 | | 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, "Indentation of a wedge into a plastic medium governed by the double-shearing model," Mech. Solids. 39 (5), 84-90 (2004) |
| Year |
2004 |
Volume |
39 |
Number |
5 |
Pages |
84-90 |
| Title |
Indentation of a wedge into a plastic medium governed by the double-shearing model |
| Author(s) |
S. E. Aleksandrov (Moscow)
E. A. Lyamina (Moscow) |
| Abstract |
The indentation of a wedge or cone into a plastic medium
is a classical problem of solid mechanics. In the present
paper the problem of indentation of a plane wedge into a plastic medium is solved for the case where the medium is governed by the double-shearing model [1]. In accordance with this model, the strain is determined by the shear along both characteristic directions of the system of static equations (the equilibrium equation and the yield equation). The most complete description of the model is given in [1] and the plane flow equations needed for solving the problem are given in [2]. The model is based on the Coulomb-Mohr yield condition. The distinctive features of this model are that the plastic flow associated law does not hold and that the characteristics for stresses and velocities coincide. In addition, for the friction law proposed in [2, 3], the solutions are singular, the asymptotic behavior of these solutions near the friction surfaces being completely similar to the case of classical plasticity [5]. In the case of the indentation of a wedge, which is considered in the present paper, there is a singular surface (a line in the plane of flow) that coincides with the surface of the wedge on which the friction law of [2, 3] is assumed to be valid. Singular velocity fields in the case of indentation of a wedge or a cone into plastic and viscous media have been studied intensely in recent time [6-9]. However, in all these studies, major attention has been given to the singularity near the vertex of the wedge or cone, which is not related to the friction law. The solutions for the case of the indentation of a cone into a rigid-plastic material, the yield condition for which depends on the mean stress, were obtained in [10, 11]. However, in [10], a rather artificial law of friction was assumed for the wall of the cone. In [11], the double-shearing model was utilized, with friction on the cone surface being absent. The solution was based on the discontinuous velocity field, which is inadmissible. Possibly, the difficulties associated with the construction of this solution can be accounted for by the fact that this solution was constructed as a generalization of the corresponding solution for a perfect rigid-plastic material. In fact, the solutions of [2, 3] directly generalize the solutions obtained for such a model of the material; see, e.g., [12]. However, in the case of the indentation of a wedge, there is an essential qualitative difference between the solutions obtained on the basis of different models, which is shown in the present paper. For example, the velocity field is continuous on the rigid-plastic boundary. |
| References |
| 1. | A. J. M. Spencer,
"Deformation of ideal granular materials,"
in H. G. Hopkins and M. J. Sewell (Editors), Solid Mechanics. The Rodney Hill 60th
Anniversary Volume, pp. 607-652, Pergamon Press, Oxford, 1982. |
| 2. | C. S. Pemberton,
"Flow of imponderable granular materials in wedge-shaped
channels,"
J. Mech. and Phys. Solids, Vol. 13, No. 6, pp. 351-360, 1965. |
| 3. | E. A. Marshall, "The compression of a slab of ideal soil between two plates,"
Acta Mech., Vol. 3, No. 2, pp. 82-92, 1967. |
| 4. | S. E. Aleksandrov and E. A. Lyamina,
"Singular solutions for plane plastic flow of materials
sensitive to the mean stress,"
Doklady AN, Vol. 383, No. 4, pp. 492-495, 2002. |
| 5. | S. Alexandrov 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. |
| 6. | 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. |
| 7. | D. Durban and O. Rand,
"Singular fields in plane-strain penetration,"
Trans. ASME. J. Appl. Mech., Vol. 58, No. 4, pp. 910-915, 1991. |
| 8. | 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. 706-770, 1992. |
| 9. | D. Durban,
"Friction and singularities in steady penetration,"
in D. Durban and J. R. A. Pearson (Editors), Proc. IUTAM Symp. on Non-linear Singularities
in Deformation and Flow, pp. 141-154, Kluwer, Dordrecht, 1999. |
| 10. | D. B. Balashov and N. V. Zvolinskii,
"The flow of a rigid-plastic medium past a cone,"
Izv. AN. MTT [Mechanics of Solids], No. 3, pp. 46-53, 1996. |
| 11. | A. J. M. Spencer,
"Plastic flow past a smooth cone,"
Acta Mech., Vol. 54, No. 1-2, pp. 63-74, 1984. |
| 12. | R. Hill, Mathematical Theory of Plasticity
[Russian translation], Gostekhizdat, Moscow, 1956. |
|
| Received |
26 August 2002 |
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