 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| I. A. Brigadnov, "Discontinuous mappings and their approximation in the nonlinear theory of elasticity," Mech. Solids. 36 (2), 36-45 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
2 |
Pages |
36-45 |
| Title |
Discontinuous mappings and their approximation in the nonlinear theory of elasticity |
| Author(s) |
I. A. Brigadnov (St. Petersburg) |
| Abstract |
Two statements of boundary value problems for the mappings of the nonlinear
theory of elasticity are considered in the form of a variational problem of
global minimization and in the form of variational (stationary and
evolutionary) equations. The interrelations of these statements,
as well as their mathematical well-posedness and physical correctness, are
discussed. The statement of the elastostatic boundary value problem in
the form of an evolutionary variational equation allows one to study a
bifurcation process for an equilibrium configuration of a deformed elastomer.
The general existence theorem is formulated for the equilibrium
configuration allowing its branching.
For highly elastic materials with an ideal saturation, the variational
problems under consideration need an extension of the set of admissible
solutions to take into account possible discontinuities. For example,
hyperelastic materials are described by potentials growing linearly
with respect to the norm of the distortion tensor, and the existence of the
limiting load is characteristic of them. An example illustrating the
existence of mappings with slip-type discontinuities is given
for the classical Bartenev-Khazanovich potential.
To take into account discontinuous mappings one can use a partial
relaxation of evolutionary variational elastostatic problem based on a
special three-dimensional finite element approximation admitting fields
with discontinuities of the slip type. As a result, the original problem
for a continuous medium is reduced to an initial value problem for a stiff
nonlinear system
of ordinary differential equations. This system is solved numerically by
using an implicit Euler scheme.
The results of computational experiments are given. They show qualitative
advantage of the suggested approach over the standard methods based
on a continuous 3D finite element approximation. |
| References |
| 1. | A. I. Lurie, Nonlinear Theory of Elasticity [in Russian], Nauka,
Moscow, 1980. |
| 2. | K. F. Chernykh and Z. N. Litvinenkova, Theory of Large Elastic
Deformations [in Russian], Izd-vo LGU, Leningrad, 1988. |
| 3. | P. G. Ciarlet, Mathematical Elasticity [Russian translation],
Mir, Moscow, 1992. |
| 4. | L. M. Zubov, "Variational principles of nonlinear theory of
elasticity," PMM [Applied Mathematics and Mechanics], Vol. 35, No. 3,
pp. 406-410, 1971. |
| 5. | J. M. Ball, "Convexity conditions and existence theorems in
nonlinear elasticity," Arch. Ration. Mech. Anal., Vol. 63, No. 4,
pp. 337-403, 1977. |
| 6. | I. A. Brigadnov, "Numerical methods in non-linear elasticity,"
in Numerical Methods in Engineering'96, pp. 158-163, Wiley, Chichester,
1996. |
| 7. | A. Kufner and S. Fučik, Non-linear Differential Equations
[Russian translation], Mir, Moscow, 1988. |
| 8. | I. A. Brigadnov, "On the existence of the limiting load in some
problems of hyperelasticity," Izv. AN. MTT [Mechanics of Solids],
No. 5, pp. 46-51, 1993. |
| 9. | I. A. Brigadnov, "Mathematical well-posedness of boundary
value problems of elastostatics for hyperelastic material," Izv. AN.
MTT [Mechanics of Solids], No. 6, pp. 37-46, 1996. |
| 10. | A. A. Tolstonogov, Differential Inclusions in Banach Space [in
Russian], Nauka, Novosibirsk, 1986. |
| 11. | I. Ekeland and R. Temam, Convex Analysis and Variational
Problems [Russian translation], Mir, Moscow, 1979. |
| 12. | A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of
Functions and Functional Analysis [in Russian], Nauka, Moscow, 1968. |
| 13. | E. Justi, Minimal Surfaces and Functions of Bounded Variation
[Russian translation], Mir, Moscow, 1989. |
| 14. | M. Giaquinta, "On the Dirichlet problem for surfaces of
prescribed mean curvature," Manuscripta Math., Vol. 12, No. 1,
pp. 73-86, 1974. |
| 15. | R. Temam, Mathematical Problems of the Theory of Plasticity
[Russian translation], Nauka, Moscow, 1991. |
| 16. | G. A. Seregin, "On the well-posedness of variational problems
of mechanics of ideal-plastic media," Doklady AN SSSR, Vol. 276, No. 1,
pp. 71-75, 1984. |
| 17. | S. I. Repin, "A variational-difference method of solving
problems with functionals of linear growth," Zhurnal Vychislitel'noi
Matematiki i Matematicheskoi Fiziki, Vol. 29, No. 5, pp. 693-708,
1989. |
| 18. | S. I. Repin, "On the solutions of mathematical plasticity
problems containing discontinuities of displacement fields," PMM
[Applied Mathematics and Mechanics], Vol. 58, No. 1, pp. 142-152,
1994. |
| 19. | I. A. Brigadnov, "Discontinuous solutions and their finite
element approximation in non-linear elasticity," in Advanced
Computational Methods in Engineering'98, pp. 141-148, Shaker
Publishing B.V., Maastricht, 1998. |
| 20. | I. A. Brigadnov, "The limited static load in finite
elasticity," in Constitutive Models for Rubber, pp. 37-43,
A. A. Balkema, Rotterdam, 1999. |
| 21. | G. M. Bartenev and T. N. Khazanovich, "On the law of
hyperelastic strains for cross-linked polymers,"
Vysokomolekulyarnye Soedineniya, Vol. 2, No. 1, pp. 20-28, 1960. |
| 22. | P. G. Ciarlet, Method of Finite Elements for Elliptic Problems
[Russian translation], Mir, Moscow, 1980. |
| 23. | I. A. Brigadnov, "On the numerical solution of boundary value
problems of elastoplastic flow," Izv. AN. MTT [Mechanics of Solids],
No. 3, pp. 157-162, 1992. |
| 24. | Yu. V. Rakitskii, S. M. Ustinov, and I. G. Chernorutskii,
Numerical Methods for Solving Stiff Systems [in Russian], Nauka,
Moscow, 1979. |
| 25. | E. Harrier, S. P. Wagner, and G. Wanner, Solving Ordinary
Differential Equations. Pt. 2. Stiff Differential-Algebraic Systems,
Springer, Berlin, 1991. |
| 26. | I. A. Brigadnov, "Numerical solution of a boundary value
problem of hyperelasticity in increments," Izv. AN. MTT [Mechanics of
Solids], No. 6, pp. 42-50, 1994. |
| 27. | I. A. Brigadnov, "Mathematical well-posedness and numerical
methods for solving initial-boundary value problems of plasticity," Izv.
AN. MTT [Mechanics of Solids], No. 4, pp. 62-74, 1996. |
|
| Received |
12 March 1998 |
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