Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters. (English) Zbl 1416.74077
Summary: In this paper, we investigate the free damped vibration of a nanobeam resting on viscoelastic foundation. Nanobeam and viscoelastic foundation are modeled using nonlocal elasticity and fractional order viscoelasticity theories. Motion equation is derived using D’Alembert’s principle and involves two retardation times and fractional order derivative parameters regarding to a nanobeam and viscoelastic foundation. The analytical solution is obtained using the Laplace transform method and it is given as a sum of two terms. First term denoting the drift of the system’s equilibrium position is given as an improper integral taken along two sides of the cut of complex plane. Two complex conjugate roots located in the left half-plane of the complex plane determine the second term describing the damped vibration around equilibrium position. Results for complex roots of characteristic equation obtained for a single nanobeam without viscoelastic foundation, where imaginary parts represent damped frequencies, are validated with the results found in the literature for natural frequencies of a single-walled carbon nanotube obtained from molecular dynamics simulations. In order to examine the effects of nonlocal parameter, fractional order parameters and retardation times on the behavior of characteristic equation roots in the complex plane and the time-response of nanobeam, several numerical examples are given.
MSC:
| 74N15 | Analysis of microstructure in solids |
Software:
MatlabReferences:
| [1] | Paulson JA, Mesbah A, Zhu X, Molaro MC, Braatz RD (2015) Control of self-assembly in micro-and nano-scale systems. J Process Control 27:38-49 · doi:10.1016/j.jprocont.2014.10.005 |
| [2] | Bae H, Chu H, Edalat F, Cha JM, Sant S, Kashyap A, Ahari AF, Kwon CH, Nichol JW, Manoucheri S, Zamanian B, Wang Y, Khademhosseini A (2014) Development of functional biomaterials with micro-and nanoscale technologies for tissue engineering and drug delivery applications. J Tissue Eng Regen Med 8(1):1-14 · doi:10.1002/term.1494 |
| [3] | Xia Y, Yang P, Sun Y, Wu Y, Mayers B, Gates B, Yin Y, Kim F, Yan H (2003) One-dimensional nanostructures: synthesis, characterization, and applications. Adv Mater 15(5):353-389 · doi:10.1002/adma.200390087 |
| [4] | Karatrantos A, Composto RJ, Winey KI, Clarke N (2011) Structure and conformations of polymer/SWCNT nanocomposites. Macromolecules 44(24):9830-9838 · doi:10.1021/ma201359s |
| [5] | Bhushan B (2007) Nanotribology and nanomechanics of MEMS/NEMS and BioMEMS/BioNEMS materials and devices. Microelectron Eng 84(3):387-412 · doi:10.1016/j.mee.2006.10.059 |
| [6] | Li M, Pernice WHP, Tang HX (2010) Ultrahigh-frequency nano-optomechanical resonators in slot waveguide ring cavities. Appl Phys Lett 97(18):183110 · doi:10.1063/1.3513213 |
| [7] | Gibson RF, Ayorinde EO, Wen YF (2007) Vibrations of carbon nanotubes and their composites: a review. Compos Sci Technol 67(1):1-28 · doi:10.1016/j.compscitech.2006.03.031 |
| [8] | Karlicic D, Murmu T, Adhikari S, McCarthy M (2015) Non-local structural mechanics. Wiley, New York · Zbl 1421.74004 · doi:10.1002/9781118572030 |
| [9] | Eringen AC (ed) (2002) Nonlocal continuum field theories. Springer, New York · Zbl 1023.74003 |
| [10] | Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10(3):233-248 · Zbl 0247.73005 · doi:10.1016/0020-7225(72)90039-0 |
| [11] | Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703-4710 · doi:10.1063/1.332803 |
| [12] | Reddy JN (2007) Nonlocal theories for buckling bending and vibration of beams. Int J Eng Sci 45:288-307 · Zbl 1213.74194 · doi:10.1016/j.ijengsci.2007.04.004 |
| [13] | Wang Q, Liew KM (2007) Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys Lett A 363:236-242 · doi:10.1016/j.physleta.2006.10.093 |
| [14] | Aydogdu M (2009) A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys E 41:1651-1655 · doi:10.1016/j.physe.2009.05.014 |
| [15] | Murmu T, Adhikari S (2010) Nonlocal effects in the longitudinal vibration of double-nanorod systems. Phys E 43:415-422 · doi:10.1016/j.physe.2010.08.023 |
| [16] | Huu-Tai T (2012) A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int J Eng Sci 52:56-64 · Zbl 1423.74356 · doi:10.1016/j.ijengsci.2011.11.011 |
| [17] | Khademolhosseini F, Phani A, Nojeh A, Rajapakse N (2012) Nonlocal continuum modeling and molecular dynamics simulation of torsional vibration of carbon nanotubes. IEEE Trans Nanotechnol 11(1):34-43 · doi:10.1109/TNANO.2011.2111380 |
| [18] | Huang LY, Han Q, Liang YJ (2012) Calibration of nonlocal scale effect parameter for bending single-layered graphene sheet under molecular dynamics. NANO 7(05):1250033 · doi:10.1142/S1793292012500336 |
| [19] | Lei Y, Murmu T, Adhikari S, Friswell MI (2013) Dynamic characteristics of damped viscoelastic nonlocal Euler-Bernoulli beams. Eur J Mech A/Solids 42:125-136 · Zbl 1406.74390 · doi:10.1016/j.euromechsol.2013.04.006 |
| [20] | Lei Y, Adhikari S, Murmu T, Friswell MI (2014) Asymptotic frequencies of various damped nonlocal beams and plates. Mech Res Commun 62:94-101 · doi:10.1016/j.mechrescom.2014.08.002 |
| [21] | Li X, McKenna GB (2012) Considering viscoelastic micromechanics for the reinforcement of graphene polymer nanocomposites. ACS Macro Lett 1(3):388-391 · doi:10.1021/mz200253x |
| [22] | Spitalsky Z, Tasis D, Papagelis K, Galiotis C (2010) Carbon nanotube-polymer composites: chemistry, processing, mechanical and electrical properties. Prog Polym Sci 35(3):357-401 · doi:10.1016/j.progpolymsci.2009.09.003 |
| [23] | Karličić D, Cajić M, Murmu T, Adhikari S (2015) Nonlocal longitudinal vibration of viscoelastic coupled double-nanorod systems. Eur J Mech A/Solids 49:183-196 · Zbl 1317.74053 · doi:10.1016/j.euromechsol.2014.07.005 |
| [24] | Karličić D, Murmu T, Cajić M, Kozić P, Adhikari S (2014) Dynamics of multiple viscoelastic carbon nanotube based nanocomposites with axial magnetic field. J Appl Phys 115(23):234303 · Zbl 1317.74053 · doi:10.1063/1.4883194 |
| [25] | Karličić D, Kozić P, Adhikari S, Cajić M, Murmu T, Lazarević M (2015) Nonlocal mass-nanosensor model based on the damped vibration of single-layer graphene sheet influenced by in-plane magnetic field. Int J Mech Sci 96:132-142 · Zbl 1317.74053 |
| [26] | Bagley RL, Torvik J (1983) Fractional calculus-a different approach to the analysis of viscoelastically damped structures. AIAA J 21(5):741-748 · Zbl 0514.73048 · doi:10.2514/3.8142 |
| [27] | Bagley RL, Torvik PJ (1986) On the fractional calculus model of viscoelastic behavior. J Rheol (1978-present) 30(1):133-155 · Zbl 0613.73034 · doi:10.1122/1.549887 |
| [28] | Koeller RC (1984) Applications of fractional calculus to the theory of viscoelasticity. J Appl Mech 51(2):299-307 · Zbl 0544.73052 · doi:10.1115/1.3167616 |
| [29] | wJ Welch S, Rorrer RA, Duren RG Jr (1999) Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials. Mech Time-Depend Mater 3(3):279-303 · doi:10.1023/A:1009834317545 |
| [30] | Imboden M, Mohanty P (2014) Dissipation in nanoelectromechanical systems. Phys Rep 534(3):89-146 · doi:10.1016/j.physrep.2013.09.003 |
| [31] | Lu H, Huang G, Wang B, Mamedov A, Gupta S (2006) Characterization of the linear viscoelastic behavior of single-wall carbon nanotube/polyelectrolyte multilayer nanocomposite film using nanoindentation. Thin Solid Films 500(1):197-202 |
| [32] | Díez-Pascual AM, Gómez-Fatou MA, Ania F, Flores A (2015) Nanoindentation in polymer nanocomposites. Prog Mater Sci 67:1-94 · doi:10.1016/j.pmatsci.2014.06.002 |
| [33] | Wright WJ, Nix WD (2009) Storage and loss stiffnesses and moduli as determined by dynamic nanoindentation. J Mater Res 24(03):863-871 · doi:10.1557/jmr.2009.0112 |
| [34] | Pathak S, Cambaz ZG, Kalidindi SR, Swadener JG, Gogotsi Y (2009) Viscoelasticity and high buckling stress of dense carbon nanotube brushes. Carbon 47(8):1969-1976 · doi:10.1016/j.carbon.2009.03.042 |
| [35] | Rossikhin YA, Shitikova MV (2001) Analysis of rheological equations involving more than one fractional parameters by the use of the simplest mechanical systems based on these equations. Mech Time-Depend Mater 5(2):131-175 · doi:10.1023/A:1011476323274 |
| [36] | Bagley RL, Torvik PJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol (1978-present) 27(3):201-210 · Zbl 0515.76012 · doi:10.1122/1.549724 |
| [37] | Wharmby AW, Bagley RL (2013) Generalization of a theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol (1978-present) 57(5):1429-1440 · Zbl 1273.65190 · doi:10.1122/1.4819083 |
| [38] | Shermergor TD (1966) On the use of fractional differentiation operators for the description of elastic-after effect properties of materials. J Appl Mech Tech Phys 7(6):85-87 · doi:10.1007/BF00914347 |
| [39] | Rossikhin YA (2010) Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Appl Mech Rev 63(1):010701 · doi:10.1115/1.4000246 |
| [40] | Bagley RL, Torvik PJ (1985) Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA journal 23(6):918-925 · Zbl 0562.73071 · doi:10.2514/3.9007 |
| [41] | Hedrih KS (2006) The transversal creeping vibrations of a fractional derivative order constitutive relation of nonhomogeneous beam. Math Problems Eng 2006(2006):1-18 · Zbl 1100.74026 |
| [42] | Di Paola M, Heuer R, Pirrotta A (2013) Fractional visco-elastic Euler-Bernoulli beam. Int J Solids Struct 50(22):3505-3510 · doi:10.1016/j.ijsolstr.2013.06.010 |
| [43] | Pirrotta A, Cutrona S, Di Lorenzo S (2015) Fractional visco-elastic Timoshenko beam from elastic Euler-Bernoulli beam. Acta Mech 226(1):179-189 · Zbl 1326.74077 · doi:10.1007/s00707-014-1144-y |
| [44] | Rossikhin YA, Shitikova MV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev 50(1):15-67 · doi:10.1115/1.3101682 |
| [45] | Rossikhin YA, Shitikova MV (2010) Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl Mech Rev 63(1):010801 · doi:10.1115/1.4000563 |
| [46] | Rossikhin YA, Shitikova MV (2001) A new method for solving dynamic problems of fractional derivative viscoelasticity. Int J Eng Sci 39(2):149-176 · doi:10.1016/S0020-7225(00)00025-2 |
| [47] | Li C, Chen A, Ye J (2011) Numerical approaches to fractional calculus and fractional ordinary differential equation. J Comput Phys 230(9):3352-3368 · Zbl 1218.65070 · doi:10.1016/j.jcp.2011.01.030 |
| [48] | Rossikhin Yu A (1970). Dynamic problems of linear viscoelasticity connected with the investigation of retardation and relaxation spectra. Ph.D. Dissertation, Voronezh Polytechnic Institute, Voronezh (in Russian). Ph.D., Voronezh Polytechnic Inst, Voronezh |
| [49] | Zelenev VM, Meshkov SI, Rossikhin YA (1970) Damped vibrations of hereditary-elastic systems with weakly singular kernels. J Appl Mech Tech Phys 11(2):290-293 · doi:10.1007/BF00908110 |
| [50] | Meshkov SI, Pachevskaya GN, Postnikov VS, Rossikhin YA (1971) Integral representations of ε γ-functions and their application to problems in linear viscoelasticity. Int J Eng Sci 9(4):387-398 · Zbl 0219.73043 · doi:10.1016/0020-7225(71)90059-0 |
| [51] | Rossikhin YA, Shitikova MV (2008) Free damped vibrations of a viscoelastic oscillator based on Rabotnov’s model. Mech Time-Depend Mater 12(2):129-149 · doi:10.1007/s11043-008-9053-x |
| [52] | Rossikhin YA, Shitikova MV, Shcheglova TA (2010) Analysis of free vibrations of a viscoelastic oscillator via the models involving several fractional parameters and relaxation/retardation times. Comput Math Appl 59(5):1727-1744 · Zbl 1189.44001 · doi:10.1016/j.camwa.2009.08.014 |
| [53] | Atanackovic TM, Bouras Y, Zorica D (2014) Nano-and viscoelastic Beck’s column on elastic foundation. Acta Mech 226(7):2335-2345 · Zbl 1325.74079 · doi:10.1007/s00707-015-1327-1 |
| [54] | Atanackovic TM, Janev M, Konjik S, Pilipovic S, Zorica D (2015) Vibrations of an elastic rod on a viscoelastic foundation of complex fractional Kelvin-Voigt type. Meccanica 50(7):1679-1692 · Zbl 1325.74080 |
| [55] | Ansari R, Oskouie MF, Sadeghi F, Bazdid-Vahdati M (2015) Free vibration of fractional viscoelastic Timoshenko nanobeams using the nonlocal elasticity theory. Phys E 74:318-327 · doi:10.1016/j.physe.2015.07.013 |
| [56] | Ansari R, Oskouie MF, Gholami R (2016) Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory. Phys E 75:266-271 · doi:10.1016/j.physe.2015.09.022 |
| [57] | Lazopoulos KA (2006) Non-local continuum mechanics and fractional calculus. Mech Res Commun 33(6):753-757 · Zbl 1192.74010 · doi:10.1016/j.mechrescom.2006.05.001 |
| [58] | Challamel N, Zorica D, Atanacković TM, Spasić DT (2013) On the fractional generalization of Eringenʼs nonlocal elasticity for wave propagation. Comptes Rendus Mécanique 341(3):298-303 · doi:10.1016/j.crme.2012.11.013 |
| [59] | Atanackovic TM, Janev M, Oparnica L, Pilipovic S, Zorica D (2015). Space-time fractional Zener wave equation. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society, vol 471(2174) · Zbl 1371.35319 |
| [60] | Cajić M, Karličić D, Lazarević M (2015) Nonlocal vibration of a fractional order viscoelastic nanobeam with attached nanoparticle. Theor Appl Mech 42(3):167-190 · Zbl 1462.74061 · doi:10.2298/TAM1503167C |
| [61] | Ansari R, Shahabodini A, Rouhi H (2015) A nonlocal plate model incorporating interatomic potentials for vibrations of graphene with arbitrary edge conditions. Curr Appl Phys 15(9):1062-1069 · doi:10.1016/j.cap.2015.06.012 |
| [62] | Ansari R, Pourashraf T, Gholami R (2015) An exact solution for the nonlinear forced vibration of functionally graded nanobeams in thermal environment based on surface elasticity theory. Thin-Walled Struct 93:169-176 · doi:10.1016/j.tws.2015.03.013 |
| [63] | Ansari R, Hasrati E, Gholami R, Sadeghi F (2015) Nonlinear analysis of forced vibration of nonlocal third-order shear deformable beam model of magneto-electro-thermo elastic nanobeams. Compos Part B Eng 83:226-241 · doi:10.1016/j.compositesb.2015.08.038 |
| [64] | Ansari R, Mohammadi V, Shojaei MF, Gholami R, Rouhi H (2014) Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory. Eur J Mech A/Solids 45:143-152 · Zbl 1406.74279 · doi:10.1016/j.euromechsol.2013.11.002 |
| [65] | Ansari R, Gholami R, Rouhi H (2015) Size-dependent nonlinear forced vibration analysis of magneto-electro-thermo-elastic Timoshenko nanobeams based upon the nonlocal elasticity theory. Compos Struct 126:216-226 · doi:10.1016/j.compstruct.2015.02.068 |
| [66] | Kazemi-Lari MA, Fazelzadeh SA, Ghavanloo E (2012) Non-conservative instability of cantilever carbon nanotubes resting on viscoelastic foundation. Phys E Low-Dimens Syst Nanostruct 44(7):1623-1630 · doi:10.1016/j.physe.2012.04.007 |
| [67] | Ghorbanpour-Arani AH, Rastgoo A, Sharafi MM, Kolahchi R, Arani AG (2016) Nonlocal viscoelasticity based vibration of double viscoelastic piezoelectric nanobeam systems. Meccanica 51(1):25-40 · Zbl 1332.74030 · doi:10.1007/s11012-014-9991-0 |
| [68] | Barretta R, Feo L, Luciano R (2015) Torsion of functionally graded nonlocal viscoelastic circular nanobeams. Compos Part B Eng 72:217-222 · doi:10.1016/j.compositesb.2014.12.018 |
| [69] | Atanackovic TM, Pilipovic S, Stankovic B, Zorica D (2014) Fractional calculus with applications in mechanics: vibrations and diffusion processes. Wiley, New York · Zbl 1291.74001 · doi:10.1002/9781118577530 |
| [70] | Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Academic Press, Cambridge · Zbl 0924.34008 |
| [71] | Petráš I (2011) Fractional derivatives, fractional integrals, and fractional differential equations in Matlab. Eng Educ Res Using MATLAB InTech kap 10:239-264 |
| [72] | Heymans N, Podlubny I (2006) Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol Acta 45(5):765-771 · doi:10.1007/s00397-005-0043-5 |
| [73] | Cajić M, Lazarević MP (2014) Fractional order spring/spring-pot/actuator element in a multibody system: application of an expansion formula. Mech Res Commun 62:44-56 · doi:10.1016/j.mechrescom.2014.08.009 |
| [74] | Pritz T (1996) Analysis of four-parameter fractional derivative model of real solid materials. J Sound Vib 195(1):103-115 · Zbl 1235.34026 · doi:10.1006/jsvi.1996.0406 |
| [75] | Atanackovic TM, Pilipovic S, Stankovic B, Zorica D (2014) Fractional calculus with applications in mechanics: wave propagation, impact and variational principles. Wiley, New York · Zbl 1293.74001 · doi:10.1002/9781118577530 |
| [76] | Ansari R, Sahmani S (2012) Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models. Commun Nonlinear Sci Numer Simul 17(4):1965-1979 · doi:10.1016/j.cnsns.2011.08.043 |
| [77] | Mitrinovic D, Keckic JD (1984) The Cauchy method of residues: theory and applications, vol 259. Springer, Berlin · Zbl 0546.30004 · doi:10.1007/978-94-015-6872-2 |
| [78] | Phadikar JK, Pradhan SC (2010) Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Comput Mater Sci 49(3):492-499 · doi:10.1016/j.commatsci.2010.05.040 |
| [79] | Reddy JN, Pang SD (2008) Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J Appl Phys 103(2):023511 · doi:10.1063/1.2833431 |
| [80] | Kempfle S, Schäfer I, Beyer H (2002) Fractional calculus via functional calculus: theory and applications. Nonlinear Dyn 29(1-4):99-127 · Zbl 1026.47010 · doi:10.1023/A:1016595107471 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
