The use of shape correction factors for elastic indentation measurements
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Elastic properties of small volumes of materials can be measured from the unloading of small indentations using the so-called nanoindenters The analytic elastic solutions for axisymmetric indenters are currently used to calculate modulus from the unloading curve, sometimes using corrections derived for flat, rigid punches. It is shown that these corrections represent an upper limit for the correction. More realistic corrections are derived for the Vickers, Berkovich, and Knoop indenter shapes, using the assumption of a uniformly loaded area. Results show that the axisymmetric solution overestimates the elastic compliance of the Vickers indenter by a factor of 1.0055, of the Berkovich indenter by a factor of 1.0226, and of the Knoop indenter by a factor of 2.682.
So-called nano-indenters are increasingly being used to characterize the hardness and modulus of materials that are not available in bulk forms. The elastic unloading portion of the nanoindenter data is used to determine the elastic properties of the material as well as to determine the size of the indentation for calculating hardness. (See the recent work by Oliver and Pharr.1) The solutions for the elastic unloading of an indentation assume that the indenter is axisymmetric, i.e., that it can be described as a body of revolution about an axis perpendicular to the indented surface. An axisymmetric indenter will always contact a flat specimen over a circular cross section. The condition of axisymmetry is satisfied explicitly for spherical or conical indenters. The general axisymmetric result2 is
be modified to 1 -
dh_ dP
(2)
P 2
where /3 = 1.000 for circular cross sections, /3 = 1.012 for square cross sections, and /3 = 1.034 for equilateral triangular cross sections. The reason that the displacement is lower for the square and triangular cross sections can be easily understood by considering the application of Love's stress function to an elastic half-space. (See Ref. 5 for a good discussion.) The displacement of the surface of the halfspace from a point load is just 1 1 - v2 P w
TV
dh_ dP
- vl
2 VI
E,
(1)
where h is the elastic displacement, P is the load, A is the area of the contact between the indenter and the specimen, E and v are Young's modulus and Poisson's ratio, respectively, assuming both specimen s and indenter i are isotropic materials. It is important to note that, except for the case of a flat punch, A is a function of h. The more commonly used Vickers (square pyramid) or Berkovich (equilateral triangular pyramid) indenters will intersect a flat specimen over a square or equilateral triangle, respectively. Reputable sources2'3 suggest using the correction factors of King4 to account for this small change in geometry. Using numerical methods, King4 analyzes the load versus displacement behavior of flat rigid indenters of circular, square, and equilateral triangular cross section, and he determines that Eq. (1) can J. Mater. Res., Vol. 10, No. 2, Feb 1995 http://journals.cambridge.org
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(3)
= E
r
where r
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