Three-dimensional mathematical problems of thermoelasticity of anisotropic bodies, I

Memoirs on Differential Equations and Mathematical Physics

Volumn 17, Issue , 1999, Pages 7-126

  Jentsch, Lothar ^a   Natroshvili, David ^b  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

^b GEORGIAN TECHNICAL UNIVERSITY   (Georgia)

Author keywords

Anisotripic bodies; Boundary value problems; Crack problems; Fundamental solutions; Interface problems; Potential theory; Pseudo oscillations; Pseudodifferen tial equations on manifold; Radiation conditions; Ther moelastic vibrations; Thermoelasticity

Indexed keywords

EID: 85070504770     PISSN: 15120015     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (85)

1. R. A. Adams, Sobolev spaces, Academic Press, New York-San Fran­cisco-London, 1975.
2. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, II. Comm. Pure Appl. Math. 12(1959), 623-727, 17(1964), 35-92.
3. A. Douglis and L. Nirenberg, Interior estimates for elliptic sys­tems of partial differential equations. Comm. Pure Appl. Math. 8(1954), 503-538.
4. M. S. Agranovich, Elliptic singular integro-differential operators. (Russian) Uspekhi Mat. Nauk 20(1965), No. 5(131), 3-10.
5. M. S. Agranovich, Spectral properties of diffraction problems. (Russian) In: N. N. Voitovich, B. Z. Katsenelenbaum, and A. N. Sivov, Generalized method of eigenoscillation in the diffraction theory (Russian), Nauka, Moscow, 1977, 289-412.
6. H. Brakhage and P. Werner, Über das Dirichletsche AuBen-raumproblem fiir die Helmholtzsche Schwingungsgleichung. Arch. Math. 16(1965), 325-329.
7. T. V. Buchukuri and T. G. Gegelia, On the uniqueness of so­lutions of the basic problems of elasticity for infinite domains. (Russian) Differentsial'nie Uravneniya 25(1988), No. 9, 1556-1565.
8. T. V. Burchuladze and T. G. Gegelia, Development of the potential method in the theory of elasticity. (Russian) Metsniereba, Tbilisi, 1985.
9. O. Chkadua, The non-classical boundary-contact problems of elas­ticity for homogeneous anisotropic media. Math. Nachr. 172(1995), 49-64.
10. D. Colton and R. Kress, Integral equation methods in scattering theory. John Wiley, New York, 1983.
11. D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory. Springer- Verlag: Berlin, Heidelberg, New York, 1992.
12. M. Costabel and E. Stephan, A direct boundary integral equa­tion method for transmission problems. J. Math. Anal. Appl. 106(1985), 376-413.
13. M. Costabel and E. Stephan, An improved boundary element Galerkin method for three-dimensional crack problems. Integral Equations Operator Theory 10(1987), 467-504.
14. CM. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch. Rational Mech. Anal. 29(1968), No. 4, 241-271.
15. R. Duduchava, On multidimensional singular integral operators, I, II. J. Oper. Theory 11(1984), 41-76, 199-214.
16. R. Duduchava, D. Natroshvili, and E. Shargorodsky, Basic boundary value problems of thermoelasticity for anisotropic bodies with cuts, I, II. Georgian Math. J. 2(1995), No. 2, 123-140, No. 3, 259-276.
17. R. Duduchava, D. Natroshvili, and Shargorodsky, Boundary-value problems of the mathematical theory of cracks. Proc. I. Vekua Inst. Appl. Math. Tbilisi Univ. 39(1990), 68-84.
18. R. Duduchava, A.-M. Sãndig, and W. L. Wendland, Asym-totics of solutions to boundary equations with applications to interface cracks in anisotropic composites. Preprint Nr. 95—9, Universitãt Stuttgart, Mathematisches Institut A, 1995.
19. R. Duduchava and W. L. Wendland, The Wiener-Hopf method for systems of pseudodifferential equations with an application to cracks problems. Integral Equations Operator Theory 23(1995), 294-335.
20. G. I. Eskin, Boundary value problems for elliptic pseudodifferential equations. Translations of Math. Monographs AMS, 52, Providence, Rode Island, 1981.
21. M. V. Fedorjuk, Seddle-point method. (Russian) Nauka, Moscow, 1977.
22. G. Fichera, Existence theorems in elasticity. Handb. der Physik, Bd. 6/2, Springer Veiiag, Heidelberg, 1973.
23. I. M. Gelfand and G. E. Shilov, Generalized functions and operations on them. (Russian) 2-nd Ed., Fismatgiz, Moscow, 1959.
24. T. G. Gegelia and L. Jentsch, Potential methods in continuum mechanics. Georgian Math. J., 1(1994), No. 6, 599-640.
25. T. Hargé, Valeurs propres d'un corps élastique. C.R. Acad. Sci. Paris, Sér. I Math. 311(1990), 857-859.
26. G. C. Hsiao, On the boundary-field equation methods for fluid-structure interactions. Proceedings of the 10. TMP, Teubner Texte zur Math-ematik, Bd. 134, 79-88, Stuttgart-Leipzig, 1994.
27. G. C. Hsiao, R. E. Kleinman, and G. F. Roach, Weak so­lution of fluid-solid interaction problems. Preprint-Nr. 1917, Technische Hochschule Darmstadt, Fachbereich Mathematik, 1997.
28. J. Ignaczak and W. Nowacki, Singular integral equations of thermoelasticity. Internai. J. Engrg. Sci. 5(1966), No. 1, 53-68.
29. L. Jentsch, Über stationáre thermoelastische Schwingungen in in-homogenen Kõrpern. Math. Nachr. 64(1974), 171-231.
30. L. Jentsch, Stationáre thermoelastische Schwingungen in stiickweise homogenen Kõrpern infolge zeitlich periodischer AuBentemperatur. Math. Nachr. 69(1975), 15-37.
31. L. Jentsch, Verallgemeinerte thermoelastische Schwingungen in Kõrpern mit gleitenden Einschliissen aus anderem Material. In: Jentsch, L. und Maul, J., Mathematical Research, Bd. 4, Akademie Veiiag Berlin, 1980.
32. L. Jentsch, Zur Existentz von reguláren Lõsungen der Elastostatik stückweise homogener Kõrper mit neuen Kontaktbedingungen an den Tren- nfláchen zwischen zwei homogenen Teilen. Abh. d. Sachs. Akad. d. Wis-sensch. zu Leipzig, Math. Naturw. Klasse, Bd. 53, H. 2, Akademie-Verlag Berlin, 1977.
33. L. Jentsch and D. Natroshvili, Thermoelastic oscillations of anisotropic bodies. Preprint 96—1, Technische Universitãt Chemnitz-Zwi-ckau, Fakultãt fur Mathematik, 1-45(1996) (see also, E. Meister (Ed.):Modern Mathematical Methods in Diffraction Theory and its Application in Engineering, (Proceedings of Sommerfeld'96 Workshop, Freudenstadt, Schwarzwald, 30 September - 4 October, 1996), Methoden und Verfahren der mathematischen Physik, Band 42, Peter Lang, Frankfurt am Main- Berlin-New York-Paris, 1997, 68-87).
34. L. Jentsch and D. Natroshvili, Non-classical interface problems for piecewise homogeneous anisotropic elastic bodies. Math. Meth. Appl. Sci. 18(1995), 27-49.
35. L. Jentsch and D. Natroshvili, Non-classical mixed interface problems for anisotropic bodies. Math. Nachr. 179(1996), 161-186.
36. L. Jentsch and D. Natroshvili, Interaction between thermoe­lastic and scalar oscillation fields. Integral Equations Operator Theory 28 (1997), 261-288.
37. L. Jentsch and D. Natroshvili, Interaction between thermoe­lastic and scalar oscillation fields (general anisotropic case). Preprint 97—5, Technische Universitãt Chemnitz-Zwickau, Fakultãt far Mathematik, 1997.
38. L. Jentsch and D. Natroshvili, Mathematical problems of the thermoelasticity theory of anisotropic bodies. Thermal Stresses *97, Pro­ceedings of the Second International Symposium on Thermal Stresses and Related Topics, (June 8-11, 1997, Rochester Institute of Technology, Ther­mal Stresses '97) Rochester Institute of Technology, Rochester, 513-516 (1997).
39. L. Jentsch and D. Natroshvili, Non-local approach in mathe­matical problems of fluid-structure interaction. Preprint 98—1, Technische Universitãt Chemnitz, Fakultãt far Mathematik, 1998.
40. L. Jentsch, D. Natroshvili, and I. Sigua, Mixed interface problems of thermoelastic pseudo-oscillations. Preprint 97—1, Technische Universitãt Chemnitz Zwickau, Fakultãt far Mathematik, 1997.
41. L. Jentsch, D. Natroshvili, and W. L. Wendland, General transmission problems in the theory of elastic oscillations of anisotropic bod­ies. DFG-Schwerpunkt Randelementmethoden, Bericht Nr. 95-7, Stuttgart University, 1-42 (1995) (to appear in Journal of Mathematical Analysis and Applications).
42. D. S. Jones, Low-frequency scattering by a body in lubricated contact. Quart. J. Mech. Appl. Math. 36(1983), 111-137.
43. R. V. Kapanadze, On some properties of singular operators in normed spaces. (Russian) Proc. Tbilisi Univ., Mat. Mech. Astronom. 129(1968), 17-26.
44. V. A. Kondratijev and O. A. Oleinik, Boundary value prob­lems for system of equations of the elasticity theory in unbounded domains. Korn's inequalities. (Russian) Uspekhi Mat. Nauk 43, 5(263)(1988), 55­98.
45. V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, and T. V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. (Russian) Nauka, Moscow, 1976; English translation: North-Holland Series in Applied Mathematics and Mechanics, V. 25, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1979.
46. R. Leis, Zur Dirichletschen Randwertaufgabe des AuBenraums der Schwingungsgleichung. Math. Z. 90(1965), 205-211.
47. J. L. Lions and E. Magenes, Problèmes aux limites non ho-mogènes et applications, vol. 1. Dunod, Paris, 1968.
48. C. J. Luke and P. A. Martin, Fluid-solid interaction: acoustic scattering by a smooth elastic obstacle. SIAM J. Appl. Math. 55, 4(1995), 904-922.
49. V. G. Maz'ya, Sobolev spaces. (Russian) Leningrad University, Leningrad, 1985.
50. V. G. Maz'ya, Boundary integral equations. In R. V. Gamkrelidze (Ed.): Encyclopedia of Mathematical Sciences, vol. 27, 127-222, Springer Verlag: Berlin, Heidelberg, 1991.
51. S. G. Mikhlin and S. Prossdorf, Singular integral operators. Springer Verlag, Berlin, 1986.
52. C. Miranda, Partial differential equations of elliptic type. 2nd ed., Springer Verlag, Berlin-Heidelberg-New York, 1970.
53. 3 and applications to electromagnetic scattering. The Journal of Fourier Analysis and Applications 3, 2(1997), 131-192.
54. S. Mizohata, Theory of partial differential equations. (Russian) Mir, Moscow, 1976.
55. D. Natroshvili, Boundary integral equation method in the steady-state oscillation problems for anisotropic bodies. Math. Methods Appl. Sci. 20, 2(1997), 95-119.
56. D. Natroshvili, Investigation of boundary value and initial bound­ary value problems of the mathematical theory of elasticity and thermoelas­ticity for homogeneous anisotropic bodies by potentials methods. (Russian) Doct. Thesis, A.Razmadze Tbilisi Math. Inst. Acad, of Sci. of Georgia, Tbilisi, 1-325, 1984.
57. D. Natroshvili, Dynamic problems of thermoelasticity of anisotro­pic bodies. (Russian) Difjerentsial'nye Uravnenia 20(1984), No. 1, 87-98.
58. D. Natroshvili, Mixed interface problems for anisotropic elastic bodies. Georgian Math. J. 2(1995), No. 6, 631-652.
59. D. Natroshvili, O. Chkadua, and E. Shargorodsky, Mixed boundary value problems of the anisotropic elasticity. (Russian) Proc. I. Vekua Inst. Appl. Math. Tbilisi Univ. 39(1990), 133-181.
60. D. Natroshvili, A. Djagmaidze, and M. Svanadze, Some linear problems of the theory of elastic mixtures. Tbilisi University Press, Tbilisi, 1986.
61. S. Nicaise, Polygonal interface problems. Methoden und Verfahren der Mathematischen Physik, Bd. 39, Verlag Peter Lang, Frankfurt am Main, 1993.
62. S. Nicaise and A.-M. Sãndig, General interface problems. Math. Meth. Appl. Sci. 17(1994), 395-450.
63. W. nowacki, Dynamic problems of thermoelasticity. PWN-Polish Scientific Publishers, Warszawa, Poland, Nordhoff International Publishing, Leyden, The Nederlands, 1975.
64. O. I. Panich, On the question of the solvability of the exterior boundary value problems for the wave equation and Maxwell's equations. (Russian) Uspekhi Mat. Nauk 20, 1(121)(1965), 221-226.
65. T. von Petersdorff, Boundary integral equations for mixed Dirichlet, Neumann and transmission conditions. Math. Meth. Appl. Sci. 11(1989), 185-213.
66. J. S. Podstrigach and Ju. M. Koliano, Generalized thermo-mechanics. Naukova Dumka, Kiev, 1976.
67. A.-M. Sãndig, Boundary integral equations for coupled structures in solid mechanics. The direct method. Universitãt Stuttgart, Mathematis-ches Institut A, Preprint Nr., 95-20, (1995).
68. R. Seeley, Analytic extention of the trace associated with elliptic boundary problems. Amer. J. Math. 91(1969), 963-983.
69. E. Shamir, Elliptic systems of singular integral operators. /. Trans. Amer. Soc. 127(1967), 107-124.
70. E. Shargorodsky, Boundary value problems for elliptic pseudo-differential operators: the halfspace case. (Russian) Proc. Razmadze Math. Inst. Razmadze), 99(1993), 44-80.
71. E. Shargorodsky, Boundary value problems for elliptic pseudo-differential operators on manifolds. (Russian) Proc. Razmadze Math. Inst. 105(1993), 108-132.
72. E. Shargorodsky, An L analogue of the Vishik-Eskin theory. Mem. Differential Equations Math. Phys. 2(1994), 41-148.
73. I. B. Simonenko, A new general method for the investigation of linear operator equations of singular integral equations type', 1,11. Izv. Akad. Nauk SSSR, Ser. Math. 29(1965), 567-586, 757-782.
74. I. Sneddon, Fourier transforms. McGraw-Hill, New York-Toronto-London, 1951.
75. 3. Habilitationsschrift, TH Darmstadt, 1984.
76. 3. Math. Nachr. 134(1987), 21-53.
77. F. Treves, Introduction to pseudodifferential and Fourier integral operators, vol. 1, Pseudodifferential operators. Plenum Press, New York-London, 1982.
78. H. Triebel, Interpolation theory, function spaces, differential oper­ators. VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.
79. H. Triebel, Theory of function spaces. Leipzig Birkhauser Verlag, Basel-Boston-Stuttgart, 1983.
80. B. R. Vainberg, The radiation, limiting absorption and limiting amplitude principles in the general theory of partial differential equations. (Russian) Uspekhi Math. Nauk21, 3(129)(1966), 115-194.
81. B. R. Vainberg, On the method of stationary phase. (Russian) Vestnik Moscow Univ., Ser. Mat., Mech. 1(1976), 50-58.
82. B. R. Vainberg, Asymptotic methods in the equations of mathe­matical physics. (Russian) Nauka, Moscow, 1982.
83. I. N. Vekua, On metaharmonic functions. (Russian) Proc. Tbilisi Math. Inst. Acad. Sci. Georgian SSR 12(1943), 105-174.
84. I. I. Vorovich, V. M. Aleksandrov, and V. A. Babeshko, Non-classical mixed problems of elasticity theory. (Russian) Nauka, Mos­cow, 1974.
85. J. T. Wloka, B. Rowley, and B. Lawruk, Boundary value problems for elliptic systems. Cambridge University Press, Cambridge-New York-Melburn, 1995.