3.1. Tensile Test
In Figure 2
and Figure 3
, the snapshots of the model at different strains during the tensile test for different strain rates and temperatures are shown. In both figures, the 1st snapshot was taken at a strain of 0.0, the 2nd in the elastic region of deformation at the strain of 0.03, and the 3rd at the strain where each structure reached peak stress. The 4th and 5th snapshots are taken in the plastic region of flow stresses at strains of 0.15 and 0.25. For analysis and visualization, OVITO’s module dislocation analysis (DXA) [34
] was used. The module allows us to observe ratios of different lattice configurations and identification of all dislocation line defects. The purple atoms are arranged in the BCC lattice while the white represents the dislocation types of <111>, <100> and <110> (see Figure 2
and Figure 3
). In all tensile simulations, the two most common deformation mechanisms were recognized, GB slip and dislocation glide [13
]. Moreover, grain rotation [17
] and growth were noticed.
The undeformed state at a strain of zero represents the structure right after the equilibrations, which was obtained after the annealing cycle. After those cycles, initial structure with 9 flawless BCC monocrystals changed to the polycrystalline structure of 9 grains containing a different density of dislocations (see numbered grains in Figure 2
a). It was observed that the highest density of dislocation in the initial structure developed in grain number 4 and the lowest in the grain 2 (using DXA tool of the OVITO software [31
]). Due to model construction, the ratio of BCC lattice before deformation in all the samples equals 84.3%, which is higher than most of the literature [3
]. In all samples, we noticed a decrease in the BCC ratio with deformation, where the ratio change in elastic compared to the plastic region is negligible. With increased strain rates, a significant decrease of BCC lattice was noticed at the same strain values. Even if only one layer of atoms, such effect is observed as a higher increase in a number of white atoms in the top layer in Figure 2
b from strain 0 to 0.25, compared to Figure 2
a. This means that the fraction of stacking faults in the grain lattice and fraction of GBs increases with increased strain rate in the flow stress region. From the tests done at different temperatures and the same values of strain rate, we found that the BCC ratio is also reduced with an increase in temperatures. Still, the effect of different strain rates was a more dominant factor in the BCC lattice ratio change (see Figure 3
During the tensile test some of the initial dislocations (yellow arrows in Figure 2
a and Figure 3
a,b) started to move to the boundaries and piled up there, presumably because of high energy GBs [13
]. Others, which ended on GBs of neighboring grains with a low difference in lattice angle, initiated dislocations in the neighboring grain. At high values of strain, some GBs started to lose their improper lattice structure. Such regions changed to the boundary at which neighboring grains share a plane of atoms. GB that went through such a process of generating twin planes from strain 0 to 0.25 are shown in Figure 2
a,b (green ellipses). The main difference in GB configurations happened after the peak stresses. The reason for such processes could be more frequent GBs slides at the high strain values and increased diffusion coefficient at the higher prestrains [22
]. Furthermore, before some grains grew in a way that the GB between neighbor grains disappeared, and grains started to share a plane of atoms. In Figure 2
b, grain growth was observed gradually from strain 0 to 0.15. Newly formed GB is marked with a black ellipse. It is seen that grain number 8 is dissolved and became part of neighboring grains 7 and 9. Therefore, both the grains sizes were increased. The grain rotation can be observed in Figure 2
a and Figure 3
b from strain 0 to 0.25 in grain number 2, which is marked with red lines.
shows the stress-strain curves at different temperatures and strain rates. All the stress-strain curves have a linear elastic region, which is followed by a stress overshoot. After the peak stresses, nearly constant values of flow stress were observed. The fluctuation of the curves in the flow stress region is dependent on the distribution, orientation and size of grains. The averaging of more stress-strain curves with a change in the parameters, as mentioned above, leads to more constant flow stress values.
In Figure 4
a, tensile tests were performed at the temperature of 300 K, and strain rates from 0.004 to 0.012 ps
with a step size of 0.002 ps
were applied. It is noticed that the change in strain rate does not affect the elastic modulus. However, the peak stresses increase significantly with strain rates. Moreover, the flow stresses increased moderately with strain rates. The reason for the increase in the flow stress is that the density of partial dislocations in the region before the strain at peak stresses decreases with increased strain rate. This leads to increased values of delayed partial dislocations with increased strain rate, causing stress overshoots after the overshoot dislocation density starts to rise, which leads to stress reduction to the constant value of flow stress. Similar observations were made by Rida et al. [28
b demonstrates the stress-strain curves of tensile tests, which were carried out at temperatures 500 and 700 K with strain rates of 0.006 and 0.008 ps
. It is observed that the elastic modulus decreases with increased temperature. Moreover, a decrease of the elastic modulus with temperature leads to a lower ratio of peak to flow stresses, which is seen as lower overshoot. Furthermore, higher temperatures lead to lower peak and flow stresses. As mentioned before, not only higher strain rates but also higher temperatures decreased the BCC lattice ratio in the model when being deformed. In addition, we compared strains at a temperature of 300, 500 and 700 K at strain rates of 0.006 and 0.008 ps
. It is found that high temperatures decrease the energy needed for GBs sliding more than for dislocation glide, making GBs dominant deformation mechanism, which is also discussed by Ghaffarian et al. [35
]. The reduction of energy needed to initiate GBs slide makes plastic deformation easier, which leads to lower peak and flow stresses.
3.2. Specimen with Void
At the nanoscale, continuum approximations based on the finite element method are no longer valid since the technique can not consider dislocation movement and grain slides. From the continuum mechanics, a problem of the plate with a hole is known, where a stress concentration is three times higher on the hole edge than the pressure applied on the plate edges [36
]. In this section, the influence of a hole on the mechanical properties of NC BCC iron is discussed. In the [37
] deformation process of NC monocrystal with a hole is analyzed, where the NC model response can be described as follows
Dislocations nucleate near the void;
With additional deformation, dislocations initiated near void propagate to the specimen edges;
Reduction of specimen stress resistance;
Necking of specimen seen at higher strain values.
In our simulations, the initial model (see Figure 1
) was taken, and a through-hole with a diameter of 30 Å was made in the center of the model, as it is shown in the most left snapshot in Figure 5
. We were interested in the influence such a hole would make on the stress-strain curve in a uniaxial tensile test. We need to be aware of periodic boundary conditions used, which means the necking will not form the same way as in the work of Potirniche et al. [37
]. Here, MD simulations were performed at a temperature of 300 K and strain rates of 0.006 and 0.008 ps
. Snapshots of the deformed structure at different strains are shown in Figure 5
In Figure 5
, at the strain value of zero, the model after minimizations, equilibration and annealing is shown. We can comprehend the structure of 9 grains in the model. The snapshot at a strain of 0.015 shows von-Misses stress of each atom in linearly elastic structure response. One can notice that the highest stresses are formed in the GBs, however, a small stress increase formed in the BCC lattice next to the hole edge (yellow ellipses) was observed. At a strain value of 0.08, we recognized stress concentration in BCC lattice near the edge on both sides of a hole increases (red ellipses). This does not only correspond to stress concentration in NC monocrystal [37
] but also to the continuum mechanics model. The next snapshot at a strain of 0.12, the structure is loaded with the highest stress. Here, we can observe that grain with number 6 starts to slide into the void. Furthermore, crack initiation is seen (black ellipse). With further increase in strain, the grain 6 kept sliding into the hole while the hole in pulling axis direction kept increasing its size.
depicts the stress-strain curves of the models with and without holes at the center. The model with a hole reached lower peak stress than the model without a hole. However, the flow stresses of both models are nearly the same values for the considered range of strain rates. The significant difference was also observed in the low strain region. The model without a hole has a linearly elastic response over an extended range of strain 0.04, while the hole model curve starts to divert from the linearity at a strain of 0.018. With further investigation of change in neighbor atom distances, we concluded that curve flattening at low strains happens due to small GBs slips, which became more frequent with further deformation, leading to additional stress-strain curve flattening.
We observed that the hole first initiated stress concentrations, leading to dislocation nucleation at smaller strains. The hole also represented a volume to where the grains could be pushed more easily, making GB slips possible at lower strains. When such a slip happened, structure relaxation occurred. The described process leads to lower overshoot compared to the one without the hole model.
3.3. Shearing Test
In this section, shearing tests on NC BCC iron are discussed, focusing on the influence of different strain rates on the mechanical properties. The MD simulations were carried out with two different strain rates of 0.004 and 0.012 ps, which defined the upper face of the simulation box’s translatory motion. The bottom part of the box was fixed. We need to be aware that the chosen simulation boundary conditions do not allow the simulation box to be deformed orthogonally to the translatory direction of the upper cell face. This means the shearing component of stress is the main load for structure deformation at lower shear strains. However, as loading continues, the influence of normal stresses increases.
shows the snapshots of the model at different shear strains during shearing test simulation at a strain rate of 0.012 ps
. As mentioned earlier, the purple atoms represent the ones arranged in the BCC lattice, and the white represents the dislocation types of <111>, <100> and <110>. The first snapshot is taken at the shear strain equal to zero, which shows the structure after the minimizations, equilibration and annealing. As in the tensile test, 9 grains were formed with different density of dislocations. The shear strain of 0.06 represents the structure in its maximum shear strain in the linear elastic response, where no dislocation activity was found. Atom configuration at a shear strain of 0.24 shows the structure at its maximum stress load. As we compared it to the undeformed state, grain boundary slips were noticed by observing a change in neighbor atom distances. We found out that the ratio of BCC lattice stays constant—it fluctuates around the initial value of 82%, which is different compared to the tensile test, where the ratio decreases. The last two snapshots represent the structure response in the shear stress flow region where grain boundary slips were the main deforming mechanism recognized.
depicts the shear stress-strain curves. The structure response is linearly elastic for small shear strains—for both strain rates up to shear strain of approximately 0.05. At the shear strain of 0.05, the curve for strain rate 0.012 ps
starts to flatten and makes a small stress overshoot over a long shear strain. In the case of a slow strain rate of 0.004 ps
, at a shear strain of 0.075, the stress suddenly drops. The reason for such a response was sudden GB slip, which happened in the model. If simulations results are averaged for different grain configurations, the curves do not have such fluctuations before the peak values of shear stress and also in the flow stress region. Both overshoots are followed by constant shear flow stresses. We concluded that higher shear strain rates lead to higher peak shear stresses and higher shear flow stresses. Moreover, the strain rate has no significant influence on the shear modulus.
The cross-section of the specimen is the same as in the tensile test. It is seen that for the same cross-section and strain rate, the structure can be plastically deformed with lower flow stresses when shear stress is applied instead of a normal one. This work can be effectively used to investigate the high-speed forming of nanocrystalline metals, e.g., iron and aluminum, using MD simulations [38