Some basic equations for the next generation of surface testers solving the problem of pileup, sink-in, and making area-
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In the present paper the formulas for the stiffnesses of a new and more general surface tester concept are given and discussed. The concept is based on the idea that the next generation of surface testers will provide the means to use all degrees of freedom of movement a probe on a sample surface could perform. Thus, in addition to the ordinary normal stiffness, lateral and tilting stiffness are measured, as well as twisting stiffness, and then used in the subsequent parameter determination of the investigated materials. It is shown in the paper that such a concept would not only solve classical problems such as “pileup” and “sink-in” completely, but it would also supersede the need of area-function calibration for the indenter tips and allow direct measurement of local intrinsic and residual stresses, anisotropy, and many other things, too. I. INTRODUCTION
Currently, a great majority of indenter machines use one degree of freedom of movement only, namely in the normal direction. (There are currently only very few nanoindenters available that also provide measurement of load and displacement in the lateral direction with an accuracy comparable to the normal direction.) These machines measure independently displacement and load and thus depend on just one basic equation for the analysis of the sample material. This basic equation concerns the stiffness of a mechanical contact in normal direction. It gives the result of the derivative of total contact load P with respect to the displacement or contact depth. @P ¼ 2aEr @h
:
ð1Þ
The symbol a denotes the contact radius, and Er gives the so-called reduced modulus defined as 1 1 n2I 1 n2S ¼ þ Er EI ES
;
ð2Þ
with E and n giving the Young’s modulus and Poisson’s ratio for Indenter and Sample, respectively. So, assuming that the material parameters of the indenter are known, there are always three unknowns in the simplest contact situation, the contact radius a, the sample Young’s modulus, and the sample Poisson’s ratio. While the contact radius can be determined using the “known” indenter geometry calibrated using the so-called area a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2009.0077
1032
http://journals.cambridge.org
J. Mater. Res., Vol. 24, No. 3, Mar 2009 Downloaded: 15 Mar 2015
function to the contact depth, either of the material parameters ES or nS must be estimated to determine the other. Usually, it is the Poisson’s ratio which is estimated.1 To overcome this problem, Lucas et al.2 used multiaxial indentation. They investigated a variety of bulk materials and achieved impressive agreement with the known Poisson’s ratios of the used sample materials. However, their approach, based on the results of Mindlin3 and Jonson4 leads to unclear boundary conditions making the subsequent parameter identification potentially inaccurate. The important paper2 is therefore discussed in more detail in the section “Hypothetical Application: An Example.” Unfortunately, the area-function procedure only works well if the
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