AbstractThis paper studies the spatial behavior of the steady state vibrations in a cylinder made of a dual-phase-lag anisotropic rigid conductor material. We analyze the influence of the lagging model upon the spatial behavior of the amplitude of vibration along the axis of the cylinder, providing the explicit expressions of the decay rate and of the corresponding critical frequency in terms of the coefficients of the considered constitutive equation (or delay times). In fact, for the amplitude of the harmonic vibrations we obtain some exponential decay estimates of Saint-Venant type, provided the frequency of vibration is lower than a critical value. This gives information on the thermal penetration depth of the steady state vibrations describing the heat affected zone. Illustrative examples are given for the class of lagging behavior models that are thermodynamically compatible.
Tzou DY. A unified approach for heat conduction from macro to micro-scales. Journal of Heat Transfer 1995; 117: 8-16. https://doi.org/10.1115/1.2822329
Tzou DY. The generalized lagging response in small-scale and high-rate heating. International Journal of Heat and Mass Transfer 1995; 38: 3231-3234. https://doi.org/10.1016/0017-9310(95)00052-B
Tzou DY. Experimental support for the lagging behavior in heat propagation. Journal of Thermophysics and Heat Transfer 1995; 9:686-693. https://doi.org/10.2514/3.725
Tzou DY. Macro-To Micro-Scale Heat Transfer: The Lagging Behavior, Second Ed. John Wiley and Sons: Chichester 2015.
Chiriţă S. On high-order approximations for describing the lagging behavior of heat conduction. Mathematics and Mechanics of Solids. https://doi.org/10.1177/1081286518758356
Chiriţă S, Ciarletta M, Tibullo V. Qualitative properties of solutions in the time differential dual-phase-lag model of heat conduction. Applied Mathematical Modelling 2017; 50: 380-393. https://doi.org/10.1016/j.apm.2017.05.023
Flavin JN, Knops RJ. Some spatial decay estimates in continuum dynamics. Journal of Elasticity 1987; 17: 249-264. https://doi.org/10.1007/BF00049455
Flavin JN, Knops RJ, Payne LE. Decay estimates for the constrained elastic cylinder of variable cross section. Quarterly of Applied Mathematics 1989; 47: 325-350. https://doi.org/10.1090/qam/998106
Flavin JN, Knops RJ, Payne LE. Energy bounds in dynamical problems for a semi-infinite elastic beam. In: Eason G, Ogden RW Eds. Elasticity: Mathematical methods and applications. The Ian N. Sneddon 70th Birthday Volume. Chichester: Ellis Horwood Limited 1990; pp. 101-111.
Chiriţă S. Spatial decay estimates for solutions describing harmonic vibrations in a thermoelastic cylinder. Journal of Thermal Stresses 1995; 18: 421-436. https://doi.org/10.1080/01495739508946311
Chiriţă S, Galeș C, Ghiba ID. On spatial behavior of the harmonic vibrations in Kelvin-Voigt materials. Journal of Elasticity 2008; 93: 81-92. https://doi.org/10.1007/s10659-008-9167-z
Galeș C, Chiriţă S. On spatial behavior in linear viscoelasticity. Quarterly of Applied Mathematics 2009; 67: 707-723. https://doi.org/10.1090/S0033-569X-09-01149-0
Passarella F, Zampoli V. Spatial estimates for transient and steady-state solutions in transversely isotropic plates of Mindlin-type. European Journal of Mechanics A/Solids 2009; 28: 868-876. https://doi.org/10.1016/j.euromechsol.2009.01.004
Flavin JN, Rionero S. Qualitative estimates for partial differential equations: An introduction. CRC Press: Boca Raton 1996.
Chiriţă S, Ciarletta M, Tibullo V. On the thermomechanic consistency of the time differential dual-phase-lag models of heat conduction. International Journal of Heat and Mass Transfer 2017; 114: 277-285. https://doi.org/10.1016/j.ijheatmasstransfer.2017.06.071