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The underlying theory behind the extraction of elastic modulus and hardness from the unloading load–displacement data obtained with a spherical indenter was explored in detail. A formal treatment of the effect of indenter elasticity was given, and the validity of the use of the reduced or combined modulus in analytical treatments was verified. The “Oliver and Pharr” method and the “Field and Swain” methods of analyses were compared in detail and shown to be equivalent.

I. INTRODUCTION

There has been considerable interest in the last decade in the mechanical characterization of thin-film systems using submicron indentation tests using either spherical or pyramidal indenters.1–5 Usually, the principal goal of such testing is to extract elastic modulus and hardness of the film specimen material from experimental readings of indenter load and depth of penetration. These readings give an indirect measure of the area of contact at full load, from which the mean contact pressure, and thus hardness, may be estimated. The test procedure, for both spheres and pyramidal indenters, usually involves an elastic–plastic loading sequence followed by an unloading. The initial portion of the unloading is assumed to be purely elastic. An analysis of the initial portion of the unloading response gives an estimate of the elastic modulus of the indented material. The validity of the results for hardness and modulus depends largely upon the analysis procedure used to process the raw data. Such procedures are concerned not only with the extraction of modulus and hardness but also correcting the raw data for various systematic errors that have been identified for this type of testing. The present work is concerned with the underlying theory behind the extraction of elastic modulus and hardness from the unloading load– displacement data obtained with a spherical indenter. Methods of correction of the raw data may be applied to account for initial penetration depth, indenter shape function, and instrument compliance. Further corrections may be required which depend upon the nature of the specimen material. The so-called indentation size effect, piling-up, and sinking-in are among the most serious of these. The purpose of this work is to focus on the analysis

a)

II. ANALYSIS OF INDENTATION TEST DATA A. General

Consider the loading of an initially flat specimen with a spherical indenter. Upon loading, there may be an initial elastic response at low loads followed by elastic and plastic deformations within the specimen material at higher loads. With reference to Fig. 1, the depth of penetration of a rigid spherical indenter beneath the original specimen free surface is h t at full load Pmax. When the load is removed, assuming no reverse plasticity, the unloading is elastic, and at complete unload, there is a residual impression of depth h r. If the load Pmax is then reapplied, then the reloading is elastic through a distance h e ⳱ h t − h r according to the “Hertz” equation: P = 4⁄3 E*R1 Ⲑ 2h e3 Ⲑ 2

(1)

.

In Eq. 1, E* is the composite m