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y Coorevits and Alain Iost Arts et Métiers ParisTech, MSMP, Lille 59800, France

Didier Chicot Univ. Lille, FRE 3723—LML—Laboratoire de Mécanique de Lille, Lille F-59000 France (Received 8 January 2017; accepted 22 March 2017)

The unloading part of a load–displacement curve from instrumented indentation tests is usually approximated by a power law (Oliver and Pharr model), where the load is the dependent variable. This approach generally fits well the data. Nevertheless, the convergence is occasionally quite questionable. In this regard, we propose a different approach for the Oliver and Pharr model, called the inverted approach, since it assigns the displacement as the dependent variable. Both models were used to fit the unloading curves from nanoindentation tests on fused silica and aluminum, applying a general least squares procedure. Generally, the inverted methodology leads to similar results for the fitting parameters and the elastic modulus (E) when convergence is achieved. Nevertheless, this approach facilitates the convergence, because it is a better conditioned problem. Additionally, by Monte Carlo simulations we found that robustness is improved using the inverted approach, since the estimation of E is more accurate, especially for aluminum. I. INTRODUCTION

The instrumented indentation test (IIT) has been largely studied due to its advantages and facilities to estimate the mechanical properties of materials from the load– displacement curve.1,2 The technique is simple to execute; however, the interpretation of the data could be rendered difficult depending on the type of system, the material, and/or the scale of measurement. The principal properties calculated from the load–displacement data are the elastic modulus, E, and the hardness, H. Additionally, the work hardening coefficient, yield stress,3–5 and fracture toughness6,7 can be calculated as well from IIT. Several approaches have been developed to compute the elastic modulus and hardness of materials. Some authors8–12 calculate these properties from the loading part of the load–displacement curve. Nevertheless, most of the studies consider the unloading curve to compute them.13–16 Besides, the properties can be calculated from the indentation work deduced from the area under the load–penetration curve.17–19 The methodology proposed by Doerner and Nix16 to determine the mechanical properties of materials represents the fundamental of the Oliver and Pharr method.15,20

Contributing Editor: George M. Pharr a) Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2017.120

Doerner and Nix used a flat punch approximation that considers a constant contact area during withdrawal of the indenter and consequently, the unloading curve is linear, therefore, the stiffness should be calculated as the reciprocal of the compliance expressed by the next relation [Eq. (1)],16 dh 1
p 1=2 1 ¼ ; ð1Þ dP 2hp 24:5 ER where hp is the plastic depth obtained as the intercept with the displacement axis of the tangent line to the unloadin