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.

使用有限元分析和分析模型，我们研究了压痕蠕变中硬度的应变率敏感性（${米}_H$）和之间的关系 ${米}_H$和流动应力，应变率敏感性米_σ，对于锥体（自相似）和球形（非自相似）的压头。现在${米}_H/{米}_{\sigma }$ 结果用描述比率的通用曲线扩展了先前的圆锥结果（Elmustafa等，2007a，b） ${米}_H/{米}_{\sigma }$ 根据 $H/{Ë}^{\ast }/{∊}_H$ 从值1开始，对于较小的值 $H/{Ë}^{\ast }/{∊}_H$ （全塑）时为零 $H/{Ë}^{\ast }/{∊}_H\approx 2.5$ （完全弹性）。

我们还研究了不同有效应变水平的影响（${∊}_H$）。对于圆锥体，应变水平由角度确定$\beta$（25.3°，22.5°，19.7°）是锥面与样品表面所成的角度。m _H / _mσ逐渐减小，并且材料经受H / E * ≈0.23和0.18的完全弹性变形。$\beta$ 分别为25.3°和19.7° $\beta$如Elmustafa等人所示，其H / E * ≈0.21的角度为22.5° 。（2007a）。米的模拟结果_ħ /米_σ对H / E *关于各种$\beta$当H / E *归一化为切线的切线时，角度折叠为一条曲线$\beta$ 角度。

对于球形压头，应变水平是压痕半径/压头半径比（$\mathrm{一种}/\mathrm{[R}）$。对于非自相似的球形压痕，对深度在0.01至0.5 R之间进行了模拟。结果表明，比米_ħ /米_σ不维持与唯一的关系H / E *，并用压头的半径增加穿透深度的比而变化，$\mathrm{一种}/\mathrm{[R}$。它也得出结论，数据晕倒类似于锥形凹陷和所产生的一个单条曲线米_ħ /米_σ为归一化的（H / E接近零*的≈0.4）/（A / R）。还发现的是，归一化的硬度数据，当米_ħ /米_σ接近于零，适合约翰逊溶液很好。使用球形压头尖端的Al ₂ O ₃样品的纳米压痕实验数据显示出与Johnson完全弹性行为解决方案相似的行为。