 | | 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: | | 11223 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8011
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| In English (Mech. Solids): | | 3212 |
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| N.A. Belov and V.A. Kadymov, "Analysis of the Problem on a Plastic Layer Flow between Rigid Plates Approaching Each Other," Mech. Solids. 46 (1), 36-46 (2011) |
| Year |
2011 |
Volume |
46 |
Number |
1 |
Pages |
36-46 |
| DOI |
10.3103/S0025654411010067 |
| Title |
Analysis of the Problem on a Plastic Layer Flow between Rigid Plates Approaching Each Other |
| Author(s) |
N.A. Belov (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str. 1, Moscow, 119526 Russia, belov@ipmnet.ru)
V.A. Kadymov (Moscow State Technical University "MAMI," Bol'shaya Semyonovskaya 38, Moscow, 107023 Russia, vkadymov@yandex.ru) |
| Abstract |
The present paper deals with the solution of the boundary value problem on a plastic layer flow between rigid plates approaching each other in A. A. Il'yushin's setting. After averaging over the layer thickness, the problem is reduced to a two-dimensional spreading problem for a domain with free boundary occupied by a plastic medium.
The plastic medium material in such a flow is well described by the viscous fluid model. But the solution obtained earlier holds only in the perfect fluid approximation. It is shown that this solution does not satisfy one of the dynamic conditions on the free boundary of the domain.
A sound analytic study of the boundary value problem is performed in the paper. The boundary layer is used to obtain a solution satisfying all boundary conditions. If the flow described by perfect fluid is directed along the normal to the boundary, then it also has a tangential velocity component in the boundary layer. Moreover, the obtained velocity field is used to derive the evolution equation for the free boundary. It is shown that, for a sufficiently smooth boundary, this equation coincides with the equation obtained earlier. |
| Keywords |
plastic flow, thin layer, boundary value problem, boundary layer |
| References |
| 1. | A. A. Il'yushin,
"Problems of the General Theory of Plasticity,"
Prikl. Mat. Mekh.
24 (3), 399-411 (1960)
[J. Appl. Math. Mech. (Engl. Transl.)
24 (3), 587-603 (1960)]
(See also
A. A. Il'yushin,
Collected Works, Vol. 2: Plasticity
(Fizmatlit, Moscow, 2004),
pp. 211-219
[in Russian]). |
| 2. | V. N. Bezukhov,
On Settlement of a Plastic Layer of Non-Circular Shape in Plane,
Candidate's Dissertation in Mathematics and Physics
(Izd-vo MGU, Moscow, 1955)
[in Russian]. |
| 3. | I. A. Kiiko,
Theory of Plastic Flow
(Izd-vo MGU, Moscow, 1978)
[in Russian]. |
| 4. | V. A. Kadymov,
Several Problems of Plastic Flow in a Thin Layer of Metal,
Candidate's Dissertation in Mathematics and Physics
(Izd-vo MGU, Moscow, 1981)
[in Russian]. |
| 5. | I. A. Kiiko,
Plastic Flows of Metals,
in Scientific Foundations of Progressive Engineering and Technology
(Mashinostroenie, Moscow, 1985)
[in Russian]. |
| 6. | V. A. Kadymov,
"Self-Similar Equations in the Problem of Spreading of a Plastic Layer on Plane
and Their Solutions,"
Vestn. TulGU
15 (3), 38-44 (2009). |
| 7. | L. Prandtl,
"Examples of Application of Hencky Theorem to Equilibrium of Plastic Bodies,"
ZAMM 3 (6), 401-406 (1923) [in German]
[in Theory of Plasticity,
Ed. by Yu. N. Rabotnov (IL, Moscow, 1948),
pp. 102-113 [in Russian]]. |
| 8. | A. N. Mokhel and R. L. Salganik,
"A Thin Ideally Plastic Layer with an Arbitrary Contour Compressed between Rigid Plates,"
Dokl. Akad. Nauk SSSR
293 (4), 809-813 (1987)
[Sov. Phys. Dokl. (Engl. Transl.)
32, 279 (1987)]. |
| 9. | S. S. Grigoryan,
"A Problem of L. Prandtl and the Theory of Flow of a Plastic Material over Surfaces,"
Dokl. Akad. Nauk SSSR
257 (5), 1075-1077 (1981)
[Sov. Phys. Dokl. (Engl. Transl.)
26, 399 (1981)]. |
| 10. | N. A. Belov and V. A. Kadymov,
On the Boundary Value Problem of Flow of a Thin Plastic Layer,
Preprint No. 928
(IPMekh RAN, Moscow, 2010)
[in Russian]. |
| 11. | N. E. Kochin, I. A. Kibel, and N. V. Roze,
Theoretical Hydrodynamics, Part 2
(Fizmatgiz, Moscow, 1963; Interscience, New York, 1964). |
| 12. | A. D. Polyanin and V. F. Zaitsev,
Handbook of Nonlinear Equations of Mathematical Physics.
Exact Solutions
(Fizmatlit, Moscow, 2002)
[in Russian]. |
|
| Received |
07 September 2010 |
| Link to Fulltext |
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