**4. Results**

#### *4.1. Global Results of Numerical Simulation*

The global results of the simulation in comparison with experimental observations are shown in Figure 8. The global results are calculated by taking an average of the results at each solution point for each increment. The data are combined in a meaningful way to produce observable trends. The stress and damage evolution with respect to the true global strain in the selected RVE is shown here in Figure 8.

**Figure 8.** (**a**) RVE chosen to run full phase simulations in IPF colors; yellow box represents one of the crack initiation zones. (**b**) Outcome of the simulation flow curve in comparison with the in situ experimental test results, showing a good match; the dips in the experimental results are the points where the test was stopped to take local pictures of the inclusions for later DIC. E1–E4 show the damage evolution around an MnS inclusion, S1–S5 inset figures show the damage evolution around a comparable MnS inclusion during numerical simulation.

A good correlation between global experimental and numerical simulation results is observed with less than 3% difference of predictions. The inset figures in Figure 8 from S1–S5 show damage evolution around an MnS inclusion. Inset figures E1–E4 show damage evolution around a similar-sized MnS inclusion captured during in situ testing. Interestingly, not only do the global results match well, but the local damage evolution is also comparable. These local deformation and damage trends are discussed later in this article.

#### *4.2. Local Results of Numerical Simulation*

The local distributions of strains, stress, triaxiality, and damage at 3%, 9%, and 15.5% of global strain are shown in Figure 9. Due to the heterogeneous distribution of non-metallic inclusions and different Schmidt factors of each ferrite grain, the local distributions of each attribute are largely heterogeneous. Therefore, the grain boundaries have been overlaid in contrasting colors to clarify the distributions of attributes within individual grains.

It is observed that at 3% global strain, the local strain distribution is relatively homogeneous within the selected RVE. As the global strain increases and reaches a maximum of 15.5%, an extreme heterogeneous distribution of strain along the grain boundaries in thick channels aligned at 45 degrees to the loading axis is observed with intermediate areas of very low local strain. On the other hand, the stress distribution is high from the beginning in the ferrite grains with a high Schmidt factor and increases significantly within those grains. The value remains similar as the global strain reaches the maximum value.

Of course, the local stress and strain distribution is largely affected by the presence and distribution of non-metallic inclusions. The behavior is discussed in more detail in the next section.

**Figure 9.** Evolution of local strain, local stress, stress triaxiality, and local damage in the selected RVE at 3%, 9%, and 15.5% global true strains, where the loading axis is horizontal to the micrographs.

Stress triaxiality is the relative degree of hydrostatic stress in each stress state, which is usually used to estimate the type of fracture that might take place in the material. In the current RVE, it is observed that the triaxiality measure in most of the grains is close to 0.1, which means that the stress (no matter how high) is predominantly hydrostatic. Therefore, the deviatoric component of stress in these grains is low and will not contribute significantly towards material flow and eventual damage. In addition, it is interesting to note that the triaxial stress remains consistent with the increase in strain and only slightly shifts to neighboring grains after the damage initiation and propagation.

Generally, it is observed that the damage initiates at the interface of the non-metallic inclusions, especially in the narrow zones locked between the cluster of inclusions. After initiations, the damage propagates at an oblique angle to the applied external load. As a result, the local voids coalesce and form larger cracks that grow faster, and material damage accelerates. It is important to mention here that the continuum mechanics simulations are intrinsically unstable due to the loss of stiffness at several solution points. Furthermore, it takes immense computing power to calculate the corresponding equilibrium for each increment after damage initiates in the RVE. Therefore, the simulations were only computed up to 15.5% of the global strain, and the results are shown in Figure 9.

Although the local results generally for the ferrite matrix are shown in Figure 9, one of the main objectives of this article is to investigate the effect of non-metallic inclusions on the material's local deformation and damage behavior. In Figure 10, the focus has specifically been shifted towards three different zones in the RVE of comparable size with different inclusion compositions and distributions. Zone-I comprises a single grain with a large MnS inclusion in the middle. Zone-II comprises several small ferrite grains with a clustered distribution of non-metallic inclusions primarily on the grain boundaries. Zone-III comprises relatively larger ferrite grains with a diffused distribution of non-metallic inclusions within the grains and on the grain boundaries. The distribution of local attributes in the three zones is selectively identified in the corresponding subplots Figure 10b–e. There is a relatively low stress concentration in zone-I and less strain due to almost nonexistent stress triaxiality, and therefore, no damage takes place in this zone. This can be due to the high Schmidt factor of this specific grain and the central position of a large hard MnS inclusion which restricts the slip planes from moving in this grain freely.

**Figure 10.** A construction figure with (**a**) showing the selected RVE with the identification of three zones. All non-metallic inclusions are displayed by white points. In other subplots, (**b**) true Mises stress, (**c**) true Mises strain, (**d**) stress triaxiality, and (**e**) damage propagation withing the ferrite matrix are shown. Zone-I has a ferrite grain with a large MnS inclusion in the middle, zone-II is a region of small ferrite grains with clustered non-metallic inclusions present on the grain boundaries, and zone-III has relatively large ferrite grains with a dispersed distribution of non-metallic inclusions within grains and on the grain boundaries. The loading axis is horizontal to the micrographs.

Small non-metallic inclusions distributed on the grain boundaries of small ferrite grains act as high-stress concentration sites with the large plastic flow in zone-II. Consequently, the stress triaxiality in this zone is relatively high, indicating a large plastic flow and evolution of ductile damage. This damage evolution due to the coalescence of small voids that form on the inclusion/matrix interface is visible in Figure 10e. It is observed that local damage incidents join together to form microcracks that propagate at maximum shear plane oriented 45 degrees to the loading axis. This can be due to two reasons: first, due to the 2D nature of the RVE, as has been pointed out in the previous work [52]; and second, due to the high amount of slip system activation in grains and areas with maximum critical resolved shear.

Zone-III comprises relatively larger ferrite grains with a relatively broader distribution of non-metallic inclusions within the grain and grain boundaries. In zone-III, although the triaxiality is comparable with zone-II, due to the lack of clustering on non-metallic inclusions, the stress distribution and strain distribution are lower. Hence, no damage occurs in the zone.

In Figure 10, a qualitative comparison of local attributes in three different zones shows how the distribution of non-metallic inclusions plays a role in affecting the local stress, strain, and damage. To quantitively compare three zones, a statistical normalized probability distribution was adopted to systematically compare the distribution of stress and strain within zone-I, zone-II, and zone-III, as shown in Figure 11.

**Figure 11.** The normalized frequency of (**a**) strain and (**b**) stress for each zone identified in Figure 10.

To quantitatively compare the stress and strain distribution in zones I to III, a statistical distribution tool is used, and the results are presented in Figure 11. It is observed that the strain distribution in all three zones, as shown in Figure 11a, is very similar. It peaks around 0.15–0.18 and then linearly drops to a strain of 0.45. The strain in zone-I is slightly higher and that in zone-II is slightly lower due to the local orientation distribution. The stress distributions in these zones reach up to 1100 MPa.

The statistical stress distribution in the three zones is presented in Figure 11b and proves the already stated hypothesis. It is observed that the stress in zone-I and zone-III peaks around 750 MPa, with a slight bias towards the lower stress side. On the other hand, the stress distribution in zone-II, due to the close clustering of several non-metallic inclusions on the grain boundaries of small ferrite grains, peaks at 850 MPa (11% higher than the other zones), and its distribution is biased towards the higher-stress side. The distribution reaches 1250 MPa (12% higher than the other zones). This high local attribute distribution in zone-II makes it more prone to local damage initiation and propagation in high deformation regimes.

#### *4.3. Local Results of the In Situ Tensile Test*

The local results of the in situ tensile tests at 3%, 5%, and 8% of global strain are shown in Figure 12 The methodology and procedure of local strain measurement is provided in Appendix B. The complete set of results, due to its extensiveness, is provided in Appendix C of this article. Readers can refer to Appendix B and C for further details about collection and frame by frame evolution of local strain measurements.

**Figure 12.** Local strain distribution in ferrite matrix and MnS inclusion at 3%, 5.5%, and 8% global strains.

In Figure 12, at a subsequent global strain of 3% and 400 MPa global stress, it can be observed that in the zones with increased elongation, the strain rate is much higher than in the zones where the strain rate has not increased much from the beginning of the test. The effect of the increase in elongation at the ferrite grain boundaries near the non-metallic inclusion is preserved. At this stress, two zones with a significant increase in strain can be seen in the middle of the inclusion. When comparing the entire image series, at a load of 450 MPa and global strain of 5%, the sulfide inclusion is cracked in the area of thinning. The inclusion in the left part of the crack is brighter due to its change in position, which significantly affects the strain values in this inclusion area, as the digital image correction software requires areas with contrast for accurate calculation. The breaking point itself is hidden at this point due to the topography.

The maximum values in the range of increased load are above 8.9%. It is observed that the two sulfide inclusions move further away from each other. The strain in the region increases, reaching a value of over 13–64%. Around the formed cavity of a small inclusion located on the upper right side of the elongated inclusion, the elongation near the pearlitic zone is greater than on the other side of the same grain. On the side of the ferrite grain around this cavity is a zone of increased deformation with a value of about 7–11%. There is a significant difference in elongation within and around the small cavities. In the middle cavity, the deformation is minimal and is between 2% and 3.5%, while in the left cavity, the deformation around and within the cavity is between 6.4% and 9.8%.

Due to further load increase and subsequent high strain values, the DIC program loses previously captured pixels for the calculation. By enlarging the gaps created, some pixels can significantly change their gray tone, causing the program to find a similar range of pixels elsewhere and greatly distorting the calculation values. At a maximum global strain of 8% (450 MPa), a large loss of pixels is observed (in areas beyond the specified deformation limit of dark red) because the topography changes too much, and the color range continues to change in many areas. In addition, the distance between the halves of the sulfide inclusion become significantly large. In the lower deformation zone in the matrix around the sulfide grain inclusion, the values are 43.6%. In contrast, the zone-I around the inclusion next to the perlite grain remains below the total deformation.

#### *4.4. Damage Evolution around Non-Metallic Inclusions*

The MnS inclusion tracked to analyze local deformation and damage behavior in Figure 12 is an exception. Usually, the inclusion size in the material user consideration is between 1–4 μm; please refer to Figure 4 for reference. Unfortunately, such small inclusions are hard to track and record local strain around due to high magnification, resulting in fast drifting during testing in the SEM chamber [53]. Despite this paradox, in the current work, during in situ testing, a few MnS inclusions of small size were tracked and recorded at different strains. The results are shown in Figures 13 and 14.

In Figures 13 and 14, several inclusions classified as MnS inclusions based on the EDS analysis were tracked at different external strains that were applied. The results shown here were at the beginning of the test when the inclusions were identified, at external load when the damage was initiated, and an external load when the damage started to evolve.

In Figure 13, it is observed that the damage initiates perpendicular to the line of action of the load at about 525 MPa global stress and propagates in that same direction. It is important to keep in mind that the inclusion is embedded in the matrix, and here only the surface is being observed. The surface matrix seems to be extremely deformed due to the internal propagation of the damage around the inclusion. It is also interesting to note that the damage initiates at the inclusion/matrix interface. In contrast, the inclusion remains free of any cracking, unlike the case of large inclusion, which was shown and discussed earlier in Figures 5 and 12. This is because the damage of the inclusion is a function of the size and distribution in the matrix, where small inclusions with an aspect ratio of less than 2.5 are generally not highly prone to damage.

**Figure 13.** (**a**) Localized single inclusion identification, (**b**) represents a round inclusion at 525 MPa and 550 MPa, respectively. (**c**) represents an elliptical inclusion at 525 MPa and 550 MPa, respectively. EDS show the maximum concentration of manganese and sulfur along the measured length verifying both to be an MnS inclusions. The white arrows point to the local damage initiation and growth.

In Figure 14, two other inclusions are tracked, again identified as MnS based on the EDS analysis shown in the same figure. Both inclusions are completely embedded in small ferrite grains and resemble the case of zone-I in Figure 10.

In Figure 14, it is observed that the damage starts to initiate at 550 MPa of external load, which is slightly higher than the damage initiation load for other inclusions but is in a similar range. At 575 MPa of external load, damage on the inclusion matrix interface grows so extensive that it is easily identifiable and is marked with the help of white arrows. In this case, it is observed that voids initiate at an oblique angle to the applied external load and not perpendicular to the load as observed earlier. This position of the damage initiation depends on several factors: i.e., orientation, size, shape, and sharpness of the inclusion edges and the place of the inclusion (inside a large grain or on the grain boundary). It would also largely depend on the orientation of the surrounding matrix grains. As mentioned earlier, all these inclusions have 3D geometry, and the third dimension would also affect the results observed on the surface of the specimen.

**Figure 14.** (**a**) Localized single inclusion identification, (**b**) represents an elliptical inclusion parallel to the applied load at 550 MPa and 575 MPa, respectively. (**c**) represents an elliptical inclusion placed 30 degree to the applied load at 550 MPa and 575 MPa, respectively. EDS show the maximum concentration of manganese and sulfur along the measured length verifying both to be an MnS inclusions. The white arrows point to the local damage initiation and growth.
