Evaluating initial unloading stiffness from elastic work-of-indentation measured in a nanoindentation experiment
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Debrupa Lahiri and Arvind Agarwal Nanomechanics and Nanotribology Laboratory, Department of Mechanical and Materials Engineering, Florida International University, Miami, Florida 33174 (Received 11 July 2012; accepted 2 January 2013)
Differentiation of the energy-based power function used to represent the nanoindentation unloading response at the peak indentation load generally overestimates the contact stiffness. This is mainly because of the larger curvature associated with this function and the proximity between the contact and maximum penetration depths. Using the nanoindentation data from ceramics and metals, we have shown that these two errors can be eliminated if the derivative is multiplied by the geometric and stiffness correction factors, respectively. The stiffness correction factor is found to be a function of the elastic energy constant and is independent of the peak indentation load. The contact stiffness evaluated by the proposed method is in excellent agreement with that obtained from the power law derivative for a wide range of elastoplastic materials and peak indentation loads. The relationship between the elastic recovery ratio and elastic energy constant developed in this study further simplifies the proposed procedure.
I. INTRODUCTION
The experimental load-displacement curve obtained by probing the surface of a material in a nanoindentation experiment are analyzed to evaluate the reduced modulus of a material according to the fundamental relation given by1 pffiffiffiffiffi 2 Su ¼ b pffiffiffi Er Ac p
;
ð1Þ
where Su is initial unloading stiffness or contact stiffness, Ac is the projected area of elastic contact, Er is the reduced modulus of a material and b is the correction factor that takes the lack of axial symmetry of the pyramidal indenter into account. The contact stiffness is defined as the slope of the unloading curve evaluated at the maximum depth of penetration. The area of contact is either measured independently from the hardness impression or derived using the contact stiffness according to the procedure developed by Oliver and Pharr.2 To evaluate the slope, one needs a mathematical description of the unloading response, which is difficult to obtain analytically owing to the complexities involved in the indentation process.3 The unloading response is usually described by an algebraic function established by curve fitting of the experimental load versus displacement data. a)
Address all correspondence to this author. e-mail: suksawan@fiu.edu DOI: 10.1557/jmr.2013.3 J. Mater. Res., Vol. 28, No. 6, Mar 28, 2013
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In the most widely used Oliver and Pharr (OP) method, the unloading curve obtained using a Berkovich indenter is represented by a power law whose parameters are determined by the least square fitting. The exponent of the power law, according to Oliver and Pharr, is slightly material-dependent and may take a value in the range 1.25–1.51, which led them to conclude that the shape of a Berkovich indenter may be described as a parabola of re
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