[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.
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