Analytical relationship among nominal hardness, reduced Young’s modulus, the work of indentation, and strain hardening exponent

Excerpt

In an instrumented indentation test, the reduced modulus is expressed as: where \( S_{\text{u}} = \left. {{\text{d}}P/{\text{d}}h} \right|_{{h = h_{\text{m}} }} \) is the initial unloading stiffness at maximum load P _m; A(h _cm), h _m, and h _cm are the corresponding projected contact area, maximum indentation depth, and maximum contact depth; and β is a correction factor. E _r is related to the Young’s modulus and Poisson’s ratio of the indented material (E, ν) and those of the indenter (E _i, ν _i) by the equation 1/E _r = (1 − ν ²)/E + (1 − ν _i ² )/E _i, from which an estimate of E is derived if E _r is first determined. Obviously Eq. 1 indicates that the accuracy of the measured value of E _r (or E) relies on the reliability of the methods used to derive S _u and A(h _cm) (or h _cm), but S _u may vary substantially according to the condition of a test. For example, at low load condition load–displacement data are scattered, such that the value of S _u derived would have great uncertainty. In addition, A(h _cm) (or h _cm) estimated by applying the well-known Oliver and Pharr method [1, 2] can have a large error when the indented material is soft and shows weak work hardening. Improvement is achieved by applying an energy-based method, obtained by combining dimensional theorem and finite element simulations as reported in our recent work [3, 4]. In this method, an approximate relationship between the ratio of a nominal hardness to reduced Young’s modulus (H _n/E _r), and the ratio of the work done during unloading to that during loading denoted as total work afterwards (W _e/W _t) was founded, in which the nominal hardness defined by H _n ≡ P _m/A(h _m) is essentially different from the real hardness [5] H ≡ P _m/A(h _cm) and can be determined accurately by fully utilizing the accuracy of the measured load and displacement data from an instrumented indentation system. Consequently, E _r (and thus E) can be determined merely from H _n, W _e, and W _t. This approach is referred to as the pure energy method [6], and has been shown to be very successful. However, in our previous approach the relationship between H _n/E _r and W _e/W _t was derived entirely based on numerical simulations, while the physical insight and the subsequent analytic formulation were not achieved yet. In this work, we derive an equation of H _n/E _r as a function of W _e/W _t and hardening exponent n based on a more physical point of view in order to consolidate the physical basis of the method. …