On Effective Indenters Used in Nanoindentation Data Analysis
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On Effective Indenters Used in Nanoindentation Data Analysis Guanghui Fu1, Ling Cao1 and Tiesheng Cao2 1 LC Dental, 43713 Boscell Road, Fremont, CA 94538, U.S.A. 2 Department of Ultrasonographic Diagnostics, Fourth Military Medical University, Xi'an, 710038, CHINA
ABSTRACT Effective indenter concept was introduced by Pharr and Bolshakov to explain nanoindentation unloading curves. This paper shows that the contact stiffness under a uniform pressure distribution is 57% higher than what is given by the fundamental relation. This is due to the fact that there is no physical indenter that gives a uniform pressure distribution during elastic contact, and the fundamental relation used in nanoindentation data analysis does not apply. INTRODUCTION Nanoindentation experiments have become a commonly used technique to investigate mechanical properties of thin films and small volumes of materials. The analysis of the experimental load ± displacement (P ± h) curve is based on the fundamental relation among contact stiffness, contact area and elastic modulus. The slope of the P-h curve, S dP dh , is defined as contact stiffness and it can be measured from nanoindentation experiments. The fundamental relation relates contact stiffness to the projected contact area ( A ), Young¶s modulus of the material ( E ), and Poisson¶s ratio of the material (Q ), as 2 E S A (1) S (1 Q 2 ) The fundamental relation is based on the analytical solution of normal indentation of an elastic half-space by a rigid smooth frictionless axisymmetric indenter [1]. Borodich and Keer [2] prove the fundamental relation by using the indentation shape to relate the load and the depth of the indentation. An alternative approach is to use the contact pressure to relate the load and the indentation depth [3]. The technique of assuming a pressure distribution between an indenter and a half space has been introduced to the nanoindentation data analysis by Pharr and Bolshakov [4] to explain nanoindentation unloading curves. They assume a uniform pressure distribution and use linear elasticity theory to obtain the deformed surface profile. The effective indenter shape is approximated from the deformed surface profile. The mechanical properties of the material are obtained through the effective indenter shape function and the fundamental relation. There is a lack of studies on the effects of assumed pressure distributions on the fundamental relation. In this paper, we show that a uniform pressure distribution overestimates the contact stiffness by 57%, and it is because there is no physical indenter that gives a uniform pressure distribution during elastic contact.
THEORY We consider a pressure distribution p(r , a) , which applies to a circular area 0 d r d a on the plane boundary of an elastic half-space z t 0 . The problem is considered in the linear theory of elasticity and the half-space is assumed to be isotropic and homogeneous. The stress components have two subscripts corresponding to the appropriate coordinates. E and Q are Young¶s modulus a
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