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Horák, Jiří V.

On solvability of one special problem of coupled thermoelasticity. I: Classical boundary conditions and steady sources. (English) Zbl 0854.35019

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 34, 39-58 (1995).
Summary: The existence and uniqueness of a weak solution of a special problem arising in linear theory of coupled thermoelasticity is proved by using the Rothe method of discretization in time. First, a model problem is derived from 3D theory of coupled thermoelasticity and then a priori estimations of Rothe vector functions and their time derivatives for the case of classical boundary conditions and steady sources are shown. Approximative properties of the Rothe vector functions and their convergence to the weak solution as well as continuous dependence of the solution on the given data are also proved.

Cited in 1 Review

MSC:

35D05 Existence of generalized solutions of PDE (MSC2000)
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K20 Plates

Keywords:

coupled thermoelasticity; Rothe method of discretization in time
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References:

[1] Boley B. A., Weiner J. H.: Theory of Thermal Stresses. J. Wiley and sons, New York, 1960 · Zbl 1234.74001
[2] Dafermos C. M.: On the Existence and the Asymptotic Stability of Solution to the Equations of Linear Thermoelasticity. Arch. Rational Mech. Anal., 29 (1968), 241-271. · Zbl 0183.37701 · doi:10.1007/BF00276727
[3] Washizu K.: Variational Methods in Elasticity and Plasticity. Pergamon Press, Oxford, 1968. · Zbl 0164.26001
[4] Kovalenko A. D.: Fundamentals of thermoelasticity. Izdatelstvo ”Naukova dumka”, Kiev, 1970
[5] Nowacki W.: Dynamical problems of thermoelasticity. Izdatelstvo ”Mir”, Moskva, 1970 · Zbl 0227.73009
[6] Aubin J. P.: Approximation of Elliptic Boundary - Value Problems. Wiley-Interscience, London, 1972. · Zbl 0248.65063
[7] Carlson D. E.: Linear Thermoelasticity. Encyklopedia of Physics S. Flüge, Volume VIa/2, Mechanics of Solids II. Springer Verlag, Berlin, 1972.
[8] Lions J. L.: Někatoryje metody rešenija nelinejnych krajevych zadač. Izdatelstvo ”Mir”, Moskva, 1972
[9] Adams R. A.: Sobolev Spaces. Academic Press, New York, 1975. · Zbl 0314.46030
[10] Michlin S. G.: Variational methods in mathematical physics. Alfa, Bratislava, 1975 · Zbl 0328.35001
[11] Truesdell C.: A first course in rational mechanics. Izdatelstvo ”Mir”, Moskva 1975 · Zbl 0342.73001
[12] Glowinski R., Lions J. L. , Trémolieres R.: Analyse numérique des inéquations variationnelles. Dunod, Paris, 1976. · Zbl 0358.65091
[13] Kufner A., John O., Fučík S.: Function Spaces. Academia, Praha, 1977.
[14] Michlin S. G.: Linějnyje uravněnija v častnych proizvodnych. Moskva, Vyššaja škola, 1977
[15] Nowacki W.: Coupled fields in mechanics of solids. W. T. Koiter: Theoretical and Applied Mechanics, Proceedings of the 14th IUTAM Congress, Delft, The Netherlands, 1976, North-Holland, Amsterdam, 1977. · Zbl 0429.73105
[16] Nowinski J. L.: Theory of thermoelasticity with applications. Sijthoff & Noordhoff international Publishers, Aplhen Aan den Tijn, 1978. · Zbl 0379.73004
[17] Aubin J. P.: Applied Functional Analysis. J. Wiley and sons, New York, 1979. · Zbl 0424.46001
[18] Day W. A.: Justification of the Uncoupled and Quasistatic Approximation in a Problem of Dynamic Thermoelasticity. Arch. Rational Mech. Anal. 77 (1981), 387-396. · Zbl 0503.73008 · doi:10.1007/BF00280644
[19] Day W. A.: Further Justification of the Uncoupled and Quasi-Static Approximations in Thermoelasticity. Arch. Rational Mech. Anal. 79 (1982), 85-95. · Zbl 0507.73003 · doi:10.1007/BF02416566
[20] Bock I., Lovíšek J., Štangl J.: Contact problem for two elastic beams. (in slovak), Strojnický časopis 35 (1984), No 3, 353-373
[21] Ženíšek A.: The existence and uniqueness theorem in Biot’s consolidation theory. Aplikace matematiky, 29 (1984), No 3, 194-211. · Zbl 0557.35005
[22] Ženíšek A.: Finite element methods for coupled thermoelasticity and coupled consolidation of clay. R. A. I. R. O. Numer. Anal. 18 (1984), 183-205. · Zbl 0539.73005
[23] Day W. A.: Heat Conduction Within Linear Thermoelasticity. Springer-Verlag, New York, 1985. · Zbl 0577.73009
[24] Horák J.: Evolution variational inequalities in thermoelasticity. MÚ ČSAV, Praha, 1985
[25] Horák J.: Solution of the problem in linear theory of coupled thermoelasticity. Ph. D. Thesis, Faculty of Natural Sciences, UP Olomouc, 1993
[26] Kačur J.: Method of Rothe in Evolution Equations. Taubner - Texte zur Mathematik, Band 80, Liepzig, 1985. · Zbl 0582.65084
[27] Rektorys K.: Method of dicretization in time and partial differential equations. TKI, SNTL, Praha, 1985
[28] Kačur J., Ženíšek A.: Analysis of approximate solution of coupled dynamical thermoelasticity and related problems. Aplikace matematiky 31 (1986), No 3, 190-223. · Zbl 0627.73002
[29] Tauchert T. R.: Thermal Stresses in Plates - Dynamical Problems, chapter 1. Richard B. Hetnarski: Thermal Stresses, vol II., North-Holland, Amsterdam, 1986.
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