Elastic loading and elastoplastic unloading from nanometer level indentations for modulus determinations
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Elastic loading and elastoplastic unloading from nanometer level indentations for modulus determinations W. W. Gerberich, W. Yu, D. Kramer, A. Strojny, D. Bahr, E. Lilleodden, and J. Nelson Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455 (Received 10 March 1997; accepted 15 October 1997)
A new method for evaluating modulus and hardness from nanoindentation load/ displacement curves is presented. As a spherical indenter penetrates an elastoplastic half-space, the elastic displacement above the contact line is presumed to diminish in proportion to the total elastic displacement under the indenter. Applying boundary conditions on the elastic and plastic displacements for elastic and rigid plastic contacts leads to an expression that can be best fit to the entire unloading curve to determine E p , the reduced modulus. Justification of the formulation is presented, followed by the results of a preliminary survey conducted on three predominantly isotropic materials: fused quartz, polycrystalline Al, and single crystal W. Diamond tips with radii ranging from 130 nm to 5 mm were used in combination with three different nanoindentation devices. Results indicate that the method gives property values consistent with accepted values for modulus and hardness. The importance of surface roughness and indentation depth are also considered.
I. INTRODUCTION
Nanoindentation evaluation of mechanical properties at ultra-light loads is rapidly becoming an important requirement for processing nanostructural features and materials. These depth-sensing instruments are now routinely utilized in evaluating micron level properties with confidence. There remains a need for nm level displacements giving properties such as hardness and modulus, particularly for films in the range of 10 nm–100 nm in thickness. Toward this goal, there recently was an elastoplastic loading approach by Hainsworth et al.1 which was later utilized as a first approach to evaluating 300 nm thick films. This addressed only the loading portion of a load, P, versus displacement, d, curve relying on an approach formulated by Loubet et al.2 which requires P ~ d 2 . While the relationship derived by force fitting this relationship is useful at large loads and displacements, there is a deviation from linearity (P ~ d 2 ) for small loads and displacements at loads under 10 mN for a Berkovich diamond tip. To reinforce this, three curves were randomly selected from the present study, one each for W, Al, and fused quartz in Fig. 1. At mN and nm level forces and displacements, the relationship is much closer to a 3y2 power law rather than a squared one. More importantly, for very small loads and displacements, the required assumption1 that the plastic depth of indentation is directly proportional to the characteristic radius of contact is not followed due to the nonsharpness of the Berkovich tip. While a “sharp Berkovich-like” tip with a 65± included angle may have tip ra
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