  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
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R. I. Nepershin, "Indentation of an Perfectly Plastic Halfspace by a Finite Wedge," Mech. Solids. 38 (4), 129135 (2003) 
Year 
2003 
Volume 
38 
Number 
4 
Pages 
129135 
Title 
Indentation of an Perfectly Plastic Halfspace by a Finite Wedge 
Author(s) 
R. I. Nepershin (Moscow) 
Abstract 
We consider unsteady plane plastic flow of a perfectly plastic halfspace indented by a wedge of finite width, with contact friction taken into account. The first stage of the indentation is described by a selfsimilar 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 halfspace. We determine the shape of the boundary of the halfspace 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 halfspace to the steady flow of an unbounded plastic medium about the wedge.
Selfsimilar solutions are known for unsteady problems of plane plastic flow of a perfectly plastic halfspace 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 halfspace 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 halfspace was studied under the assumption that the rigidplastic boundary moves along the initial boundary of the halfspace with the formation of a curvilinear rigid region near the wedge. There are some finiteelement calculations of the initial indentation of rigid punches into an elasticplastic halfspace [57]. 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 elasticplastic modeling.
In [8], the unsteady indentation of a plane smooth punch into a perfectly plastic halfspace 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 halfspace. Such a model can be used for studying the entire process of unsteady indentation up to the steadystate 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 halfspace up to its steady motion in the unbounded perfectly plastic medium, with contact friction taken into account. 
References 
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 elasticplastic bodies with variable boundaries," Izv. AN SSSR. MTT [Mechanics of Solids], No. 2, pp. 7178,
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. 349370, 1970. 
6.  C. Hardy, C. N. Baronet, and G. V. Tordion,
"The elastoplastic indentation of a halfspace by rigid sphere,"
Intern. J. Num. Mech. Engng., Vol. 3, No. 3, pp. 451462, 1971. 
7.  G. Z. Voyiadjis and N. E. Buckner,
"Indentation of a halfspace with a rigid indentor,"
Intern. J. Num. Mech. Engng, Vol. 19, No. 10, pp. 15551578. 
8.  R. I. Nepershin,
"Indentation of a rigidplastic halfspace by plane punch,"
PMM [Applied Mathematics and Mechanics], Vol. 66, No. 1, pp. 140146, 2002. 

Received 
25 September 2001 
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