Theoretical analysis of the relationships between hardness, elastic modulus, and the work of indentation for work-harden
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Theoretical analysis of the relationships between hardness, elastic modulus, and the work of indentation for work-hardening materials Rong Yang State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China; and Graduate University of Chinese Academy of Sciences, Beijing 100049, China
Taihua Zhanga) State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China
Yihui Feng State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China; and Graduate University of Chinese Academy of Sciences, Beijing 100049, China (Received 17 May 2010; accepted 11 August 2010)
In our previous paper, the expanding cavity model (ECM) and Lame´ solution were used to obtain an analytical expression for the scale ratio between hardness (H) to reduced modulus (Er) and unloading work (Wu) to total work (Wt) of indentation for elastic-perfectly plastic materials. In this paper, the more general work-hardening (linear and power-law) materials are studied. Our previous conclusions that this ratio depends mainly on the conical angle of indenter, holds not only for elastic perfectly-plastic materials, but also for work-hardening materials. These results were also verified by numerical simulations. Over the past two decades, instrumented indentation has gradually evolved into a conventional testing method for measuring mechanical properties of materials at small-scale and in the process has advanced our understanding of mechanical behavior of materials.1 The most frequently used analytic method, developed by Oliver and Pharr,2 is based on the solutions of elastic contact; thus its estimation of contact area has been found to be incapable of accounting for pileup behavior of material. Another method based on dimensional analysis and finite element calculations, proposed by Cheng and Cheng3 in 1998, uses an approximate linear relation of (H/Er)/ (Wu/Wt) to overcome the dependence on the uncertain contact area. Recently, many questions have been raised in regard to this scale ratio relation. Alkorta et al.4 and Malzbender5 indicated that this method can incur significant error for soft materials, while Chen and Bull6 found that when H/Er is larger than 0.1, significant deviation occurs in this relation. In our previous paper,7 analytical approaches were adopted to uncover the physical nature of this scale ratio, and it was found that for elastic perfectly-plastic case, the scale ratio has the form [2(1 n) a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2010.0267
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J. Mater. Res., Vol. 25, No. 11, Nov 2010
cot a]/3, where n is Poisson’s ratio and a is the halfincluded angle of the conical indenter; in the linear elastic regime, this ratio reduces to the form (cot a)/2. In this paper, the same approach is taken with certain relaxed constraints to investigate work-hardened materials. The leading order of the
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