**1. Introduction**

Nanocrystalline (NC) materials, especially NC metals and alloys, have attracted much attention due to their novel properties, such as improved wear resistance, high yield and high fracture strength [1–3]. It is well known that with the Hall–Petch rule the yield stress is increased following the grain size decrease from millimeter to submicron in coarse grained metals [4,5]. Interestingly, the range of application of the Hall–Petch rule about the yield stress and grain size can be expanded to nanoscale in NC metals such that the hardness and yield stress can increase 5–10 times, compared with their partners of coarse grain [6]. However, the micromechanism of the deformation processes in both regimes is considered to be different. In coarse grained metals, dislocations are generated from intragranular sources and they are stored and rearranged by the interaction between dislocation–dislocation. In NC metals, grain boundary mediated deformation is considered to control the strengthening [6–11].

The plastic deformation of NC metals may be related to many factors. Using grain size as the sole parameter to characterize its mechanical properties may be overly simplified, and thus sometimes give rise to uncertainties [12]. It has been revealed that the Hall–Petch rule breaks down when grain size decreases down to some critical size. To understand these novel results in experiments, even controversial findings [13,14], computer simulations at atomic level, especially molecular dynamics (MD), are expected to offer key insights. Indeed, MD is very helpful in understanding the deformation processes including plastic and elastic deformation, since it can provide real-time behavior and uncover the transient responses which are difficult to detect in experiments [15]. A lot of work has been taken to explore the critical size quantitatively and it is considered that the strength decreases generally as grain size decreases down to 20–10 nm. For example, the MD simulation showed that the critical size of NC Cu was about 10–15 nm [9].

It is well known that the mechanical response is always rate-sensitive [16–18]. For example, in the stainless steel, it was found that high strain rates (such as 104–105 s−1) could produce twin bundles with high density and nanoscale thickness [19]. Under high strain rates, such as pulsed shocking wave loading, it was found the shear and tensile strengths in metals could have very high values [18]. It is also noticed that the process of plastic deformation in NC metals is very sensitive to the loading rate [20,21]. In NC metals, the strain rate sensitivity (*m*) is an order higher than that of coarse grain [16,22,23]. It is considered that the large value of *m* is related to the interaction between grain boundaries (GBs) and dislocations in the plastic process. Another parameter, activation volume (*V*\*) which is related to the *m*, is considered to affect the rate-controlling mechanism. The *V*\* of NC metal is about two orders smaller than that of coarse grain [24,25]. Recently, many studies have found the *m* and *V*\* are very helpful to quantify the deformation mechanism in NC metals further [10,11,17,26,27]. Experimental measuring [27] indicated the *m* of NC gold with a grain size of 30 nm was 0.01 under the strain rate above 10−<sup>4</sup> s<sup>−</sup>1. Asaro et al. [17] showed theoretically that the value of *m* increased with the decrease of average grain size in NC metals, while the *V*\* was increased with the increase of average grain size. Wang et al. found that the *V*\* decreased with the increase of temperature in the NC Ni experiment with a grain size of 15 nm [10].

MD with its inherent constraints makes the time scale of simulation limited. The dynamics of the system is probed over just a few nanoseconds. Even with the quick development of computational techniques, the time period for the dynamics of a system with intermediate size (about 106 atoms) can be probed to be about 103 ns and thus the deformation of the system is simulated under very high strain rates, such as the typically used 1 ns−<sup>1</sup> for deformation processes, corresponding to the strain of 0.1 in 0.1 ns. Through this limit of simulation time, we can modulate the strain rate to explore the deformation processes to provide some insights into the atomic mechanism. Some simulation works about strain rate on NC metals with small grain size have been performed [11,15,28–30]. For example, in simulations of NC copper with grain sizes of 2.1–11.5 nm, it was found that grain coarsening was closely related to strain rate [31]. The grain size of grain coarsening increased with the decrease of strain rate. The simulation on 2D NC copper with a grain size of 9 nm [26] showed Young's modulus was kept almost constant at strain rates below 5 <sup>×</sup> 105 s−1. When the strain rate was more than <sup>5</sup> <sup>×</sup> <sup>10</sup><sup>5</sup> <sup>s</sup><sup>−</sup>1, Young's modulus increased with the increase of strain rate. However, there is still a lot of work needing to be undertaken. For instance, the mechanism of the change of elastic modulus with the strain rate isn't fully understood. The temperature effect combined with the strain rate on the deformation processes needs to be explored further. The effect of strain rate on grain growth also needs to be studied in depth.

In this work, we use MD simulations to study the effect of strain rate on the mechanical behaviors and deformation mechanisms in NC gold with small grain size. Two models of NC gold are constructed. One is with an average grain size of 6 nm, and the other is with 18 nm. For the deformation under applied tensile strain, the strain rate is modulated from 0.01 per ns to 10 per ns. Two temperatures including 300 K and 800 K are adopted to combine the change of strain rate to explore the deformation mechanism. The simulation results clearly show strain rate effects on Young's modulus, flow stress and grain growth of NC gold in tensile deformation and also reveal the atomic mechanism to some extent by combing the results from known experiments.
