Critical examination of the two-slope method in nanoindentation
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n this paper, we derive corrected analytical expressions for calculating the hardness and modulus by the two-slope method. This method relies on the determination of the slopes of the loading and unloading curves rather than the indenter displacement as an input. These expressions take into account the correction factor ␣ in the fundamental relations among contact stiffness, elastic modulus, and contact area, which is frequently forgotten or misused in the literature. It is shown that these corrected expressions allow measurements of the hardness and modulus in very good agreement with the commonly used technique based on the determination of the contact area. Additionally, the correction factor ␣ can be easily determined if Young’s modulus of the material is known.
I. INTRODUCTION
An analysis technique, which is generally not used in the literature, to our knowledge, has been formulated by Oliver1 for calculating hardness, modulus, and contact area from indentation data. This technique will be referred to here as the two-slope method because it relies on the calculation of the slopes of the loading and unloading curves, which represent the loading and unloading contact stiffness, respectively, when they are calculated at the maximum penetration depth hmax. The twoslope method is based on the same set of equations and assumptions used in the Oliver and Pharr (O&P) analysis,2 and on the P-h2 relationship describing the loading curve as follows1 P = Er
冉 冑 1
公C
Er ⑀ + H 
冑冊
公 2
H Er
−2 2
h
,
(1)
where Er is the reduced elastic modulus and H the hardness. The factor C comes from the relation Ac ⳱ Ch2c between the contact area Ac and the contact depth hc, where C ⳱ tan2 for a conical indenter. In the case of a Berkovich indenter, the angle that gives the same depth-to-area ratio as the perfect cone is ⳱ 70.32° and thus C ⳱ 24.56. The factor ⑀ comes from the relation hc ⳱ hmax − ⑀Pmax/Su used in the O&P method, in which Su is the unloading contact stiffness measured at the
a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2005.0272 2194
http://journals.cambridge.org
J. Mater. Res., Vol. 20, No. 8, Aug 2005 Downloaded: 24 Mar 2015
maximum depth of penetration and is given in Oliver’s paper1 by the following expression Su = 
2
公
Er公Ac .
(2)
The factor  appearing in Eq. (2), according to Oliver, is supposed to take the constant value  ⳱ 1.034 for correcting the fact that the Berkovich indenter is not axially symmetric.3 By differentiating Eq. (1) with respect to the displacement, the slope Sl of the loading curve is obtained and finally the expressions of the reduced elastic modulus and hardness formulated by Oliver are1
冉
Er =
冑
H=
SuSl 1 CP 2Su − ⑀Sl
1 S2uSl C 2P 2Su − ⑀Sl
冉
冊
冊
,
(3)
2
.
(4)
Therefore, calculating the loading and unloading slopes at maximum load, which are the loading and unloading contact stiffness, respectively, allows direct calculation of elastic modulus and hardness without obligation to determine the
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