Strain rate sensitivity of hardness in indentation creep with conical and spherical indenters taking into consideration elastic deformations

Abstract

Using finite element analysis and analytical modeling, we investigate the strain rate sensitivity of the hardness in indentation creep ($m_H$) and the relationship between $m_H$ and the strain rate sensitivity of the flow stress, m_σ, for cone (self-similar) and spherical (non-self-similar) indenters. The present $m_H/m_{\sigma }$ results extend previous results (Elmustafa et al, 2007a, b) for cones in terms of a universal curve that describes the ratio $m_H/m_{\sigma }$ as a function of $H/E^{\ast }/{∊}_H$ starting from a value of 1, for small values of $H/E^{\ast }/{∊}_H$ (fully plastic) to zero at $H/E^{\ast }/{∊}_H\approx 2.5$ (fully elastic).

We also investigated the effect of varying effective strain levels (${∊}_H$). For the cone, the strain level is determined by the angle $\beta$ (25.3°, 22.5°, 19.7°) which is the angle the cone face makes with the specimen surface. m_H/m_σ becomes vanishingly small and the material undergoes full elastic deformation for H/E*≈ 0.23 and 0.18 for the $\beta$ angles of 25.3° and 19.7°, respectively, as compared to $\beta$ angle of 22.5° for which H/E*≈ 0.21 as shown by Elmustafa et al. (2007a). The simulation results of m_H/m_σ versus H/E* for various $\beta$ angles collapsed to a single curve when H/E* is normalized to the tangent of the $\beta$ angles.

In the case of the spherical indenter, the strain level is a function of indent radius/indenter radius ratio ($a/R)$. Simulations were performed for depths between 0.01 and 0.5 R for the non-self similar spherical indentations. The results indicate that the ratio m_H/m_σ does not maintain a unique relation with H/E* and varies with the increase in the ratio of depth of penetration to the radius of the indenter,$a/R$. It is also concluded that the data collapsed to a single curve similar to the one produced for conical indentation and that m_H/m_σ approaches zero for a normalized (H/E*)/(a/R) of ≈ 0.4. It is also found that the normalized hardness data, when m_H/m_σ approaches zero, fits Johnson solution well. Nanoindentation experimental data of Al₂O₃ samples using a spherical indenter tip displayed identical behavior similar to Johnson fully elastic behavior solution.