Abstract
An analytical model extending classical thermal elasticity is presented. It allows to introduce a correction to the attenuation of the mechanical waves at the higher frequency range. A material data set taken from experimental studies can be used to identify the attenuation rate as a function of frequency. An example is provided. The particular solution of the developed equations system in the form of the traveling monochromatic wave is obtained.
Similar content being viewed by others
Notes
\({\varvec{E}}{\varvec{E}}={\varvec{e}}_k {\varvec{e}}_k {\varvec{e}}_n {\varvec{e}}_n\), \({\varvec{I}}=\frac{1}{2}\left( {\varvec{e}}_k {\varvec{e}}_n {\varvec{e}}_n {\varvec{e}}_k+{\varvec{e}}_k {\varvec{e}}_n {\varvec{e}}_k {\varvec{e}}_n\right) \).
References
Nowacki, W.: Thermoelasticity. Elsevier, Amsterdam (2013)
Müller, I., Müller, W.H.: Fundamentals of Thermodynamics and Applications: With Historical Annotations and Many Citations from Avogadro to Zermelo. Springer, New York (2009)
Ignaczak, J., Ostoja-Starzewski, M.: Thermoelasticity with Finite Wave Speeds. Oxford University Press, Oxford (2010)
Jou, D., Lebon, G., Casas-Vázquez, J.: Extended Irreversible Thermodynamics. Springer, New York (2010)
Papenfuss, C., Forest, S.: Walter de Gruyter. J. Non-Equilib. Thermodyn. 31(4), 319 (2006)
Ivanova, E.A. , Vilchevskaya, E.N.: Description of thermal and micro-structural processes in generalized continua: Zhilin’s method and its modifications. In: Generalized Continua as Models for Materials, pp. 179–197. Springer, New York (2013)
Ivanova, E.A., Vilchevskaya, E.N.: Zhilin’s Method and Its Modifications,"Encyclopedia of Continuum Mechanics", vol. Chap 7, pp. 1–9. Springer, Berlin (2018)
Zhilin, P.: Phase transitions and general theory of elasto-plastic bodies. In: Proceedings of XXIX Summer School-Conference. Advanced Problems in Mechanics, pp. 36–48 (2002)
Zhilin, P.: Advanced Problems in Mechanics, vol. 2. Institute for Problems in Mechanical Engineering, St. Petersburg (2006)
Indeitsev, D., Naumov, V., Semenov, B.: Dynamic effects in materials of complex structure. Mech. Solids 42(5), 672 (2007)
Indeitsev, D., Meshcheryakov, Y.I., Kuchmin, A.Y., Vavilov, D.: Multi-scale model of steady-wave shock in medium with relaxation. Acta Mechanica 226(3), 917 (2015)
Ivanova, E.A.: Derivation of theory of thermoviscoelasticity by means of two-component medium. Acta Mechanica 215(1–4), 261 (2010)
Ivanova, E.A.: Description of mechanism of thermal conduction and internal damping by means of two-component Cosserat continuum. Acta Mechanica 225(3), 757 (2014)
Krivtsov, A.: Heat transfer in infinite harmonic one-dimensional crystals. Doklady Phys. 60(9), 407 (2015)
Krivtsov, A.M., Kuzkin, V.A.: Discrete and continuum thermomechanics (2017). arXiv:1707.09510
Sokolov, A.A., Krivtsov, A.M., Müller, W.H., Vilchevskaya, E.N.: Change of entropy for the one-dimensional ballistic heat equation: sinusoidal initial perturbation. Phys. Rev. E 99(4), 042107 (2019)
Babenkov, M.B., Krivtsov, A.M., Tsvetkov, D.V.: Unsteady heat conduction processes in a harmonic crystal with a substrate potential (2017). arXiv:1802.02037
Babenkov, M., Krivtsov, A., Tsvetkov, D.: Heat propagation in the one-dimensional harmonic crystal on an elastic foundation. Phys. Mesomech. (2019)
Gavrilov, S.N., Krivtsov, A.M., Tsvetkov, D.V.: Heat transfer in a one-dimensional harmonic crystal in a viscous environment subjected to an external heat supply. Contin. Mech. Thermodyn. 31(1), 255 (2019)
Kuzkin, V.A.: Unsteady ballistic heat transport in harmonic crystals with polyatomic unit cell. Contin. Mech. Thermodyn. 31(6), 1573 (2019)
Yu, Y.J., Hu, W., Tian, X.G.: A novel generalized thermoelasticity model based on memory-dependent derivative. Int. J. Eng. Sci. 81, 123 (2014)
El-Karamany, A.S., Ezzat, M.A.: Modified Fourier’s law with time-delay and kernel function: application in thermoelasticity. J. Therm. Stress. 38(7), 811 (2015)
Povstenko, Y.: Fractional Thermoelasticity, vol. 219. Springer, New York (2015)
Szabo, T.L.: Time domain wave equations for lossy media obeying a frequency power law. J. Acous. Soc. Am. 96(1), 491 (1994)
Fellah, Z.E.A., Berger, S., Lauriks, W., Depollier, C.: Time domain wave equations for lossy media obeying a frequency power law: application to the porous materials. In: Acoustics, Mechanics, and the Related Topics of Mathematical Analysis, pp. 143–149. World Scientific, Singapore (2002)
Grigoriev, I.S., Meilikhov, E.Z.: Handbook of Physical Quantities. CRC Press, Boca Raton (1996)
Mathews, J., Walker, R.L.: Mathematical Methods of Physics, vol. 501. WA Benjamin, New York (1970)
Pervozvansky, A.A.: Theory Course of the Automatic Control. Moscow Izdatel Nauka, Moscow (1986)
Smith, C.A., Corripio, A.B.: Principles and Practice of Automatic Process Control, 2nd edn. Wiley, New York (1997)
Ivanova, E.A., Vilchevskaya, E.N.: Truesdell’s and Zhilin’s Approaches: Derivation of Constitutive Equations Encyclopedia of Continuum Mechanics, pp. 1–11. Springer, Berlin (2017). https://doi.org/10.1007/978-3-662-53605-6_58-1
Hütter, G.: An extended Coleman–Noll procedure for generalized continuum theories. Contin. Mech. Thermodyn. 28(6), 1935 (2016)
Truesdell, C.: Rational Thermodynamics. Springer, New York (1984)
Nikol’skii, S.: A course of calculus. Nauka, Moscow (1991)
Sobolev, S.L., Browder, F.E.: Applications of Functional Analysis in Mathematical Physics. American Mathematical Society, New York (1963)
Vladimirov, V.S.: Equations of Mathematical Physics. Moscow Izdatel Nauka, Moscow (1976)
Polyanin, A.D., Manzhirov, A.V.: Handbook of Integral Equations. CRC Press, Boca Raton (1998)
Altenbach, H., Forest, S., Krivtsov, A.: Generalized Continua as Models for Materials, with Multi-scale Effects or Under Multifield Actions. Springer, New York (2013)
Kunin, I.A.: Elastic Media with Microstructure I: One-Dimensional Models, vol. 26. Springer, New York (2012)
Ayzenberg-Stepanenko, M., Cohen, T., Osharovich, G., Timoshenko, O.: Waves in periodic structures (mathematical models and computer simulations). Manuscript (2005, Beer-Shev)
Mysik, S.V.: Analyzing the acoustic spectra of sound velocity and absorption in amphiphilic liquids. St. Petersburg Polytech. Univ. J.: Phys. Math. 1(3), 325 (2015)
Landau, L., Lifshitz, E., Pitaevskij, L.: Course of Theoretical Physics. vol. 10: Physical Kinetics. Oxford (1981)
Mashinskii, E.: Amplitude-frequency dependencies of wave attenuation in single-crystal quartz: Experimental study. J. Geophys. Res.: Solid Earth 113(B11), (2008)
Upadhyay, M.V.: On the thermo-mechanical theory of field dislocations in transient heterogeneous temperature fields. Working paper or preprint (2020). https://hal.archives-ouvertes.fr/hal-02439503
Acknowledgements
The author is grateful to E.N. Vilchevskaya, E.A. Ivanova, D.A. Indeitsev, A.M. Krivtsov, and an anonymous referee for valuable discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Öchsner.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Babenkov, M.B. A model of the thermoelastic medium absorbing a part of the acoustic spectrum. Continuum Mech. Thermodyn. 33, 789–802 (2021). https://doi.org/10.1007/s00161-020-00957-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-020-00957-2