Effects of elastic indenter deformation on spherical instrumented indentation tests: the reduced elastic modulus

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Although the Hertz theory is not applicable in the analysis of the indentation of elastic-plastic materials, it is common practice to incorporate the concept of indenter/specimen combined modulus to consider indenter deformation. The appropriateness was assessed of the use of reduced modulus to incorporate the effect of indenter deformation in the analysis of the indentation with spherical indenters. The analysis based on finite element simulations considered four values of the ratio of the indented material elastic modulus to that of the diamond indenter, E/Ei (0, 0.04, 0.19, 0.39), four values of the ratio of the elastic reduced modulus to the initial yield strength, Er/Y (0, 10, 20, 100), and two values of the ratio of the indenter radius to maximum total displacement, R/δmax (3, 10). Indenter deformation effects are better accounted for by the reduced modulus if the indented material behaves entirely elastically. In this case, identical load–displacement (P − δ) curves are obtained with rigid and elastic spherical indenters for the same elastic reduced modulus. Changes in the ratio E/Ei , from 0 to 0.39, resulted in variations lower than 5% for the load dimensionless functions, lower than 3% in the contact area, Ac , and lower than 5% in the ratio H/Er . However, deformations of the elastic indenter made the actual radius of contact change, even in the indentation of elastic materials. Even though the load dimensionless functions showed only a little increase with the ratio E/Ei , the hardening coefficient and the yield strength could be slightly overestimated when algorithms based on rigid indenters are used. For the unloading curves, the ratio δe/δmax , where δe is the point corresponding to zero load of a straight line with slope S from the point (Pmax, δmax ), varied less than 5% with the ratio E/Ei . Similarly, the relationship between reduced modulus and the unloading indentation curve, expressed by Sneddon's equation, did not reveal the necessity of correction with the ratio E/Ei . The most affected parameter in the indentation curve, as a consequence of the indentation deformation, was the ratio between the residual indentation depth after complete unloading and the maximum indenter displacement, δr/δmax (up to 26%), but this variation did not significantly decrease the capability to estimate hardness and elastic modulus based on the ratio of the residual indentation depth to maximum indentation depth, hr/hmax . In general, the results confirm the convenience of the use of the reduced modulus in the spherical instrumented indentation tests.