Illustrative analysis of load-displacement curves in nanoindentation
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The nature of the elastic unloading after an elastic-plastic contact with a conical or Berkovich indenter is studied. Three representative specimens having different mechanical properties were tested. Finite-element results for the pressure distribution beneath the indenter during unloading suggest that the effective indenter is in fact very closely approximated by a sphere in the case of fused silica (a material with a relatively low value of E/H) and a more uniform pressure distribution in the case of silicon and sapphire (materials with higher values of E/H). The proposed reason for these observations is the extent and influence of an elastic enclave directly beneath the indenter as revealed by finite-element analysis. The results also show that the pressure distribution retains its form during the entire unloading. The work seeks to provide a physical reason for the value of the fitting exponent m as used in popular nanoindentation data analysis procedures.
I. INTRODUCTION
II. LOADING
1
Ever since 1992 the exponent of a power-law fit to the unloading data from a nanoindentation test has been known to be less than that expected on the basis of the routinely used elastic equations for contact with a conical indenter. Despite this, it is common practice to use either a second-degree polynomial or a linear fit to the initial portion of the unloading data to determine the contact stiffness at maximum load and also the depth of the circle of contact, as these quantities are required for a calculation of indentation hardness and modulus of the specimen material. Recently,2 it has been demonstrated that the exponent that best fits the unloading data observed comes from the upward curvature of shape of the residual impression in the specimen surface as the indenter was withdrawn. As well, dimensional analysis3 reveals that the nature of the observed loading and unloading responses follow certain scaling laws that may be applied to nearly all indentation experiments within the range of material properties usually studied. In this paper, an illustrative connection between theoretical analysis, finiteelement results, and dimensional analysis is presented for the purpose of gaining a more complete understanding of the nature of the load-displacement curves used in nanoindentation testing.
If we assume a condition of full plasticity within the specimen only and an absence of strain-hardening and indentation size effects, then the area of contact A at full load Pmax for a spherical, conical, or pyramidal indenter in contact with a semi-infinite flat specimen can be easily calculated from the hardness H of the specimen material.4 Pmax . (1) A= H For a conical indenter of half angle ␣, the contact depth hc is thus determined by geometry since the radius of the circle of contact a ⳱ hc tan ␣, and thus: hc =
公Pmax cot ␣ 公H
J. Mater. Res., Vol. 22, No. 11, Nov 2007
http://journals.cambridge.org
Downloaded: 14 Jul 2014
(2)
For a fully elastic unloading, where the unloading stiffness is given by dP/dh, elastic theory sho
Data Loading...
