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.
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Kossman, S., Coorevits, T., Iost, A. et al. A new approach of the Oliver and Pharr model to fit the unloading curve from instrumented indentation testing. Journal of Materials Research 32, 2230–2240 (2017). https://doi.org/10.1557/jmr.2017.120