Mechanics of Solids (about journal) Mechanics of Solids
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in January 1966
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IssuesArchive of Issues2003-4pp.129-135

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R. I. Nepershin, "Indentation of an Perfectly Plastic Half-space by a Finite Wedge," Mech. Solids. 38 (4), 129-135 (2003)
Year 2003 Volume 38 Number 4 Pages 129-135
Title Indentation of an Perfectly Plastic Half-space by a Finite Wedge
Author(s) R. I. Nepershin (Moscow)
Abstract We consider unsteady plane plastic flow of a perfectly plastic half-space indented by a wedge of finite width, with contact friction taken into account. The first stage of the indentation is described by a self-similar solution which preserves geometrical similarity of the plastic region. The second stage of indentation is modelled by a field of slip lines and a velocity hodograph with a slip line that corresponds to pure shear and continues up to the boundary of the half-space. We determine the shape of the boundary of the half-space near the wedge and find how the force acting on the wedge depends on the indentation depth, from the point of its contact with the half-space to the steady flow of an unbounded plastic medium about the wedge.

Self-similar solutions are known for unsteady problems of plane plastic flow of a perfectly plastic half-space indented by an infinite smooth wedge (the cases of rectilinear and curvilinear boundaries were considered in [1] and [2], respectively). An unsteady problem for a thin smooth blade indenting an ideal plastic half-space was solved in [3] by the small parameter method.

In [2, 4], the initial stage of the indentation of a smooth wedge of finite width into an perfectly plastic half-space was studied under the assumption that the rigid-plastic boundary moves along the initial boundary of the half-space with the formation of a curvilinear rigid region near the wedge. There are some finite-element calculations of the initial indentation of rigid punches into an elastic-plastic half-space [5-7]. However, the large distortion of the deformed boundary near the edge of the punch, because of singular nature of plastic flow in that region, is a great obstacle to the application of elastic-plastic modeling.

In [8], the unsteady indentation of a plane smooth punch into a perfectly plastic half-space is modeled by a field of slip lines and the hodograph of velocities with an isolated line of pure shear which reaches the boundary of the half-space. Such a model can be used for studying the entire process of unsteady indentation up to the steady-state stage of motion of the punch in an unbounded medium. This approach is taken in the present paper in order to model indentation of a finite wedge from its initial point contact with the half-space up to its steady motion in the unbounded perfectly plastic medium, with contact friction taken into account.
1.  R. Hill, The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1985.
2.  G. I. Bykovtsev and D. D. Ivlev, Theory of Plasticity [in Russian], Dal'nauka, Vladivostok, 1998.
3.  D. D. Ivlev, Theory of Ideal Plasticity [in Russian], Nauka, Moscow, 1966.
4.  G. I. Bykovtsev and A. I. Khromov, "Plane deformation of perfectly elastic-plastic bodies with variable boundaries," Izv. AN SSSR. MTT [Mechanics of Solids], No. 2, pp. 71-78, 1979.
5.  C. H. Lee and S. Kobayashi, "Elastoplastic analysis of plane strain and axisymmetric flat punch indentation by the finite element method," Intern. J. Mech. Sci., Vol.  12, No. 4, pp. 349-370, 1970.
6.  C. Hardy, C. N. Baronet, and G. V. Tordion, "The elasto-plastic indentation of a half-space by rigid sphere," Intern. J. Num. Mech. Engng., Vol. 3, No. 3, pp. 451-462, 1971.
7.  G. Z. Voyiadjis and N. E. Buckner, "Indentation of a half-space with a rigid indentor," Intern. J. Num. Mech. Engng, Vol. 19, No. 10, pp. 1555-1578.
8.  R. I. Nepershin, "Indentation of a rigid-plastic half-space by plane punch," PMM [Applied Mathematics and Mechanics], Vol. 66, No. 1, pp. 140-146, 2002.
Received 25 September 2001
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