Generalized Three-Dimensional Theory of Elastic Bodies Stability. Part 1: Finite Deformations

A problem of analysis of elastic structure stability is one of main problems in mechanics of deformable solids. Traditional methods of structure stability analysis are based on applying the theory of two-dimensional shell structures, as a rule, the classical Kirchhoff-Love theory. The development of methods for solving threedimensional problems of the stability theory could allow us to expand a range of stability problems which can be solved and to increase the accuracy of obtained solutions. V.V. Novozhilov was one of the first investigators who derived equations of the stability theory from the general nonlinear elasticity theory for the particular case of a continuum. The purpose of the present work is the derivation of generalized three-dimensional equations of the stability theory of nonlinearly elastic bodies with finite deformations for a wide class of nonlinear elasticity models. In order to achieve this aim, two approaches have been applied: the method of a varied configuration (used by A.I. Lurier), which is promising from the viewpoint of generality and universality, as well as the universal method (developed by the author of this paper earlier) for representation of nonlinearly elastic continua models on the basis of energetic pairs ofstress and strain tensors. It is shown that for two ofthe tensor pairs, the stability-theory relationships admit explicit analytical representation without calculation of eigenvalues of the stretch tensor. The results of the study expand the knowledge on fundamental relations of mechanics of deformable continua and represent a theoretical basis for analysis of stability of complex structures including those which are not thin-walled structures.

References

[1] Timoshenko S.P., Gere J.M. Theory of elastic stability. N.Y.-Toronto-London, McGraw-Hill, 1961. 356 p.

[2] Vol’mir A.S. Ustoychivost’ deformiruemykh sistem [Stability of deformable systems]. Мoscow, Nauka Publ., 1967. 964 p.

[3] Simitses G.J. An Introduction to the elastic stability of structures. New Jersey, Prentice Hall, 1976. 256 p.

[4] Bazant Z.P., Cedolin L. Stability of structures. Oxford, Oxford Univ. Press., 1990. 316 p.

[5] Iyengar N.G.R. Structural stability of columns and plates. New Delhi, Affiliated East-West Press. 1986. 284 p.

[6] Vasil’ev V.V. Mekhanika kompozitsionnykh materialov [Mechanics of composite materials]. Moscow, Mashinostroenie Publ., 1984. 272 p.

[7] Grigolyuk E.I., Chulkov P.P. Ustoychivost’ i kolebaniya trekhsloynykh obolochek. [Stability and oscillations of sandwich shells]. Moscow, Mashinostroenie Publ., 1973. 172 p.

[8] Novozhylov V.V. Osnovy nelineynoy teorii uprugosti [Fundamentals of the nonlinear theory of elasticity]. Moscow, URsS Publ., 2003. 208 p.

[9] Lur’e A.I. Nelineynaya teoriya uprugosti [Non-linear theory of elasticity]. Moscow, NaukaPubl., 1980. 512 p.

[10] Dimitrienko Yu.I. Nelineynaya mekhanika sploshnoy sredy [Nonlinear continuum mechanics]. Moscow, Fizmatlit Publ., 2009. 624 p.

[11] Guz’ A.N. Osnovy trekhmernoy teorii ustoychivosti deformiruemykh tel [Fundamentals of the three-dimensional theory of stability of deformable bodies]. Kiev, Vishcha Shkola Publ., 1986. 512 p.

[12] Kokhanenko Yu.V. The three-dimensional stability of a cylinder on the heterogeneous initial state. Dopov. Nats. Akad. Nauk Ukr. [Rep. Natl. Acad. Sci. Ukr.], 2009, no. 1, pp. 60-62 (in Russ.).

[13] Bazant Z.P. Stability of elastic, anelastic and disintegrating structures: a conspectus of main results. Appl. Math. Mech., ZAMM, 2000, vol. 80, no. 11-12, pp. 709-732.

[14] Dimitrienko Yu.I. Novel viscoelastic models for elastomers under finite strains. Eur. J. Mech. A: Solids, 2002, vol. 21, no. 1, pp. 133-150.

[15] Dimitrienko Yu.I., Dashtiev I.Z. Models of viscoelastic behavior of elastomers at finite deformations. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2001, no. 1, pp. 21-41 (in Russ.).

[16] Dimitrienko Yu.I. Mekhanika sploshnoy sredy. T. 2: Universal’nye zakony mekhaniki i elektrodinamiki sploshnoy sredy [Continuum mechanics. Vol. 2: Universal laws of mechanics and electrodynamics of continuous media]. Moscow, MGTU im. N.E. Baumana Publ., 2011. 464 p.