The critical load on a punch

Abstract

A PUNCH UDC 622. 002.3 622,23 : 621.9 The problem of impression of a punch at a boundary formed by two or three free planes can be used to determine the forces required to fracture a materiaI (solid rock) by a cutting tool. This problem is a particular case of the impression of a punch into a half-space, and is distinguished only by the presence of additional free planes on which all components of the stress tensor vanish. This enables us to use a solution of the well-known contact problem, on which we impose certain boundary conditions. We will use the approximate method of Schleicher [1,2]: the whole area of the foot of the punch is divided into a series of dements, and the load acting on each of them is assumed to be uniformly distributed with a mean pressure PX"Corresponding to the general formulation of the problem, we consider the impression of rectangular punches only, with side ratios h : b = 1 : 3, 1 : 1, and 3 : 1 (Fig. 1). The figure is divided up into nine elements. The results of determinations of the pressures on the foot of the punches for a half-space are listed in Table 1.Using the solution of Boussinesq for a point force, we calculated the normal stresses in planes passing through the perimeters of the punches perpendicular to their feet ( Fig. 1) and those occurring when these punches are impressed at the boundary of the free planar exposed surfaces of a bench ( Table 2).The results of the calculation enabled us to obtain the relation between the pressures on the elements of the figure, within the adopted precision of satisfaction of the boundary conditions at the stress-free surface [4] (Table3). But the sum of the average pressures over all the elements of the foot of each punch must be X~9 V Px = 9P,n-Therefore, to obtain the actual pressure distribution over the feet of the punches when they are impressed into an elastic quarter-space with an average pressure of Pro, it is necessary to multiply the data in Table 3 by the corresponding ratio for each punch, 9Pm/Ep' x, i.e., t where PX is the pressure in the X-th element in Table 3.