*3.1. Suppressive E*ff*ects of the Coating on the Operation of PSBs (Distribution of Plastic Shear Strain in the PSBs)*

Comparing the first column of Figure 4a,c, it is apparent that the thicker coating (tcoat = 2 μm) inhibited the slip activities (γprimary) within the PSB, especially at locations near the coating–substrate interfaces. As expected, this suppression effect was attenuated with the increase in the strain amplitude applied (compare Figure 4a,b, which represented higher and lower applied strain amplitudes, respectively). To quantitatively compare the suppressive effect of the coatings, the plastic shear strain on the primary slip system within the PSBs are plotted in Figure 5 as a function of the distance from the coating–substrate interface along the x-direction. The plots were generated using the binning analysis, described in Section 2.2. Each panel of the figure corresponds to a different PSB spacing (dPSB) and, accordingly, a different applied plastic strain range (Δγpl). The plots were obtained at the point of loading history which corresponds to an average of ~1% plastic shear strain over the entire PSBs.

**Figure 5.** Distribution of shear strain along x on the primary slip system within the PSBs for various PSB spacings and equivalent applied plastic strain ranges: (**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.

The qualitative observations made in Figure 4 are echoed in Figure 5. Coatings with increasing thickness had stronger suppression effects on the operation of PSBs. This is reflected by the consistently lower shear strain magnitudes in PSBs that interacted with thicker coatings. Comparing Figure 5a–d), the suppression effects were attenuated significantly with the increasing applied strain amplitude. This is concomitant with the increasing plastic shear strain in the PSBs near the coating–substrate interfaces as the loading amplitude increases. For instance, at a lower applied strain range (such as at 0.002 for the case of dPSB = 8 μm as shown in Figure 5d), the coatings had a profound effect suppressing the plastic shear strains near the coating–substrate interface. With a coating thickness of 0.5 μm, the shear strain within the distance of 0.5 μm from the interface was lower than the prescribed 1% magnitude. With a coating thickness of 2 μm, this distance extended beyond 2 μm. However, at a higher applied plastic shear strain range of 0.01, this distance reduced to 0.13 μm for the thinner coating and ~1.3 μm for the thicker coating, shown in Figure 5a. The attenuation at higher strain amplitudes (smaller dPSB) can be ascribed to the close coupling of neighboring PSB–substrate interactions, which induced larger elastic deflections within the coatings. As will be demonstrated in the later sections, this coupling also influences the stresses at/near the coating–substrate interfaces.
