 | | 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
|
| In English (Mech. Solids): | | 3212 |
|
| << Previous article | Volume 48, Issue 1 / 2013 | Next article >> |
| D.V. Georgievskii, "Asymptotic Integration of the Prandtl Problem in Dynamic Statement," Mech. Solids. 48 (1), 79-85 (2013) |
| Year |
2013 |
Volume |
48 |
Number |
1 |
Pages |
79-85 |
| DOI |
10.3103/S0025654413010081 |
| Title |
Asymptotic Integration of the Prandtl Problem in Dynamic Statement |
| Author(s) |
D.V. Georgievskii (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992 Russia, georgiev@mech.math.msu.su) |
| Abstract |
The dynamic statement of the problem on the compression of a thin ideally rigid-plastic layer by absolutely rigid plates moving at constant velocities towards each other contains two characteristic dimensionless parameters. One of them-the small geometric parameter α defined as the layer thickness-to-length ratio-explicitly depends on time, and its order of smallness with respect to the other dimensionless parameter-the time-independent reciprocal Euler number-increases with time. The second parameter is assumed to be much less than unity as well. An asymptotic integration procedure is used to construct the solutions of this problem as expansions in integer powers of α; this procedure depends on the parameter ratio, i.e., is different on different time intervals. The possibility of seeking the solution in this form is justified. It is also shown that the asymptotic expansions can be matched smoothly in time.
The parameter ratio at which the correction due to inertial terms in the expression for the pressure turns out to be of the same order as the terms occurring in the classical Prandtl solution of the quasistatic problem is determined. |
| Keywords |
ideally rigid-plastic body, dynamics, Prandtl problem, spreading, compression, asymptotic expansion, Euler number |
| References |
| 1. | L. Prandtl, "Anwendungsbeispiele zu einem Henckyschen Satz über
das plastische Gleichgewicht," ZAMM 3 (6), 401-406
(1923) [in Theory of Plasticity (Izd-vo Inostr. Lit.,
Moscow, 1948), pp. 102-113]. |
| 2. | A. A. Il'yushin,
"Complete Plasticity in Processes of Flow between Rigid Surfaces,
an Analogy with Sand Embarkment and Several Applications,"
Prikl. Mat. Mekh.
19 (6), 693-713 (1955). |
| 3. | V. V. Sokolovskii,
The Theory of Plasticity
(Vysshaya Shkola, Moscow, 1969)
[in Russian]. |
| 4. | A. Yu. Ishlinskii and D. D. Ivlev,
The Mathematical Theory of Plasticity
(Fizmatlit, Moscow, 2001)
[in Russian]. |
| 5. | 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)]. |
| 6. | R. Hill,
The Mathematical Theory of Plasticity
(Clarendon, Oxford, 1950; Gostekhizdat, Moscow, 1956). |
| 7. | A. A. Il'yushin, Collected Works, Vol. 4: Modeling of Dynamic Processes in Solids and Engineering Applications
(Fizmatlit, Moscow, 2009) [in Russian]. |
| 8. | G. I. Bykovtsev, "On Compression of a Plastic Layer by Rigid Rough
Plates with Forces of Inertia Taken into Account," Izv. Akad. Nauk
SSSR. OTN. Mekh. Mashinostr., No. 6, 140-142 (1960). |
| 9. | B. D. Annin, "Symmetry Analysis of Equations of Mises Plastic Flow," in Elasticity and Inelasticity (Izd-vo MGU, Moscow, 2011), pp. 101-105 [in Russian]. |
| 10. | I. A. Kiiko and B. A. Kadymov, "Generalization of the Prandtl
Problem on the Compression of a Strip," Vestnik Moskov. Univ. Ser.
I. Mat. Mekh., No. 4, 50-56 (2003) [Moscow Univ. Mech. Bull.
(Engl. Transl.) 58 (4), 31-36 (2003)]. |
| 11. | D. V. Georgievskii, "Asymptotic Expansions and the Possibilities
to Drop the Hypotheses in the Prandtl Problem," Izv. Akad. Nauk.
Mekh. Tverd. Tela, No. 1, 83-93 (2009) [Mech. Solids (Engl.
Transl.) 44 (1), 70-78 (2009)]. |
| 12. | E. Nayar,
"Several Plane Inertial Flows of Plastic Materials,"
in Continuum Mechanics
(Izd-vo Bolgar. Akad. Nauk, Sofiya, 1968),
pp. 269-277. |
| 13. | A. L. Gol'denveizer,
"Derivation of an Approximate Theory of Bending of a Plate
by the Method of Asymptotic Integration of the Equations of the Theory of Elasticity,"
Prikl. Mat. Mekh.
26 (4), 668-686 (1962)
[J. Appl. Math. Mech. (Engl. Transl.)
26 (4), 1000-1025 (1962)]. |
| 14. | D. V. Georgievskii,
"Axisymmetric Analog of the Prandtl Problem,"
Dokl. Ross. Akad. Nauk 422 (3), 331-333 (2008)
[Dokl. Phys. (Engl. Transl.)
53 (9), 504-506 (2008)]. |
| 15. | D. V. Georgievskii,
"Asymptotic Analysis of Plastic Flow along a Generating Element in a Thin Cylindrical Layer,"
Zh. Prikl. Mekh. Tekhn. Fiz.
51 (5), 111-119 (2010). |
|
| Received |
03 March 2011 |
| Link to Fulltext |
|
| << Previous article | Volume 48, Issue 1 / 2013 | Next article >> |
|
 If you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
|
|
Russian 
English