弹性模量与硬度关系

1.5
v=0.15
1.4
v=0.20
1.3
v=0.25
v=0.30
1.2
E / Er
1.1
1
0.9 0
100
200
300
400
Er (GPa )
Fig. 1. Calculated elastic modulus as a function of the reduced modulus for a diamond indenter (Ei = 1141 GPa, vi = 0.07), varying with the PoissonÕs ratio.
Received 7 April 2004; accepted 3 August 2004
Abstract
An analytical relationship between the reduced modulus Er and hardness H for solid materials is established based on the conventional depth-sensing indentation method of Oliver and Pharr. It is found that the two properties are related through a material parameter that is defined as the recovery resistance Rs. This parameter is shown to represent the energy dissipation dupriffinffiffigffiffiffi indentation. Based on indentation measurements with the use of a Berkovich indenter, the relationship is given as Er ¼ 0:6647 HRs. Also presented is a simple set of procedures to determine the area of indent. The procedures require three measured quantities, i.e., the peak load and corresponding displacements as well as the depth of residual indentation, but do not require complicated curve fitting process and regression analysis which themselves involve the specimen material. Nano-indentation tests were conducted using a Berkovich indenter on five materials spanning a wide range of hardness and plasticity. Experimental results revealed two important features: (a) the reduced modulus predicted by the new Er–H relationship is the same as that obtained by the conventional method; (b) the elastic modulus and hardness values determined by the simple set of procedures are comparable to those obtained by using the conventional method. Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
H ¼ ð2=3Þ½1 þ lnðE cos h=3ryÞry:
ð1Þ
Since its theoretical basis is not clear, the range of applicability of Eq. (1) might be limited. A simple examination was made in the case of Ti3SiC2 whose hardness (4 GPa), compressive yield strength (1 GPa) and modulus (310 GPa) were known [4]. By Eq. (1), a modulus value as high as 1188 GPa was obtained, which was much higher than the measured value of 310 GPa. The unsatisfactory prediction of elastic modulus from hardness leads to two essential questions: (a) Does there exist a theoretical and generally valid relationship between elastic modulus and hardness? (b) If there is an inherent E–H relationship for solid materials, what is it?
1359-6454/$30.00 Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.08.002
5398
Y.W. Bao et al. / Acta Materialia 52 (2004) 5397–5404
Keywords: Nano-indentation; Er–H relationship; Recovery resistance; Energy dissipation
1. Introduction
Elastic modulus E and hardness H are two essential parameters of structural materials, and the relationship between them is of keen interest to material scientists. From statistical trend, elastic modulus is usually considered to be an increasing function of hardness [1], but this rule neither has analytical support nor is generally obeyed. For example, layered ternary ceramics that have low hardness and high modulus are exceptions to this rule. Another E–H relation, often used in evaluating the elastic modulus of coatings, involves the compressive
Acta Materialia 52 (2004) 5397–5404
www.actamat-journals.com
Investigation of the relationship between elastic modulus and hardness based on depth-sensing indentation measurements
The advancement of indentation technique has made it possible to evaluate conveniently the elastic modulus and hardness by means of accurately measured loaddepth data [5–7]. Furthermore, the depth-sensing indentation technique documented by Oliver and Pharr [5] has provided a clue for establishing the theoretical relationship between elastic modulus and hardness. In this technique the elastic modulus of the specimen is related to the measured reduced modulus Er by
* Corresponding author. Tel.: +86 24 23971763; fax: +86 24 23971215.
E-mail address: ywbao@imr.ac.cn (Y.W. Bao).
yield stress ry and the half angle of indenter h (68° for VickersÕ indenter) [2,3]
Y.W. Bao *, W. Wang, Y.C. Zhou
Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, PR China
1 ¼ 1 À m2 þ 1 À m2i ;
ð2Þ
Er
Eபைடு நூலகம்
Ei
where E and m are the elastic modulus and PoissonÕs ratio of the specimen, and Ei and mi are those of the indenter. Thus, measuring the reduced modulus becomes the key for determining the elastic modulus of the test specimen. Since Ei and mi are known, the ratio of the elastic modulus to the reduced modulus, E/Er is a function of both the reduced modulus and PoissonÕs ratio, as shown in Fig. 1. The calculation implies that either the increase of the Er-value or the decrease of the PoissonÕs ratio of the test sample would lead to an increase in the value of E/Er. It is interesting that, for the reduced modulus, there is a threshold value (Er = E) depending on the value of the PoissonÕs ratio, below which the reduced modulus is higher than the elastic modulus of the test sample, rather than reduced.