Abstract
The flow of perfectly rigid-plastic material in a thin layer subjected to a load was considered in numerous studies, including the classical ones [1–12]. In the case of rigid plates coinciding with the layer face surfaces and approaching each other in a prescribed way, we deal with the Prandtl problem, which was considered in [1], or with its numerous generalizations (e.g., see [13, 14]). Traditionally, the process of obtaining the Prandtl solution is based on the hypothesis that the tangential stress is linear in thickness and hence the tangential stress attains its maximum absolute value on the surfaces of the rough plates.
The asymptotic analysis carried out in [15] with a natural small geometric parameter without any original static or kinematic hypotheses led to a solution coinciding with the generalization of the Prandtl solution to the case of plates with an arbitrary roughness factor. This solution is exact in the sense that there are finitely many nonzero terms in the asymptotic series. The ill-posedness of the expansions chosen near the middle cross-section of the layer rigorously follows from the loss of the asymptotic property in the sense of the Poincaré of the series for the velocity longitudinal component in this region. Another internal expansion constructed in [15] also exactly and physically corresponds to compression of a thin vertical strip in the middle of the layer.
The present paper is a generalization of [15] to the case of an arbitrary region occupied with a layer in horizontal projection. We present an algorithm for constructing the asymptotic solution of the problem and consider the possibility of perfectly rigid-plastic flow along one of a family of coordinate lines. To this end, it is necessary that the roughness of the pressing plates depend on the coordinates in a certain way. We also perform a detailed study of the axisymmetric analogue of the Prandtl problem (compression of a circular layer) and the kinematics of an elliptic layer spreading.
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References
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].
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.
R. Hill, The Mathematical Theory of Plasticity (Clarendon, Oxford, 1950; Gostekhizdat, Moscow, 1956).
A. Nadai, Theory of Flow and Fracture of Solids (Wiley, New York, 1950; Izd-vo Inostr. Lit., Moscow, 1954).
P. G. Hodge, “Approximate Solutions of Problems of Plane Plastic Flow,” J. Appl. Mech. 17(3), 257–264 (1950).
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).
W. Prager and P. G. Hodge, Theory of Perfectly Plastic Solids (Wiley, New York, 1951; Izd-vo Inostr. Lit., Moscow, 1956).
V. V. Sokolovskii, The Theory of Plasticity (Vysshaya Shkola, Moscow, 1969) [in Russian].
B. D. Annin, V. O. Bytev, and S. I. Senashov, Group Properties of the Equations of Elasticity and Plasticity (Nauka, Novosibirsk, 1985) [in Russian].
M. A. Zadoyan, Spatial Problems of Plasticity Theory (Nauka, Moscow, 1992) [in Russian].
A. Yu. Ishlinskii and D. D. Ivlev, The Mathematical Theory of Plasticity (Fizmatlit, Moscow, 2001) [in Russian].
I. A. Kiiko, Theory of Plastic Flow (Izd-vo MGU, Moscow, 1978) [in Russian].
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.
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.)].
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)].
D. Persivale, “Perfectly Plastic Plates: A Variational Definition,” J. Reine Angew. Math. 411(6), 39–50 (1990).
M. V. Fedoryuk, Saddle-Point Method (Nauka, Moscow, 1977) [in Russian].
V. F. Kravchenko, G. A. Nesenenko, and V. I. Pustovoit, Poincaré Asymptotics of Solution to Problems of Irregular Heat and Mass Transfer (Fizmatlit, Moscow, 2006) [in Russian].
L. I. Sedov, Continuum Mechanics, Vol. 1 (Lan’, St. Petersburg, 2004) [in Russian].
R. I. Nepershin, “Plastic Flow in the Case of a Disk Compression between Parallel Plates,” Mashinoved., No. 1, 97–100 (1968).
B. A. Druyanov and R. I. Nepershin, Theory of Technological Plasticity (Mashinostroenie, Moscow, 1990) [in Russian].
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)].
D. V. Georgievskii, “Perfect Rigid-Plastic Spreading of an Asymptotically Thin Cylindrical Layer,” Dokl. Ross. Akad. Nauk 429(3), 328–331 (2009) [Dokl. Phys. (Engl. Transl.) 54 (11), 516–519 (2009)].
L. V. Yakhno, Use of Symmetries in Construction of New Solutions of Equations of Plane Perfect Plasticity, Author’s Abstract of Candidate’s Dissertation in Mathematics and Physics (Krasnoyarsk, 2009) [in Russian].
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Original Russian Text © D.V. Georgievskii, 2011, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2011, No. 4, pp. 104–115.
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Georgievskii, D.V. Saint-Venant flow in a thin layer under plastic compression. Mech. Solids 46, 579–588 (2011). https://doi.org/10.3103/S002565441104008X
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DOI: https://doi.org/10.3103/S002565441104008X