Indentation Analysis for General Shapes of Surface Boundaries and Punch Profiles
Authors: Schneider W.C. , Cheatham J.B. 
Published: May 2022

Abstract
When a punch indents a plastic material, the surrounding surface is frequently of an irregular shape and the surface of the punch is often curved or rounded rather than sharp and straight. In this paper equations are presented for the determination of the pressure presented for the determination of the pressure beneath a punch of general profile. The usual procedure for determining the pressure beneath procedure for determining the pressure beneath a punch indenting a plastic material is to assume a slip-line field and then compute the pressure on the basis of that assumption. pressure on the basis of that assumption. In this paper we wish to free the designer from the need to assume a slip-line field when calculating the pressure beneath a punch. To this end, we present Theorem I, which reveals that the pressure is independent of the slipline shape but depends only on its terminal points. For many practical problems, points. For many practical problems, information about the terminal points is known even though the slip-line shape is unknown. Equations for the pressure distribution as well as the external loads needed to cause incipient plastic flow are given. Examples are presented to illustrate their use. presented to illustrate their use. For nonsymmetrical problems the concept of a slip-line "triple point" is presented to determine whether the right or left boundary should be used to determine pressure at the point in question. A Tresca's yield condition point in question. A Tresca's yield condition is assumed in the examples considered; however, extension to a Coulomb plastic should present no difficulties.

Introduction
In past analyses most of the plane-strain plasticity solutions for the indentation of plasticity solutions for the indentation of plastic materials such as rocks under pressure, plastic materials such as rocks under pressure, soils and metals have applied-to situations for which the pressure on the indenter is uniform. For example, sharp wedges, either smooth or rough, indenting a smooth flat half-space or acting adjacent to a straight inclined edge have uniform pressure at the interface between the punch and the plastic material. If the indenter is curved and/or the material being indented has a curved or irregular, shape, then the pressure distribution will be nonuniform in most cases. In the present paper theorems are presented that permit the analysis of presented that permit the analysis of indentation problems for which the indentation pressure is nonuniform. pressure is nonuniform. In Theorem I we prove that the indentation pressure at a point on a punch depends only on pressure at a point on a punch depends only on the angle between the tangents to the punch surface and the boundary at the two ends of a slip-line, through the point in question. Fig. 1A schematically illustrates the slip-line field adjacent to a curved punch which is indenting a curved boundary of plastic material We are concerned here only with incipient flow for the configuration shown. After the punch has penetrated an incremental distance into the plastic body it will be necessary to calculate plastic body it will be necessary to calculate the new surface configuration by use of the appropriate velocity field. The angle Theta denote the inclination of a given surface with respect to the horizontal axis while the angle Psi determines the angle of inclination of a slip line with respect to a given surface at a point. On the right side of the punch the point. On the right side of the punch the tangent to the punch surface at A forms an angle Theta A to the horizontal axis and the a slip line is inclined at the angle Psi A to the tangent plane at A. Likewise at the free surface, B, plane at A. Likewise at the free surface, B, the tangent-plane angle's Theta B and slip-line direction is Psi B. Thus the a slip line turns through an angle [Psi B - Theta B + Psi A - Theta A] from Point B to Point A. The same procedure can be applied to the left side of the punch from Point D to C.