Analysis of nanoindentation load-displacement loading curves

Abstract

Nanoindentation load-displacement curves provide a “mechanical fingerprint” of a materials response to contact deformation. Over the last few years, much attention has been focused on understanding the factors controlling the detailed shape of unloading curves so that parameters such as true contact area, Young's modulus, and an indentation hardness number can be derived. When the unloading curve is well behaved (by which we mean approximating to linear behavior, or alternatively, fitting a power-law relationship), then this approach can be very successful. However, when the test volume displays considerable elastic recovery as the load is removed [e.g., for many stiff hard materials and many inhomogeneous systems (e.g., those employing thin hard coatings)], then the unloading curve fits no existing model particularly well. This results in considerable difficulty in obtaining valid mechanical property data for these types of materials. An alternative approach, described here, is to attempt to understand the shapes of nanoindentation loading curve and thus quantitatively model the relationship between Young's modulus, indentation hardness, indenter geometry, and the resultant maximum displacement for a given load. This paper describes the development and refinement of a previous approach by Loubet et al1 originally suggested for a Vickers indenter, but applied here to understand the factors that control the shape of the loading curve during nanoindentation experiments with a pointed, trigonal (Berkovich) indenter. For a range of materials, the relationship P = Kmδ2 was found to describe the indenter displacement, δ, in terms of the applied load P. For each material, Km can be predicted from the Young's modulus (E) and the hardness (H). The result is that if either E or H is known, then the other may be calculated from the experimental loading curve. This approach provides an attractive alternative to finite element modeling and is a tractable approach for those cases where analysis of unloading curves is infeasible.

Further analysis of indentation loading curves: Effects of tip rounding on mechanical property measurements

“…9 However, since Eqs. (1) and (12) are the same in form, the results for conical indentation should capture the essential features of pyramidal indentation within the context of elastic-plastic deformation. Furthermore, previous finite element calculations for conical indention have shown good agreements with experimental indentation results obtained using pyramidal indenters, provided that the indented volume to indenter displacement relationship is the same in both cases.…”

“…For example, several empirical formulae have been proposed for the loading curves in terms of Young's modulus and hardness. 11,12 Loading curves have also been discussed using energetic considerations of reversible and irrea) Electronic mail: Yang T. Cheng@notes.gmr.com b) Electronic mail: zhengzm@LNM.imech.ac.cn versible parts of the indentation-induced deformations. 13 Using dimensional analysis and finite element calculations, we have recently derived scaling relationships for indentation into elastic-perfectly plastic solids using conical indenters.…”

Mechanical property determination of bone through nano- and micro-indentation testing and finite element simulation

“…Various methods have been developed to estimate the elastic modulus and hardness of a material (King, 1987;Hainsworth et al, 1996;Cheng and Cheng, 2000;Hay et al, 1998;Field and Swain, 1993) via indentation methods. Here, the method of Oliver and Pharr (1992) was used to determine the reduced Young's modulus E r and hardness H for each indentation:…”

Relationships between hardness, elastic modulus, and the work of indentation

“…Several experimental data points obtained using Berkovich diamond indenters from literature for Cu, 18 W, Al, fused silica, and sapphire 5 are also included in Fig. 3.…”