Critical Examination of the Two-slope Method in Nanoindentation

In this paper, we derive corrected analytical expressions for calculating the hardness and modulus by the two-slope method. This method relies on the determination of the slopes of the loading and unloading curves rather than the indenter displacement as an input. These expressions take into account the correction factor α in the fundamental relations among contact stiffness, elastic modulus, and contact area, which is frequently forgotten or misused in the literature. It is shown that these corrected expressions allow measurements of the hardness and modulus in very good agreement with the commonly used technique based on the determination of the contact area. Additionally, the correction factor α can be easily determined if Young's modulus of the material is known.