 | | 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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| D. V. Georgievskii, "Asymptotic Expansions and the Possibilities to Drop the Hypotheses in the Prandtl Problem," Mech. Solids. 44 (1), 70-78 (2009) |
| Year |
2009 |
Volume |
44 |
Number |
1 |
Pages |
70-78 |
| DOI |
10.3103/S0025654409010075 |
| Title |
Asymptotic Expansions and the Possibilities to Drop the Hypotheses in the Prandtl Problem |
| Author(s) |
D. V. Georgievskii (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992, Russia, georgiev@mech.math.msu.su) |
| Abstract |
The plane problem on the quasistatic compression of a thin perfectly plastic layer between undeformable rough plates (the Prandtl problem) has a well-known analytic solution at all points sufficiently far from the midsection and endpoints of the layer. Both the static and the kinematic component of this solution were obtained on the basis of the Prandtl hypothesis [1] stating that the tangential stress is linear along the layer thickness and is maximal in absolute value on the plate surfaces. (If the plates are perfectly rough, then this maximum value coincides with the shear yield stress.) The Prandtl hypothesis was widely confirmed in experiments carried out after the paper [1] had been published.
At the same time, it is natural to ask whether one can construct a classical solution of this problem without imposing any static or kinematic hypotheses on the unknown variables and whether there exist any other mathematical solutions in which these hypotheses do not hold and which themselves are not observed in experiments.
In the present paper, we use asymptotic analysis with a natural small geometric parameter and uniquely determine an exact solution (in the sense of finiteness of the number of terms in the asymptotic expansion), which coincides with the Prandtl solution generalized to the case of an arbitrary roughness coefficient of the plates. We rigorously show that such asymptotics cannot hold near the layer midsection, where we construct another, internal asymptotic expansion. In the abovementioned sense, the solution corresponding to the internal expansion is also exact and models the compression of a thin vertical strip in the middle of the layer. We realize two possible versions of matching of the two expansions in the cross-section whose distance from the midsection is equal to the layer thickness. |
| References |
| 1. | L. Prandtl, "Anwendungsbeispiele zu Einem Henckyschen Satz über das PlastischeGleichgewicht," ZAMM 3(6), 401-406 (1923) [in Theory of Plasticity (Izd-vo Inostr. Lit.,Moscow, 1948), pp. 102-113].
CrossRef |
| 2. | R. Hill, E. H. Lee, and S. J. Tupper, "A Method of Numerical Analysis of Plastic Flow in Plane Strain and Its Application to the Compression of a Ductile Material between Rough Plates," J. Appl. Mech. 18(1), 46-52 (1951); Mechanics. Collection of Translations and Reviews of the Foreign Periodic Literature 3 (19), 114-126. |
| 3. | R. Hill, The Mathematical Theory of Plasticity (Clarendon, Oxford, 1950; Gostekhizdat, Moscow, 1956). |
| 4. | A. Nadai, Theory of Flow and Fracture of Solids (Wiley, New York, 1950; Izd-vo Inostr. Lit., Moscow, 1954). |
| 5. | P. G. Hodge, "Approximate Solutions of Problems of Plane Plastic Flow," J. Appl. Mech. 17(3), 257-264 (1950). |
| 6. | 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). |
| 7. | W. Prager and P. G. Hodge, Theory of Perfectly Plastic Solids (Wiley, New York, 1951; Izd-vo Inostr. Lit., Moscow, 1956). |
| 8. | V. V. Sokolovskii, The Theory of Plasticity (Vysshaya Shkola, Moscow, 1969) [in Russian]. |
| 9. | M. A. Zadoyan, Spatial Problems of Plasticity Theory (Nauka, Moscow, 1992) [in Russian]. |
| 10. | A. Yu. Ishlinskii and D. D. Ivlev, The Mathematical Theory of Plasticity (Fizmatlit, Moscow, 2001) [in Russian]. |
| 11. | D. D. Ivlev, "On the 3D Flow of an Ideally Plastic Material Compressed by Rough Plates," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 1, 5-12 (1998) [Mech. Solids (Engl. Transl.) 33 (1), 3-9 (1998)]. |
| 12. | B.E. Pobedrya and I. L. Guzei, "Mathematical Modeling of Deformation of Composites with Thermodiffusion Taken into Account," in Mathematical Modeling of Systems and Processes, No. 6 (RIO PGTU, Perm, 1998), pp. 82-91. |
| 13. | I. A. Kiiko, "Generalization of the L. Prandtl Problemabout the Strip Upsetting to the Case of a Compressed Material," Vestnik Moskov. Univ. Ser. I. Mat. Mekh., No. 4. 47-52 (2002) [Moscow Univ. Math. Bull. (Engl. Transl.)]. |
| 14. | I. A. Kiiko and V. A. Kadymov, "Generalization of the L. Prandtl Problem about the Strip Compression," Vestnik Moskov. Univ. Ser. I. Mat. Mekh., No. 4, 50-56 (2003) [Moscow Univ. Math. Bull. (Engl. Transl.)]. |
| 15. | B. A. Druyanov and R. I. Nepershin, Theory of Technological Plasticity (Mashinostroenie, Moscow, 1990; Elsevier, London, 1994). |
| 16. | V. L. Kolmogorov, Mechanics of Metal Treatment by Compression (Izd-vo UrGTU-UPI, Ekaterinburg, 2001) [in Russian]. |
| 17. | D. V. Georgievskii, "The Prandtl Problem for a Plastic Layer Weakly Inhomogeneous with Respect to the Yield Strength," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 1, 47-59 (2006) [Mech. Solids (Engl. Transl.) 41 (1), 35-45 (2006)]. |
| 18. | 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)]. |
| 19. | M. Van Dyke, Perturbation Methods in Fluid Mechanics (Acad. Press, New York, 1964; Mir, Moscow, 1967). |
| 20. | A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981; Mir,Moscow, 1984). |
|
| Received |
19 February 2008 |
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