Validity of the reduced modulus concept to describe indentation loading response for elastoplastic materials with sharp
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Oliver Kraft and Ruth Schwaiger Forschungszentrum Karlsruhe, Institute for Materials Research II, 76344 Karlsruhe, Germany; and Universita¨t Karlsruhe, Institut fu¨r Zuverla¨ssigkeit von Bauteilen und Systemen, 76131 Karlsruhe, Germany (Received 7 August 2008; accepted 31 October 2008)
Recent computational parametric studies have developed reverse algorithms to extract material properties of elastoplastic materials using experimental sharp nanoindentation. These methods used reduced modulus in their parameters to include the effect of indenter compliance. To investigate the validity of using reduced modulus, we conducted experimental indentation of a couple of representative cases for elastoplastic metals with a diamond and a sapphire Berkovich tip. Then, we performed a finite element study for sharp indentation of the same material systems. Both computational and experimental results indicate that the use of reduced modulus is invalid to describe indentation loading response for elastoplastic materials in a certain material regime. Our results show that indenter compliance is overestimated by the previous predictions using reduced modulus. This overestimation leads to underestimation of indenter curvature and causes error in extracting material properties by reverse algorithms.
I. INTRODUCTION
Depth-sensing indentation has been widely used to extract material properties from its force and displacement curves. The Oliver–Pharr method1 has been the most common analysis to get Young’s modulus and hardness from its unloading curves. The analysis method was developed assuming that the indenter was rigid. Thus, the effect of indenter compliance was taken into account by the reduced modulus Er that is defined as: 1 1 n2i 1 n2s ¼ þ Er Ei Es
;
ð1Þ
where Ei is Young’s modulus of the indenter, Es is Young’s modulus of the sample, ni is Poisson’s ratio of the indenter, and ns is Poisson’s ratio of the sample. It is assumed that the indentation response for unloading is the simple sum of the reciprocal Young’s modulus of nonrigid indenter and sample material. Using the reduced modulus is similar to the common and phenomenological practice without any theoretical justification. This practice is conceptually acceptable for simple elastic contact problems, but for more complicated problems such as sharp indentation, this concept may not always be applicable. A recent study by Lim and a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2009.0120
998
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J. Mater. Res., Vol. 24, No. 3, Mar 2009 Downloaded: 05 Dec 2014
Chaudhri2,3 indicated the possible mistake of using the reduced modulus, and then Cao et al.4 found that the application of reduced modulus may induce significant errors when a sample has comparable modulus with an indenter. The reduced modulus concept is also central to reverse algorithms, which were developed to extract materials properties of elastoplastic materials from both experimental loading and unloading curves.5–11 Reverse alg
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