On the usage of the effectively shaped indenter concept for analysis of yield strength
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Using the extended Hertzian approach (EHA), the “effectively shaped indenter” corresponding to Pharr’s concept is described in terms of a parameter set {d0,d2,d4,d6}, which can be determined by a fitting procedure from the unloading curve of an indentation experiment. Owing to the limited accuracy of measurement, a given experimental curve may in principle correspond to more than one such parameter set. Based on indentation experiments with a Berkovich indenter into fused silica, we have investigated the influence of the fitting procedure itself on the results. We suggest a certain manual fitting procedure, which delivered a yield strength Y = (7.1 0.1) GPa independent of the maximum load. Manual fitting always includes some degree of subjectivity, however, both Y and the elastic field as a whole proved to be relatively robust against modifications of the parameter set. We also suggest a preliminary objective procedure, which delivered Y = 6.8 7.1 GPa. In addition, we have performed finite element method (FEM) simulations of elastic–plastic indentations of a conical indenter into a von Mises solid with a yield strength of Y = 7.0 GPa. The simulated unloading curve was analyzed using the EHA in the same manner as the experimental curves, and yield strength of 6.95 GPa was obtained being very close to the input value of the FEM.
I. INTRODUCTION
Nanoindentation at submicron scale as a suitable technique to measure mechanical properties of materials has been established over the last decade. In particular, this method has been widely adopted to determine Young’s modulus and hardness of bulk materials and thin films. A major step in the understanding of the indentation response of materials has been brought about by the introduction of the concept of the “effectively shaped indenter” by Pharr and Bolshakov.1,2 In this concept, the real indenter penetrating into the plastically deformed hardness impression is substituted by an imaginary so-called “effectively shaped indenter,” which delivers the same unloading curve when penetrating into a flat elastic half space. The shape of the “effective indenter” is derived from the inspection of the unloading curve, which is assumed to be wholly elastic. With this approach, the theoretical treatment of nanoindentation was enormously extended, so that elastic–plastic contact situations—the most likely situation in the case of sharp indenter tips—can be assessed on the basis of a linearelastic contact theory. Oliver and Pharr1 were able to derive the normal displacement and the contact pressure distribution at the surface due to the acting effective a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2009.0125
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J. Mater. Res., Vol. 24, No. 3, Mar 2009
indenter by using the results of Sneddon.3 Later, this approach was extended in the form of Schwarzer’s extended Hertzian approach (EHA), which made it possible to compute the complete elastic stress and strain fields within the indented body.4,5 Moreover, Schwarze
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