Implications of the idea of effective tip shape on nanoindentation unloading curves: AFM measurements and FE simulation
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Flavio A. Bonilla Asylum Research, Santa Barbara, California 93117 (Received 5 May 2011; accepted 7 July 2011)
The analysis of nanoindentation force data are based on Sneddon’s solution for a linear elastic half space with a rigid axisymmetric indenter. Berkovich indenters commonly used in indentation experiments are normally modeled as cones. The idea of effective tip shape was presented to better explain the behavior of the unloading curve and pressure distribution under the tip in real experiments. We examine the concept of effective tip in three dimensions by importing real indenter metrology by atomic force microscope directly into finite element analysis and simulate fused silica indentation experiments. We show that fitting the elastic reloading curves overestimates the elastic modulus of fused silica. This is explained by studying the pressure distribution at maximum depth under the effective tip. While the effective tip describes the problem geometrically, it is believed that neglecting the deformed zone in the indented material is responsible for over estimating the modulus value. I. INTRODUCTION
Nanoindentation is an important tool to measure the mechanical properties of materials at submicron scale.1 Instrumented indenters are able to monitor the variation of load with the indentation depth during loading and unloading. The extracted force curve contains information about the mechanical properties of the indented material. The most frequent quantity that has been evaluated from the unloading section of the curve is the elastic modulus.1–7 Elastic modulus evaluation is fundamentally based on the solution provided by Sneddon8,9 for an axisymmetric rigid punch indenting a linear elastic half space.1 Nanoindentation experiments generally involve elastic as well as plastic deformation of materials. It has been proven for most materials that the unloading portion of the force curve is elastic.2,3,10 Such finding enabled the incorporation of the analytical closed-form solutions of contact mechanics to estimate the elastic modulus of indented materials. The analytical solutions of indentation problems are only derived for axisymmetric shapes and due to the wide use of Berkovich indenters, it was intuitive to approximate the three-sided pyramid tip into a cone and use the analytical solution to estimate the elastic modulus. Oliver and Pharr 2 proposed a power law function to describe the unloading section of the force curve in the form F ¼ aðh hr Þm
;
ð1Þ
where hr is the residual depth after unloading. a and m are fitting parameters. Interestingly, values regularly obtained for m are between 1.2 and 1.6. This is different from the analytical solution provided by Sneddon8,9 which results in m of 2 for conical tips. To explain this discrepancy, Bolshakov et al.10 utilized the idea of effective tip shape. By utilizing twodimensional (2D) finite element analysis (FEA), they showed that a better representation for the problem of a conical tip unloaded from a plastic indent is a parabola of revolution elastically in
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