Effect of elastic surface deformation on the relation between hardness and yield strength
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The use of an analytical approach to determine the relation between hardness and yield strength for materials with a high ratio of yield strength to Young’s modulus is re-examined. It is shown that predictions using the analogy of the spherical cavity fail to reproduce experimental and finite element results because the surface deflection that occurs during loading is not taken into account. A modification is proposed to allow this. This gives a greatly improved prediction of the relationship between the hardness and yield strength of a material. It also enables the effect of the indenter shape on the measured hardness to be incorporated and explains why in some very hard materials, indentation is observed to be completely elastic.
I. INTRODUCTION
Measuring the hardness of a material is a potentially simple way of determining its flow properties, particularly where these vary with position, if the sample is extremely small or the material is hard and brittle. Where materials strain harden, the hardness is not proportional to the flow stress at the onset of yielding, but to the flow stress at a characteristic strain,1–4 which for Berkovich and Vickers indenters is of the order of 0.08–0.10. Hence even when materials strain harden, their hardness can be related to a single value for the flow stress Y. Experiments show that the ratio of the hardness H to that of the flow stress Y decreases as the ratio of the flow stress to that of Young’s modulus E increases, consistent with finite element calculations.2,3,5 The effect is generally attributed to an increasing amount of the material displaced by the indenter being accommodated elastically within the body. Marsh suggested that the situation was similar to that of a gas-filled spherical cavity expanding in a body.6 Using existing solutions to this problem,7 the pressure at the cavity surface P is related to Y by the expression P=
冉冊
c 2 Y + 2Y ln 3 a
,
(1)
where a is the cavity radius and c is the radius of the plastic zone surrounding the cavity. In this case the radial movement of material du(r) varies with the distance r from the center of the cavity
DOI: 10.1557/JMR.2004.0473 3704
http://journals.cambridge.org
J. Mater. Res., Vol. 19, No. 12, Dec 2004 Downloaded: 18 Jan 2015
and is related to the expansion of the plastic zone dc, according to
冉冊
Y c du共r兲 = 3 共1 − 兲 dc E r
冉冊
2
Y r E c
− 2 共1 − 2兲
,
(2)
where is the Poisson ratio. At the cavity surface, where r ⳱ a, the rate of material movement in the outward direction is equal to the rate at which the cavity grows; that is du(r = a) ⳱ da, so that
冉冊
Y c da = 3 共1 − 兲 dc E a
2
− 2 共1 − 2兲
冉冊
Y a E c
.
(3)
This predicts that as yielding begins the plastic zone grows rapidly into the material, faster than the rate at which the cavity is expanding. However as the plastic zone spreads out, its rate of growth decreases until both plastic zone and cavity grow such that the ratio of their sizes remains constant and is given by (for Y/E Ⰶ 1)
冋
册
1
E c = a 3 共1 − 兲 Y
3
(4)
.
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