Determination of Young's modulus by spherical indentation

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Determination of Young’s modulus by spherical indentation N. Huber, D. Munz, and Ch. Tsakmakis Forschungszentrum Karlsruhe, Technik und Umwelt, Institut f¨ur Materialforschung II, Postfach 3640, D-76021 Karlsruhe, Germany (Received 16 November 1995; accepted 25 February 1997)

In this paper we consider elastic plastic materials that are tested by spherical indentation. Finite element calculations, which take into account nonlinear geometry properties, are carried out in order to determine the influence of the plastic history on the unloading response of the material. Two different iterative methods are proposed for determining Young’s modulus under the assumption of a bilinear plasticity law. The first method deals with loading and unloading parts of the indentation test, whereas the second one deals only with unloading parts of the indentation test.

I. INTRODUCTION

The indentation test is often applied in order to characterize materials with respect to their mechanical properties. The need of such a complex, inhomogeneous deformation process results, e.g., from the microsystems technique in which it is very difficult to carry out tensile tests. Moreover, by means of the indentation test in situ measurements can be carried out without extensive preparation of the specimens. In recent years indentation testing machines have been developed, which make use of the depth-sensing technique: the load and displacement data are collected throughout the indentation process.1,2 Using such continuously measured data, the need of imaging to determine the contact radius is obviated and mechanical properties can be investigated. Figure 1 shows an example of depthsensing indentation data which are typical for a rate independent material behavior. That is, the depth-load diagram in Fig. 1 can be thought to be obtained by finite element calculations using rate-independent constitutive properties or to be obtained by a depth controlled indentation experiment in which the material under consideration is assumed to exhibit nearly rate-independent constitutive properties. With increasing load, elastic and plastic deformations are induced. During unloading the indentation depth decreases from the total depth ht to the residual depth hr due to elastic and probably additional plastic deformations. Some authors deal with the loading part which contains information about the resistance to plastic deformation, due to the plastic properties of the material.3,4 Commonly, only the unloading part of the indentation process is considered, from which the unloading stiffness S can be calculated, and thus, the elastic properties of the material can be determined.1,5 Also the residual deformation may be taken into account to analyze unloading data.6 The purpose of this paper is to show how the finite element method may be applied conveniently in order to evaluate the indentation J. Mater. Res., Vol. 12, No. 9, Sep 1997

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