Influence of Nanoindenter Tip Radius on the Estimation of the Elastic Modulus
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Influence of Nanoindenter Tip Radius on the Estimation of the Elastic Modulus Karim R. Gadelrab and Matteo Chiesa. Laboratory for Energy and NanoScience (LENS), Masdar Institute, United Arab Emirates (UAE) ABSTRACT Nanoindentation results are very sensitive to tip rounding and neglecting the value of the tip radius produces erroneous estimation of the material elastic properties. In this study we investigate the effect of tip radius on the estimation of the Elastic modulus by means of finite element analysis of Berkovich and conical tips with different tip radii. Our numerical results were already supported by an experimental study on fused silica with Berkovich tips with different tip radii. The use of classical Oliver Pharr equation overestimated the Elastic modulus. A new analytical model that modifies the Oliver Pharr equation to consider the value of the tip radius is employed to derive the Elastic modulus from load displacement curves yielding improved results compared to the classical Oliver Pharr equation. INTRODUCTION Nanoindentation is widely used for characterizing the elastic properties of nano scale materials. The analytical models that are commonly used to extract these properties are based on Sneddon’s assumptions [1]. Oliver and Pharr [2] used the mathematical solution derived by Sneddon to estimate the Elastic modulus of the indented material from the elastic unloading curve. Tip rounding is a deviation from the perfect sharp tip assumption used by Sneddon. This results in an overestimation to the Elastic modulus of the material and this becomes even more pronounced at shallow indentations under 200nm[3]. In previous work [4], indentation in elastic solids by a Berkovich indenter was studied to show the effect of tip rounding on the estimated modulus. A new analytical model that takes into account for the tip radius was proposed. The model reads
Er
S 2
S
Aber 2 R hc
hc
hmax H
1/ 2
P S max ,
2
S
(1)
Ablunt
, (2)
where S is the slope of the unloading curve at the max force. Aber is the projected area at the contact depth hc. The contact depth hc is calculated by knowing the maximum depth hmax, maximum force Pmax, and the slope at maximum force. The value of ε was shown in the previous study [4] to be 0.785 in accordance to the work done by Martin and Troyon [5]. R is the tip
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radius and Ablunt is the projected area at depth hb. hb is the vertical distance between perfect sharp tip and a blunted one and can be calculated as follows[6] § 1 · R¨ 1¸ , © sin T ¹
(3)
Ablunt S (tanT ) 2 hb2 .
(4)
hb
and
θ is the cone half angle for a conical indenter. The model was verified on fused silica with different tip radii and different indentation depths. In the present work we investigate the applicability of the new model for conical tips. The conical indentation process is modeled using finite element analysis and maintaining the previous conditions. The study revealed interesting observations where the model proved to consider the tip rounding but slightly underestimated the tru
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