Abstract
An asymptotic analysis of equations of an axisymmetric flow of a rigid-plastic material obeying the double shear model in the vicinity of surfaces with the maximum friction is performed. It is shown that the solution is singular if the friction surface coincides with the envelope of the family of characteristics. A possible character of the behavior of singular solutions in the vicinity of surfaces with the maximum friction is determined. In particular, the equivalent strain rate in the vicinity of the friction surface tends to infinity in an inverse proportion to the square root from the distance to this surface. Such a behavior of the equivalent strain rate is also observed in the classical theory of plasticity of materials whose yield condition is independent of the mean stress.
Similar content being viewed by others
REFERENCES
A. J. M. Spencer, “Deformation of ideal granular materials,” in: Mechanics of Solids. The Rodney Hill 60th Anniversary Volume, Pergamon Press, Oxford (1982), pp. 607–652.
A. Yu. Ishlinskii, “Plane motion of sand,” Ukr. Mat. Zh., 6, No.4, 430–441 (1954).
J. Ostrowska-Maciejewska and D. Harris, “Three-dimensional constitutive equations for rigid/perfectly plastic granular materials,” Math. Proc. Cambr. Philos. Soc., 108, 153–169 (1990).
W. A. Spitzig, R. J. Sober, and O. Richmond, “The effect of hydrostatic pressure on the deformation behavior of maraging and HY-80 steels and its implications for plasticity theory,” Metallurg. Trans., 7A, No.11, 1703–1710 (1976).
A. S. Kao, H. A. Kuhn, W. A. Spitzig, and O. Richmond, “Influence of superimposed hydrostatic pressure on bending fracture and formability of a low carbon steel containing globular sulfides,” Trans. ASME, J. Eng. Mater. Technol., 112, No.1, 26–30 (1990).
V. V. Sokolovskii, “Equations of a plastic boundary-layer flow,” Prikl. Mat. Mekh., 20, No.3, 328–334 (1956).
S. E. Alexandrov, “Discontinuous velocity fields in an arbitrary deformation of an ideal rigid-plastic body,” Dokl. Ross. Akad. Nauk, 324, No.4, 769–771 (1992).
S. Alexandrov and O. Richmond, “Singular plastic flow fields near surfaces of maximum friction stress,” Int. J. Non-Linear Mech., 36, No.1, 1–11 (2001).
C. S. Pemberton, “Flow of imponderable granular materials in wedge-shaped channels,” J. Mech. Phys. Solids, 13, 351–360 (1965).
E. A. Marshall, “The compression of a slab of ideal soil between rough plates,” Acta Mech., 3, 82–92 (1967).
S. Alexandrov, “Comparison of double-shearing and coaxial models of pressure-dependent plastic flow at frictional boundaries,” Trans. ASME, J. Appl. Mech., 70, No.2, 212–219 (2003).
S. E. Alexandrov and E. A. Lyamina, “Singular solutions for a plane plastic flow of materials sensitive to the mean stress,” Dokl. Ross. Akad. Nauk, 383, No.4, 492–495 (2002).
S. Alexandrov, “Interrelation between constitutive laws and fracture criteria in the vicinity of friction surfaces,” in: Physical Aspects of Fracture, Kluwer, Dordrecht (2001), pp. 179–190.
S. E. Alexandrov, R. V. Goldstein, and E. A. Lyamina, “Development of the concept of the strain-rate intensity coefficient in the plasticity theory,” Dokl. Ross. Akad. Nauk, 389, No.2, 180–183 (2003).
Author information
Authors and Affiliations
Additional information
__________
Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 180–186, September–October, 2005.
Rights and permissions
About this article
Cite this article
Alexandrov, S.E. Singular Solutions in an Axisymmetric Flow of a Medium Obeying the Double Shear Model. J Appl Mech Tech Phys 46, 766–771 (2005). https://doi.org/10.1007/s10808-005-0133-2
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10808-005-0133-2