Mechanics of elastic micropolar shells

L. M. Zubov ; V. A. Eremeyev

Dalʹnevostočnyj matematičeskij žurnal, Tome 4 (2003) no. 2, pp. 182-225.

Résumé

The general static theory of micropolar shells under finite deformations is presented. The micropolar shell or Cosserat's shell is a material surface each point of which have six degrees of freedom of the rigid body. The various statements of boundary value problems of a nonlinear statics of elastic shells are given and their variational statements are formulated. The six variational principles are considered. The nonlinear equations of compatibility of strains of elastic Cosserat's shells are obtained and deformation boundary conditions are introduced. The torsion and bending of micropolar shell are considered by using semi-inverse method. The mathematical definition of the property of surface anisotropy is given. The universal deformations of micropolar shell are introduced. These universal deformations are solutions of static problem which satisfy the equilibrium equations for any constitutive equation of orthotrophic or isotropic shell. The theory of isolated and continuously distributed dislocations in elastic micropolar shell is developed. The stress-induced phase transitions of martensitic type are considered within the framework of continuum mechanics methods. The thermodynamical equilibrium relations are investigated. The phase equilibrium conditions are established by using Lagrange's variational principle. These relations consist of static balance equations of impulse and angular moment on a phase separation line and additional thermodynamical relation. The latter is necessary to determine an a priori unknown phase line. For elastic shell of Cosserat type, the expressions of energy-impulse tensors are given. From the linear thermodynamic of irreversible processes point of view the kinetic equation of propagating phase line are formulated. This equation describes also the motion of linear defects of other nature in shells. For equilibrium deformations, energy changes are determined with regard to phase line motion. The application of theory of the micropolar shells to the the mathematical modelling of the biological or lipidic membranes is discussed. From the mechanical properties of cellular membranes point of view the constitutive equations of liquid elastic micropolar shell are proposed. The obtained governing equations are equations of two-dimensional liquid which have a property of orientation elasticity and resist to bending. The presented model is compared with the smectic liquid crystals.

Bibliographie

[1] E. L. Aero, E. V. Kuvshinskii, “Osnovnye uravneniya teorii uprugosti sred s vraschatelnym vzaimodeistviem chastits”, FTT, 2:7 (1960), 1399–1409 | MR

[2] A. A. Vakulenko, “Svyaz mikro- i makrosvoistv v uprugoplasticheskikh sredakh”, Itogi nauki i tekhniki. Mekhanika deformiruemogo tverdogo tela, 22, VINITI, M., 1991, 3–54

[3] V. S. Boiko, R. I. Garber, A. M. Kosevich, Obratimaya plastichnost kristallov, Nauka, M., 1991, 280 pp.

[4] K. Z. Galimov, Osnovy nelineinoi teorii tonkikh obolochek, Kazan, 1975 | MR | Zbl

[5] R.Gennis, Biomembrany. Molekulyarnaya struktura i funktsii, Mir, M., 1997, 624 pp.

[6] A. L. Goldenveizer, Teoriya uprugikh tonkikh obolochek, Nauka, M., 1976, 512 pp. | MR

[7] M. A. Grinfeld, Metody mekhaniki sploshnykh sred v teorii fazovykh prevraschenii, Nauka, M., 1990, 312 pp. | MR

[8] I. Dyarmati, Neravnovesnaya termodinamika. Teoriya polya i variatsionnye printsipy, Mir, M., 1974

[9] V. A. Eremeev, “Fazovye prevrascheniya v obolochkakh Kossera”, Izv. vuzov. Severo-Kavkaz. region. Estestv. nauki, 2001, Spetsvypusk. Matematicheskoe modelirovanie, 64–67

[10] V. A. Eremeev, L. M. Zubov, “Usloviya fazovogo ravnovesiya v nelineino-uprugikh sredakh s mikrostrukturoi”, Dokl. RAN, 326:6 (1992), 968–971 | MR | Zbl

[11] V. A. Eremeev, L. M. Zubov, “Ob ustoichivosti uprugikh tel s momentnymi napryazheniyami”, Izv. RAN. MTT, 1994, no. 3, 181–190

[12] V. A. Eremeev, L. M. Zubov, “Teoriya uprugikh i vyazkouprugikh mikropolyarnykh zhidkostei”, PMM, 63:5 (1999), 801–815 | MR | Zbl

[13] P. de Zhen, Fizika zhidkikh kristallov, Mir, M., 1982, 304 pp.

[14] P. A. Zhilin, “Osnovnye uravneniya neklassicheskoi teorii uprugikh obolochek”, Tr. Leningr. politekhn. in-ta, 386, 1982, 29–46

[15] L. M. Zubov, “Statiko-geometricheskaya analogiya i variatsionnye printsipy v nelineinoi bezmomentnoi teorii obolochek”, Tr. 12 Vses. konf. po teorii obolochek i plastin, v. 2, Izd-vo Erev. un-ta, Erevan, 1980, 171–176

[16] L. M. Zubov, Metody nelineinoi teorii uprugosti v teorii obolochek, Izd-vo RGU, Rostov n/D, 1982

[17] L. M. Zubov, “Nelineinaya teoriya izolirovannykh dislokatsii i disklinatsii v uprugikh obolochkakh”, Izv. AN SSSR. MTT, 1989, no. 4, 139–145

[18] L. M. Zubov, “Variatsionnye printsipy i invariantnye integraly dlya nelineino-uprugikh tel s momentnymi napryazheniyami”, Izv. AN SSSR. MTT, 1990, no. 6, 10–16

[19] L. M. Zubov, “Nepreryvno raspredelennye dislokatsii i disklinatsii v uprugikh obolochkakh”, Izv. RAN. MTT, 1996, no. 6, 102–110 | MR

[20] L. M. Zubov, “Obschie resheniya nelineinoi statiki uprugikh obolochek”, Doklady RAN, 382:1 (2002), 58–61 | MR

[21] L. M. Zubov, “O teorii ravnovesiya nelineino uprugikh obolochek”, Izv. vuzov. Severo-Kavkaz. region. Estestv. nauki, 2001, Spetsvypusk. Matematicheskoe modelirovanie, 85–89

[22] L. M. Zubov, “O bolshikh deformatsiyakh izgiba i krucheniya uprugikh obolochek, imeyuschikh formu vintovoi poverkhnosti”, Probl. mekh. deform. tverd. tela, Mezhvuz. sb-k k 70-letiyu akad. N. F. Morozova, izd-vo SPbGU, SPb, 2002, 130–136

[23] L. M. Zubov, M. I. Karyakin, “Dislokatsii i disklinatsii v nelineino uprugikh telakh s momentnymi napryazheniyami”, PMTF, 1990, no. 3, 160–167 | MR

[24] L. M. Zubov, L. M. Filippova, “Teoriya obolochek s nepreryvno raspredelennymi dislokatsiyami”, Doklady RAN, 344:5 (1995), 619–622 | MR | Zbl

[25] I. Ivens, R. Skeilak, Mekhanika i termodinamika biologicheskikh membran, Mir, M., 1982, 304 pp.

[26] Kagava Yasuo, Biomembrany, Vysshaya shkola, M., 1985, 303 pp.

[27] E. Kartan, Rimanova geometriya v ortogonalnom repere, M., 1960

[28] E. I. Kats, V. V. Lebedev, Dinamika zhidkikh kristallov, Nauka, M., 1988, 144 pp.

[29] E. Krener, Obschaya kontinualnaya teoriya dislokatsii i sobstvennykh napryazhenii, Mir, M., 1965, 103 pp.

[30] A. I. Lure, Teoriya uprugosti, Nauka, M., 1970

[31] A. I. Lure, Nelineinaya teoriya uprugosti, Nauka, M., 1980 | MR

[32] V. V. Novozhilov, K. F. Chernykh, E. I. Mikhailovskii, Lineinaya teoriya obolochek, Politekhnika, L., 1991, 656 pp. | MR

[33] V. V. Novozhilov, V. A. Shamina, “O kinematicheskikh granichnykh usloviyakh v zadachakh nelineinoi teorii uprugosti”, Izv. AN SSSR. MTT, 1975, no. 5, 63–74 | MR

[34] V. G. Osmolovskii, Variatsionnaya zadacha o fazovykh perekhodakh v mekhanike sploshnoi sredy, SPb, 2000

[35] V. A. Palmov, “Osnovnye uravneniya teorii nesimmetrichnoi uprugosti”, PMM, 28:3 (1964), 401–408 | MR

[36] K. Trusdell, Pervonachalnyi kurs ratsionalnoi mekhaniki sploshnoi sredy, Mir, M., 1975

[37] K. F. Chernykh, V. A. Shamina, “Nekotorye voprosy nelineinoi klassicheskoi teorii tonkikh sterzhnei i obolochek”, Tr. IX Vsesoyuznoi konf. po teorii obolochek i plastin, L., 1975, 99–103

[38] L. I. Shkutin, Mekhanika deformatsii gibkikh tel, Nauka, Novosibirsk, 1988, 127 pp.

[39] S. S. Antman, Nonlinear Problems of Elasticity, Springer-Verlag, Berlin, Heidelberg, New-York et al, 1995, 751 pp. | MR

[40] Cosserat {E. et F.}, Théorie des corps deformables, Paris, 1909 | Zbl

[41] A. C. Eringen, Microcontinuum Field Theories, v. I, Foundations and Solids, Springer-Verlag, Berlin, Heidelberg, New-York et al, 1999, 325 pp. | MR | Zbl

[42] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Clarendon-Press, Oxford, 1993, 149 pp. | MR | Zbl

[43] M. E. Gurtin, Configurational Forces as Basic Concepts of Continuum Physics, Springer-Verlag, Berlin, Heidelberg, New-York et al, 2000, 249 pp. | MR

[44] W. T. Koiter, “Couple–stresses in the theory of elasticity”, Pt I–II, Proc. Koninkl. Neterland. Akad. Wetensh (V), 67, no. 1, 1964, 17–44 | MR

[45] F. M. Leslie, J. C. Laverty, “Continuum theory for biaxial nematic liquid crystals”, Nonlinear Elasticity and Theoretical Mechanics, In honour of A. E. Green, eds. P. M. Naghdi, A. J. M. Spencer, A. H. England, Oxford University Press, Oxford, New York, Tokyo, 1994, 79–89 pp. | Zbl

[46] W. Nowacki, Theory of Asymmetric Elasticity, Pergamon-Press, Oxford,New-York, Toronto et al, 1986, 383 pp. | MR

[47] W. Pietraszkiewicz, Finite Rotations and Langrangian Description in the Non-Linear Theory of Shells, Warszawa, 1979

[48] W. Pietraszkiewicz, “Geometrically nonlinear theories of thin elastic shells”, Advances in Mechanics, 12:1 (1989), 51–130 | MR

[49] H. Pleiner, H. R. Brand, “Nonlinear hydrodynamics of strongly deformed smectic C and C$^{*}$ liquid crystals”, Physica A., 265 (1999), 62–77 | DOI

[50] R. A. Toupin, “Theories of elasticity with couple–stress”, Arch. Ration. Mech. Anal., 17:2 (1964), 85–112 | DOI | MR | Zbl

[51] L. M. Zubov, Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies, Springer-Verlag, Berlin, Heidelberg, New-York et al, 1997, 205 pp. | MR

[52] L. M. Zubov, “Nonlinear Theory of Isolated and Comtinuosly Distributed Dislocations in Elastic Shells”, Archives of Civil Engineering, XLV:2 (1999), 385–396

[53] L. M. Zubov, “Semi-inverse solutions in nonlinear theory of elastic shells”, Arch. Mech., 53:4–5 (2001), 599–610 | MR | Zbl