 | | 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 |
Archive of Issues
| Total articles in the database: | | 11223 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8011
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| In English (Mech. Solids): | | 3212 |
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| D. D. Ivlev, A. Yu. Ishlinskii, and R. I. Nepershin, "On the indentation of a rigid pyramid into an ideally plastic half-space," Mech. Solids. 37 (4), 47-51 (2002) |
| Year |
2002 |
Volume |
37 |
Number |
4 |
Pages |
47-51 |
| Title |
On the indentation of a rigid pyramid into an ideally plastic half-space |
| Author(s) |
D. D. Ivlev (Moscow)
A. Yu. Ishlinskii (Moscow)
R. I. Nepershin (Moscow) |
| Abstract |
A self-similar solution is presented for the problem of the indentation
of a rigid pyramid with square or triangular base into an ideally plastic
half-space in the case of complete plasticity, with contact friction
on the pyramid faces being taken into account. This problem models material
hardness testing by the indentation of a rigid pyramid. The pressure on the
pyramid and the shape of the plastic impression are in satisfactory agreement
with experimental data.
Complete plasticity of an ideally plastic body corresponds to the edges
of the Treska-Saint-Venant prism in the space of principal stresses.
It is only in the case of complete plasticity that the spatial shear
deformation along two slip planes in which the tangential stress
attains the shear yield limit is possible. The spatial problem of ideal
plasticity in the case of complete plasticity is statically determinate,
hyperbolic, and can be useful in applications [1-5].
The characteristic relations for stresses and displacement rates
for the spatial problem in the case of complete plasticity are presented
in [6] and it is shown that the familiar relations for the plane and
axisymmetric deformation are particular cases of the relations
of the general spatial problem. These relations are applied in [6]
for determining the pressure of flat punches with various shapes
of contact surfaces on an ideally plastic half-space.
In the present paper, we apply the characteristic relations of [6]
to solve a spatial self-similar problem of the indentation of a rigid
pyramid into an ideally plastic half-space, with contact friction
on the pyramid faces being taken into account. This problem models
hardness tests of metals by indenting a rigid pyramid. The shape
of the plastic impression and the pressure on the pyramid satisfactorily
agree with experiments [7]. |
| References |
| 1. | A. Yu. Ishlinskii, "Spatial deformation of incompletely
elastic and viscoplastic bodies,"
Izv. AN SSSR. OTN, No. 3, pp. 250-260, 1945. |
| 2. | A. Yu. Ishlinskii, "Axisymmetric problem of plasticity
and Brinnel tests," PMM [Applied Mathematics and Mechanics],
Vol. 8, No. 3, pp. 201-224, 1944. |
| 3. | A. Yu. Ishlinskii, Applied Problems of Mechanics. Volume 1 [in
Russian], Nauka, Moscow, 1986. |
| 4. | A. Yu. Ishlinskii and D. D. Ivlev, "Mathematical Theory
of Plasticity [in Russian], Fizmatlit, Moscow, 2001. |
| 5. | D. D. Ivlev and A. Yu. Ishlinskii, "Complete plasticity
in the theory of an ideally plastic body,"
Doklady AN, Vol. 368, No. 3, pp. 333-334, 1999. |
| 6. | D. D. Ivlev, A. Yu. Ishlinskii, and R. I. Nepershin,
"On the characteristic relations for stresses and displacement rates
in the case of complete plasticity,"
Doklady AN, Vol. 381, No. 5, pp. 616-622, 2001. |
| 7. | D. S. Dugdale, "Experiments with pyramidal indenters,"
J. Mech. and Phys. Solids, Vol. 3, No. 3, pp. 197-205, 1955. |
|
| Received |
30 April 2002 |
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