A generalized theory of tensor strain and stress measures in the classical continuum mechanics

G. L. Brovko

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2018), pp. 46-57

Résumé

A generalized theory of tensor measures of strains and stresses in classical continuum mechanics is discussed: the main axioms of the theory are proposed, the general formulas of new tensor measures are derived, the theorem of energy conjugation is established to separate the complete Lagrangean class of the measures. As a subclass, the simple Lagrangean class of energy conjugated measures of stresses and finite strains is constructed in which the families of holonomic and corotational measures are distinguished. By comparison of measures of the simple Lagrangean class with one another and by matching them with logarithmic measures, the characteristics of holonomic and corotational measures are studied. For the simple Lagrangean class and its families, their completeness and closure are established relative to any choice of a generating pair of energy conjugated measures. The applications of the new tensor measures in modeling the properties of plasticity, viscoelasticity, and shape memory are mentioned.

Bibliographie

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

[2] Sedov L.I., Mekhanika sploshnoi sredy, v. 1, 2, Nauka, M., 1973

[3] Truesdell C., Noll W., Handbuch der Physik, v. III/3, The non-linear field theories of Mechanics, Springer Verlag, Berlin, 1965 ; 3d ed., Springer-Verlag, Berlin–Heidelberg–New York, 2004 | MR | Zbl

[4] Trusdell K., Pervonachalnyi kurs ratsionalnoi mekhaniki sploshnykh sred, Mir, M., 1975

[5] Zhermen P., Mekhanika sploshnykh sred, Vysshaya shkola, M., 1983

[6] Jaric J., Mehanika Kontinuuma, IRO Gradevinska knjiga, Beograd, 1988

[7] Pobedrya B.E., Georgievskii D.V., Osnovy mekhaniki sploshnoi sredy. Kurs lektsii, Fizmatlit, M., 2006

[8] Gurtin M.E., Fried E., Anand L., The mechanics and thermodynamics of continua, Cambridge University Press, Cambridge–N. Y.–Melbourne–Madrid–Cape Town–Singapore–São Paolo–Delhi–Dubai–Tokyo, 2010 | MR

[9] Brovko G.L., Osnovy mekhaniki sploshnoi sredy (kratkii konspekt lektsii, zadachi, uprazhneniya), v. 1, Izd-vo Popechitelskogo soveta mekh.-mat. f-ta MGU, M., 2011; т. 2, 2013

[10] Eglit M.E., Lektsii po osnovam mekhaniki sploshnykh sred, Knizhnyi dom “Librokom”, M., 2013

[11] Astarita Dzh., Marruchchi Dzh., Osnovy gidromekhaniki nenyutonovskikh zhidkostei, Mir, M., 1978

[12] Tolokonnikov L.A., Mekhanika deformiruemogo tverdogo tela, Vysshaya shkola, M., 1979

[13] Lure A.I., Nelineinaya teoriya uprugosti, Nauka, M., 1980

[14] Pozdeev A.A., Trusov P.V., Nyashin Yu.I., Bolshie uprugoplasticheskie deformatsii, Nauka, M., 1986 | MR

[15] Levitas V.I., Bolshie uprugoplasticheskie deformatsii materialov pri vysokom davlenii, Naukova dumka, Kiev, 1987

[16] Kondaurov V.I., Nikitin L.V., Teoreticheskie osnovy reologii geomaterialov, Nauka, M., 1990

[17] Chernykh K.F., Nelineinaya uprugost (teoriya i prilozheniya), Solo, SPb., 2004

[18] Korneev S.A., Ponyatiya i osnovy lokalno-neravnovesnoi termodinamiki sploshnoi sredy, Izd-vo OmGTU, Omsk, 2009

[19] Markin A.A., Sokolova M.Yu., Termomekhanika uprugoplasticheskogo deformirovaniya, Fizmatlit, M., 2013

[20] Seth B.R., “Finite strain in elastic problems”, Phil. Trans. Roy. Soc. London. Ser. A, 234 (1935), 231–264 | DOI | Zbl

[21] Seth B.R., “Generalized strain measures with applications to physical problems”, Second Order Effects in Elasticity, Plasticity and Fluid Dynamics, eds. M. Reiner, D. Abir, Pergamon Press, Oxford, 1964, 162–172 | MR

[22] Setkh B.R., “Ponyatie mery deformatsii v tekhnike vysokoskorostnogo deformirovaniya”, Uspekhi mekhaniki deformiruemykh sred, Nauka, M., 1975, 528–531

[23] Hill R., “Aspects of invariance in solid mechanics”, Adv. Appl. Mech., 18 (1978), 1–75 | MR | Zbl

[24] Man C.S., Guo Z.H., “A basis-free formula for time rate of Hill's strain tensors”, Int. J. Solid and Struct., 30 (1993), 2819–2842 | DOI | MR | Zbl

[25] Hencky H., “The elastic behavior of vulcanized rubber”, J. Appl. Mech., 1 (1933), 45–53

[26] Scrzypek J., Wroblewski A., “Application of logarithmic strains to changing principal directions via progressing transformations”, J. Struct. Mech., 13:3–4 (1985), 283–299 | DOI

[27] Zhao Z., “Logarithmic strain and plastic constitutive equation at finite strain”, Proc. Int. Conf. Nonlin. Mech. (Shanghai, Oct. 28–31, 1985), Beijing, 1985, 651–656

[28] Trusov P.V., Obobschenie teorii uprugoplasticheskikh protsessov na sluchai bolshikh plasticheskikh deformatsii, Dokt. dis., M., 1986

[29] Xiao H., Bruhns O.T., Meyers A., “Large strain responses of elastic-perfect plasticity and kinematic hardening plasticity with the logarithmic rate: Swift effect in torsion”, Int. J. Plasticity, 17 (2001), 211–235 | DOI | Zbl

[30] Brovko G.L., “Nekotorye podkhody k postroeniyu opredelyayuschikh sootnoshenii plastichnosti pri bolshikh deformatsiyakh”, Uprugost i neuprugost, Izd-vo MGU, M., 1987, 68–81

[31] Markin A.A., Tolokonnikov L.A., “Mery i opredelyayuschie sootnosheniya konechnogo uprugoplasticheskogo deformirovaniya”, Prikladnye problemy prochnosti i plastichnosti, Gorkii, 1987, 32–37

[32] Markin A.A., Variant opredelyayuschikh sootnoshenii i postanovka granichnykh zadach pri konechnykh uprugoplasticheskikh deformatsiyakh, Dokt. dis., M., 1988

[33] Levitas V.I., Bolshie uprugoplasticheskie deformatsii materialov pri vysokom davlenii, Dokt. dis., M., 1988

[34] Novozhilov V.V., Chernykh K.F., “Ob “istinnykh” merakh napryazhenii i deformatsii v nelineinoi mekhanike deformiruemogo tela”, Izv. AN SSSR. Mekhan. tverdogo tela, 1987, no. 5, 73–80

[35] Brovko G.L., “Ponyatiya obraza protsessa i pyatimernoi izotropii svoistv materialov pri konechnykh deformatsiyakh”, Dokl. AN SSSR, 308:3 (1989), 565–570 | MR | Zbl

[36] Brovko G.L., “Materialnye i prostranstvennye predstavleniya opredelyayuschikh sootnoshenii deformiruemykh sred”, Prikl. matem. i mekhan., 54:5 (1990), 814–824 | MR | Zbl

[37] Brovko G.L., “Svoistva i integrirovanie nekotorykh proizvodnykh po vremeni ot tenzornykh protsessov v mekhanike sploshnoi sredy”, Izv. AN SSSR. Mekhan. tverdogo tela, 1990, no. 1, 54–60 | MR

[38] Brovko G.L., “Ob odnom semeistve golonomnykh tenzornykh mer deformatsii i napryazhenii”, Vestn. Mosk. un-ta. Matem. Mekhan., 1992, no. 4, 86–91 | MR | Zbl

[39] Brovko G.L., Razvitie matematicheskogo apparata i osnov obschei teorii opredelyayuschikh sootnoshenii mekhaniki sploshnoi sredy, Dokt. dis., M., 1996

[40] Brovko G.L., “Invariance types of tensors, tensor processes and their transforms in classical continuum mechanics”, Proc. Int. Seminar on Geometry, Continua Microstructure (Sinaia, Romania, September 26–28, 2001), Bucharest, 2002, 13–24 | Zbl

[41] Brovko G.L., “Osnovy obobschennoi teorii tenzornykh mer deformatsii i napryazhenii”, Problemy nelineinoi mekhaniki, K vosmidesyatiletiyu L. A. Tolokonnikova, TulGU, Tula, 2003, 123–132

[42] Brovko G.L., “Podkhody k postroeniyu ratsionalnoi mekhaniki klassicheskikh i neklassicheskikh sred”, X Vseros. s'ezd po fundamentalnym problemam teoreticheskoi i prikladnoi mekhaniki, Izbrannye tezisy dokladov (Nizhnii Novgorod, 24–30 avgusta 2011 g.), Izd-vo Nizhegorod. gos. un-ta im. N.I. Lobachevskogo, Nizhnii Novgorod, 2011, 27–28

[43] Brovko G.L., “Matematicheskie osnovy mekhaniki sploshnykh sred: klassicheskie i neklassicheskie teorii”, Uprugost i neuprugost, Izd-vo MGU, M., 2012, 20–43

[44] Brovko G.L., “On general principles of the theory of constitutive relations in classical continuum mechanics”, J. Eng. Math., 78 (2013), 37–53 | DOI | MR | Zbl

[45] Brovko G.L., “Razvitie obschikh printsipov teorii opredelyayuschikh sootnoshenii sploshnykh sred”, Izv. TulGU. Estestvennye nauki, 2013, no. 2–2, 43–58

[46] Brovko G.L., “Elementy nelineinoi mekhaniki sploshnoi sredy v sovremennoi teorii”, Izv. MGTU “MAMI”, 4:2(24) (2015), 34–43

[47] Brovko G.L., Elementy matematicheskogo apparata mekhaniki sploshnoi sredy, Fizmatlit, M., 2015

[48] Finoshkina A.S., “Ispolzovanie novykh ob'ektivnykh proizvodnykh v prosteishikh modelyakh gipouprugosti i plasticheskogo techeniya s kinematicheskim uprochneniem”, Izv. TulGU. Matematika. Mekhanika. Informatika, 6:2 (2000), 160–166

[49] Finoshkina A.S., “Usage of the new objective derivatives in models of plasticity at finite strains: the theory and numerical experiments”, V Int. Congr. Math. Modeling, Book of Abstracts, v. 1, JINR, Dubna, 2002, 35

[50] Finoshkina A.S., Modeli plastichnosti pri konechnykh deformatsiyakh, Kand. dis., M., 2003

[51] Brovko G.L., Ivanova O.A., Finoshkina A.S., “On geometrical and analytical aspects in formulations of problems of classic and non-classic continuum mechanics”, Operator Theory: Advances and Applications, 191, Birkhäuser Verlag, Basel/Switzerland, 2009, 51–79 | MR | Zbl

[52] Shutkin A.S., “Podkhody k obobscheniyu opredelyayuschikh sootnoshenii deformiruemykh tverdykh tel na oblast konechnykh deformatsii”, Mekhan. kompozitsionnykh materialov i konstruktsii, 16:2 (2010), 166–180

[53] Shutkin A.S., Modeli materialov s pamyatyu formy pri konechnykh deformatsiyakh, Kand. dis., M., 2011

[54] Muravlev A.V., “Obobschenie teorii uprugoplasticheskikh protsessov A.A. Ilyushina na sluchai konechnykh deformatsii”, Vestn. Nizhegorod. un-ta im. N.I. Lobachevskogo. Mekhan. deformiruemogo tverdogo tela, 2011, no. 4(4), 1642–1644

[55] Muravlev A.V., Devyatov A.S., “Razvitie teorii uprugoplasticheskikh protsessov A.A. Ilyushina i eksperimentalno-teoreticheskikh metodov issledovaniya vyazkoplasticheskikh svoistv materialov pri konechnykh deformatsiyakh”, Probl. mashinostr. i avtomatiz., 2016, no. 1, 84–90 | MR

[56] Martynova E.D., Stetsenko N.S., “Ispolzovanie odnoparametricheskogo semeistva ob'ektivnykh proizvodnykh Gordona–Shouoltera dlya opisaniya konechnykh deformatsii vyazkouprugikh tel”, Vestn. Mosk. un-ta. Matem. Mekhan., 2017, no. 6, 64–68 | Zbl

[57] Ovchinnikova N.V., “O tenzornykh merakh napryazhenii i deformatsii, ispolzuemykh v ANSYS dlya resheniya uprugoplasticheskikh zadach pri konechnykh deformatsiyakh”, Vestn. Mosk. un-ta. Matem. Mekhan., 2017, no. 5, 31–36

[58] Tunguskova Z.G., “Analiticheskoe predstavlenie tenzora napryazhenii v zadache o sdvige gipouprugogo tela s ispolzovaniem korotatsionnykh proizvodnykh opredelennogo vida”, Uprugost i neuprugost, Izd-vo MGU, M., 2016, 260–262

[59] Lankaster P., Teoriya matrits, Nauka, M., 1982 | MR