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Paper 263

Obtaining Material Properties from Indentation Loading-Unloading Curves using Simplified Equations

J.J. Kang, M. Boris, A.A. Becker and W. Sun
Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, United Kingdom

Keywords: indentation, optimisation, elastic-plastic properties.

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Indentation testing is a successful technique for measuring the hardness and elastic properties from loading-unloading force-displacement curves. The instrumentation of indentation tests and the analysis of the resulting data have been extensively developed in the last two decades.

In previous work, the authors have focused on the evaluation of elastic-plastic material properties based on dimensional analysis [1] and finite element (FE) simulations [2]; both methods are based on non-linear optimisation techniques. However, extensive computational times are required for the FE analyses due to the fact that the optimisation procedure is based on iterative FE computations [2]. Since the characteristic indentation loading-unloading response can be described by dimensional analysis [1], more simplified approaches which do not require iterative FE analyses can be devised to obtain the elastic-plastic mechanical properties from experimental loading-loading curves, based on optimisation techniques.

The objective of this study is to present an improved approach for extracting the elastic-plastic material properties from the indentation loading-unloading curves. Based on the Olive-Pharr method [3], the complicated mathematical equations based on dimensional analysis can be simplified. The mathematical functions involve material parameters (Young's modulus, yield stress and post-yield work hardening exponent). The simulated target load-displacement curves using different indenter geometries (conical and Vickers indenters) and different indenter angles (60° and 70.3°) can be obtained from FE analysis. For predicting the sets of material properties from the loading-unloading curves, a MATLAB nonlinear least square routine with practical equality and inequality constraints is used to produce the best fit between the simulated target load-displacement curve and the predicted optimised curves.

The mathematical fitting functions are based on using a number of key factors (called k factors). In order to obtain the best optimised results, three types of 'focus tests' have been introduced. Once the three focus tests have been performed, the average of the optimised values can be used as the initial guess values for the final test with new bounds. Then, at every optimisation loops, the bounds are reduced based on the recently optimised results of the previous loop. Therefore, the final focus test usually converges much quicker than the other tests and gives better estimated results. Nevertheless, some limitations of this algorithm still exist due to the k factors. Therefore, a parametric study of the k factors and further work to investigate the relationships between the material properties of the specimen and the geometry of indenter are required. This further study can help to reduce the number of unknowns or the bounds of the parameters and improve the optimisation results.


J.J. Kang, A.A. Becker, W. Sun, "A combined dimensional analysis and optimisation approach for determining elastic-plastic properties from indentation tests", J. Strain. Analysis, 46(8), 749-759, 2011.
J.J. Kang, A.A. Becker, W. Sun, "Determining Elastic-plastic properties from Indentation data obtained from Finite Element Simulations and Experimental Results", Int. J. Mech. Sci. (Submitted)
W.C. Oliver, G.M. Pharr, "An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments", J. Mater Res, 7(6), 1564-1583, 1992.