On the strain rate sensitivity of size-dependent plasticity in BCC iron at elevated temperatures: Discrete dislocation dynamics investigation

Highlights

• MDDP simulations were carried out on BCC iron samples to study the size effect under high temperature/ high strain rates.

• Strong size effect is manifested irrespective of temperature.

• Size effect is weakly dependent on the applied rate of deformation but is dependent on temperature.

Abstract

Multiscale discrete dislocation plasticity (MDDP) simulations are carried out to investigate the mechanical response and microstructure evolution of BCC iron micropillars under combined high temperature and strain rate deformation. The simulations are conducted at sizes ranging between 0.25 μm and 2 μm under an applied deformation rate ranging between 10³s⁻¹ and 10⁷s⁻¹ and subjected to different temperatures. MDDP based constitutive equation interrelating the size effect exponent to strain rate and temperature is also proposed indicating that the exponent is relatively sensitive to temperature and at a lesser degree to strain rate.

Detailed investigation of the microstructure shows that self-multiplication of dislocations is responsible for the strengthening mechanism in BCC iron micropillars. At low temperatures and small sizes, screw dislocations have a weak effect on plasticity for a certain period of time but subsequently control the self-multiplication process. At larger sizes, the motion of screw dislocations is responsible for plasticity at low temperatures. Due to the large volume size, screw dislocations are entangled inside the sample leading to a self-multiplication of dislocations via cross slip and other dislocation–dislocation interactions. At higher temperatures and for all sample sizes, mixed dislocations control plasticity via the multiplication of a complex network of dislocations. MDDP generated results are in good agreement with previous experimental studies on BCC metals.

Introduction

Over the last decade, unraveling the mechanical properties and deformation mechanisms of micro and nano pillars has been a major concern for scientists and researchers. Unlike bulk materials, the yield strength of micro and submicron pillars increases as the sample size decreases. However, this “smaller is stronger” trend differs in an essential way between materials. At the micro and nano scales, the size effect is quantified by an empirical power-law that relates the yield strength to the size of the pillar such that:

$${\sigma }_y=k{d}^{-n}$$

where σ_y is the yield strength, d is the sample size, k and n are material constants determined experimentally. The power dependence expressed in Eq. (1) is highly correlated with the crystal structure of the material. Experimental (Greer et al., 2008; Greer and Nix, 2006; Jennings et al., 2011; Kim et al., 2010; Rogne and Thaulow, 2015; Schneider et al., 2013, 2009) and simulation studies (Greer et al., 2008; Kim et al., 2009) show that, unlike FCC metals where size effect is significant with the exponent n ranging between 0.6 and 1 (Frick et al., 2008; Greer and Nix, 2006; Kim and Greer, 2009; Volkert and Lilleodden, 2006), size effect in BCC metals is less pronounced at temperatures below the critical temperature with an exponent ranging between 0.2 and 0.5. The critical temperature is the temperature above which the movement of dislocations becomes athermal, i.e., screw and edge dislocations have the same mobility and move by phonon drag. As the temperature exceeds the critical temperature, size effect is more pronounced and the behavior is similar to that of FCC metals (Greer et al., 2008; Rogne and Thaulow, 2015; Schneider et al., 2013, 2009; Torrents et al., 2016). This behavior is attributed to the high lattice friction of screw dislocations which is size independent but temperature dependent. Lattice friction is the resolved shear stress required to move a dislocation at a temperature greater than 0 K. In BCC metals, the complex core structure of screw dislocations (Domain and Monnet, 2005; Gilbert et al., 2011) is responsible for the high lattice friction as well as the difference in mobility between screw and edge dislocations. Below the critical temperature, screw dislocations have high lattice friction, thus, they move by thermal activation at a mobility less than their edge counterparts.

Experimental studies on different types of BCC pillars emphasize the dependency of the size exponent on the critical temperature. Schneider et al. (2009) performed a series of compression experiments at room temperature on four BCC pillars and found that the size exponent is inversely proportional to the respective critical temperature of the micropillars. The higher the critical temperature, the less pronounced the size effect. Kim et al. (2010) performed uniaxial compression and tension experiments and found that the yield strength and flow stresses for the metals that have the lowest critical temperature are strongly size-dependent. Besides the high dependency on critical temperature, Torrents et al. (2016) showed that size effect is also affected by dislocation density. Atomistic simulations on Tungsten, performed by Cui et al. (2016) revealed that even at temperatures below the critical temperature, the mobility of screw dislocations becomes dominated by phonon drag as the sample size decreases below 200 nm. Hence, mixed dislocations dominate plasticity as screw dislocations are unable to be stored in the limited sample volume.

Apart from the effect of the critical temperature, the power exponent is strain rate affected (Huang et al., 2015; Schneider et al., 2013; Wei et al., 2004). Schneider et al. (2013) investigated the effect of strain rate sensitivity on submicron Molybdenum pillars and found that these micropillars resemble their bulk counterparts. Huang et al. (2015) reported that the strain rate sensitivity decreases as the sample size is decreased and becomes negligible as the pillar is reduced below 200 nm. They attributed this behavior to the competing effect between lattice friction and bow out stresses. As the pillar size is reduced to a few nanometers, bow out stresses dominate the deformation mechanisms because few dislocations are present, leading to a decrease in strain rate sensitivity.

Even though the size exponent of BCC pillars is similar to FCC pillars at temperatures greater than the critical temperature, the deformation mechanisms in FCC and BCC pillars are not the same. Extensive studies on FCC pillars proposed two models to account for the size effect at the microstructure level. One model is the dislocation starvation model followed by surface nucleation. Greer and Nix (2006) were the first to elaborate the dislocation starvation model, where they observed that the dislocation density is less than the initial dislocation density when the sample is plastically deformed. This was attributed to the annihilation of dislocations at the free surfaces. After annihilation and to ensure plasticity, dislocations are generated and multiplied via surface nucleation (Greer et al., 2011; Jennings et al., 2011; Zhu and Li, 2010). The other model is the single arm source (SAS) model (Parthasarathy et al., 2007; Rao et al., 2008). In this model, dislocations are multiplied when single arm sources are truncated at the surface. On the contrary to FCC structure, these models are not found in BCC even at high temperatures where BCC and FCC size effects are similar. In BCC metals, dislocation multiplication is detected during plastic deformation (Weinberger and Cai, 2008). At temperatures below the critical temperature, screw dislocations are sessile and don't annihilate at the free surface, therefore they have enough time to interact between each other, cross slip and therefore multiply. At temperatures greater than the critical temperature, previous atomistic simulations (Greer et al., 2008; Weinberger and Cai, 2008) revealed that dislocations form segments inside the pillars interacting with each other's and with the free surface leading to dislocations multiplication.

Despite all the studies that have been developed in an effort to understand the deformation mechanisms and mechanical properties of the size effect in BCC metals, there are still some unresolved issues related to the effect of high temperature/ high strain rate on the size exponent and the dynamic evolution of dislocations in BCC iron. In order to get a comprehensive analysis of the combining effects of high temperature and high strain rate on size effect in iron single crystals, multiscale dislocation dynamics plasticity (MDDP) simulations have been utilized. The simulations are conducted to mimic loading conditions at temperatures ranging between 300 K and 900 K and strain rates ranging between 10³s⁻¹ and 10⁷s⁻¹ at different pillar sizes ranging between 0.25 μm and 2 μm under monotonic compressive loading conditions.

This paper is organized as follows. In Section 2, the MDDP approach, the simulation set up and the data used for the simulations are described. In Section 3, the results are discussed and a dislocation dynamics constitutive equation, based on MDDP, is established correlating the exponent with temperature and strain rate under high rate of deformation over a wide range of temperatures.