 | | 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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| A. G. Zalazinskii, A. A. Polyakov, and A. P. Polyakov, "On plastic compression of a porous body," Mech. Solids. 38 (1), 101-110 (2003) |
| Year |
2003 |
Volume |
38 |
Number |
1 |
Pages |
101-110 |
| Title |
On plastic compression of a porous body |
| Author(s) |
A. G. Zalazinskii (Ekaterinburg)
A. A. Polyakov (Ekaterinburg)
A. P. Polyakov (Ekaterinburg) |
| Abstract |
When solving boundary value problems in the mechanics of
pressure shaping of noncompact materials, it is necessary to take into
account irreversible deformations of volume compression (extension). A.
Coulomb [1] considered the limit state pyramid in the theory of soil
pressure. For plastic compressible materials, yield conditions in the form
of Coulomb's pyramid bounded by a plane of constant hydrostatic pressure
(the Coulomb-Mohr) pyramid were used in [2, 3].
Von Mises and Schlechter proposed a yield condition expressed by
a circular cone in the space of stresses bounded by a plane of constant
hydrostatic pressure (von Mises-Schlechter's cone) [1].
This condition was considered in [3, 5]. The said yield conditions
are of piecewise smooth type. Some models of ideal compressible
plastic media with piecewise smooth yield surfaces were investigated by
D. D. Ivlev and G. I. Bykovtsev [6], who showed that
the relations of such models can be regarded as a theory of a
hardening plastic bodies with singular load surfaces [6].
R. Green proposed a yield condition of elliptic type [7].
This condition is expressed in terms of
an ellipsoid of revolution in the space of principal stresses,
the symmetry axis of the ellipse coinciding with the hydrostatic axis.
There is also a yield condition of cylindrical type [3].
This condition, in contrast to those mentioned above, involves no
dilatancy relation between the characteristics of volume and shear
strains [3]. Some boundary value problems in the mechanics of
shaping of noncompact materials were solved in [3, 5, 8-14], mostly on the
basis of elliptic yield conditions or those with the Coulomb-Mohr
pyramid or the von Mises-Schlechter cone. A number of problems
were solved in [3] with the help of cylindrical yield conditions.
In the present paper, cylindrical yield conditions are used for solving
boundary value problems in the mechanics of plastic flow of porous media
in the cases where dilatancy is absent or its effect is insignificant.
For instance, no compacting is observed in the zone of slippage on
the delivery side [8] during rolling or in the matrix opening during
extrusion of a porous material [9]. Note also that yield conditions of
cylindrical type simplify the solution of boundary value problems with
discontinuous fields of velocities and strains. In such cases,
cylindrical yield conditions are convenient and effective
for the evaluation of energy-force parameters of the processes of
plastic deformation of noncompact materials and are useful for calculations of
porosity variation.
For the numerical solution of boundary value problems, the equation of the
yield surface should be supplemented by relations between the compression
yield stress, the shear yield stress, and the current porosity.
A review of most common relation of this type can be found in [15].
In general, the relations given in the present paper allow us to take into account
the geometry of the pores. As an illustration of these relations,
combined with the cylindrical yield conditions, we solve the problem of
double-action compacting of a porous axially symmetric blank, and
we also use kinematically admissible velocity fields with strong discontinuity surfaces
for solving boundary value problems related to lateral semicontinuous extrusion of
a porous mass. |
| References |
| 1. | A. M. Freudental and H. Geiringer,
The Mathematical Theories of the Inelastic Continuum [Russian translation], Fizmatgiz,
Moscow, 1962. |
| 2. | D. D. Ivlev and T. N. Martynova,
"On the theory of compressible ideally plastic media,"
PMM [Applied Mathematics and Mechanics], Vol. 27, No. 3, pp. 589-592, 1963. |
| 3. | B. A. Druyanov,
Applied Theory of Plasticity of Solids [in Russian], Mashinostroenie, Moscow, 1989. |
| 4. | S. S. Grigoryan,
"On basic concepts of soil dynamics," PMM [Applied Mathematics and Mechanics], Vol. 24, No. 4,
pp. 1057-1062, 1960. |
| 5. | T. Tabata, S. Masaki, and Y. Abe,
"A yield criterion for porous materials and analysis of axis-symmetric
compression of porous disks," Jap. Soc. Technol. Plast.,
Vol. 18, No. 196, pp. 373-380, 1977. |
| 6. | D. D. Ivlev and G. I. Bykovtsev,
Theory of Hardening Plastic Bodies [in Russian], Nauka, Moscow, 1971. |
| 7. | R. J. Green,
"Plasticity theory for porous media," Int. J. Mech. Sci., Vol. 14, No. 4, pp. 215-224,
1972. |
| 8. | S. E. Aleksandrov,
"On yield laws for porous and powder bodies,"
Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 107-112, 1994. |
| 9. | L. I. Zhivov, V. A. Pavlov, V. I. Makagon, and Yu. G. Oresov,
"Technological regimes of hot extrusion of powdered titanium," in
Theory and Practice of Powder Compaction [in Russian], pp. 146-150, Naukova
Dumka, Kiev, 1975. |
| 10. | U. Stahlberg and H. Keife,
"A powder-compacting model and its application to extrusion,"
I. Mater. Proc. Technol., Vol. 30, pp. 143-157, 1992. |
| 11. | M. B. Shtern,
"Special features of plane deformation of compacting materials,"
Poroshkovaya Matallurgiya, No. 3, pp. 14-21, 1982. |
| 12. | I. S. Degtyarev and V. L. Kolmogorov,
"Dissipation of power and kinematic relations on the velocity discontinuity surface
in a compressible plastic material," Zh. Prikl. Mekhaniki i Teoret.
Fiziki, No. 5, pp. 167-173, 1972. |
| 13. | A. G. Zalazinskii, V. I. Novozhonov, V. L. Kolmykov, and M. V. Sokolov,
"Modelling of compacting of blocks and extrusion of rods from a titanium
sponge," Metally, No. 6, pp. 64-68, 1997. |
| 14. | V. M. Segal, V. I. Reznikov, and V. F. Malyshev,
"Measurement of density of porous materials in plastic shaping,"
Poroshk. Metallurgiya, No. 7, pp. 6-11, 1979. |
| 15. | M. B. Shtern,
"On the theory of plasticity for porous bodies and powder compacting,"
in Rheologic Models and Deformation Processes of Porous Powdered and
Composite Materials [in Russian], pp. 12-23, Naukova Dumka, Kiev, 1985. |
| 16. | B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko,
Modern Geometry [in Russian], Nauka, Moscow, 1979. |
| 17. | A. G. Zalazinskii,
"Application of extremal theorems for the determination of stresses and
strains in the developed plastic flow of a composite material," Izv. AN.
MTT [Mechanics of Solids], No. 6, pp. 106-113, 1984. |
| 18. | A. G. Zalazinskii,
`Mathematical Modelling of Shaping of Structurally Inhomogeneous
Materials [in Russian], Izd-vo UrO AN SSSR, Sverdlovsk, 1990. |
| 19. | V. V. Skorokhod,
Rheologic Foundations of the Theory of Fritting [in Russian],
Naukova Dumka, Kiev, 1972. |
| 20. | P. C. T. Chen,
"Upper bound solutions to plane strain extrusion problems,"
Trans. ASME. Ser. B., Vol. 91, p. 109, 1970. |
|
| Received |
30 March 2000 |
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