A critical examination of the fundamental relations used in the analysis of nanoindentation data
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A critical examination of the fundamental relations used in the analysis of nanoindentation data Jack C. Hay IBM Research, T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598
A. Bolshakov Baker Hughes Inteq, P.O. Box 670968, Houston, Texas 77267-0968
G. M. Pharra) Department of Materials Science and Engineering, The University of Tennessee, 434 Dougherty Engineering Building, Knoxville, Tennessee 37996-2200 and Oak Ridge National Laboratory, Metals and Ceramics Division, P.O. Box 2008, Oak Ridge, Tennessee 37831-6116 (Received 14 September 1998; accepted 8 March 1999)
Methods for analyzing nanoindentation load-displacement data to determine hardness and elastic modulus are based on analytical solutions for the indentation of an elastic half-space by rigid axisymmetric indenters. Careful examination of Sneddon’s solution for indentation by a rigid cone reveals several largely ignored features that have important implications for nanoindentation property measurement. Finite element and analytical results are presented that show corrections to Sneddon’s equations are needed if accurate results are to be obtained. Without the corrections, the equations underestimate the load and contact stiffness in a manner that leads to errors in the measured hardness and modulus, with the magnitudes of the errors depending on the angle of the indenter and Poisson’s ratio of the half-space. First order corrections are derived, and general implications for the interpretation of nanoindentation data are discussed.
I. INTRODUCTION
In the past two decades, a great deal of effort has been directed toward the development of techniques for characterizing the mechanical properties of thin films and small volumes of material. Load and depth sensing indentation, commonly referred to as nanoindentation, is one means by which this has been achieved.1–12 Through the combined use of high resolution testing instrumentation and simple principles of analysis based on elastic and elastic/plastic contact theory, nanoindentation is now used routinely for small-scale mechanical property measurements, sometimes at indentation depths of only a few nanometers.7–10 Several analytical approaches have been developed to measure mechanical properties from indentation loaddisplacement data,2–5,9–12 most of which have focused on the elastic modulus, E (Young’s modulus), and the hardness, H. Central to these approaches are the methods by which experimentally measurable quantities such as the indentation load, P, the indenter penetration depth, h, and the indentation contact stiffness, S dPydh, are related to the projected contact area, A, and the elastic constants of the material, E and n (n Poisson’s ratio).
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Address all correspondence to this author at the University of Tennessee.
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http://journals.cambridge.org
J. Mater. Res., Vol. 14, No. 6, Jun 1999
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In one form or another, most methods make use of the relation13 p 2 E S p A. (1) 2 p s1 2 n d This
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