Relation between the ratio of elastic work to the total work of indentation and the ratio of hardness to Young's modulus for a perfect conical tip

A linear relationship between the ratio of elastic work to the total indentation work and hardness to reduced modulus, i.e., W_e/W_t = λ H/E_r, has been derived analytically and numerically in a number of studies and has been widely accepted. However, the scaling relationship between W_e/W_t and H/E_r has recently been questioned, and it was found that λ is actually not a constant but is related to material properties. In this study, a new relationship between W_e/W_t and H/E_r has been derived, which shows excellent agreement with numerical simulation and experimental results. We also propose a method for obtaining the elastic modulus and hardness of a material without invoking the commonly used Oliver and Pharr method. Furthermore, it is demonstrated that this method is less sensitive to tip imperfections than the Oliver and Pharr approach is.