 | | 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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| In English (Mech. Solids): | | 4810 |
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| I. A. Brigadnov, "Evaluation of the bearing capacity of nonlinearly elastic bodies," Mech. Solids. 36 (1), 3-10 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
1 |
Pages |
3-10 |
| Title |
Evaluation of the bearing capacity of nonlinearly elastic bodies |
| Author(s) |
I. A. Brigadnov (St. Petersburg) |
| Abstract |
For a nonlinearly elastic solid whose stored energy function has linear
growth with respect to the modulus of the distortion tensor, a variational
problem of limit analysis is formulated and examined numerically. In this
setting, we estimate the limit load, i.e., the magnitude of external
"dead" loads beyond which there is no statically determinate deformed
configuration that would be stable with respect to finite variations of the
mapping. Moreover, we formulate the problem of optimizing the shape of a
nonlinearly elastic body to ensure its maximal bearing capacity under
given external loads.
These are variational problems with
an integral functional of linear growth and their treatment requires
that the set of admissible fields be extended, so as to incorporate
possible discontinuities.
For the variational problem of limit analysis considered
in this paper, we use a partial relaxation based on
a special finite element approximation allowing for fields with
discontinuities of slip type. As a result, the original problem is
reduced to a nonlinear system of algebraic equations whose global
stiffness matrix may happen to be ill-conditioned.
To solve the determining system of algebraic equations numerically,
we use the method of adaptive block relaxation, which is insensitive to
the condition number of the global stiffness matrix.
The results of
numerical experiments given here show the qualitative advantage of the
proposed method as compared with standard methods based on continuous
finite element approximations. |
| References |
| 1. | A. I. Lur'e, 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. | J. M. Ball, "Convexity conditions and existence theorems in
nonlinear elasticity," Arch. Rat. Mech. Anal., Vol. 63, No. 4,
pp. 337-403, 1977. |
| 5. | I. A. Brigadnov, "On the existence of the limit load in some
problems of hyperelasticity," Izv. AN. MTT [Mechanics of Solids], No. 5, pp. 46-51, 1993. |
| 6. | I. A. Brigadnov, "Existence theorems for boundary value problems
in hyperelasticity," Matem. Sbornik, Vol. 187, No. 1, pp. 3-16, 1996. |
| 7. | I. A. Brigadnov, "Mathematical well-posedness of boundary value
problems of elastostatics for hyperelastic materials,"
Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 37-46, 1996. |
| 8. | I. A. Brigadnov, "Numerical methods in nonlinear elasticity," in
Numerical Methods in Engineering'96, pp. 158-163, Wiley,
Chichester, 1996. |
| 9. | I. A. Brigadnov, "Discontinuous solutions and their finite element
approximation in nonlinear elasticity," in ACOMEN'98 - Advanced
Computational Methods in Engineering, pp. 141-148, Shaker Publ. B. V.,
Maastricht, 1998. |
| 10. | I. A. Brigadnov, "The limited static load in finite elasticity,"
in Constitutive Models for Rubber, pp. 37-43, Balkena, Rotterdam, 1999. |
| 11. | A. Kufner and S. Fuchik, Nonlinear Differential Equations [Russian
translation], Nauka, Moscow, 1988. |
| 12. | I. Ekeland and R. Temam, Convex Analysis and Variational Problems
[Russian translation], Mir, Moscow, 1979. |
| 13. | A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of
Functions and Functional Analysis [in Russian], Nauka, Moscow, 1968. |
| 14. | E. Giusti, Minimal Surfaces and Functions of Bounded Variation
[Russian translation], Mir, Moscow, 1989. |
| 15. | R. Temam, Mathematical Problems in Plasticity [Russian
translation], Nauka, Moscow, 1991. |
| 16. | S. I. Repin, "A variational difference method for solving
problems with a functional of linear growth," Zh. Vychisl. Matem. i Mat.
Fiziki, Vol. 29, No. 5, pp. 693-708, 1989. |
| 17. | G. M. Bartenev and T. N. Khazanovich, "On the highly elastic
deformation law for cross-linked polymers," Vysokomolek. Soedineniya,
Vol. 2, No. 1, pp. 20-28, 1960. |
| 18. | P. G. Ciarlet, Finite Element Method for Elliptic Problems
[Russian translation], Mir, Moscow, 1980. |
| 19. | I. A. Brigadnov, "On the numerical solution of boundary value
problems of elastic-plastic flow," Izv. AN. MTT [Mechanics of Solids], No. 3, pp. 157-162, 1992. |
| 20. | I. A. Brigadnov, "Numerical solution of a difference boundary
value problem in hyperelasticity," Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 42-50, 1994. |
| 21. | 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 |
23 December 1999 |
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