**67**, 3, pp. 347–367, 2019

**10.24423/EngTrans.1001.20190426**

### 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

nano-beam using the general form of the Eringen theory generalized based on the fractional derivatives.

In this paper, using the conformable fractional derivative (CFD) definition the generalized form of the Eringen nonlocal theory (ENT) is used to consider the effects of integer and noninteger stress gradients in the constitutive relation and also to consider small-scale effect in the vibration of a Timoshenko nano-beam. The governing equation is solved by the Galerkin method.

Free vibration and buckling of a Timoshenko simply supported (S) nano-beam is investigated, and the influence of the fractional and nonlocal parameters is shown on the frequency ratio and buckling ratio. In this sense, the obtained formulation allows for an easier mapping of experimental results on nano-beams.

The new theory (fractional parameter) makes the modeling more flexible. The model can conclude all of the integer and non-integer operators and is not limited to the special operators such as ENT. In other words, it allows to use more sophisticated/flexible mathematics to model physical phenomena.

**Keywords**: fractional calculus; nonlocal fractional derivative model; free vibration; Timoshenko beam; Galerkin method; buckling

**Full Text:**PDF

#### References

Koiter W.T., Couple stresses in the theory of elasticity, I and II, Nederlandse Akademie van Wetenschappen Proceedings Series B, 67: 17–29, 1964.

Mindlin R.D., Tiersten H.F., Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11(1): 415–448, 1962.

Yang F., Chong A.C.M., Lam D.C.C., Tong P., Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39(10): 2731–2743, 2002.

Aifantis E., On the role of gradients in the localization of deformation and fracture, International Journal of Engineering Science, 30: 1279–1299, 1992.

Mindlin R.D., Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures, 1: 417–438, 1965.

Mindlin R.D., Eshel N.N., On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures, 4: 109–124, 1968.

Eringen A.C., Nonlocal Polar Field Models, Academic, New York, 1976.

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.

Gurtin M.E., Murdoch A.I., A continuum theory of elastic material surfaces, Archive for Rational Mechanics and Analysis, 57(4): 291–323, 1975.

Challamel N., Zorica D., Atanacković T.M., Spasić D.T., On the fractional generalization of Eringen’s nonlocal elasticity for wave propagation, Comptes Rendus Mécanique, 341(3): 298–303, 2013.

Rahimi Z., Sumelca W., Yang X.J., Linear and non-linear free vibration of nano beams based on a new fractional nonlocal theory, Engineering Computations, 34(4): 1754–1770, 2017, DOI:10.1108/EC-07-2016-0262.

Davis G.B., Kohandel M., Sivaloganathan S., Tenti G., The constitutive properties of the brain paraenchyma: Part 2. Fractional derivative approach, Medical Engineering & Physics, 28(5): 455–459, 2006.

Bagley R.L., Torvik P.J., A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology, 27(3): 201–210, 1983.

Ahmad W.M., El-Khazali R., Fractional-order dynamical models of love, Chaos, Solitons and Fractals, 33(4): 1367–1375, 2007.

Lima M.F., Machado J.A.T., Crisóstomo M.M., Experimental signal analysis of robot impacts in a fractional calculus perspective, Journal of Advanced Computational Intelligence and Intelligent Informatics, 11(9): 1079–1085, 2007.

Torvik P.J., Bagley R.L., On the appearance of the fractional derivative in the behavior of real materials, Journal of Applied Mechanics, 51(2): 294–298, 1984.

De Espındola J.J., da Silva Neto J.M., Lopes E.M., A generalised fractional derivative approach to viscoelastic material properties measurement, Applied Mathematics and Computation, 164(2): 493–506, 2005.

Podlubny I., Petráš I., Vinagre B.M., O’leary P., Dorčák Ľ., Analogue realizations of fractional-order controllers, Nonlinear Dynamics, 29(1–4): 281–296, 2002.

Silva M.F., Machado J.T., Lopes A.M., Fractional order control of a hexapod robot, Nonlinear Dynamics, 38(1–4): 417–433, 2004.

Sommacal L., Melchior P., Oustaloup A., Cabelguen J.M., Ijspeert A.J., Fractional multi-models of the frog gastrocnemius muscle, Journal of Vibration and Control, 14(9–10): 1415–1430, 2008.

Heymans N., Dynamic measurements in long-memory materials: fractional calculus evaluation of approach to steady state, Journal of Vibration and Control, 14(9–10): 1587–1596, 2008.

Lazopoulos K.A., Non-local continuum mechanics and fractional calculus, Mechanics Research Communications, 33(6): 753–757, 2006.

Sumelka W., Non-local continuum mechanics based on fractional calculus, Advances in Applied Mechanics, 33: 295–361, 1997.

Sumelka W., Non-local Kirchhoff–Love plates in terms of fractional calculus, Archives of Civil and Mechanical Engineering, 15(1): 231–242, 2015.

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.

Atanackovic T.M., Stankovic B., Generalized wave equation in nonlocal elasticity, Acta Mechanica, 208(1–2): 1–10, 2009.

Demir D.D., Bildik N., Sinir B.G., Linear vibrations of continuum with fractional derivatives, Boundary Value Problems, 1: 1–15, 2013.

Demir D.D., Bildik N., Sinir B.G., Application of fractional calculus in the dynamics of beams, Boundary Value Problems, 1: 1–13, 2012.

Carpinteri A., Cornetti P., Sapora A., A fractional calculus approach to nonlocal elasticity, The European Physical Journal-Special Topics, 193(1): 193–204, 2011.

Yang X.J., Baleanu D., Srivastava H.M., Local fractional integral transforms and their applications, Academic Press, 2015.

Yang X.J., Local Fractional Functional Analysis & Its Applications, Hong Kong: Asian Academic Publisher Limited, 2011.

Khalil R., Al Horani M., Yousef A., Sababheh M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264: 65–70, 2014.

Wang C.M., Zhang Y.Y., He X.Q., Vibration of nonlocal Timoshenko beams, Nanotechnology, 18(10): 105401, 2007.

Wang C.M., Kitipornchai S., Lim C.W., Eisenberger M., Beam bending solutions based on nonlocal Timoshenko beam theory, Journal of Engineering Mechanics, 134(6): 475–481, 2008.

Wang C.M., Zhang Y.Y., Ramesh S.S., Kitipornchai S., Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory, Journal of Physics D: Applied Physics, 39(17): 3904, 2006.

Ghannadpour S.A.M., Mohammadi B., Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory using Chebyshev polynomials, Advanced Materials Research, 123: 619–622, 2010.

Reddy J.N., Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science, 45(2): 288–307, 2007.

Aydogdu M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E: Low-dimensional Systems and Nanostructures, 41(9): 1651–1655, 2009.

Thai H.T., A nonlocal beam theory for bending, buckling, and vibration of nanobeams, International Journal of Engineering Science, 52: 56–64, 2012.

Bhrawy A.H., Alofi A.S., The operational matrix of fractional integration for shifted Chebyshev polynomials, Applied Mathematics Letters, 26(1); 25–31, 2013.

Secer A., Alkan S., Akinlar M.A., Bayram M., Sinc-Galerkin method for approximate solutions of fractional order boundary value problems, Boundary Value Problems, 1: 281, 14 pages, 2013.

Kazem S., An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations, Applied Mathematical Modelling, 37(3): 1126–1136, 2013.

Doha E.H., Bhrawy A.H., Ezz-Eldien S.S., A new Jacobi operational matrix: an application for solving fractional differential equations, Applied Mathematical Modelling, 36(10): 4931–4943, 2012.

Bhrawy A.H., Alghamdi M.M., Taha T.M., A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line, Advances in Difference Equations, 2012: 179, 12 pages, 2012.

DOI: 10.24423/EngTrans.1001.20190426

Copyright © 2014 by Institute of Fundamental Technological Research

Polish Academy of Sciences, Warsaw, Poland