Engineering Transactions, Online first

Nonlocal State-Space Strain Gradient Approach to the Vibration of Piezoelectric Functionally Graded Nanobeam

Karunya Institute of Technology and Sciences Coimbatore

Karunya Institute of Technology and Sciences Coimbatore

Imam Khomeini International University

In this work, the state-space nonlocal strain gradient theory is used for the vibration analysis of piezoelectric functionally graded material (FGM) nanobeam. Power law relations are used to describe the computing analysis of FGM constituent properties. The refined higher-order beam theory and Hamilton’s principle are used to obtain the motion of equations of the piezoelectric nanobeam. Besides, the governing equations of the piezoelectric nanobeam are extracted by the developed nonlocal state-space theory, and the analytical wave dispersion method is used to solve wave propagation problems. The real and imaginary solutions for wave frequency, loss factor and wave number are obtained and presented in graphs.

Keywords: wave propagation; functionally graded materials (FGMs); nonlocal strain gradient state-space theory; piezoelectric nanobeam
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Ke L.-L., Wang Y.-S., Wang Z.-D., Nonlinear vibration of piezoelectric nanobeams based on the nonlocal theory, Composite Structures, 94(6): 2038–2047, 2012,

Ke L.-L., Wang Y.-S., Thermo electric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory, Smart Materials and Structures, 21(2): 025018, 2012,

Ebrahimi F., Khosravi K., Dabbagh A., Wave dispersion in viscoelastic FG nanobeam via a novel spatial-temporal nonlocal strain gradient framework, Waves in Random and Complex Media, 2021,

Lazopoulos K.A., Lazopoulos A.K., On the fractional deformation of a linearly elastic bar, Journal of the Mechanical Behavior of Materials, 29(1): 9–18, 2020,

Lazopoulos K.A., Lazopoulos A.K., On fractional bending of beams, Archive of Applied Mechanics, 86: 1133–1145, 2016,

Alotta G., Failla G., Zingales M., Finite element formulation of a nonlocal hereditary fractional-order Timoshenko beam, Journal of Engineering Mechanics, 143(5): D4015001, 2017,

Sumelka W., Blaszczyk T., Liebold C., Fractional Euler-Bernoulli beams: Theory, numerical study and experimental validation, European Journal of Mechanics – A/Solids, 54: 243–251, 2015,

Sidhardh S., Patnaik S., Semperlotti F., Geometrically nonlinear response of a fractional-order nonlocal model of elasticity, International Journal of Non-Linear Mechanics, 125: 103529, 2020,

Stempin P., Sumelka W., Space-fractional Euler-Bernoulli beam model – Theory and identification for silver nanobeam bending, International Journal of Mechanical Sciences, 186: 105902, 2020,

Oskouie M.F., Ansari R., Rouhi H., Bending analysis of functionally graded nanobeams based on the fractional nonlocal continuum theory by the variational Legendre spectral collocation method, Meccanica, 53(4): 1115–1130, 2018,

Romano G., Barretta R., Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams, Composites Part B: Engineering, 114: 184–188, 2017,

Barretta R., Fabbrocino F., Luciano R., Sciarra F.M. de, Closed-form solutions in stress-driven two-phase integral elasticity for bending of functionally graded nano-beams, Physica E: Low-Dimensional Systems and Nanostructures, 97: 13–30, 2018,

Beda P.B., Dynamical systems approach of internal length in fractional calculus, Engineering Transactions, 65(1): 209–215, 2017.

Mohammadi F.S., Rahimi Z., Sumelka W., Yang X.-J., Investigation of free vibration and buckling of Timoshenko nano-beam based on a general form of Eringen theory using conformable fractional derivative and Galerkin method, Engineering Transactions, 67(3): 347–367, 2019, http://

Nguyen T.-T., Kim N.-I., Lee J., Free vibration of thin-walled functionally graded open-section beams, Composites Part B: Engineering, 95: 105–116, 2016,

Ebrahimi F., Barati M.R., Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magneto-electrical field in thermal environment, Journal of Vibration and Control, 24(3): 549–564, 2018,

Zaoui F.Z., Ouinas D., Tounsi A., New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations, Composites Part B:Engineering, 159: 231–247, 2019, doi:

Alibeigi B., Tadi Beni Y., Mehralian F., On the thermal buckling of magneto-electro-elastic piezoelectric nanobeams, The European Physical Journal Plus, 133: 133, 2018,

Shariati A., Ebrahimi F., Karimiasl M., Selvamani R., Toghroli A., On bending characteristics of smart magneto-electro-piezoelectric nanobeams system, Advances in Nano Research, 9(3): 183–191, 2020,

Ebrahimi F., Karimiasl M., Selvamani R., Bending analysis of magneto-electro piezoelectric nanobeams system under hygro-thermal loading, Advances in Nano Research, 8(3): 203–214, 2020,

Ebrahimi F., Dabbagh A., Wave Propagation Analysis of Smart Nanostructures, CRC Press, 2019,

Li S-R., Su H-D., Cheng C-J., Free vibration of functionally graded material beams with surface- bonded piezoelectric layers in thermal environment, Applied Mathematics and Mechanics, 30(8): 969–982, 2009,

Kiani Y., Eslami M.R., Thermal buckling analysis of functionally graded material beams, International Journal of Mechanics and Materials in Designs, 6(3): 229–238, 2010,

Sun D., Luo S-N., Wave propagation of functionally graded material plates in thermal environments, Ultrasonics, 51(8): 940–952, 2011,

Thai H-T., Choi D-H., A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation, Composites Part B: Engineering, 43(5): 2335–2347, 2012,

Thai H-T., Park T., Choi D-H., An efficient shear deformation theory for vibration of functionally graded plates, Archive of Applied Mechanics, 83(1): 137–149, 2013,

Shahsavari D., Shahsavari M., Li L., Karami B., A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation, Aerospace Science and Technology, 72: 134–149, 2018,

Eringen A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54(9): 4703–4710, 1983,

Civalek Ö., Demir Ç., Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory, Applied Mathematical Modelling, 35(5): 2053–2067, 2011,

Arani A.G., Amir S., Shajari A.R., Mozdianfard M.R., Electro-thermo-mechanical buckling of DWBNNTs embedded in bundle of CNTs using nonlocal piezoelasticity cylindrical shell theory, Composites Part B: Engineering, 43(2): 195–203, 2012,

Reddy J.N., El-Borgi S., Eringen’s nonlocal theories of beams accounting for moderate rotations, International Journal of Engineering Science, 82: 159–177, 2014,

Zhang L.L., Liu J.X., Fang X.Q., Nie G.Q., Effects of surface piezoelectricity and nonlocal scale on wave propagation in piezoelectric nanoplates, European Journal of Mechanics - A/Solids, 46: 22–29, 2014,

Nejad M.Z., Hadi A., Eringen’s non-local elasticity theory for bending analysis of bi-directional functionally graded Euler–Bernoulli nano-beams, International Journal of Engineering Sciences, 106: 1–9, 2016,

Nejad M.Z., Hadi A., Non-local analysis of free vibration of bi-directional functionally graded Euler–Bernoulli nano-beams, International Journal of Engineering Sciences, 105: 1–11, 2016,

Nejad M.Z., Hadi A., Rastgoo A., Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory, International Journal of Engineering Sciences, 103: 1–10, 2016,

Ebrahimi F., Barati M.R., Size-dependent vibration analysis of viscoelastic nanocrystalline silicon nanobeams with porosities based on a higher order refined beam theory, Composite Structures, 166: 256–267, 2017,

Lim C.W., Zhang G., Reddy J.N., A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids, 78: 298–313, 2015,

Farajpour A., Haeri Yazdi M.R., Rastgoo A., Mohammadi M., A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment, Acta Mechanica, 227(7): 1849–1867, 2016,

Li L., Li X., Hu Y., Free vibration analysis of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Sciences, 102: 77–92, 2016,

Ebrahimi F., Dabbagh A., Wave propagation analysis of embedded nanoplates based on a nonlocal strain gradient-based surface piezoelectricity theory, The European Physical Journal Plus, 132(11): 449, 2017,

Hadi A., Zamani Nejad M., Hosseini M., Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Sciences, 128: 12–23, 2018, doi:

DOI: 10.24423/EngTrans.2242.20221020