The Indentation Elastic Response e- Indentation Shape and the Stress Distribution
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ABSTRACT The unloading of an indentation provides information about the shape of the indentation and the elastic properties of the materials. The assumptions of axisymmetry and material isotropy are critically examined, and a model for transversely isotropic materials is compared to measurements on single crystals. The methods used to infer the area of the indentation from the unloading curve are examined. The area is a fundamental value for the determination of hardness, modulus, and other mechanical properties in the so-called nano-indentor and other continuously monitored indentor techniques. The models of elastic recovery which are currently used are found to lack the flexibility to model the parameters which determine indentation depth. If the current self-consistent model is extended to cover the important aspects of the unloading, the area of the indentation is still not determined uniquely. Guidelines for further development of a unique model are suggested.
INTRODUCTION Since the demonstration of sub-micron indentation [1], the technique of continuously monitored indentation has been used extensively to characterize small volumes and thin films of
materials. Although indentation measurements have the potential for measuring properties much more far-reaching, hardness and modulus are the most simple, and by far the most common, measurements to made. Both measurements depend upon knowing the area of the indentation, and one of the current challenges is to calculate this area of the indentation from the elastic recovery curve and the shape of the unloaded indentor. Recent work by Oliver and Pharr [2] outlines the techniques involved. In general, the loading of an indentation is considered to reflect both elastic and plastic deformation of the material. The unloading of the indentation, in the ideal case, reflects the elastic properties of the material and the shape and size of the indentation. In analyzing the unloading curve, it is generally assumed that the deformation is controlled by linear, isotropic elasticity, and the general load, P, displacement, h, relationship is given by [3] ah -
aP
I
-
,+-w-, where E =
P2 v
2
(1)
E
A refers to the area of the indentation; the isotropic elastic properties are identified as belonging to the specimen, s, or the indentor, i; # = 1 for a round contact area and is a constant depending
on the shape of the contact area and the stress distribution for non-circular contact areas. King [4] calculated what has been shown [5] to be the upper limit of 3 for some common indentor shapes and shows that the error introduced by using a triangular indentor is less than 3.4%. A more 699 Mat. Res. Soc. Symp. Proc. Vol. 356 0 1995 Materials Research Society
realistic calculation [5] puts #3at 1.023 for the triangle (Berkovich) or 1.0055 for the square (Vickers) pyramids. However, for a Knoop shape indentor f3 is 2.5-3, indicating that Knoop indentors be used cautiously in the evaluation of elastic properties. ANISOTROPIC MATERIALS
For the case of highly anisotropic materi
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