Materials indented at small scales may simultaneously exhibit indentation size and strain rate effects which complicate the identification of the mechanisms that control deformation and strength. Here, we explore the possibility that indentation size and rate effects in some materials can be decoupled in a simple way. Nanoindentation tests with various load-time histories were carried out to measure the hardness of a tungsten single crystal over a wide range of indentation depths (∼500 - 3600 nm) and indentation strain rates (∼5×10^–5 - 2×10^–1 s^–1). Under these conditions, this material exhibits significant indentation size and rate effects, but the size effect is, to a good approximation, independent of strain rate. It is shown that this behavior can be understood by the Nix-Gao model for the indentation size effect modified to include the effects of a strain rate dependent friction stress. As a consequence, the size and rate dependencies of the hardness can be expressed as the sum of two independent terms:

$H\left({\dot{\epsilon }}_i,h_c\right)=H_f\left({\dot{\epsilon }}_i\right)+\left(H_0-H_f\right)\sqrt{1+\frac{h^{*}}{h_c}}$,

where $H\left({\dot{\epsilon }}_i,h_c\right)$ is the hardness at given indentation strain rate (${\dot{\epsilon }}_i$) and contact depth ($h_c$), $H_f\left({\dot{\epsilon }}_i\right)$ is the hardness contributed by the rate dependent friction stress, and $\left(H_0-H_f\right)$ and $h^{*}$ are size and rate independent constants that follow from the Nix-Gao analysis. This formula, together with an expression for the rate dependence of $H_f$, was successfully applied to decouple the indentation size and rate effects observed in tungsten. In addition, the physics underlying the rate independence of indentation size effect is discussed, which provides guidance for application of the proposed approach to other materials.

以小尺度压痕的材料可能同时表现出压痕尺寸和应变率效应，这使控制变形和强度的机制的识别变得复杂。在这里，我们探讨了一些材料中的压痕尺寸和速率效应可以以一种简单的方式解耦的可能性。进行了具有各种加载时间历史的纳米压痕测试，以测量钨单晶在很宽的压痕深度（~500 - 3600 nm）和压痕应变率（~5×10 ^–5 - 2×10 ^{– 1}秒^–1）。在这些条件下，这种材料表现出显着的压痕尺寸和速率效应，但尺寸效应在良好的近似下与应变速率无关。结果表明，这种行为可以通过 Nix-Gao 模型来理解，该模型将压痕尺寸效应修改为包括应变率相关摩擦应力的影响。因此，硬度的大小和速率相关性可以表示为两个独立项的总和：

$H\left({\dot{\epsilon }}_{\mathrm{一世}},H_C\right)=H_F\left({\dot{\epsilon }}_{\mathrm{一世}}\right)+\left(H_0-H_F\right)\sqrt{1+\frac{H^{*}}{H_C}}$,

在哪里$H\left({\dot{\epsilon }}_{\mathrm{一世}},H_C\right)$是给定压痕应变率下的硬度 (${\dot{\epsilon }}_{\mathrm{一世}}$) 和接触深度 ($H_C$),$H_F\left({\dot{\epsilon }}_{\mathrm{一世}}\right)$是由速率相关的摩擦应力贡献的硬度，和$\left(H_0-H_F\right)$和$H^{*}$是从 Nix-Gao 分析得出的与尺寸和速率无关的常数。这个公式，连同一个表示速率依赖性的表达式$H_F$，已成功应用于解耦在钨中观察到的压痕尺寸和速率效应。此外，还讨论了压痕尺寸效应与速率无关的物理原理，为将所提出的方法应用于其他材料提供了指导。