*3.2. Fracture Tendency within the Coatings (Distribution of* σ*zz in the Coatings)*

Hard coatings, especially ceramic ones, may fracture from pre-existing flaws due to tensile stresses parallel to the coating layer. As shown in the third column of Figure 4a,b, the tensile stress (σzz) in the coating appear to rise at the locations where PSBs intersect with the coating for specimens at a small coating thickness of tcoat = 0.5 μm. In addition, due to the load transfer from the substrates to the coatings, σzz is discontinuous across the interface. Due to the superposition of the neighboring PSB–coating interactions, the maximum σzz may be higher at a smaller PSB spacing (higher applied strain ranges). As an example, the profiles of σzz in the coating near the interface along the z-direction (i.e., values on the dashed lines shown in Figure 4) are plotted in Figure 6 for the 0.5-μm thick coating.

The observations made from Figure 4 are confirmed in the σzz profiles shown in Figure 6, noting the values of the stress within the blue shades (locations on the interfaces where PSBs intersect with the coatings). At higher applied strain amplitudes (such as the 0.01 plastic strain range reflected by dPSB = 1 μm, Figure 6a), both the overall and the peak values of the σzz are higher compared to lower applied strain amplitudes, indicating a higher tendency to develop tensile fracture, as expected. Figure 7a shows the variation of the peak σzz values as a function of dPSB (reflecting the applied strain amplitude Δγpl), demonstrating a similar decreasing trend of the peak σzz with increasing dPSB for all tcoat values.

**Figure 6.** Stress profiles, including the σzz in the coating near the interface, σxx at the interface, and τxz at the interface, of the computational cells with 0.5-μm coating thickness. The blue shades indicate the locations where the PSBs intersect with the coatings. Note that the horizontal axes are not of the same scale, which led to their different appearance in thickness. The four panels respectively show data for (**a**) dPSB = 1 μm and Δγpl = 0.01, (**b**) dPSB = 2 μm and Δγpl = 0.007, (**c**) dPSB = 4 μm and Δγpl = 0.004, and (**d**) dPSB = 8 μm and Δγpl = 0.002.

**Figure 7.** The maximum stresses, including (**a**) σzz in the coating at the interface, as well as (**b**) σxx and **(c)** τxz at the interface, as functions of the increasing PSB spacing, dPSB. Note that the applied plastic shear strain range is inversely related to dPSB—i.e., dPSB = 1 μm corresponds to Δγpl = 0.01, dPSB = 2 μm corresponds to Δγpl = 0.007, dPSB = 4 μm corresponds to Δγpl = 0.004, and dPSB = 8 μm corresponds to Δγpl = 0.002.

Comparing the third column of Figure 4c with that of the Figure 4a,b, it is interesting to note that when the thickness of the coating is large, the distribution of the σzz within the coating is less influenced by the presence of the PSBs. For instance, Figure 8 shows the profiles of σzz in the coating along the z-direction at both the surface and the interface. In both coating thicknesses shown, the profiles of σzz in the coatings at the interface were nearly identical (see the thick and the thin dashed lines). However, for the case of a thin coating (tcoat = 0.5 μm), significant variations in the stress can be observed on the surface (thin solid line). However, when the coating is thick (tcoat = 2 μm), σzz is approximately invariant at the surface (thick solid line). The combined observations made in the third column of Figure 4 and in Figure 8 imply that a coating's sensitivity to the presence of a potential surface flaw in the coating is different for different coating thicknesses. For instance, a thinner coating only experiences higher tensile stresses near the PSBs, a flaw at other locations may still be relatively safe and may not lead to early onset of fracture. On the other hand, the tensile stress on the surface of thicker coatings is uniform which makes thick coatings more susceptible to tensile fracture from surface flaws.

**Figure 8.** The profiles of σzz along the z-direction on both sides of the coating for two simulations: (1) tcoat = 0.5 μm, dPSB = 1 μm and (2) tcoat = 2 μm, dPSB = 1 μm. The corresponding stress and strain contours of these two simulations have been shown in Figure 4a,c. The blue shade indicates the location where the PSBs intersect with the coating.

To further investigate the variation of σzz at different locations within the coating, its standard deviation (SD) along the z-direction for all model geometries was calculated. Figure 9 shows the SD of σzz at interface and the coating surface with respect to the ratio of dPSB to the coating thickness tcoat (i.e., λ = dPSB/tcoat). It is evident that, as the ratio λ increases (i.e., thickness of the coating decreases with respect to the PSB spacing), the variation of σzz along the z-direction at the coating surface significantly increases. On the other hand, the variation of σzz at the interface is always quite significant and is not affected by λ. This agrees with the observations made earlier in Figure 8.

**Figure 9.** The standard deviation (SD) of σzz at two locations—i.e., at the coating–substrate interface and coating surface, versus the ratio λ = dPSB/tcoat.

## *3.3. Delamination Tendency at the Coating–Substrate Interface*

The tendency for coating–substrate delamination was assessed by evaluating the interfacial stresses σxx and τxz, which were perpendicular and parallel to the interfaces, respectively. As shown in Figure 4, due to the model setup of a perfectly bonded interface, both σxx and τxz stress components were continuous across the interface. Similar to the behavior of σzz, σxx appeared to slightly increase when the loading amplitude increased (i.e., when dPSB decreased). Interestingly, the magnitude of τxz showed an opposite trend—it appeared to decrease when the loading amplitude increased (i.e., when dPSB decreased). This can be seen by comparing Figure 4a, 4b in the second and the fourth columns. The opposite trends observed here may be ascribed to the "symmetries" in the σxx and τxz values on both sides of the location where the PSBs intersect with the coating.

In Figure 6, the profiles of σxx and τxz at the interface along the z-direction (i.e., values on the dashed lines shown in Figure 4) are plotted for the 0.5-μm coating thickness. At the PSB spacing of 8 μm, the interfacial stresses induced by an individual PSB is clear (see Figure 6d). The sign of the τxz component is opposite at the locations left and right of the intersection between the PSB and coating (see the red and blue arrows marking the ±τxz). On the other hand, the sign of the σxx is the same on the left and right of this intersection—i.e., both values are positive (see the red arrows marking the +σxx in Figure 6d). As a result, when the density of the PSB increases, the superposition of the stress fields from neighboring PSB–coating interactions increases the maximum magnitude of σxx and reduces the maximum magnitude of the τxz. The variation of the maximum σxx and τxz stresses are discernable from Figure 6 by comparing the four panels. In addition, the magnitude of both stress components also increased with increasing coating thickness (compare Figure 4a and Figure 4c in the second and fourth columns). The σxx and τxz appeared to somewhat signify the "suppressive" action of the coatings—as a function of the thickness—on the operations of the PSBs (compare Figures 4 and 5). In other words, the suppressive effects tend to be stronger when the values of these stresses increase.

For a clearer comparison, the peak values of these stresses for all the simulations have been obtained and plotted in Figure 7b,c. The substantial increases in the peak σxx and τxz values due to the increase in the coating thickness are also evident. The combined action of the two stresses may be responsible for the delamination of the coating–substrate interface. This is in line with the critical plane approach put forth by Fatemi and Socie [47], which stated that the planes with large plastic shear strain and large normal stress tend to initiate fatigue cracks.

Figure 5 has shown that the suppression imposed by the coating on the operation of the PSBs is more effective at larger coating thicknesses. Nevertheless, excessively thick coatings are associated with the risk of early crack initiation due to coating tensile fracture (with pre-existing surface flaws) and coating–substrate delamination (due to the combined action of both tensile and shear stresses at the interface). Therefore, thicker coatings are only preferred if a coating with higher fracture toughness as well as an ideal coating–substrate adhesion can be achieved. Otherwise, thicker coatings may be detrimental to the fatigue performance of the coated parts.

On the other hand, while the applied plastic shear strain range is expected to monotonically affect the tendency of tensile fracture in the coating (i.e., higher amplitude leads to easier fracture), its effect on coating–substrate delamination may be more complex. Since the increase in the applied strain range decreases τxz but increases σxx, there may exist an intermediate plastic shear strain amplitude that favors delamination the most, assuming that the interfacial delamination is driven by the combined action of normal and shear stresses.
