Boundary conditions method for solving linear mechanics problems

V. B. Pen'kov ; V. V. Pen'kov

Dalʹnevostočnyj matematičeskij žurnal, Tome 2 (2001) no. 2, pp. 115-137.

Résumé

A new method for solving linear mechanics problems is proposed. The solveability of the basic problems of isotropic elasticity and the convergence are strictly proofed. A methodology for creating medium basic of states is developed. Testing and new problems for a circle, a parallelepiped, L-shaped straight prism are solved. The aspects of convergence are studied in a numeric experiment. 9 figs., 58 refs.

Bibliographie

[1] I. S. Arzhanykh, Integralnye uravneniya osnovnykh zadach teorii polya i teorii uprugosti, Izd-vo AN UzbSSR, Tashkent, 1954

[2] V. I. Blokh, “Funktsii napryazhenii v teorii uprugosti”, PMM, XIV:4 (1950)

[3] Yu. A. Brychkov, A. P. Prudnikov, “Sfericheskie funktsii”, Matematicheskaya entsiklopediya, v. 5, Sovetskaya entsiklopediya, M., 1977, 293–294

[4] B. G. Galerkin, Sobranie sochinenii, v. 1, 2, M., 1952 | Zbl

[5] Z. Domanskii, A. Piskorek, Z. Roek, “O primenenii metoda Fishera-Rissa-Kupradze dlya resheniya pervoi zadachi Fure”, Rocz. Pol. Tow. Mat. 1: Prace Mat., 16 (1972), 137–147 | MR

[6] E. F. Emelyanov, “Garmonicheskii mnogochlen”, Matematicheskaya entsiklopediya, 1, Sovetskaya entsiklopediya, M., 1977, 886–887

[7] L. A. Igumnov, N. M. Khutoryanskii, “Chislennoe issledovanie polei peremeschenii i napryazhenii v vyazkouprugoi srede ot sosredotochennykh impulsnykh istochnikov”, Prikladnye problemy prochnosti i plastichnosti, Algoritmizatsiya i avtomatizatsiya reshenii zadach uprugosti i plastichnosti, Gork. Un-t, Gorkii, 1983, 42–51

[8] V. N. Ionov, G. A. Vvedenskii, “O vozmozhnykh formakh obschego resheniya uravnenii ravnovesiya v krivolineinykh koordinatakh”, Izvestiya VUZov. Matematika, 1964, no. 6, 59–66 | MR

[9] V. N. Ionov, P. M. Ogibalov, Prochnost prostranstvennykh elementov konstruktsii, v. 1, Vysshaya shkola, M., 1972, 752 pp.

[10] V. N. Ionov, P. M. Ogibalov, Prochnost prostranstvennykh elementov konstruktsii, v. 2, Vysshaya shkola, M., 1972, 536 pp.

[11] N. A. Kilchevskii, Elementy tenzornogo ischisleniya i ego prilozheniya k mekhanike, GITTL, M., 1954

[12] R. Kristensen, Vvedenie v teoriyu vyazkouprugosti, Mir, M., 1974, 338 pp.

[13] Yu. A. Krutkov, Tenzor funktsii napryazhenii i obschie resheniya v statike teorii uprugosti, Izd. AN SSSR, M., 1949

[14] V. D. Kupradze, T. V. Burchuladze, “Granichnye zadachi termouprugosti”, Differentsialnye uravneniya, 5:1 (1969), 3–43 | MR | Zbl

[15] V. D. Kupradze, T. G. Gegeliya, M. O. Basheleishvili, T. V. Burchuladze, Trekhmernye zadachi matematicheskoi teorii uprugosti, Nauka, M., 1976, 664 pp.

[16] R. Kurant, D. Gilbert, Metody matematicheskoi fiziki, v. 1, GTTL, M.–L., 1933, 527 pp.

[17] S. G. Lekhnitskii, Anizotropnye plastinki, GITTL, M., 1957, 463 pp.

[18] S. G. Lekhnitskii, “O nekotorykh voprosakh, svyazannykh s teoriei izgiba tonkikh plit”, Prikladnaya matematika i mekhanika, II:2 (1938), 181–210

[19] A. I. Lure, Teoriya uprugosti, Nauka, M., 1970, 940 pp.

[20] A. Lyav, Matematicheskaya teoriya uprugosti, M., 1936

[21] Mekhanika v SSSR za 50 let, v. 3, Mekhanika deformiruemogo tverdogo tela, Nauka, M., 1972, 480 pp.

[22] S. G. Mikhlin, Variatsionnye metody v matematicheskoi fizike, Nauka, M., 1970, 512 pp. | MR | Zbl

[23] S. G. Mikhlin, Chislennaya realizatsiya variatsionnykh metodov, Nauka, M., 1966, 432 pp. | MR

[24] N. I. Muskhelishvili, Nekotorye osnovnye zadachi matematicheskoi teorii uprugosti, Nauka, M., 1966, 707 pp.

[25] D. G. Natroshvili, “O fundamentalnykh matritsakh uravnenii ustanovivshikhsya kolebanii i psevdokolebanii anizotropnoi teorii uprugosti”, Soobsch. AN Gruz.SSR, 96:1 (1979), 49–53 | MR

[26] V. Novatskii, Dinamicheskie zadachi termouprugosti, Mir, M., 1970

[27] V. Novatskii, Teoriya uprugosti, Mir, M., 1975, 872 pp. | MR

[28] P. F. Papkovich, Teoriya uprugosti, Oborongiz, M., 1939

[29] V. B. Penkov, V. V. Penkov, “Prostranstva sostoyanii v zadachakh mekhaniki kontinuuma”, Mezhdunarodnaya konferentsiya “Teoriya priblizhenii i garmonicheskii analiz”, Tezisy dokladov (Rossiya, Tula, 26–29 maya 1998 g.)

[30] V. B. Penkov, V. V. Penkov, “Metod granichnykh sostoyanii v reshenii osnovnykh zadach dlya uprugogo parallelepipeda”, Sovremennye metody v teorii kraevykh zadach, Voronezhskaya vesennyaya matematicheskaya shkola “Pontryaginskie chteniya – Kh” (Voronezh, mai, 1999 g.), Voronezh, 1999, 194 pp.

[31] V. B. Penkov, “Teorema vzaimnosti dlya kvazistaticheskoi nyutonovskoi sredy”, II mezhdunarodnaya nauchno-tekhnicheskaya konferentsiya “Problemy plastichnosti v tekhnologii”, Tezisy dokladov, OGTU, Orel, 1998, 10–11

[32] V. V. Penkov, “Metod granichnykh sostoyanii: formirovanie bazisa prostranstva vnutrennikh sostoyanii sredy”, Izvestiya TulGU. Seriya Matematika. Mekhanika. Informatika, 4:2 (1998), 128–134

[33] V. B. Penkov, “Metod granichnykh sostoyanii dlya nyutonovskoi sredy”, II mezhdunarodnaya nauchno-tekhnicheskaya konferentsiya “Problemy plastichnosti v tekhnologii”, Tezisy dokladov, OGTU, Orel, 1998, 11–12

[34] V. B. Penkov, Zimnyaya shkola po mekhanike sploshnykh sred (dvenadtsataya), Tezisy dokladov (Perm, 25–31.01.99), Perm, 1999, 250 pp.

[35] V. B. Penkov, “Asimptotiki parallelepipeda”, Yubileinaya nauchno-prakticheskaya konferentsiya “Prikladnaya matematika–99”, Tezisy dokladov (Tula, 03–05.05.99), TulGU, Tula, 1999, 92–93

[36] Yu. N. Rabotnov, Mekhanika deformiruemogo tverdogo tela, Nauka, M., 1979, 744 pp.

[37] Yu. N. Rabotnov, Elementy nasledstvennoi mekhaniki tverdykh tel, Nauka, M., 1977, 384 pp. | MR

[38] Dzh. R. M. Radok, “Ploskie zadachi lineinoi teorii vyazkouprugosti”, Problemy mekhaniki sploshnoi sredy, K semidesyatiletiyu akademika N. I. Muskhelishvili, Izd-vo AN SSSR, M., 1961, 318–327

[39] M. G. Slobodyanskii, “Obschie formy reshenii uravnenii uprugosti dlya odnosvyaznykh i mnogosvyaznykh oblastei, vyrazhennykh cherez garmonicheskie funktsii”, PMM, 18 (1954), 55–74 | MR

[40] I. Stigan, “Funktsii Lezhandra”, Spravochnik po spetsialnym funktsiyam, per. s angl., Nauka, M., 1979, 153–156 | MR

[41] A. I. Tikhonov, A. A. Samarskii, Uravnenie matematicheskoi fiziki, Nauka, M., 1972, 763 pp.

[42] L. A. Tolokonnikov, V. B. Penkov, “Nekotorye effektivnye resheniya zadachi o skolzhenii metalla v sloe”, Prikladnaya mekhanika, 26(36):9 (1990), 75–82 | MR | Zbl

[43] A. G. Ugodchikov, N. M. Khutoryanskii, Metod granichnykh elementov v mekhanike deformiruemogo tverdogo tela, KGU, Kazan, 1986, 295 pp. | MR

[44] G. Fikera, Teoremy suschestvovaniya v teorii uprugosti, Mir, M., 1974, 160 pp.

[45] M. M. Filonenko-Borodich, Teoriya uprugosti, Fizmatgiz, M., 1959

[46] R. Khorn, Ch. Dzhonson, Matrichnye analiz, Mir, M., 1989, 655 pp. | MR

[47] N. M. Khutoryanskii, “Privedenie metoda potentsiala v zadachakh teorii uprugosti i vyazkouprugosti”, Prikladnye problemy prochnosti i plastichnosti, 10, GGU, Gorkii, 1974, 122–135

[48] N. M. Khutoryanskii, “Nestatsionarnyi dinamicheskii tenzor Grina dlya trekhmernoi transversalno izotropnoi uprugoi sredy”, Prikladnye problemy prochnosti i plastichnosti, 44, NGU, Nizhnii Novgorod, 1990, 30–34 | MR

[49] N. M. Khutoryanskii, “Tenzor Grina nestatsionarnoi dinamicheskoi teorii uprugosti dlya anizotropnoi odnorodnoi bezgranichnoi sredy”, Statika i dinamika deformiruemykh sistem, Prikladnye problemy prochnosti i plastichnosti, 30, Gork. Un-t, Gorkii, 1985, 23–31 | MR

[50] N. M. Khutoryanskii, Teoremy vzaimnosti v teorii vyazkouprugosti nestabilnykh materialov i ikh primenenie, Nauka, Alma-Ata, 1981, 350 pp.

[51] N. M. Khutoryanskii, L. A. Igumnov, “Postroenie nestatsionarnoi dinamicheskoi teorii vyazkouprugosti dlya nekotorykh differentsialnykh modelei stabilnoi izotropnoi odnorodnoi sredy”, Statika i dinamika deformiruemykh sistem, Prikladnye problemy prochnosti i plastichnosti, Gork. Un-t, Gorkii, 1982, 12–20

[52] N. M. Khutoryanskii, Kh. A. Sosa, V. Zu, “Metod granichnykh elementov dlya ploskikh zadach elektrouprugosti”, Prikladnye problemy prochnosti i plastichnosti, M., 1997, 183–195

[53] D. G. Ionescu, “La theorie des fonctions analytiques et l'hydrodinamique subsonque”, Prilozhenie teorii funktsii v mekhanike sploshnoi sredy, Trudy mezhdunarodnogo simpoziuma v Tbilisi, 17-23.09.1963, v. 2, Mekhanika zhidkosti i gaza, matematicheskie metody, Nauka, M., 1965, 235–251

[54] N. Khutoryansky , H. Sosa, “Dynamic representation formulas and fundamental solution for piezoelasticity”, Int. J. Solids Struct., 32:22 (1995), 3307–3325 | DOI | MR | Zbl

[55] H. Neuber, “Ein neuer Anzatz zur Losung raumlicher Probleme der Elastizitetstheorie”, Zeith. fur angew. Math. und Mech., 14:4 (1934), 203–212 | Zbl

[56] W. Nowacki, “On some problems of thermoelasticity”, Problems of continum mechanics, Philadelphia, 1961

[57] W. Nowacki, Theoria niesymetrycney sprezystosci, PWN, Warszawa, 1971

[58] W. Thomson, “Note on the Integration of the Equations of Equilibrium of an Elastic Solid”, Mathematical and Physical Papers, 1, Cambridge, 1882