iran)
Keywords: Free vibration, Two directional FGM, Nanobeam, Nonlocal theory, Thermal load,
Abstract :
This paper represented a numerical technique for discovering the vibrational behavior of a two-directional FGM (2-FGM) nanobeam exposed to thermal load for the first time. Mechanical attributes of two-directional FGM (2-FGM) nanobeam are changed along the thickness and length directions of nanobeam. The nonlocal Eringen parameter is taken into the nonlocal elasticity theory (NET). Uniform temperature rise (UTR), linear temperature rise (LTR), non-linear temperature rise (NLTR) and sinusoidal temperature rise (STR) during the thickness and length directions of nanobeam is analyzed. Third-order shear deformation theory (TSDT) is used to derive the governing equations of motion and associated boundary conditions of the two-directional FGM (2-FGM) nanobeam via Hamilton’s principle. The differential quadrature method (DQM) is employed to achieve the natural frequency of two-directional FGM (2-FGM) nanobeam. A parametric study is led to assess the efficacy of coefficients of two-directional FGM (2-FGM), Nonlocal parameter, FG power index, temperature changes, thermal rises loading and temperature rises on the non-dimensional natural frequencies of two-directional FGM (2-FGM) nanobeam.
[1] Wang S.S., 1983, Fracture mechanics for delamination problems in composite materials, Journal of Composite Materials 17(3): 210-223.
[2] Mohammadi K., Mahinzare M., Ghorbani K., Ghadiri M., 2017, Cylindrical functionally graded shell model based on the first order shear deformation nonlocal strain gradient elasticity theory, Microsystem Technologies 24: 1133-1146.
[3] Shafiei N., Ghadiri M., Mahinzare M., 2019, Flapwise bending vibration analysis of rotary tapered functionally graded nanobeam in thermal environment, Mechanics of Advanced Materials and Structures 26: 139-155.
[4] Ghadiri M., Mahinzare M., Shafiei N., Ghorbani K., 2017, On size-dependent thermal buckling and free vibration of circular FG Microplates in thermal environments, Microsystem Technologies 23: 4989-5001.
[5] Tejaswini N., Ramesh Babu K., Sai Ram K.S., 2012, Functionally graded material: An overview, International Journal of Advanced Engineering Science and Technological 3: 5-7.
[6] Hosseini-Hashemi S., Nazemnezhad R., Bedroud M., 2014, Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity, Applied Mathematical Modelling 38(14): 3538-3553.
[7] Anjomshoa A., 2013, Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory, Meccanica 48(6): 1337-1353.
[8] Shojaeefard M.H., Googarchin H.S., Ghadiri M., Mahinzare M., 2017, Micro temperature-dependent FG porous plate: Free vibration and thermal buckling analysis using modified couple stress theory with CPT and FSDT, Applied Mathematical Modelling 50: 633-655.
[9] Tornabene F., Fantuzzi N., Bacciocchi M., 2016, The local GDQ method for the natural frequencies of doubly-curved shells with variable thickness: A general formulation, Composites Part B: Engineering 92: 265-289.
[10] Tornabene F., Fantuzzi N., Bacciocchi M., Viola E., 2015, Static and dynamic analyses of doubly-curved composite thick shells with variable radii of curvatures, Conference: XXII Convegno Italiano dell’Associazione Italiana di Meccanica Teorica e Applicata (AIMETA2015).
[11] Mohammadi K., Mahinzare M., Rajabpour A., Ghadiri M., 2017, Comparison of modeling a conical nanotube resting on the Winkler elastic foundation based on the modified couple stress theory and molecular dynamics simulation, European Physical Journal Plus 132(3): 115.
[12] Şimşek M., 2010, Vibration analysis of a functionally graded beam under a moving mass by using different beam theories, Composite Structures 92(4): 904-917.
[13] Ghadiri M., Shafiei N., Safarpour H., 2017, Influence of surface effects on vibration behavior of a rotary functionally graded nanobeam based on Eringen’s nonlocal elasticity, Microsystem Technologies 23(4): 1045-1065.
[14] Tornabene F., Fantuzzi N., Ubertini F., Viola E., 2015, Strong formulation finite element method based on differential quadrature: A survey, Applied Mechanics Reviews 67(2): 20801.
[15] Eringen A.C., Edelen D.G.B., 1972, On nonlocal elasticity, International Journal of Engineering Science 10(3): 233-248.
[16] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54(9): 4703-4710.
[17] Peddieson J., Buchanan G.R., McNitt R.P., 2003, Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science 41(3-5): 305-312.
[18] Reddy J.N., 2007, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science 45(2-8): 288-307.
[19] Wang Q., Liew K.M., 2007, Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures, Physics Letters, Section A: General, Atomic and Solid State Physics 363(3): 236-242.
[20] Aydogdu M., 2009, A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E: Low-Dimensional Systems and Nanostructures 41(9): 1651-1655.
[21] Fernández Sáez J., Zaera R., Loya J.A., Reddy J.N., 2016, Bending of Euler-Bernoulli beams using Eringen’s integral formulation: A paradox resolved, International Journal of Engineering Science 99: 107-116.
[22] Phadikar J.K., Pradhan S.C., 2010, Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates, Computational Materials Science 49(3): 492-499.
[23] Pradhan S.C., Murmu T., 2010, Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever, Physica E: Low-Dimensional Systems and Nanostructures 42(7): 1944-1949.
[24] Mahinzare M., Ranjbarpur H., Ghadiri M., 2018, Free vibration analysis of a rotary smart two directional functionally graded piezoelectric material in axial symmetry circular nanoplate, Mechanical Systems and Signal Processing 100: 188-207.
[25] Ebrahimi F., Salari E., 2015, Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method, Composites Part B: Engineering 79: 156-169.
[26] Ebrahimi F., Salari E., 2015, A semi-analytical method for vibrational and buckling analysis of functionally graded nanobeams considering the physical neutral axis position, CMES-Computer Modeling in Engineering & Sciences 105(2): 151-181.
[27] Ebrahimi F., Ghadiri M., Salari E., Hoseini S.A.H., Shaghaghi G.R., 2015, Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams, Journal of Mechanical Science and Technology 29(3): 1207-1215.
[28] Ke L.L., Wang Y.S., 2011, Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory, Composite Structures 93(2): 342-350.
[29] Thai H.T., 2012, A nonlocal beam theory for bending, buckling, and vibration of nanobeams, International Journal of Engineering Science 52: 56-64.
[30] Eltaher M.A., Emam S.A., Mahmoud F.F., 2012, Free vibration analysis of functionally graded size-dependent nanobeams, Applied Mathematics and Computation 218(14): 7406-7420.
[31] Eltaher M.A., Alshorbagy A.E., Mahmoud F.F., 2013, Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams, Composite Structures 99: 193-201.
[32] Eltaher M.A., Emam S.A., Mahmoud F.F., 2013, Static and stability analysis of nonlocal functionally graded nanobeams, Composite Structures 96: 82-88.
[33] Mahinzare M., Mohammadi K., Ghadiri M., Rajabpour A., 2017, Size-dependent effects on critical flow velocity of a SWCNT conveying viscous fluid based on nonlocal strain gradient cylindrical shell model, Microfluidics Nanofluidics 21(7): 123.
[34] Malekzadeh P., Shojaee M., 2013, Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams, Composites Part B: Engineering 52: 84-92.
[35] Ebrahimi F., Barati M.R., 2016, A nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams, Arabian Journal for Science and Engineering 41(5): 1679-1690.
[36] Huang Y., Li X.F., 2010, A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, Journal of Sound and Vibration 329(11): 2291-2303.
[37] Şimşek M., Cansiz S., 2012, Dynamics of elastically connected double-functionally graded beam systems with different boundary conditions under action of a moving harmonic load, Composite Structures 94(9): 2861-2878.
[38] Rezaiee-Pajand M., Hozhabrossadati S.M., 2016, Analytical and numerical method for free vibration of double-axially functionally graded beams, Composite Structures 152: 488-498.
[39] Tounsi A., Semmah A., Bousahla A.A., 2013, Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory, Journal of Nanomechanics Micromechanics 3(3): 37-42.
[40] Al-Basyouni K.S., Tounsi A., Mahmoud S.R., 2015, Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position, Composite Structures 125: 621-630.
[41] Mahinzare M., Barooti M.M., Ghadiri M., 2017, Vibrational investigation of the spinning bi-dimensional functionally graded (2-FGM) micro plate subjected to thermal load in thermal environment, Microsystem Technologies 24(3): 1695-1711.
[42] Sahmani S., Bahrami M., Ansari R., 2014, Nonlinear free vibration analysis of functionally graded third-order shear deformable microbeams based on the modified strain gradient elasticity theory, Composite Structures 110(1): 219-230.
[43] Khorshidi K., Fallah A., 2016, Buckling analysis of functionally graded rectangular nano-plate based on nonlocal exponential shear deformation theory, International Journal of Mechanical Sciences 113: 94-104.
[44] Asghari M., Rahaeifard M., Kahrobaiyan M.H., Ahmadian M.T., 2011, The modified couple stress functionally graded Timoshenko beam formulation, Materials and Design 32(3): 1435-1443.
[45] Asghari M., Ahmadian M.T., Kahrobaiyan M.H., Rahaeifard M., 2010, On the size-dependent behavior of functionally graded micro-beams, Materials and Design 31(5): 2324-2329, 2010.
[46] Alshorbagy A.E., Eltaher M.A., Mahmoud F.F., 2011, Free vibration characteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling 35(1): 412-425.
[47] 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(1): 221-228.
[48] Simsek M., Yurtcu H.H., 2013, Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory, Composite Structures 97: 378-386.
[49] Thai H.T., Vo T.P., 2012, Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories, International Journal of Mechanical Sciences 62(1): 57-66.
[50] Niknam H., Aghdam M.M., 2015, A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation, Composite Structures 119: 452-462.
[51] Ghiasian S.E., Kiani Y., Eslami M.R., 2013, Dynamic buckling of suddenly heated or compressed FGM beams resting on nonlinear elastic foundation, Composite Structures 106: 225-234.
[52] Ghiasian S.E., Kiani Y., Eslami M.R., 2015, Nonlinear thermal dynamic buckling of FGM beams, European Journal of Mechanics - A/Solids 54: 232-242.
[53] Ebrahimi F., Salari E., 2015, Effect of various thermal loadings on buckling and vibrational characteristics of nonlocal temperature-dependent FG nanobeams, Mechanics of Advanced Materials and Structures 23(12): 1-58.
[54] Alibeigloo A., 2010, Thermoelasticity analysis of functionally graded beam with integrated surface piezoelectric layers, Composite Structures 92(6): 1535-1543.
[55] Su Z., Jin G., Ye T., 2016, Vibration analysis and transient response of a functionally graded piezoelectric curved beam with general boundary conditions, Smart Materials and Structures 25(6): 65003.
[56] Li Y., Shi Z., 2009, Free vibration of a functionally graded piezoelectric beam via state-space based differential quadrature, Composite Structures 87(3): 257-264.
[57] Xiang H.J., Shi Z.F., 2009, Static analysis for functionally graded piezoelectric actuators or sensors under a combined electro-thermal load, European Journal of Mechanics - A/Solids 28(2): 338-346.
[58] Ebrahimi F., Barati M.R., 2016, Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium, Journal of the Brazilian Society of Mechanical Sciences and Engineering 39: 937-952.
[59] Ebrahimi F., Reza Barati M., 2016, Vibration analysis of nonlocal beams made of functionally graded material in thermal environment, European Physical Journal Plus 131(8): 1-22.
[60] Ghadiri M., Shafiei N., 2016, Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method, Microsystem Technologies 22(12): 2853-2867.
[61] Shafiei N., Kazemi M., Ghadiri M., 2016, Nonlinear vibration of axially functionally graded tapered microbeams, International Journal of Engineering Science 102: 12-26.
[62] Shafiei N., Kazemi M., Ghadiri M., 2016, On size-dependent vibration of rotary axially functionally graded microbeam, International Journal of Engineering Science 101: 29-44.
[63] Ghadiri M., Shafiei N., 2016, Vibration analysis of rotating functionally graded Timoshenko microbeam based on modified couple stress theory under different temperature distributions, Acta Astronautica 121: 221-240.
[64] Shafiei N., Mousavi A., Ghadiri M., 2016, Vibration behavior of a rotating non-uniform FG microbeam based on the modified couple stress theory and GDQEM, Composite Structures 149: 157-169.
[65] 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.
[66] Ebrahimi F., Barati M.R., 2016, Dynamic modeling of a thermo–piezo-electrically actuated nanosize beam subjected to a magnetic field, Applied Physics A 122(4): 1-18.
[67] Reddy J.N., Chin C.D., 1998, Thermomechanical analysis of functionally graded cylinders and plates, Journal of Thermal Stresses 21(6): 593-626.