Browse > Article
http://dx.doi.org/10.12989/sem.2020.73.3.287

A GN-based modified model for size-dependent coupled thermoelasticity analysis in nano scale, considering nonlocality in heat conduction and elasticity: An analytical solution for a nano beam with energy dissipation  

Hosseini, Seyed Mahmoud (Industrial Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad)
Publication Information
Structural Engineering and Mechanics / v.73, no.3, 2020 , pp. 287-302 More about this Journal
Abstract
This investigation deals with a size-dependent coupled thermoelasticity analysis based on Green-Naghdi (GN) theory in nano scale using a new modified nonlocal model of heat conduction, which is based on the GN theory and nonlocal Eringen theory of elasticity. In the analysis based on the proposed model, the nonlocality is taken into account in both heat conduction and elasticity. The governing equations including the equations of motion and the energy balance equation are derived using the proposed model in a nano beam resonator. An analytical solution is proposed for the problem using the Laplace transform technique and Talbot technique for inversion to time domain. It is assumed that the nano beam is subjected to sinusoidal thermal shock loading, which is applied on the one of beam ends. The transient behaviors of fields' quantities such as lateral deflection and temperature are studied in detail. Also, the effects of small scale parameter on the dynamic behaviors of lateral deflection and temperature are obtained and assessed for the problem. The proposed GN-based model, analytical solution and data are verified and also compared with reported data obtained from GN coupled thermoelasticity analysis without considering the nonlocality in heat conduction in a nano beam.
Keywords
nano-sized resonator; nonlocal heat conduction; Green-Naghdi theory; analytical solution; nonlocal coupled thermoelasticity; energy dissipation;
Citations & Related Records
Times Cited By KSCI : 25  (Citation Analysis)
연도 인용수 순위
1 Berezovski, A., Engelbrecht, J. and Van, P. (2014), "Weakly nonlocal thermoelasticity for microstructured solids: microdeformation and microtemperature", Arch. Appl. Mech., 84(9-11), 1249-1261. https://doi.org/10.1007/s00419-014-0858-6.   DOI
2 Bostani, M. and Karami Mohammadi, A. (2018), "Thermoelastic damping in microbeam resonators based on modified strain gradient elasticity and generalized thermoelasticity theories", Acta Mechanica, 229(1), 173-192. https://doi.org/10.1007/s00707-017-1950-0.   DOI
3 Bougoffa, L., Al-Jeaid, H.K. and Khanfer, A. (2010), "On the solutions of a boundary value problem of linear thermoelasticity system with nonlocal conditions", Appl. Math. Comput., 217(8), 4227-4233. https://doi.org/10.1016/j.amc.2010.10.037.   DOI
4 Clark, H.R. and Guardia R.R. (2016), "On a nonlinear thermoelastic system with nonlocal coefficients", J. Math. Anal. Appl., 433(1), 338-354. https://doi.org/10.1016/j.jmaa.2015.07.018   DOI
5 Dhaliwal, R.S. and Jun, W. (1994), "Some theorems in generalized nonlocal thermoelasticity", J. Eng. Sci., 32(3), 473-479. https://doi.org/10.1016/0020-7225(94)90135-X.   DOI
6 Dong, Y., Cao, B.Y. and Guo, Z.Y. (2014), "Size dependent thermal conductivity of Si nanosystems based on phonon gas dynamics", Physica E, 56, 256-262. https://doi.org/10.1016/j.physe.2013.10.006.   DOI
7 Ebrahimi, F., and Haghi, P. (2017), "Wave propagation analysis of rotating thermoelastically-actuated nanobeams based on nonlocal strain gradient theory", Acta Mechanica Solida Sinica, 30(6), 647-657. https://doi.org/10.1016/j.camss.2017.09.007.   DOI
8 Ebrahimi, F., Mahmoodi, F., and Barati, M.R. (2017), "Thermo-mechanical vibration analysis of functionally graded micro/nanoscale beams with porosities based on modified couple stress theory", Adv. Mater. Res., 6(3), 279-301. https://doi.org/10.12989/amr.2017.6.3.279.   DOI
9 Tzou, D.Y., and Guo, Z.Y. (2010), "Nonlocal behavior in thermal lagging", J. Thermal Sci., 49(7), 1133-1137. https://doi.org/10.1016/j.ijthermalsci.2010.01.022.   DOI
10 Tzou, D.Y. (2015), Macro-to Micro-scale Heat Transfer: The Lagging Behavior, John Wiley and Sons Ltd, United Kingdom.
11 Yang, F., Chong, A.C.M., Lam, D.C.C., and Tong, P., (2002), "Couple stress based strain gradient theory for elasticity", J. Solids Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X.   DOI
12 Yu, Y.J., Tian, X.-G., and Liu, X.-R. (2015), "Size-dependent generalized thermoelasticity using Eringen's nonlocal model", European J. Mech. A/Solids, 51, 96-106. https://doi.org/10.1016/j.euromechsol.2014.12.005.   DOI
13 Yu, Y.J., Tian, X.-G., and Xiong Q.-L. (2016), "Nonlocal thermoelasticity based on nonlocal heat conduction and nonlocal elasticity", European J. Mech. A/Solids, 60, 238-253. https://doi.org/10.1016/j.euromechsol.2016.08.004.   DOI
14 Yu, Y.J., Tian, X.-G., and Liu, J. (2017), "Size-dependent damping of a nanobeam using nonlocal thermoelasticity: extension of Zener, Lifshitz, and Roukes' damping model", Acta Mechanica, 228(4), 1287-1302. https://doi.org/10.1007/s00707-016-1769-0.   DOI
15 Zakeri, M., Attarnejad, R. and Ershadbakhsh, A.M. (2016), "Analysis of Euler-Bernoulli nanobeams: A mechanical-based solution", J. Comput. Appl. Mech., 47(2), 159-180. https://doi.org/10.22059/JCAMECH.2017.140165.97.
16 Zenkour, A., Abouelregal, A., Alnefaie, K., Abu-Hamdeh, N., and Aifantisb, E. (2014), "A refined nonlocal thermoelasticity theory for the vibration of nanobeams induced by ramp-type heating", Appl. Math. Comput., 248, 169-183. https://doi.org/10.1016/j.amc.2014.09.075.   DOI
17 Zenkour, A.M., and Abouelregal, A.E. (2014), "Vibration of FG nanobeams induced by sinusoidal pulse-heating via a nonlocal thermoelastic model", Acta Mechanica, 225(12), 2407-2415. https://doi.org/10.1007/s00707-014-1146-9.
18 Eringen, A.C. (1974), "Theory of nonlocal thermoelasticity", J. Eng. Sci., 12(12), 1063-1077. https://doi.org/10.1016/0020-7225(74)90033-0.   DOI
19 El-Nabulsi, R.A. (2018), "Nonlocal approach to nonequilibrium thermodynamics and nonlocal heat diffusion processes", Continuum Mech. Thermodynam., 30(4), 889-915. https://doi.org/10.1007/s00161-018-0666-2.   DOI
20 Elsibai, K.A. and Youssef, H.M. (2011), "State-Space Approach to Vibration of Gold Nano-Beam Induced by Ramp Type Heating without Energy Dissipation in Femtoseconds Scale", J. Thermal Stress., 34(3), 244-263. https://doi.org/ 10.1080/01495739.2010.545737.   DOI
21 Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer-Verlag, New York, USA.
22 Ezzat, M.A., and El-Bary, A.A. (2017), "Fractional magneto-thermoelastic materials with phase-lag Green-Naghdi theories", Steel Compos. Struct., 24(3), 297-307. https://doi.org/10.12989/scs.2017.24.3.297.   DOI
23 Green, A.E., and Lindsay, K. (1972), "Thermoelasticity", J. Elasticity, 2, 1-7. https://doi.org/10.1007/BF00045689.   DOI
24 Fabrizio, M., Lazzari, B., and Nibbi, R. (2011), "Thermodynamics of non-local materials: extra fluxes and internal powers", Continuum Mech. Thermodynam., 23, 509. https://doi.org/10.1007/s00161-011-0193-x.   DOI
25 Fang, Y., Yan, B., and Tee, K.F. (2017), "Probabilistic reliability of micro-resonators with thermoelastic coupling", Earthq. Struct., 12(2), 213-221. https://doi.org/10.12989/eas.2017.12.2.000.   DOI
26 Guyer, R.A., and Krumhansl, J.A. (1966), "Solution of the linearized phonon boltzmann equation", Phys. Rev., 148(2), 765-778. https://doi.org/10.1103/PhysRev.148.766.
27 Green, A.E., and Naghdi, P.M. (1992), "On undamped heat waves in an elastic solid", J. Thermal Stress., 15, 253-264. https://doi.org/10.1080/01495739208946136.   DOI
28 Hosseini, M., Shishesaz, M., and Hadi, A. (2019), "Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness", Thin Wall. Struct., 134, 508-523. https://doi.org/10.1016/j.tws.2018.10.030.   DOI
29 Green, A.E., and Naghdi, P.M. (1993), "Thermoelasticity without energy dissipation", J. Elasticity, 31, 189-208. https://doi.org/10.1007/BF00044969.   DOI
30 Hetnarski, R.B., and Eslami, M.R. (2009), Thermal Stresses - Advanced Theory and Applications, Springer, Dordrecht, Austria.
31 Zenkour A.M. (2018), "A novel mixed nonlocal elasticity theory for thermoelastic vibration of nanoplates", Compos. Struct., 185, 821-833. https://doi.org/10.1016/j.compstruct.2017.10.085.   DOI
32 Zenkour, A.M., Abouelregal, A.E., Alnefaie, K.A., Abu-Hamdeh, N.H., Aljinaidi, A.A., and Aifantis, E.C. (2015), "State space approach for the vibration of nanobeams based on the nonlocal thermoelasticity theory without energy dissipation", J. Mech. Sci. Technol., 29(7), 2921-2931. https://doi.org/10.1007/s12206-015-0623-y.   DOI
33 Zenkour, A.M., and Abouelregal, A.E. (2015), "Thermoelastic interaction in functionally graded nanobeams subjected to time-dependent heat flux", Steel Compos. Struct., 18(4), 909-924. https://doi.org/10.12989/scs.2015.18.4.909.   DOI
34 Zenkour, A.M. (2017), "Nonlocal thermoelasticity theory without energy dissipation for nano-machined beam resonators subjected to various boundary conditions", Microsyst. Technol., 23(1), 55-65. https://doi.org/10.1007/s00542-015-2703-4.   DOI
35 Zenkour, A. and Abouelregal, A. (2019), "Thermoelastic Vibration of Temperature-Dependent Nanobeams Due to Rectified Sine Wave Heating-A State Space Approach". J. Appl. Comput. Mech., 5(2), 299-310. https://doi.org/10.22055/JACM.2018.26311.1323.
36 Abouelregal, A.E. and Zenkour, A.M. (2017), "Variability of thermal properties for a thermoelastic loaded nanobeam excited by harmonically varying heat", Smart Struct. Syst., 20(4), 451-460. https://doi.org/10.12989/sss.2017.20.4.451.   DOI
37 Abbas, I.A. (2014), "A GN model based upon two-temperature generalized thermoelastic theory in an unbounded medium with a spherical cavity", Appl. Math. Comput., 245, 108-115. https://doi.org/10.1016/j.amc.2014.07.059.   DOI
38 Abbondanza, D., Battista, D., Morabito, F., Pallante, C., Barretta, R., Luciano, R., Marotti de Sciarra, F. and Ruta, G. (2016), "Linear dynamic response of nanobeams accounting for higher gradient effects", J. Appl. Comput. Mech., 2(2), 54-64. https://doi.org/10.22055/JACM.2016.12330.
39 Abouelregal, A.E. and Zenkour, A.M. (2018), "Nonlocal thermoelastic model for temperature-dependent thermal conductivity nanobeams due to dynamic varying loads", Microsyst. Technol., 24(2), 1189-1199. https://doi.org/ 10.1007/s00542-017-3485-7.   DOI
40 Aifantis, E.C. (1999), "Gradient deformation models at nano, micro, and macro scales", Journal of Engineering Materials and Technology, 121(2), 189-202. https://doi.org/ 10.1115/1.2812366.   DOI
41 Akbas, S.D. (2016a), "Analytical solutions for static bending of edge cracked micro beams" Struct. Eng. Mech., 59: 579-599. https://doi.org/10.12989/sem.2016.59.3.579.   DOI
42 Akbas, S.D. (2016b), "Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium", Smart Struct. Syst., 18(6), 1125-1143. https://doi.org/10.12989/sss.2016.18.6.1125.   DOI
43 Hosseini, S.M. (2018), "Analytical solution for nonlocal coupled thermoelasticity analysis in a heat-affected MEMS/NEMS beam resonator based on Green-Naghdi theory", Appl. Math. Model., 57, 21-36. https://doi.org/10.1016/j.apm.2017.12.034.   DOI
44 Hosseini, S.M., Sladek, J., and Sladek, V. (2011), "Meshless local Petrov-Galerkin method for coupled thermoelasticity analysis of a functionally graded thick hollow cylinder", Eng. Anal. Boundary Elements, 35(6), 827-835. https://doi.org/10.1016/j.enganabound.2011.02.001.   DOI
45 Hosseini, S.M. (2014a), "Application of a hybrid meshless technique for natural frequencies analysis in functionally graded thick hollow cylinder subjected to suddenly thermal loading", Appl. Math. Model., 38(2), 425-436. https://doi.org/10.1016/j.apm.2013.06.034.   DOI
46 Hosseini, S.M (2014b), "Application of a hybrid mesh-free method for shock-induced thermoelastic wave propagation analysis in a layered functionally graded thick hollow cylinder with nonlinear grading patterns", Eng. Anal. Boundary Elements, 43, 56-66. https://doi.org/10.1016/j.enganabound.2014.03.007.   DOI
47 Hosseini, S.M., and Zhang, C. (2018), "Coupled thermoelastic analysis of an FG multilayer graphene platelets-reinforced nanocomposite cylinder using meshless GFD method: A modified micromechanical model", Eng. Anal. Boundary Elements, 88, 80-92. https://doi.org/10.1016/j.enganabound.2017.12.010.   DOI
48 Akbas, S.D. (2017a), "Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory", J. Struct. Stability Dynam., 17(3), 1750033. https://doi.org/10.1142/S021945541750033X.   DOI
49 Zhang, H., Kim, T., Choi, G., and Cho, H.H. (2016), "Thermoelastic damping in micro- and nanomechanical beam resonators considering size effects", J. Heat Mass Transfer, 103, 783-790. https://doi.org/ 10.1016/j.ijheatmasstransfer.2016.07.044.   DOI
50 Inan, E., & Eringen, A.C. (1991), "Nonlocal theory of wave propagation in thermoelastic plates", J. Eng. Sci., 29(7), 831-843. https://doi.org/10.1016/0020-7225(91)90005-N.   DOI
51 Akbas, S.D. (2017b), "Forced vibration analysis of functionally graded nanobeams", International Journal of Applied Mechanics, 9(7), 1750100. https://doi.org/10.1142/S1758825117501009.   DOI
52 Akbas, S.D. (2018a), "Forced vibration analysis of cracked functionally graded microbeams", Adv. Nano Res., 6(1), 39-55. https://doi.org/10.12989/anr.2018.6.1.039.   DOI
53 Akbas, S.D. (2018b), "Forced vibration analysis of cracked nanobeams", Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40(8), 392. https://doi.org/10.1007/s40430-018-1315-1.   DOI
54 Akbas, S.D. (2018c), "Bending of a Cracked Functionally Graded Nanobeam", Adv. Nano Res., 6(3), 219-242. https://doi.org/10.12989/anr.2018.6.3.219.   DOI
55 Akbas, S.D. (2019), "Axially Forced Vibration Analysis of Cracked a Nanorod", J. Comput. Appl. Mech., 50(1), 63-68. https://doi.org/10.22059/JCAMECH.2019.281285.392.
56 Ansari, R., Rouhi, S. and Ahmadi, M. (2018), "On the thermal conductivity of carbon nanotube/polypropylene nanocomposites by finite element method", J. Comput. Appl. Mech., 49(1), 70-85. https://doi.org/10.22059/JCAMECH.2017.243530.195.
57 Arash, B., Jiang J.W. and Rabczuk, t. (2015), "A review on nanomechanical resonators and their applications in sensors and molecular transportation", Appl. Phys. Rev., 2, 021301. https://doi.org/10.1063/1.4916728.   DOI
58 Ardito, R., Comi, C., Corigliano, A. and Frangi, A. (2008a), "Solid damping in micro electro mechanical systems", Meccanica, 43(4), 419-428. https://doi.org/10.1007/s11012-007-9105-3.   DOI
59 Jou D., Lebon G., and Criado-Sancho, M. (2010b), "Variational principles for thermal transport in nanosystems with heat slip flow", Phys. Rev. E, 82: 031128. https://doi.org/10.1103/PhysRevE.82.031128.   DOI
60 Jou D., Casas-Vazquez J., and Lebon G. (2010a), Extended Irreversible Thermodynamics, Springer Netherlands, Netherlands.
61 Kiani, K. (2015), "Free vibrations of elastically embedded stocky single-walled carbon nanotubes acted upon by a longitudinally varying magnetic field", Meccanica, 50, 3041-3067. https://doi.org/10.1007/s11012-015-0184-2.   DOI
62 Kumar, R., and Devi, S. (2017), "Thermoelastic beam in modified couple stress thermoelasticity induced by laser pulse", Comput. Concrete, 19(6), 701-710. https://doi.org/10.12989/cac.2017.19.6.701.   DOI
63 Li, D., and He, T. (2018), "Investigation of generalized piezoelectric-thermoelastic problem with nonlocal effect and temperature-dependent properties", Helion, 4(10), e00860. https://doi.org/10.1016/j.heliyon.2018.e00860.
64 Lord, H.W., and Shulman, Y. (1967), "A generalized dynamical theory of thermoelasticity", J. Mech. Phys. Solids, 15, 299-309. https://doi.org/10.1016/0022-5096(67)90024-5.   DOI
65 Ma, Y. (2012), "Size-dependent thermal conductivity in nanosystems based on non-Fourier heat transfer", Appl. Phys. Lett., 101(21), 211905. https://doi.org/10.1063/1.4767337.   DOI
66 Malikan, M. (2019), "On the buckling response of axially pressurized nanotubes based on a novel nonlocal beam theory", J. Appl. Comput. Mech., 5(1), 103-112. https://doi.org/10.22055/JACM.2018.25507.1274.
67 Ignaczak, J., and Ostoja-Starzewski, M. (2010), Thermoelasticity with Finite Wave Speeds, Oxford University Press, Uk.
68 Barretta, R., Canadija, M., Luciano, R. and Marotti de Sciarra, F. (2018), "Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams", J. Eng. Sci., 126, 53-67. https://doi.org/10.1016/j.ijengsci.2018.02.012.   DOI
69 Ardito, R., Comi, C., Corigliano, A. and Frangi, A. (2008b), "Errata-corrige to "Solid damping in micro electro mechanical systems".", Meccanica, 43, 557. https://doi.org/ 10.1007/s11012-008-9137-3.   DOI
70 Balta, F. and Suhubi, E.S. (1977), "Theory of nonlocal generalised thermoelasticity", J. Eng. Sci., 15(9-10), 579-588. https://doi.org/ 10.1016/0020-7225(77)90054-4.   DOI
71 Bensaid, I., Abdelmadjid, C., Mangouchi, A. and Kerboua, B. (2017), "Static deflection and dynamic behavior of higher-order hyperbolic shear deformable compositionally graded beams", Adv. Mater. Res., 6(1), 13-26. https://doi.org/10.12989/amr.2017.6.1.013.   DOI
72 Bensaid, I. and Guenanou, A. (2017), "Bending and stability analysis of size-dependent compositionally graded Timoshenko nanobeams with porosities", Adv. Mater. Res., 6(1), 45-63. https://doi.org/10.12989/amr.2017.6.1.045.   DOI
73 Bensaid, I., Bekhadda, A. and Kerboua, B. (2018a), "Dynamic analysis of higher order shear-deformable nanobeams resting on elastic foundation based on nonlocal strain gradient theory", Adv. Nano Res., 6(3), 279-298. https://doi.org/10.12989/anr.2018.6.3.279.   DOI
74 Bensaid, I., Bekhadda, A., Kerboua, B. and Abdelmadjid, C. (2018b), "Investigating nonlinear thermal stability response of functionally graded plates using a new and simple HSDT", Wind Struct., 27(6), 369-380. https://doi.org/10.12989/was.2018.27.6.369.   DOI
75 Polizzotto, C. (2003), "Unified thermodynamic framework for nonlocal/gradient continuum theories", European J. Mech. A/Solids, 22(5), 651-668. https://doi.org/10.1016/S0997-7538(03)00075-5.   DOI
76 Meric, R.A. (1988), "Sensitivity analysis of functionals with respect to shape for dynamically loaded nonlocal thermoelastic solids", J. Eng. Sci., 26(7), 703-711. https://doi.org/10.1016/0020-7225(88)90089-4.   DOI
77 Moradi-Dastjerdi, R., and Payganeh, G. (2017), "Thermoelastic dynamic analysis of wavy carbon nanotube reinforced cylinders under thermal loads", Steel Compos. Struct., 25(3), 315-326. https://doi.org/10.12989/scs.2017.25.3.315.   DOI
78 Bensaid, I. and Bekhadda, A. (2018), "Thermal stability analysis of temperature dependent inhomogeneous size-dependent nano-scale beams", Adv. Mater. Res., 7(1), 363-378. https://doi.org/10.12989/amr.2018.7.1.001.   DOI
79 Bensaid, I. and Kerboua, B. (2019), "Improvement of thermal buckling response of FG-CNT reinforced composite beams with temperature-dependent material properties resting on elastic foundations", Adv. Aircraft Spacecraft Sci., 6(3), 207-223. https://doi.org/10.12989/aas.2019.6.3.207.   DOI
80 Olofinkua, J. (2018), "On The Effect of Nanofluid Flow and Heat Transfer with Injection through an Expanding or Contracting Porous Channel", J. Comput. Appl. Mech., 49(1), 1-8. https://doi.org/10.22059/JCAMECH.2018.255680.264.
81 Polizzotto, C. (2014), "Stress gradient versus strain gradient constitutive models within elasticity", J. Solids Struct., 51(9), 1809-1818. https://doi.org/10.1016/j.ijsolstr.2014.01.021.   DOI
82 Rana, G.C., Chand, R., Sharma, V. and Sharda A. (2016), "On the onset of triple-diffusive convection in a layer of nanofluid", J. Comput. Appl. Mech., 47(1), 67-77. https://doi.org/10.22059/JCAMECH.2016.59256.
83 Rezazadeh, G., Sheikhlou, M., and Shabani, R. (2015), "Analysis of bias DC voltage effect on thermoelastic damping ratio in short nano-beam resonators based on nonlocal elasticity theory and dual-phase-lagging heat conduction model", Meccanica, 50(12), 2963-2976. https://doi.org/10.1007/s11012-015-0171-7.   DOI
84 Sobolev, S.L. (1994), "Equations of transfer in non-local media", J. Heat Mass Transfer, 37(14), 2175-2182. https://doi.org/10.1016/0017-9310(94)90319-0.   DOI
85 Tan, Z.-Q., and Chen, Y.-C. (2019), "Size-dependent electro-thermo-mechanical analysis of multilayer cantilever microactuators by Joule heating using the modified couple stress theory", Compos. Part B Eng., 161, 183-189. https://doi.org/10.1016/j.compositesb.2018.10.067.   DOI