An analytical approach for relating hardness and yield strength for materials with high ratio of yield strength to Young
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An analytical approach for relating hardness and yield strength for materials with high ratio of yield strength to Young modulus Luc J. Vandeperre, Finn Giuliani and William J. Clegg Ceramics Laboratory, Department of Materials Science and Metallurgy, University of Cambridge, Cambridge, CB2 3QZ, United Kingdom ABSTRACT For materials with a high ratio of Y to the elastic modulus, E, experimental data show that the ratio of the hardness to the flow stress decreases from 3 toward 1 as Y / E increases. This behaviour is predicted by finite element calculations but to date analytical expressions have not been able to correctly predict the relation between Y and H nor have they been able to show how the geometry of the indenter is important. Therefore, in this paper the correlation between H and Y for such materials is re-examined using an analytical approach to provide a physical interpretation, which explains the trends observed.
INTRODUCTION Hardness testing is a straightforward way to characterise the mechanical properties of hard, brittle materials. For rigid plastic materials, Tabor [1] showed that the hardness, H, is about three times the yield strength, Y, consistent with experimental results. However, for materials with a high ratio of Y to the elastic modulus, E, experimental data show that this is incorrect with the ratio of the hardness to the flow stress decreasing toward 1 as Y / E increases (see figure 1). This behaviour is predicted by finite element calculations but to date analytical expressions have not been able to correctly predict the relation between Y and H nor have they been able to show why the geometry of the indenter is important. Therefore, in this paper the correlation between H and Y for such materials is re-examined using an analytical approach to provide a physical interpretation, which explains the trends observed.
ANALYSIS The deformation of an elastic substrate when a rigid cone is pressed into it has been considered by Sneddon [6] who showed that the mean pressure, Pm, at which equilibrium is reached between cone and substrate is given by Pm =
E 2 (1 − ν 2 ) tan α
(1)
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where α is the included semi-angle of the cone. Sneddon’s treatment did not include the possibility of plastic flow, although some plastic deformation would be expected beneath a sharp indenter. However, even when plastic flow occurs, an equilibrium pressure, the hardness, is reached at which the indenter does not sink further into the substrate. As E and ν are material properties, this might be considered as equivalent to indenting an elastic material with these properties at a mean pressure equal to the measured hardness with an indenter whose included semi-angle can be obtained from the expression H=
E 2 (1 − ν 2 ) tan α ∗
(2)
where H = Pm.
Figure 1. Variation of H / E with Y / E as determined experimentally [2] or with finite element calculations [3] compared with various analytical models: (i) Tabor’s [1] analysis for rigid plastic materials, (ii) Marsh’s [2] approach using Hill’s [4] ex
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