Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

The solution to the Lamb’s problem for nonlocal thermo-visco-elastic medium

Treść / Zawartość
Warianty tytułu
Rozwiązanie zagadnienia Lamba w nielokalnej termo-lepko-sprężystości
Języki publikacji
In our work we constructed the solution of the initial-boundary value problem so-called Lamb’s problem for the half space occupied with thermo-visco-elastic medium. The visco-elastic medium was described by the Biot model, whereas thermal interactions were described by the Gurtin-Pipkin model. Using the Cagniard de Hoop method we obtained the solution of the Lamb’s problem to the considered system of integro-differential equations. Based on the constructed solution of the above mentioned problem, we described the type of waves which propagate in the thermo-visco-elastic medium and the domain of their influence. The propagation of the Rayleigh’s wave was investigated. We discovered the new Rayleigh’s wave in thermo-visco-elastic medium (second Rayleigh’s wave) and named it GRL-wave (Gawinecki-Rafa-Lazuka–wave).
W pracy skonstruowano rozwiązanie zagadnienia brzegowo-początkowego typu Lamba dla półprzestrzeni wypełnionej ośrodkiem termo-lepko-sprężystym. Ośrodek lepko-sprężysty został opisany przez model Biota, natomiast oddziaływania termiczne zostały opisane przez model Pipkina-Gurtina. Rozważano również zagadnienie brzegowo-początkowe typu Daniłowskiej. Rozważano propagacje fal typu Rayleigha w badanym ośrodku termo-lepko-sprężystym.
Opis fizyczny
Bibliogr. 58 poz.
  • Military University of Technology, Faculty of Cybernetics, Institute of Mathematics and Cryptology, 2 Gen.W. Urbanowicza Str., 00-908 Warsaw, Poland
  • Military University of Technology, Faculty of Cybernetics, Institute of Mathematics and Cryptology, 2 Gen.W. Urbanowicza Str., 00-908 Warsaw, Poland
  • Military University of Technology, Faculty of Cybernetics, Institute of Mathematics and Cryptology, 2 Gen.W. Urbanowicza Str., 00-908 Warsaw, Poland
  • [1] Gawinecki J.A., Sikorska B., Nakamura G. and Rafa J., Mathematical and physical interpretation of the solution to the initial-boundary value problem in linear hyperbolic thermoelasticity theory, Z. Angew. Rath. Math., 87, 11, 2007, 715–746.
  • [2] Gawinecki J., Global existence of solution for non small data to non-linear spherically symmetric thermoviscoelasticity, Math. Meth. in Appl. Sci. 20, 2003, 907–936.
  • [3] Gurtin M.E., Pipkin A.C., A General Theory of Heat Conduction with Finite Wave Speeds, Arch. Rational Mech. Anal. 31, 1968, 113–128.
  • [4] Ackerman C.C., Bentman B., Fairbank H.A. and Guyer R.A., Second sound in solid helium, Phys. Rev. Lett., 16, 1966, 789–791.
  • [5] Ackerman C.C. and Overton W.C., Second sound in solid helium – 3, Phys. Rev. Lett., 22, 1969, 764–766.
  • [6] Adams R.A., Sobolev Spaces, Academic Press, New York, 1975.
  • [7] Atkins K.R. and Osborne D.V., The velocity of second sound below 1°K, Philos. Mag., 41, 1950, 1078–1081.
  • [8] Beckenbach E.F., Modern Mathematics for the Engineer, McGraw-Hill, New York, 1961.
  • [9] Bertman B. and Sandiford R.J., Second sound in solid helium, Sci. Amer., 222, No. 5, 1970, 92–101.
  • [10] Cagniard L., Reflection and Refraction of Progressive Seismic Waves, McGraw-Hill, New York, 1962.
  • [11] Cattaneo C., On the condiction of heat, Atti Sem. Mat. Fis. Univ. Modena 3, 1948, 3–21.
  • [12] Cattaneo C., A form of heat equation which eliminates the paradox of instantaneous propagation, C. R. Acad. Sci. Paris, 247, 1958, 431–433.
  • [13] Chandrasekharaiah D.S., Thermoelasticity with second sound: A review, Appl. Mech. Rev., 39, 1986, 355–376.
  • [14] Chester M., Second sound in solids, Phys. Rev., 131, 1963, 2013–2015.
  • [15] Coleman B.D. and Noll W., An aproximation theorem for functionals, with applications in continuum mechanics, Arch. Ration. Mech. Anal., 6, 1960, 355–370.
  • [16] Dingle R.B., Remarks on two-liquid model of Helium II, Philos. Mag., 42, 1951, 1080–1088.
  • [17] Dingle R.B., Derivation of the velocity of second sound from Maxwell’s equation of transfer, Proc. Phys. Soc., A65, 1952, 364–376.
  • [18] Fiodor G.D., Laplace Transforms in Engineering, Akad´emiai Kiad´o, Budapest, 1965.
  • [19] Francis P.H., Thermo-mechanical effects in elastic wave propagation. A survey, J. Sound Vib., 21, 1972, 181–192.
  • [20] Gawinecki J., Global solution to the Cauchy problem in non-linear hyperbolic thermoelasticity, Math. Methods Appl. Sci., 15, 1992, 223–237.
  • [21] Gawinecki J., Global solutions to initial value problems in nonlinear hyperbolic thermoelasticity, Dissertationes Math., 344, 1995, 66.
  • [22] Gawinecki J. and Duc Hung Do, Global existence of solution to the initial value problem for nonlinear hyperbolic heat equation, Bull. Polish Acad. Sci. Math., 39, 1991, 21–29.
  • [23] Gawinecki J., Rafa J. and Włodarczyk E., L∞ − L1-time decay estimate to the solution of the Cauchy problem of the system of equations describing nonlocal model of the propagation of heat with finite speed, Biuletyn WAT, 42, 1993, no. 12, 4–20.
  • [24] Gradshte˘I.S. and Ryzhik I.I., Tables of Integrals, Sums, Series and Products, Nauka, 1971 (in Russian).
  • [25] von Gutfeld R.J. and Nethercot A.H., Temperature dependence of heat-pulse propagation in sapphire, Phys. Rev. Lett., 17, 1966, 868–871.
  • [26] de Hoop A.T., A modification of Cagniard’s methods of solving seismic pulse problems, Appl. Sci. Res. Sec., B 8, 1959, 349–356.
  • [27] Hörmander L., The Analysis of Linear Partial Differential Operators, Vol. 2, Differential Operators with constant Coefficients, Springer, Berlin, 1989.
  • [28] Hörmander L., Estimates for translation invariant operators in Lp spaces, Acta Math., 104, 1960, 93–140.
  • [29] Jackson H.E. and Walker C.T., Thermal conductivity, second sound and phononphonon interactions in NaF, Phys. Rev., B 3, 1971, 1428–1439.
  • [30] Jackson H.E., Walker C.T. and McNelly T.W., Second sound in NaF, Phys. Rev. Lett., 25, 1970, 26–28.
  • [31] Kaliski S., Wave equation of heat conduction, Bull. Acad. Polon. Sci. S´er. Sci. Tech., 13, 1965, 211–219.
  • [32] Kaliski S., Wave equations of thermoelasticity, Bull. Acad. Polon. Sci. S´er. Sci. Tech., 13, 1965, 253–260.
  • [33] Lambermont J. and Lebon G., On a generalization of the Gibbs equation for heat conduction, Phys. Lett., A 42, 1973, 499–500.
  • [34] Landau L.D., The theory of superfluidity of helium II, J. Phys., USSR 5, 1941, 71–90.
  • [35] Lifshitz E.M., Superfluidity, Sci. Amer., 198, no. 6, 1958, 30–36.
  • [36] London F., Superfluids, Wiley, Now York, vol. II, 1954, p. 101.
  • [37] Luikov A.V., Application of methods of thermodynamics of irreversible process to investigation of heat and mass exchange, J. Engrg. Phys., 9, 1965, 287–304.
  • [38] Luikov A.V., Application of irreversible thermodynamics methods to investigation of heat and mass transfer, Int. J. Heat Mass Transfer, no. 9, 1966, 139–152.
  • [39] Maurer M.J., Relaxation model for heat conduction in metals, J. Appl. Phys., 40, 1969, 5123–5130.
  • [40] Maurer R.D. and Herlin M.A., Second sound velocity in helium II, Phys. Rev., 76, 1949, 948–950.
  • [41] Maxwell J.D., On the dynamic theory of gases, Philos. Trans. Roy. Soc., London, 157, 1967, 49–88.
  • [42] McNelly T.F., Rogers S.J., Channin D.J., Rollefson R.J., Goubau W.M., Schmidt G.E., Krumhansl J.A., and Pohl R.O., Heat pulses in NaF: Onset of second sound, Phys. Rev. Lett., 24, 1970, 100-102.
  • [43] Nettleton R.E., Relaxation theory of thermal conduction in liquids, Phys. Fluids 3, 1960, 216–225.
  • [44] Nowacki W., Dynamic Problems of Thermoelasticity, Noordhoff, Leyden, 1975.
  • [45] Nunziato J.W., On heat conduction in materials with memory, Quart. Appl. Math., 29, 1971, 187–204.
  • [46] Pellam J.R. and Scott R.B., Second sound velocity in paramagnetically cooled liquid helium II, Phys. Rev., 76, 1949, 869–870.
  • [47] Peshkov V., Second sound in helium II, J. Phys. USSR 8, 1944, 131.
  • [48] Prudnikov A.P., Brychkov Yu.A. and Marichev O.I., Integrals and Series. Special Functions, Nauka, 1983 (in Russian).
  • [49] Rogers S.J., Transport of heat and approach to second sound in some isotopically pure alkali-halide crystals, Phys. Rev., 3, 1971, 1440–1457.
  • [50] Schwartz L., Theorie des distributions, Nouvelle ed., Hermann, Paris, 1966.
  • [51] Sobolev S.L., Applications of Functional Analysis in Mathematical Physics, Nauka, Moscow, 1988 (in Russian).
  • [52] Tisza L., The theory of liquid helium, Phys. Rev., 72, 1947, 838–839.
  • [53] Triebel H., Theory of Function Spaces, Birkhauser, Basel, 1983.
  • [54] Vernotte P., The true heat equation, C. R. Acad. Sci., Paris, 247, 1958, 103.
  • [55] Vernotte P., Paradoxes in the continuous theory of the heat equation, C. R. Acad. Sci., Paris, 246, 1958, 3154–3158.
  • [56] Vladimirov V., Mikhailov V., Chabounine M., Karimov Kh., Sidorov Y. et Vacharine A., Recueil de Problemes d’Equations de Physique Mathematique, Mir, Moscow, 1976.
  • [57] Ward J.C. and Wilks J., The velocity of second sound in liquid helium near absolute zero, Philos. Mag., 42, 1951, 314–316.
  • [58] Ward J.C. and Wilks J., Second sound and the thermomechanical effect at very low temperatures, Philos. Mag., 43, 1952, 48–50.
Funding source of the work – statuary activity of the Military University of Technology.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.