 | | 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 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
|
| In English (Mech. Solids): | | 4760 |
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| << Previous article | Volume 36, Issue 4 / 2001 | Next article >> |
| B. A. Zhukov, "A modification of the Signorini method for plane strains in an incompressible material," Mech. Solids. 36 (4), 47-53 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
4 |
Pages |
47-53 |
| Title |
A modification of the Signorini method for plane strains in an incompressible material |
| Author(s) |
B. A. Zhukov (Volgograd) |
| Abstract |
One of the methods for solving static problems of nonlinear elasticity
is the method of expansion of the solution into absolutely
convergent series with respect to a small parameter [1], suggested
by Signorini for the case where the specific strain energy is represented
by a polynomial of the strain tensor invariants. In the case of plane strain,
this method allows the application of the powerful means of the complex
analysis [2]. If the material is incompressible, the terms in the
expansion of the incompressibility condition (beginning from the second one)
are nonlinear and cannot be solved by Kolosov-Muskhelishvili potentials.
Hence, problems for displacements and mixed boundary-value problems contain
an equation to be satisfied in a plane domain. A method permitting one to take
into account the incompressibility conditions has been suggested in [3].
This method is based on the original statement of the plane elasticity
problem that has been formulated by L. A. Tolokonnikov [4] on the basis
of additional hypotheses as regards the relation between stresses and strains.
In the present paper, a modification of the Signorini method permitting
one to eliminate the incompressibility condition is suggested. Second-order
effects are singled out. An example of the solution of a problem for
displacements is given. |
| References |
| 1. | A. E. Green and J. E. Adkins, Large Elastic Deformations
and Nonlinear Mechanics of a Continuous Medium [Russian translation],
Mir, Moscow, 1965. |
| 2. | N. I. Muskhelishvili, Some Basic Problems of Mathematical
Theory of Elasticity [in Russian], Nauka, Moscow, 1966. |
| 3. | V. G. Gromov and L. A. Tolokonnikov, "On the calculation
of approximations in the problem of finite plane deformations of
an incompressible material," Izv. AN SSSR. OTN. Mekhanika
i Mashinostroenie, No. 2, pp. 81-86, 1963. |
| 4. | L. A. Tolokonnikov, "On the stress-strain relation in
nonlinear elasticity," PMM [Applied Mathematics and Mechanics],
Vol. 20, No. 3, pp. 439-444, 1956. |
| 5. | V. I. Arnold, Mathematical Methods in Classical Mechanics
[in Russian], Nauka, Moscow, 1974. |
| 6. | A. I. Lur'e, Nonlinear Theory of Elasticity [in Russian],
Nauka, Moscow, 1980. |
| 7. | B. A. Zhukov, "Comparison of Griffith and Sneddon
tensile crack models
with respect to second-order effects in an incompressible
hyperelastic material," in Modern Problems of Continuum Mechanics.
Proceedings of the Third International Conference. Volume 1 [in Russian],
pp. 154-158, MP Kniga, Rostov-on-Don, 1997. |
|
| Received |
15 June 1999 |
| << Previous article | Volume 36, Issue 4 / 2001 | Next article >> |
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