On the constraint factor associated with the indentation of work-hardening materials with a spherical ball
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I.
INTRODUCTION
IT is well known
that the indentation hardness of ductile materials is about 2.8 to 3 times their uniaxial flow stress values. [1,21The above increase in the resistance to plastic flow, under indentation conditions, arises mainly because the plastic zone underneath the indentor is confined within a larger volume of the material which is either elastic or rigid. The stress state in the plastic zone is one of hydrostatic compressive stress plus shear, with only the shear component being responsible for the plastic flow. Thus, the plastic deformation underneath the indentor is constrained unlike in the uniaxial tension or compression tests. The factor by which the resistance to plastic flow under indentation conditions (i.e., hardness) is higher than the uniaxial flow stress value is defined as the constraint factor (CF). The present investigation is concerned with the indentation of metallic materials with spherical indentors, i.e., balls. The CF for a spherical indentor indenting a flat, semi-infinite material has been estimated by a number of investigators. I3-9] While some investigators like Ishlinsky t3] and Richmond et al. [41 have considered the spherical indenter to be rigid and the indented material to be rigid, perfectly plastic, Johnson [5'9~ has modeled the indentation of an elastic, ideally plastic half-space by an elastic indenter. The numerical modeling of a similar problem has been carried out by Hardy et al.t61 The problem of indentation of a fully plastic and a strainhardening half-space by a rigid/elastic indentor appears analytically intractable. However, finite element modeling (FEM) analyses of this problem have been performed by Lee et al. [71 and Follansbee and Sinclair. [8] An approximate analysis of the indentation by a sphere due to Matthews tl~ has the ability to predict the CF values Y. T I R U P A T A / A H and G. S U N D A R A R A J A N , Scientists, are with the Defence Metallurgical Research Laboratory, Hyderabad 500 258, India. Manuscript submitted July 10, 1990. METALLURGICAL TRANSACTIONS A
for all types of half-space, i.e., elastic, elastic-plastic, and rigid-plastic. The models which assume the indented half-space to be rigid, perfectly plastic, or fully plastic strain hardening predict a constant CF value in the range of 2.8 to 3.1 in the case of spherical indenters. [1'3,4'7's] In contrast, the models based on elastic-plastic half-space predict a CF value in the range of 1 to 3, depending on the magnitude of the parameter ( E e / Y ) " ( a / r ) , where Ee is the effective elastic modulus of the ball-indented material system, Y is the yield stress of the indented material, a is the radius of the circular indentation formed, and r is the ball radius. 15,91 Experimental data on the constraint factor and on the nature of deformation underneath a spherical indenter have been provided by Tabor, [IJ O'Neill, [2] Samuels and Mulhearn, [l~ Francis, ti2~ and Tirupataiah and Sundararajan. [~3] The above investigations have clearly revealed that in the case of spherical
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