*3.2. Evolution and Related Failure Mechanisms of GBs under the External Strain Effect*

In this section, the stress–strain curves for GBs and related single perfect crystal cases are calculated firstly, as shown in Figure 3. It is clear from Figure 3a, ∑3(111) and ∑17(410) GBs have the maximum and minimum peak tensile stress, respectively. In fact, a similar conclusion could also be made for single perfect crystal cases, as shown in Figure 3b. The peak tensile stresses of all cases studied in this work are listed in Table S1, in the Supplementary Materials. Compared with the results of single crystals, it is clear that the appearance of GBs induces the decrease in peak tensile stress.

**Figure 3.** (**a**) Strain–stress curves of ∑3(111), ∑3(112), ∑5(012), ∑5(013), ∑9(221), ∑ 11(113), ∑17(410). (**b**) Strain–stress curves of single crystals corresponding to each grain boundary.

In order to explore the reason behind the peak value decrease induced by GB defect, the atomic stress state and related atomic structure were analyzed for these GB systems. Based on the analyses, the following mechanisms have been explored.

#### 3.2.1. Phase Transformation Induced Grain Boundary Failure

Figure 4 depicts the structural evolution of the ∑3(112) and ∑5(013) GB system under the effect of external strain. From these results, it can be found that with increasing strain, the phase transition occurs initially in the GB region and extends to the grain interior, that is, from an initial bcc phase to fcc phase, as shown in Figure 4a. In the present work, this transition occurs with strain around 10% and related stress around 12.67 GPa, as shown in Figure 3a, and is also confirmed by the atomic potential energy and displacement, that is, atoms in the phase-transition region have higher potential energy and larger displacement distance, as shown in Figure S2 in the Supplementary Materials. In fact, the phase transition from bcc to fcc under external stress has also been reported in polycrystalline Fe systems with stress values up to around 13 GPa under shock wave [38]. It is clear that ∑3(112) GB

has a similar effect on the phase transformation of Fe system. An example of a formed fcc structure is also shown in Figure S3 in the Supplementary Materials with a lattice constant around 3.7 Å, which is slightly larger than the value (3.6 Å) of fcc-Fe under the normal condition [39]. Together with this phase transition, a new interface is formed between the newly formed fcc phase and the original bcc phase, as shown in Figure 4b, whose position varies with the phase transition until the slip system {123}<111> becomes activated near the interface from the bcc phase side. Dislocation nucleation is then initiated from this activated slip system and finally the dislocation network is formed with increasing strain. One possible reason for the slip system activation in bcc instead of fcc phase for ∑3(112) GB can be explained by different Schmid factors (*μ*) in these two phases with external stress along <112> direction. In this case, the maximum Schmid Factor *μ* of bcc and fcc slip systems is 0.4115 and 0.4082, respectively. The larger Schmid factor in bcc phase indicates the it has higher probability to initiate the slip system in bcc phase. The value of 0.4115 in bcc phase is related to the {123}<111> slip system, same as the present simulation results, as shown in Figure 4b. In addition to above results, the larger displacement related to the phase transition is also confirmed to result in a larger free volume near the bcc-fcc interface, which is suspected as a possible reason for the change in stress field of the GB system. For example, comparing the maximum free volume in GB at states of 0 strain and 15% strain, as shown in Figure 5, it is clear that without strain, the maximum free volume (FVmax) is around 17.99 Å3 and the average value of free volume is around 7.99 Å3. While with 15% strain, FVmax reaches around 25.71 Å<sup>3</sup> with an average value around 11.98 Å3 at the bcc-fcc interface induced by the phase transition. The stress field around this maximum free volume is around 21.48 GPa, resulting in the local stress concentration increasing and related activation of the slip system. Thus, the present results indicate that the increase in free volume induced by phase transition may be also one possible reason to induce the local stress to its critical value and the failure of GB system in bcc Fe.

**Figure 4.** Snapshots of ∑3(112) and ∑5(013) GB under external strain effect. (**a**) Shows the state of ∑3(112) GB at which the phase transition starts with a strain value of 10% and (**b**) is the state at which the slip system in bcc phase near bcc-fcc interface is activated for ∑3(112). (**c**) Shows the state of ∑5(013) GB at which the phase transition starts with a strain value of 8.5% and (**d**) is the state at which the slip system is activated for ∑5(013) in fcc phase near the bcc-fcc interface. In the figure, the green, blue and white points are atoms in fcc, bcc and other phase states, respectively.

In fact, a similar process has also been observed in the ∑5(013) case, as shown in Figures S4 and S5 in the Supplementary Materials. The difference between these two cases is that in ∑5(013) case, there are still three atomic layers at the GB center without

going through the phase transition process, as shown in Figure 4c. The phase transition occurs with strain up to 8%, at which the maximum free volume is also observed near the bcc-fcc interface with FVmax up to 24.43 Å3. The local stress concentration is also observed above the maximum free volume region with a value around 21.65 GPa, resulting in the activation of slip system and related dislocation nucleation, as shown Figure 4d. Different to the activation of slip system initially in bcc phase for ∑3(112) GB, the activation of slip system under external stress along the <013> direction occurs initially in fcc phase in the {111} plane along <110> direction. Following the same method, the Schmid factor is also calculated for this case with external stress along <013> direction. The maximum Schmid factor for bcc and fcc phases are 0.4115 and 0.4899, respectively, in this case, which is the main reason for slip system activation near the interface from fcc phase side, as shown in Figure 4d.

**Figure 5.** The maximum free volume and related stress distribution near the bcc-fcc interface when the stress is up to the peak value (shown by stress-strain curve in Figure 2 for (**a**) ∑3(112), (**b**) ∑5(013) and (**c**) ∑5(012) GB respectively. The blue and green balls are atoms in bcc and fcc state respectively. The larger red balls are free volume higher than 20 Å3 in GB region.

3.2.2. Mechanical Failure Induced by Activation of Slip System from GB Plane

The second phenomenon accompanying the failure of GB explored in this work is the activation of slip systems directly at GB region without going through the phase transformation, as observed in the ∑5(012) and ∑3(111) ∑9(221), ∑11(113) and ∑17(410) GB cases. One example of ∑5(012) case is shown in Figure 6. As shown in Figure 6, the slip systems are activated from the GB plane to the grain interior at a time of around 16 ps after applying the external strain, at which the strain is around 8%. The Schmid factor is also calculated for this case. The results indicated the *μmax* is up to 0.4625, which is related to the slip in {123} plane along <111> direction, as shown in Figure 6c. Furthermore, careful analysis of the local structure indicates local disordered regions in the GB region, which have high potential energy and high stress along the normal direction of GB plane. In fact, the local stress concentration is also observed in these regions with maximum stress around 27.4 GPa, as shown Figure S6 in the Supplementary Materials. Following the analysis method in Section 2, the atomic displacements were calculated around GB, indicating the maximum displacement distance also observed in these regions, as shown in the Supplementary Materials. The free volume change around these regions is then calculated. When the strain is 0, FVmax is around 31.24 Å3, which is then increased to

33.36 Å<sup>3</sup> above the disordered region, as shown in Figure 5c Therefore, the maximum free volume change is also one possible factor relating to the failure of ∑5(012) GB from the activation of slip system directly in local GB plane region. Further analysis of ∑3(111) ∑9(221), ∑11(113) and ∑17(410) GB reaches a similar conclusion. The example of ∑3(111) GB has been shown in Figures S7 and S8 in the Supplementary Materials. Based on these results, the derivative of critical stress along the normal direction of GB plane of GB failure, Δσ, can be described as a function of GB formation energy, *EGB*, as shown in Figure 7a and the following equation:

$$
\Delta \sigma = 1.241 - 1.297 \exp\left[ -0.5 \left( \frac{E\_{GB} - 4.073}{0.3886} \right)^2 \right] \tag{4}
$$

**Figure 6.** Snapshots of ∑5(012) GB evolution under external stress at different simulation times (t): (**a**) t = 16 ps, (**b**) t = 17 ps, (**c**) t = 19 ps and (**d**) t = 21 ps, respectively.

**Figure 7.** The Gaussian Fit of the relationship between GB energy and Δ*σ* (**a**). The relationship between free volume and fit stress. Fit stress equals the sum of the stress peak value and Δ*σ* (**b**).

Furthermore, the dependence of critical stress (*σ* + Δ*σ*) of GB failure on free volume change (Δ*FV*) has also been explored, as shown in Figure 7b, which can be described by a new equation (Equation (5)). All of these results indicate that once the free volume change has been identified, the mechanical properties of GB can be estimated.

$$
\sigma + \Delta \sigma = 8.21 \,\Delta F V^{0.2421} \tag{5}
$$
