Abstract
Dynamic solutions of an analogue of the Prandtl problem in the case of a cylindrical layer, including terms with \(\alpha^{-1}\) and \(\alpha^{0}\), for various configurations of cylinders are obtained and analyzed on the basis of asymptotic analysis with a natural small geometric parameter \(\alpha\) without any static or kinematic hypotheses.
Similar content being viewed by others
REFERENCES
L. Prandtl, ‘‘Anwendungsbeispiele zu einem Henckyschen Satz uber das plastische Gleichgewicht,’’ ZAMM 3, 401–405 (1923). doi 10.1002/zamm.19230030601
R. Hill, The Mathematical Theory of Plasticity (Clarendon Press, Oxford, 1950).
I. A. Kiyko, ‘‘Generalizations of Prandtl’s problem on contraction of a strip to the case of compressible materials,’’ Moscow Univ. Mech. Bull. 57 (4), 16–21 (2002).
A. I. Kuznetsov, ‘‘The problem of the inhomogeneous layer,’’ Arch. Mech. Stos. 12, 163–172 (1960).
R. I. Nepershin, ‘‘Plastic flow in the case of a disk compression between parallel plates,’’ Mashinovedenie, No. 1, 97–100 (1968).
D. V. Georgievskii and R. R. Shabaikin, ‘‘Quasi-static and dynamic compression of a planar circular ideally plastic layer between rigid plates,’’ in Mathematical Modeling and Experimental Mechanics of Deformable Solids (Tver State Univ., Tver, 2017), pp. 56–83.
R. I. Nepershin, ‘‘Effect of the heat transfer on non-isothermal plane plastic flow at compression of a thin billet between flat rigid dies,’’ Probl. Mashinostr. Nadezhnosti Mash., No. 1, 96–103 (1997).
G. I. Bykovtsev, ‘‘On compression of a plastic layer by rigid rough plates with forces of inertia taken into account,’’ Izv. Akad. Nauk SSSR, Mekh. Mashinostr., No. 6, 140–142 (1960).
I. A. Kiĭko and V. A. Kadymov, ‘‘Generalizations of a problem of L. Prandtl on the compression of a band,’’ Moscow Univ. Math. Bull. 58 (4), 31–36 (2003).
E. Nayar, ‘‘Some planar inertial flows of plastic materials,’’ in Continuum Mechanics (Bolg. Akad. Nauk, Sofia, 1968), pp. 269–277.
D. V. Georgievskii, ‘‘Asymptotic integration of the Prandtl problem in dynamic statement,’’ Mech. Solids 48, 79–85 (2013). doi 10.3103/S0025654413010081
D. V. Georgievskii, ‘‘Asymptotic analysis of plastic flow along the generatrix in a thin cylindrical layer,’’ J. Appl. Mech. Tech. Phys. 51, 713–720 (2010). doi 10.1007/s10808-010-0091-1
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by I. Obrezanova
About this article
Cite this article
Shabaykin, R.R. Dynamic Deformation of a Thin Plastic Layer between Converging Rigid Cylinders. Moscow Univ. Mech. Bull. 75, 87–95 (2020). https://doi.org/10.3103/S0027133020040068
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0027133020040068