On the regularity of solutions of a thermoelastic system under noncontinuous heating regimes. (English) Zbl 0754.73021

Jarušek, Jiří

Apl. Mat. 35, No. 6, 426-250 (1990).

Problems in quasi-linear thermoelasticity are considered for a homogeneous isotropic body; the quasi-linearity is derived from the possible dependence on temperature of modulus of elasticity and specific heat; a stronger non-linearity is also admitted at the boundary through a radiation term.
The question studied and answered in the affirmative (within appropriate qualifications) is whether the stress field remains continuous (and bounded) even though there is a jump in the temperature of the furnace (discontinuous boundary datum). Essential hypothesis is the smoothness of the boundary of the heated body, or at least a local condition of convexity at an isolated singularity of the boundary. The paper will not make easy reading for an engineer who needs to be convinced that no damage will ensue from a boundary temperature jump.

Reviewer: G.Capriz (Pisa)

MSC:

74A15	Thermodynamics in solid mechanics
74B99	Elastic materials
80A20	Heat and mass transfer, heat flow (MSC2010)

Keywords:

temperature shock; quasi-linear thermoelasticity; homogeneous isotropic body; radiation term; stress field; smoothness of the boundary

References:

[1]	S. Agmon A. Douglis L. Nirenberg: Estimates near boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Part II. Comm. Pure Appl. Math. 17 (1964) 35-92. · Zbl 0123.28706
[2]	O. V. Běsov V. P. lljin S. M. Nikolskij: Integral Transformations of Functions and Imbedding Theorems. (in Russian). Nauka, Moskva 1975.
[3]	P. Grisvard: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Math. 24, Pitman, Boston-London-Melbourne 1985. · Zbl 0695.35060
[4]	P. Grisvard: Problemes aux limites dans les polygones, Mode d’emploi. EDF Bull. Direct. Etud. Rech. Ser. C. Math. Inform. (1986) 1, 21 - 59. · Zbl 0623.35031
[5]	J. Jarušek: Contact problems with bounded friction. Coercive case. Czech. Math. J. 33 (108) (1983), 237-261. · Zbl 0519.73095
[6]	J. Jarušek: Remark to the generalized gradient method for the optimal large-scale heating problem. Probl. Control Inform. Theory 16 (1987) 2, 89-99. · Zbl 0647.73047
[7]	J. Jarušek: Optimal control of thermoelastic processes III. (in Czech). Techn. rep. Inst. Inform. Th. Autom. No. 1561, Praha 1988.
[8]	V. A. Kondratěv: Elliptic boundary value problems with conical or angular points. (in Russian). Trudy Mosk. Mat. Obšč., Vol. 16 (1967), 209-292.
[9]	A. Kufner O. John S. Fučík: Function Spaces. Academia, Praha 1977.
[10]	A. Kufner A. M. Sändig: Some Applications of Weighted Sobolev Spaces. Teubner-Texte Math., Band 100, Teubner V., Leipzig 1987. · Zbl 0662.46034
[11]	O. A. Ladyženskaja V. A. Solonnikov N. N. Uralceva: Linear and Quasilinear Equations of Parabolic Type. (in Russian). Nauka, Moskva 1967.
[12]	J. L. Lions E. Magenes: Problèmes aux limites non-homogènes et applications. Dunod, Paris 1968. · Zbl 0212.43801
[13]	J. Nečas: Solution of the biharmonic problem for an infinite angle. (in Czech). Časopis pěst. mat. 83 (1958), Part I, 257-286, Part. II, 399-424.
[14]	J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Praha 1967. · Zbl 1225.35003
[15]	J. Nečas: Introduction to the Theory of Nonlinear Elliptic Equations. Teubner-Texte Math., Band 52, Teubner V., Leipzig 1983. · Zbl 0526.35003
[16]	R. H. Nochetto: Error estimates for two-phases Stefan problem in several space variables. Part II: Nonlinear flux conditions. Preprint No. 416, 1st. Anal. Numer, CNR. Pavia, Pavia 1984.
[17]	A. M. Sändig U. Richter R. Sändig: The regularity of boundary value problems for the Lame equations in polygonal domain. Rostock. Math. Kolloq. 36 (1989), 21-50. · Zbl 0674.35024
[18]	A. Visintin: Sur le problème de Stefan avec flux non-lineaire. Preprint No. 230, Ist. Anal. Numer. C. N. R. Pavia, Pavia 1981. · Zbl 0478.35084