Vibration of FG viscoelastic nanobeams due to a periodic heat flux via fractional derivative model

Document Type : Research Paper


1 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

3 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt


In this work, the vibrations of viscoelastic functionally graded Euler–Bernoulli nanostructure beams are investigated using the fractional-order calculus. It is assumed that the functionally graded nanobeam (FGN) is due to a periodic heat flux. FGN can be considered as nonhomogenous composite structures; with continuous structural changes along the thick- ness of the nanobeam usually, it changes from ceramic at the bottom of the metal at the top. Based on the nonlocal model of Eringen, the complete analytical solution to the problem is established using the Laplace transform method. The effects of different parameters are illustrated graphically and discussed. The effects of fractional order, damping coefficient, and periodic frequency of the vibrational behavior of nanobeam was investigated and discussed. It also provides a conceptual idea of the FGN and its distinct advantages compared to other engineering materials. The results obtained in this work can be applied to identify of many nano-structures such as nano-electro mechanical systems (NEMS), nano-actuators, etc.


Main Subjects

[1]    Rahaeifard M, Kahrobaiyan MH, Ahmadian MT, Firoozbakhsh K. Strain gradient formulation of functionally graded nonlinear beams. Int J Eng Sc 2013; 65: 49–63.
[2]    Koizumi M. The concept of FGM. Ceramic Trans 1993; 34: 3–10.
[3]    Zenkour AM, Abouelregal AE. Effect of harmonically varying heat on FG nanobeams in the context of a nonlocal two-temperature thermoelasticity theory. Euro J Comp Mech 2014; 23(1–2): 1–14.
[4]    Abouelregal AE, Zenkour AM. Thermoelastic problem of an axially moving microbeam subjected to an external transverse excitation. J Theor Appl Mech 2015; 53(1): 167–178 Warsaw.
[5]    Sankar BV. An elasticity solution for functionally graded beams. J Compos Sci Technol2001; 61(5): 689–696.
[6]    Aydogdu M, Taskin V. Free vibration analysis of functionally graded beams with simply supported edges. J Mater Des2007; 28(5): 1651–1656.
[7]    Chakraborty A, Gopalakrishnan S, Reddy JN. A new beam finite element for the analysis of functionally graded materials. Int J Mech Sci 2003; 45(3): 519-539.
[8]    Zenkour AM, Abouelregal AE. Effect of ramp-type heating on the vibration of functionally graded microbeams without energy dissipation. Mech Advan Mat Struc 2016; 23(5): 529–537.
[9]    Alibeigloo A. Thermoelasticity analysis of functionally graded beam with integrated surface piezoelectric layers. Comp Struc 2010; 92(6): 1535–1543.
[10]     Allam MNM, Abouelregal AE. The thermoelastic waves induced by pulsed laser and varying heat of inhomogeneous microscale beam resonators. J Therm Stres 2014; 37(4), 455-470.
[11]    Carrera E, Abouelregal AE, Abbas IA, Zenkour AM. Vibrational analysis for an axially moving microbeam with two temperatures. J. Therm Stres 2015; 38: 569–590.
[12]    Uymaz, B. Forced vibration analysis of functionally graded beams using nonlocal elasticity. Comp Struct 2013; 105: 227-239.
[13]    Abouelregal AE, Zenkour AM. Effect of phase lags on thermoelastic functionally graded microbeams subjected to ramp-type heating. IJST, Trans Mech Eng 2014; 38(M2): 321–335.
[14]    Eringen AC. Nonlocal polar elastic continua. Inte J Eng Sci 1972; 10: 1–16.
[15]    Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 1983; 54: 4703–4710.
[16]    Eringen AC, Edelen DGB. On nonlocal elasticity. Int J Eng Sci 1972; 10: 233–248.
[17]    Adhikari S, Mrumu T, McCarthy MA. Dynamic finite element analysis of axially vibrating nonlocal rods. Fin Elem Analy Design 2013; 63: 42–50.
[18]    Benzair A, Tounsi1 A, Besseghier A, Heireche H, Moulay N, Boumia L. The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. J Phys D: Appl Phys 2008; 41(22): 225404-1-10.
[19]    Wang, Q, Liew KM. Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys Lett A 2007; 363(3): 236–242.
[20]    Togun N. Nonlocal beam theory for nonlinear vibrations of a nanobeam resting on elastic foundation. Bound Val Prob 2016; 1: 1-14.
[21]    Zenkour AM, Abouelregal AE. Vibration of FG nanobeams induced by sinusoidal pulse-heating via a nonlocal thermoelastic model. Acta Mech 2014; 225(12): 3409–3421.
[22]    Zenkour AM, Abouelregal AE. Effect of harmonically varying heat on FG nanobeams in the context of a nonlocal two-temperature thermoelasticity theory, Europ J Comput Mech 2014; 23(1-2): 1-14.
[23]    Abouelregal AE, Zenkour AM. Thermoelastic response of nanobeam resonators subjected to exponential decaying time varying load. J Theo App Mech 2017; 55(3): 937-948 Warsaw.
[24]    A Abouelregal AE, Zenkour AM. Dynamic response of a nanobeam induced by ramp-type heating and subjected to a moving load. Micro Tech 2017; 23(12): 5911-5920.
[25]    Povstenko YZ. Thermoelasticity that uses fractional heat conduction equation, J Math Sci 2009; 162(2): 296–305.
[26]    Miller K, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[27]    Podlubny I. Fractional Differential Equations, Academic Press, San Diego, 1999.
[28]    Mandelbrot BB. The Fractal Geometry of Nature, Macmillan, 1983.
[29]    Klimek M. Fractional sequential mechanics-models with symmetric fractional derivative. Czechoslov J. Phys. 2001; 51: 1348–1354.
[30]    Riewe F. Mechanics with fractional derivatives. Phys Rev E 1997; 55: 3581.
[31]    Mainardi F. Fractional Calculusand Wavesin Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, Singapore, 2010.
[32]    Sumelka W. Fractional viscoplasticity. Mech Res Commun 2014; 56: 31–36.
[33]    Rossikhin YA, Shitikova MV. Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl Mech Rev 2010; 63: 010801.
[34]    Pouresmaeeli S, Ghavanloo E, Fazelzadeh SA. Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium. Compos Struct 2013; 96: 405-410.
[35]    Lei Y, Adhikari S, Friswell MI. Vibration of nonlocal Kelvin–Voigt viscoelastic damped Timoshenko beams. Int J Eng Sci 2013; 66-67: 1–13.
[36]    Morland LW, Lee EH. Stress analysis for linear viscoelastic materials with temperature variation. Trans Soc Rheol 1960; 4: 233–263.
[37]    Biot MA. Theory of stress–strain relations in an isotropic viscoelasticity, and relaxation phenomena. J Appl Phys 1965;18: 27–34.
[38]    Enelund M, Olsson P. Damping described by fading memory analysis and application to fractional derivative models. Int J Sol Struc 1999; 36: 939–970.
[39]    Bagley R. On the equivalence of the Riemann-Liouville and the Caputo fractional order derivatives in modeling of linear viscoelastic materials. Fract Calc Appl Analy 2007; 10(2): 123-126.
[40]    Caputo M, Mainardi F. Linear models of dissipation in anelastic solids. Rivista del Nuovo Cimento 1971; 1(2): 161–198.
[41]    Hosseini SM, Kalhori H, Shooshtari A,  Mahmoodi SN. Analytical solution for nonlinear forced response of a viscoelastic piezoelectric cantilever beam resting on a nonlinear elastic foundation to an external harmonic excitation. Composites Part B: Engineering, 2014; 67: 464-471.
[42]    Mainardi F. Fractional calculus and waves in linear viscoelastisity: An introduction to mathematical models, London, Imperial College Press, 2009.
[43]    Bagley RL, Torvik PJ. Fractional calculus-a different approach to the analysis of viscoelastically damped structures. AIAA J. 1983; 21(5), 741–748.
[44]    Bagley RL, Torvik PJ. On the fractional calculus model of viscoelastic behavior. J. Rheol. 1986; 30: 133–155.
[45]    Lord H, Shulman Y. A generalized dynamical theory of thermoelasticity. J Mech Phys Solid 1967; 15: 299-309.
[46]    Honig G, Hirdes U. A method for the numerical inversion of the Laplace transform. J Comput Appl Math 1984;10: 113-132.
Volume 50, Issue 1
June 2019
Pages 148-156
  • Receive Date: 02 February 2019
  • Revise Date: 02 March 2019
  • Accept Date: 16 March 2019