A numerical approach to evaluation of elastic modulus using conical indenter with finite tip radius
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Geometrical self-similarity is a feature of mathematically sharp indenters such as conical and Berkovich indenters. However, self-similarity is considered inappropriate for practical use because of inevitable indenter tip blunting. In this study, we analyze the load–depth curves of conical indenters with various tip radii via finite element analyses. Based on the numerical data, we propose a method of restoring the Kick’s law coefficient C of finite tip-radius indenter to that of zero tip-radius indenter, thereby retaining the self-similarity of the sharp indenter. We then regress the unloading slope for the evaluation of elastic modulus in several ways. Finally, we establish a method to evaluate elastic modulus, which successfully provides the value of the elastic modulus with a maximum error of less than 5%, regardless of tip radius and material properties of both indenter and specimen.
I. INTRODUCTION
Instrumented indentation test is a well-directed method to extract material characteristics from the measured indentation load–depth curves. Analyses of indentation load–depth data from sharp and spherical indentation tests were attempted to identify the stress-strain relationship and the corresponding material properties,1–10 residual stresses,11–14 creep properties,15,16 and even fracture characteristics.17,18 On the basis of Sneddon’s1 theory for the indentation of an elastic half-space, Doerner and Nix2 and Oliver and Pharr3 suggested the methods for evaluation of Young’s modulus using the initial unloading curves. Many researchers then developed their own techniques to determine elastic–plastic properties using sharp5–7,10 and spherical4,8–10 indenters. Contrary to spherical indentation test, sharp indenters such as conical and Berkovich indenters have selfsimilarity irrespective of indentation depth. However, the actual tip of a sharp indenter is not mathematically sharp but invariably dull because of manufacturability and possible wear from use. The blunted indenter tip may cause nontrivial problems in evaluating material properties, especially when shallowly indented load–depth data are
a)
Address all correspondence to this author. e-mail: [email protected] Present address: Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996. b) Present address: Hyundai & Kia Research & Development Division, Hwaseong, Gyeonggi, 445-706, Republic of Korea. DOI: 10.1557/JMR.2008.0314 2528
J. Mater. Res., Vol. 23, No. 9, Sep 2008
used. It is thus essential to analyze the effect of tip radius on the evaluation of material properties with self-similar indenters.19–23 The present work begins with a review of some prior indentation methods. We then perform finite element (FE) analyses using ABAQUS24 FE code to study the effects of tip radius and material properties on the load– depth curve. Comparing the load–depth curves of conical indenters with various tip radii, we introduce a way of eliminating tip-radius effect. By regressing the unloading slope of load–depth curve in sever
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