A modified fractional-order thermo-viscoelastic model and its application to a polymer micro-rod heated by a moving heat source

doi:10.1007/s10483-022-2835-9

Abstract

Abstract: Classical thermo-viscoelastic models may be challenged to predict the precise thermo-mechanical behavior of viscoelastic materials without considering the memory-dependent effect. Meanwhile, with the miniaturization of devices, the size-dependent effect on elastic deformation is becoming more and more important. To capture the memory-dependent effect and the size-dependent effect, the present study aims at developing a modified fractional-order thermo-viscoelastic coupling model at the microscale to account for two fundamentally distinct fractional-order models which govern the memory-dependent features of thermal conduction and stress-strain relation, respectively. Then, the modified theory is used to study the dynamic response of a polymer micro-rod heated by a moving heat source. The governing equations are obtained and solved by the Laplace transform method. In calculation, the effects of the fractional-order parameter, the fractional-order strain parameter, the mechanical relaxation parameter, and the nonlocal parameter on the variations of the considered variables are analyzed and discussed in detail.

Key words: size-dependent effect, fractional-order strain, fractional-order thermal conduction, generalized thermo-viscoelasticity, moving heat source

2010 MSC Number:

74F05

Wei PENG, Like CHEN, Tianhu HE. A modified fractional-order thermo-viscoelastic model and its application to a polymer micro-rod heated by a moving heat source. Applied Mathematics and Mechanics (English Edition), 2022, 43(4): 507-522.

TrendMD

References

[1] LI, X. Y., QIN, Q. H., and TIAN, X. G. Thermo-viscoelastic analysis of biological tissue during hyperthermia treatment. Applied Mathematical Modelling, 79, 881–895(2020)
[2] RAO, M. D. Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes. Journal of Sound and Vibration, 262, 457–474(2003)
[3] ZHANG, N. H. and CHENG, C. J. A time domain method for quasi-static analysis of viscoelastic thin plates. Applied Mathematics and Mechanics (English Edition), 22(10), 1109–1117(2001) https://doi.org/10.1007/BF02436446
[4] ALHARBI, A. M., OTHMAN, M. I. A., and ATEF, H. M. Thomson effect with hyperbolic two-temperature on magneto-thermo-visco-elasticity. Applied Mathematics and Mechanics (English Edition), 42(9), 1311–1326(2021) https://doi.org/10.1007/s10483-021-2763-7
[5] LYU, Q., LI, J. J., and ZHANG, N. H. Quasi-static and dynamical analyses of a thermoviscoelastic Timoshenko beam using the differential quadrature method. Applied Mathematics and Mechanics (English Edition), 40(4), 549–562(2019) https://doi.org/10.1007/s10483-019-2470-8
[6] EOM, K., PARK, H. S., YOON, D. S., and KWON, T. Nanomechanical resonators and their applications in biological/chemical detection: nanomechanics principles. Physics Report, 503(4/5), 115–163(2011)
[7] CURRANO, L. J., YU, M., and BALACHANDRAN, B. Latching in an MEMS shock sensor: modeling and experiments. Sensors and Actuators A-Physical, 159(1), 41–50(2010)
[8] TORII, A., SASAKI, M., HANE, K., and OKUMA, S. Adhesive force distribution on microstructures investigated by an atomic force microscope. Sensors and Actuators A-Physical, 44(2), 153–158(1994)
[9] LYU, Q., ZHANG, N. H., ZHANG, C. Y., WU, J. Z., and ZHANG, Y. C. Effect of adsorbate viscoelasticity on dynamical responses of laminated microcantilever resonators. Composite Structures, 250, 112553(2020)
[10] SOBHY, M. and ZENKOUR, A. M. The modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on visco-Pasternak foundations. Mechanics of Advanced Materials and Structure, 27(7), 525–538(2020)
[11] ATTIA, M. A. and ABDEL RAHMAN, A. A. On vibrations of functionally graded viscoelastic nanobeams with surface effects. International Journal of Engineering Science, 127, 1–32(2018)
[12] FLECK, N., MULLER, G., ASHBY, M., and HUTCHINSON, J. Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia, 42(2), 475–487(1994)
[13] ZHANG, Y. H., HONG, J. W., LIU, B., and FANG, D. N. Strain effect on ferroelectric behaviors of BaTiO₃ nanowires: a molecular dynamics study. Nanotechnology, 21, 015701(2010)
[14] ERINGEN, A. C. Nonlocal Continuum Field Theories, Springer-Verlag, New York (2002)
[15] LAM, D. C. C., YANG, F., CHONG, A. C. M., WANG, J., and TONG, P. Experiments and theory in strain gradient elasticity. Journal of Mechanics and Physics of Solids, 51(8), 1477–1508 (2003)
[16] YANG, F., CHONG, A. C. M., LAM, D. C. C., and TONG, P. Couple stress based strain gradient theory for elasticity. International Journal of Solids Structure, 39(10), 2731–2743(2002)
[17] ANSARI, R., SHOJAEI, M. F., MOHAMMADI, V., GHOLAMI, R., and ROUHI, H. Buckling and postbuckling of single-walled carbon nanotubes based on a nonlocal Timoshenko beam model. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik, 95(9), 1–13(2014)
[18] REDDY, J. N. Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. International Journal of Engineering Science, 48, 1507–1518(2010)
[19] LIM, C. W., LI, C., and YU, J. L. Dynamic behaviour of axially moving nanobeams based on nonlocal elasticity approach. Acta Mechanica Sinica, 26(5), 755–765(2010)
[20] YU, Y. J., TIAN, X. G., and LIU, X. R. Size-dependent generalized thermoelasticity using Eringen’s nonlocal model. European Journal of Mechanics A-Solids, 51, 96–106(2015)
[21] BIOT, M. A. Thermoelasticity and irreversible thermodynamics. Journal of Applied Physics, 27(3), 240–253(1956)
[22] PESHKOV, V. Second sound in helium. Journal of Physical, 8, 381–386(1944)
[23] CATTANEO, C. A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Comptes Rendus Physique, 247, 431–433(1958)
[24] VERNOTTE, P. M. and HEBD, C. R. Paradoxes in the continuous theory of the heat conduction. Comptes Rendus de l’Académie des Sciences de Paris, 246, 3154–3155(1958)
[25] LORD, H. W. and SHULMAN, Y. A. A generalized dynamical theory of thermoelasticity. Journal of the Mechanics and Physics of Solids, 15, 299–309(1967)
[26] EZZAT, M. A. and EL-KARAMANY, A. S. The uniqueness and reciprocity theorems for generalized thermoviscoelasticity for anisotropic media. Journal of Thermal Stresses, 25(6), 507–522 (2002)
[27] GURTIN, M. E. and STERNBERG, E. On the linear theory of viscoelasticity. Archive for Rational Mechanics and Analysis, 11, 291–356(1962)
[28] DI PAOLA, M., PIRROTTA, M., and VALENZA, A. Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results. Mechanics of Materials, 43, 799– 806(2011)
[29] CAPUTO, M. and MAINARDI, F. A new dissipation model based on memory mechanism. Pure and Applied Geophysics, 91, 134–147(1971)
[30] YOUSSEF, H. M. Theory of generalized thermoelasticity with fractional order strain. Journal of Vibration and Control, 22, 3840–3857(2016)
[31] LI, C. L., GUO, H. L., TIAN, X. G., and HE, T. H. Generalized thermoviscoelastic analysis with fractional order strain in a thick viscoelastic plate of infinite extent. Journal of Thermal Stresses, 42(8), 1051–1070(2019)
[32] ZHUANG, Q., YU, B., and JIANG, X. Y. An inverse problem of parameter estimation for timefractional heat conduction in a composite medium using carbon-carbon experimental data. Physica B: Condensed Matter, 456, 9–15(2015)
[33] YOUSSEF, H. M. Theory of fractional order generalized thermoelasticity. Journal of Heat Transfer, 132(6), 061301(2010)
[34] ZHANG, P. and HE, T. H. A generalized thermoelastic problem with nonlocal effect and memorydependent derivative when subjected to a moving heat source. Wave in Random and Complex Media, 30(1), 142–156(2020)
[35] PENG, W., MA, Y. B., LI, C. L., and HE, T. H. Dynamic analysis to the fractional order thermoelastic diffusion problem of an infinite body with a spherical cavity and variable material properties. Journal of Thermal Stresses, 43(1), 38–54(2020)
[36] NARAYAN, O. and RAMASWAMY, S. Anomalous heat conduction in one-dimensionalmomentum-conserving systems. Physical Review Letters, 89(20), 200601(2002)
[37] BRANCIK, L. Programs for fast numerical inversion of Laplace transforms in Matlab language environment. Proceedings of the 7th Conference MATLAB 99, Czech Republic, Prague, 27–39 (1999)