The general mathematical theory of plasticity and the Ilyushin postulates of macroscopic definahility and isotropy

V. G. Zubchaninov

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2018), pp. 29-46

Résumé

The physical laws of the connection between stresses and strains of the general modern theory of the processes of elastoplastic deformation and its postulates of macroscopic definiteness and isotropy of initially isotropic continuous media are considered and analyzed. The foundations of this theory in continuum mechanics were developed in the middle of 20th century by the outstanding Russian mechanicist, corresponding member of the Russian Academy of Sciences, academician of Russian academy of rocket and artillery sciences, head of the department of the theory of elasticity of Moscow university of M. V. Lomonosov, by A. A. Ilyushin. His theory of small elastoplastic deformations under simple loading became a generalization of Genky's deformation theory of flow, and his theory of elastoplastic processes which are close to simple loading became a generalization of flow theory of Saint-Venant–Mises for reinforcing medium. In these works, the concepts of simple and complex loading processes, the concepts of directing deviation tensors of form change were introduced. The Bridgman's law of elastic variation of the volume and the universal laws of the single curve of hardening of Roche and Eichinger under simple loading, and the universal law of hardening of Odquist for plastic deformations generalized to strengthening elastic-plastic media for processes close to simple loading, but without taking into account the specific history of deformation for trajectories of small and medium curvature were adopted. The question of the possibility of applying the postulate of isotropy to the evaluation of the influence of parameters of the form of the stress-strain state arising due to deformation anisotropy when the internal structure of materials changes is discussed. Also, the question of the legitimacy of representing symmetric tensors of the second rank of stresses and deformations in the form of vectors of the coordinate linear Euclidean six-dimensional space is discussed. The corresponding principle of the identity of tensors and vectors is proposed.

Bibliographie

[1] Ilyushin A.A., Ob osnovakh obschei matematicheskoi teorii plastichnosti, Izd-vo AN SSSR, M., 1963

[2] Ilyushin A.A., Mekhanika sploshnoi sredy, Izd-vo MGU, M., 1990

[3] Ilyushin A.A., “O svyazi mezhdu napryazheniyami i deformatsiyami v mekhanike sploshnoi sredy”, Prikl. matem. i mekhan., 18:6 (1954), 641–666

[4] Ilyushin A.A., “Nekotorye voprosy teorii plasticheskikh deformatsii”, Prikl. matem. i mekhan., 7:4 (1943), 245–272

[5] Ilyushin A.A., “Teoriya plastichnosti pri prostom nagruzhenii tel, material kotorykh obladaet uprochneniem”, Prikl. matem. i mekhan., 11:2 (1947), 293–296 | MR | Zbl

[6] Ilyushin A.A., Plastichnost, v. 1, Uprugoplasticheskie deformatsii, Gostekhizdat, M.–L., 1948; Логос, М., 2004

[7] Ilyushin A.A., “Voprosy obschei teorii plastichnosti”, Prikl. matem. i mekhan., 24:3 (1960), 399–411 | Zbl

[8] Ilyushin A.A., “Ob osnovakh obschei matematicheskoi teorii plastichnosti”, Voprosy teorii plastichnosti, Izd-vo AN SSSR, M., 1961, 399–411

[9] Ilyushin A.A., Trudy, v. 1, (1935–1945), Fizmatlit, M., 2003

[10] Ilyushin A.A., Trudy, v. 2, Plastichnost (1946–1966), Fizmatlit, M., 2004

[11] Ilyushin A.A., Lenskii V.S., Soprotivlenie materialov, Fizmatlit, M., 1954

[12] Yu.N. Rabotnov (red.), Teoriya plastichnosti, Sb. statei, IIL, M., 1948

[13] Nadai A., Plastichnost i razrushenie tverdykh tel, IIL, M., 1956

[14] Khill R., Matematicheskaya teoriya plastichnosti, Gostekhizdat, M., 1956

[15] Ilin V.A., Poznyak E.G., Lineinaya algebra, Fizmatlit, M., 2002

[16] Gelfand I.M., Lektsii po lineinoi algebre, GITTL, M.–L., 1948 | MR

[17] Sokolnikov I. S., Tenzornyi analiz, Nauka, M., 1971 | MR

[18] Ivlev D.D., “O postulate izotropii v teorii plastichnosti”, Izv. FN SSSR. OTN. Mekhan. i mashinostr., 1960, no. 2, 125–127

[19] Novozhilov V.V., “Ob odnom napravlenii v teorii plastichnosti”, Izv. AN SSSR. OTN. Mekhan. i mashinostr., 1961, no. 3, 176–181

[20] Ilyushin A.A., “Esche o postulate izotropii”, Izv. AN SSSR. OTN. Mekhan. i mashinostr., 1962, no. 1, 201–204

[21] Novozhilov V.V., “I esche o postulate izotropii”, Izv. AN SSSR. OTN. Mekhan. i mashinostr., 1962, no. 1, 205–208

[22] Zubchaninov V.G., Matematicheskaya teoriya plastichnosti, Izd-vo Tver. gos. tekhn. un-ta, Tver, 2002

[23] Zubchaninov V.G., Mekhanika protsessov plasticheskikh sred, Fizmatlit, M., 2010

[24] Zubchaninov V.G., Mekhanika sploshnykh deformiruemykh sred, Izd-vo Tver. gos. tekhn. un-ta, Tver, 2000

[25] Zubchaninov V.G., Ustoichivost i plastichnost, v. 1, Ustoichivost, Fizmatlit, M., 2007

[26] Zubchaninov V.G., Ustoichivost i plastichnost, v. 2, Plastichnost, Fizmatlit, M., 2008

[27] Zubchaninov V.G., “Gipoteza ortogonalnosti i printsip gradientalnosti v teorii plastichnosti”, Izv. RAN. Mekhan. tverdogo tela, 2008, no. 5, 68–73

[28] Zubchaninov V.G., “Postulat izotropii i zakon slozhnoi razgruzki sploshnykh sred”, Izv. RAN. Mekhan. tverdogo tela, 2011, no. 1, 27–37

[29] Zubchaninov V.G., “Teoriya protsessov i postulat izotropii A.A. Ilyushina”, Uprugost i neuprugost, Izd-vo MGU, M., 2011, 73–79

[30] Zubchaninov V.G., “Teoriya uprugoplasticheskogo deformirovaniya materialov A.A. Ilyushina, ee kriterii i obschii postulat izotropii”, Uprugost i neuprugost, Izd-vo MGU, M., 2016, 83–93

[31] Zubchaninov V.G., “Tenzornaya forma postulata izotropii v teorii plastichnosti”, Matematicheskoe modelirovanie i eksperimentalnaya mekhanika deformiruemogo tverdogo tela, Izd-vo Tver. gos. tekhn. un-ta, Tver, 2017, 64–75

[32] Lenskii V.S., “Eksperimentalnaya proverka zakonov izotropii i zapazdyvaniya pri slozhnom nagruzhenii”, Izv. AN SSSR. OTN, 1958, no. 11, 15–24

[33] Lenskii V.S., “Eksperimentalnaya proverka osnovnykh postulatov obschei teorii uprugoplasticheskikh deformatsii”, Voprosy plastichnosti, Izd-vo AN SSSR, M., 1961, 58–82

[34] Korovin I.M., “Eksperimentalnoe opredelenie zavisimosti napryazhenii i deformatsii s odnoi tochkoi izloma”, Inzh. zhurn. Mekhan tverdogo tela, 4:3 (1964), 592–600

[35] Dao-Zui-Bik, “O gipoteze lokalnoi opredelennosti v teorii plastichnosti”, Vestn. Mosk. un-ta. Matem. Mekhan., 1965, no. 2, 67–75

[36] Vasin R.A., “Nekotorye voprosy svyazi napryazhenii i deformatsii pri slozhnom nagruzhenii”, Uprugost i neuprugost, 1, Izd-vo MGU, M., 1971, 59–126

[37] Kravchuk A.S., O teorii plastichnosti dlya traektorii deformirovaniya srednei krivizny, 2, Izd-vo MGU, M., 1971, 91–100

[38] Malyi V.I., “Ob uproschenii funktsionalov teorii uprugoplasticheskikh protsessov”, Prikl. mekhan., 14:2 (1978), 48–53 | MR

[39] Muravlëv A.V., “Nekotorye obschie svoistva svyazi napryazhenii i deformatsii v teorii plastichnosti”, Izv. AN SSSR. Mekhan. tverdogo tela, 1984, no. 6, 178–179

[40] Lenskii V.S., Lenskii E.V., “Trekhchlennoe sootnoshenie teorii plastichnosti”, Izv. AN SSSR. Mekhan. tverdogo tela, 1985, no. 4, 111–115

[41] Okhashi I., Tokuda M., Kurita I., Suzuki T., “Nekotorye dannye ob obschem zakone plastichnosti Ilyushina”, Izv. AN SSSR. Mekhan. tverdogo tela, 1981, no. 6, 53–64

[42] Ohashi Y., Tokuda M., Yamashita N., “Effect of third invariant of stress deviator on plastic deformation of mild steel”, J. Mech. and Phys. Solids, 23:4–5 (1975), 295–323 | DOI

[43] Zubchaninov V.G., Okhlopkov N.L., Garanikov V.V., Eksperimentalnaya plastichnost, v. 1, Protsessy slozhnogo deformirovaniya, Izd-vo Tver. gos. tekhn. un-ta, Tver, 2003

[44] Zubchaninov V.G., Okhlopkov N.L., Garanikov V.V., Eksperimentalnaya plastichnost, v. 2, Protsessy slozhnogo nagruzheniya, Izd-vo Tver. gos. tekhn. un-ta, Tver, 2004

[45] Zubchaninov V.G., Alekseev A.A., Gultyaev V.I., “Chislennoe modelirovanie protsessov slozhnogo deformirovaniya po dvuzvennym traektoriyam”, Problemy prochnosti i plastichnosti, 76:1 (2014), 18–25

[46] Zubchaninov V.G., Alekseev A.A., Gultyaev V.I., “Modelirovanie slozhnogo deformirovaniya po ploskim krivolineinym traektoriyam”, Problemy prochnosti i plastichnosti, 77:2 (2015), 113–123

[47] Zubchaninov V.G., “Teoriya uprugoplasticheskogo deformirovaniya materialov i obschii postulat izotropii A. A. Ilyushina”, Problemy prochnosti, plastichnosti i ustoichivosti, Mat-ly VIII Mezhdunar. nauch. simp., Izd-vo Tver. gos. tekhn. un-ta, Tver, 2015, 9–24

[48] Alekseev A.A., Zubchaninov V.G., “Protsessy uprugoplasticheskogo deformirovaniya materialov po ploskim traektoriyam”, Uprugost i neuprugost, Izd-vo MGU, M., 2016, 132–135

[49] Zubchaninov V.G., “Teoriya protsessov, polnaya i nepolnaya plastichnost sploshnykh sred i postulat izotropii A.A. Ilyushina”, Problemy prochnosti, plastichnosti i ustoichivosti v mekhanike deformiruemogo tverdogo tela, Mat-ly VII Mezhdunar. nauch. simp., Izd-vo Tver. gos. tekhn. un-ta, Tver, 2011, 30–49

[50] Georgievskii D.V., Ustoichivost protsessov deformirovaniya vyazkoplasticheskikh tel, Izd-vo URSS, M., 1998

[51] Treschëv A.A., Teoriya deformirovaniya i prochnosti materialov, chuvstvitelnykh k vidu napryazhennogo sostoyaniya. Opredelyayuschie sootnosheniya, Izd-vo Tul. gos. un-ta, Tula, 2008

[52] Brovko G.L., Bykov D.L., Georgievskii D.V., Kiiko I.A., Molodtsov I.M., Pobedrya B.E., “Nauchnoe nasledie A.A. Ilyushina i razvitie ego idei v mekhanike”, Izv. RAN. Mekhan. tverdogo tela, 2011, no. 1, 5–18

[53] Vasin R.A., “Teoriya uprugoplasticheskikh protsessov i issledovanie strukturno-mekhanicheskikh svoistv materialov”, Izv. RAN. Mekhan. tverdogo tela, 2001, no. 1, 19–26

[54] Markin A.A., Sokolova M.Yu., Khristich D.V., “Postulat A.A. Ilyushina dlya anizotropnykh materialov i variant opredelyayuschikh sootnoshenii”, Izv. RAN. Mekhan. tverdogo tela, 2001, no. 1, 38–45