 | | 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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| S. E. Aleksandrov and E. A. Lyamina, "Compression of a mean-stress sensitive plastic material by rotating plates," Mech. Solids. 38 (6), 40-48 (2003) |
| Year |
2003 |
Volume |
38 |
Number |
6 |
Pages |
40-48 |
| Title |
Compression of a mean-stress sensitive plastic material by rotating plates |
| Author(s) |
S. E. Aleksandrov (Moscow)
E. A. Lyamina (Moscow) |
| Abstract |
The theory of materials which are plastic, incompressible but sensitive to the mean stress is a common tool in the mechanics of granular media [1, 2]. However, the dependence of the yield condition on the mean stress is observed also for some alloys, although less pronounced than in granular materials [3-5]. Forming of such alloys is performed by the traditional processes of plastic working, which gives rise to specific boundary-value problems which do not occur in the models of granular medium flows. In the present paper, we consider a material layer obeying equations of the model of [2]. The layer is subjected to compression by rotating plates with rough surfaces. Major attention is given to the behavior of the solution near the friction surface. It is known [6, 7] that in the case of classical plasticity, the velocity field is singular near the surface of maximal friction (on such surfaces, the specific friction forces are equal to the shear yield stress). This friction law, when used in other models of materials, causes considerable difficulties, if one tries to generalize the classical plasticity problems. As an illustration, we note that formal generalizations of the Prandtl solution of the problem of compression of a plastic material with rough surface between two parallel plates with rough surfaces (this solution can be found, for instance, in [8]) to viscoplastic materials [9] or to plastic hardening materials [10] are unjustified in the case of the maximal friction law (in both cases, the average hardening per unit length tends to infinity as the friction forces approach their maximal values). Such a behavior of solutions takes place for any deviation from the model of ideal rigid-plastic materials. However, the solution of a similar problem obtained in [11] within the framework of the model from [2] exists in the case of limit friction law proposed in [11, 12], and this solution predicts a finite mean force per unit length. Direct verification shows that this solution is singular and the asymptotic behavior of the velocity field near the friction surface is quite similar to that observed in the case of classical plasticity [6, 7]. In the present paper, we use similar assumptions, the model of material from [2], and the friction law from [11, 12] and obtain a solution of the problem for a layer compressed between two rotating plates. The problem is reduced to a linear nonhomogeneous second-order ordinary differential equation with a regular singular point corresponding to the friction surface. |
| References |
| 1. | V. V. Sokolovskii,
Statics of Granular Media [in Russian], Fizmatlit, Moscow, 1960. |
| 2. | A. J. M. Spencer, "Deformation of ideal granular materials," in H. G. Hopkins and M. J. Sewell (Editors), Mechanics of Solids.
The Rodney Hill 60th Anniversary Volume, pp. 607-652,
Pergamon Press, Oxford, 1982. |
| 3. | 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,"
Metallurgical Trans., Vol. 7A, No. 11, pp. 1703-1710, 1976. |
| 4. | W. A. Spitzig, "Effect of hydrostatic pressure on plastic-flow properties of iron single crystals," Acta Metallurg., Vol. 27, No. 4,
pp. 523-534, 1979. |
| 5. | 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. Engng. Mater. Technol., Vol. 112, No. 1, pp. 26-30, 1990. |
| 6. | S. E. Aleksandrov,
"Velocity field near its discontinuity surface for an arbitrary flow of an ideal rigid-plastic material,"
Izv. AN. MTT [Mechanics of Solids], No. 5, pp. 116-122, 1995. |
| 7. | S. Aleksandrov and O. Richmond,
"Singular plastic flow fields near surfaces of maximum friction stress,"
Intern. J. Non-Linear Mech., Vol. 36, No. 1, pp. 1-11, 2001. |
| 8. | L. M. Kachanov, Basic Principles of Plasticity Theory [in Russian],
Gostekhizdat, Moscow, 1956. |
| 9. | M. J. Adams, B. J. Briscoe, G. M. Corfield, C. J. Lawrence, and T. D. Papathanasiou,
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| 10. | I. F. Collins and S. A. Meguid, "On the influence of hardening and anisotropy on the plane-strain compression of thin metal strip,"
Trans. ASME. J. Mech., Vol. 44, No. 2, pp. 271-278, 1977. |
| 11. | E. A. Marshall,
"The compression of a slab of ideal soil between rough plates,"
Acta Mech., Vol. 3 No. 2, pp. 82-92, 1967. |
| 12. | C. S. Pemberton, "Flow of imponderable granular materials in wedge-shaped channels," J. Mech. Phys. Solids, Vol. 13, No. 6, pp. 351-360, 1965. |
| 13. | M. A. Zadoyan, "A particular solution of the ideal plasticity equations in cylindrical coordinates," Doklady AN SSSR, Vol. 157, No. 1, pp. 73-75, 1964. |
| 14. | S. Aleksandrov, "Interrelation between constitutive laws and fracture in the vicinity of friction surfaces," in E. Bouchaud et al. (Editors), pp. 179-190, Physical Aspects of Fracture, Kluwer, Dordrecht,
2001. |
| 15. | N. A. Fleck and D. Durban,
"Steady penetration of a rigid cone with a rough wall into a power-law viscous solid," Trans. ASME. J. Appl. Mech., Vol. 58, No. 4, pp. 872-880,
1991. |
| 16. | D. Durban and O. Rand, "Singular fields in plane-strain penetration," Trans. ASME. J. Appl. Mech., Vol. 58, No. 4, pp. 910-915,
1991. |
| 17. | D. Durban and N. A. Fleck, "Singular plastic fields in steady penetration of a rigid cone,"
Trans. ASME. J. Appl. Mech., Vol. 59, No. 4, pp. 706-710, 1992. |
|
| Received |
19 December 2001 |
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