Abstract

Nanoindentation is a non-destructive and simple method to measure important mechanical properties of materials. According to the analysis of Oliver and Pharr e.g. Young’s modulus and hardness can determine directly from a measured load-displacement curve [1]. Indirectly the loaddisplacement curve contains the whole stress-strain behavior of the material although it is not directly accessible [2]. Thus the determination of the stress-strain curve from a given load-displacement curve leads to an inverse problem. Several approaches and methods have been developed in order to solve the inverse indentation problem. On the one hand there are approaches based on dimensional analysis [3–7]. On the other hand optimization algorithms were developed. A main problem is that the inverse problem is ill-posed and thus the uniqueness of the inverse solution is often not granted [8,9]. Different material parameter can lead to indistinguishable load-displacement curves. This problem occurs although multiple indenter algorithms are used [10]. In a first step this paper shows for a power-law material behavior (σ=Kεn) the problem of non-uniqueness in inverse analysis using an optimization algorithm. In case of a single indenter optimization process the inverse solution is not unique and there is an infinite number of material parameter combinations leading to indistinguishable load-displacement curves. In a second step an energy based mathematical analysis of the problem is introduced, which shows that a mathematical relationship between all possible inverse solutions exists. In order to extract a unique solution a second indenter with a different geometry (different apex angle) is used. The second indenter leads due to changed applied strain field to a second set of inverse solutions and a second mathematical relationship. In case of the two parameters of the power-law the unique solution can be calculated from the two derived equations. This procedure was subsequently checked and verified with finite-element calculations.

Keywords: Indentation; FEM; Stress/strain relationship; Inverse analysis; Uniqueness; Ill-posed problem