**1. Introduction**

Fatigue cracks for many metals often initiate from surface markings (i.e., intrusions and extrusions) formed due to cyclic slip localization [1,2]. This localized slip activity is associated with permanent changes in the microstructure of the material developed during the cyclic loading, and the slip markings can re-emerge at the same locations upon reapplication of the load even after surface polishing [3]. Due to their persistent nature, these surface markings are commonly referred to as persistent slip markings (PSMs), and the localized deformation volume is referred to as persistent slip bands (PSBs). Cyclic plastic deformation in the PSBs is typically accommodated by the motion of dislocations on the primary slip system [4–6].

For FCC materials, these dislocations have the same Burgers vectors and form very organized bundles known as veins (see Figure 1a). When these bundles collapse along the primary slip direction, PSBs are formed [7]. A typical dislocation microstructure (i.e., the ladder structure) of the PSBs constitutes regularly spaced dislocation walls separating dislocation channels, in which screw "runner" dislocation propagates in a to-and-fro manner. The width of the dislocation channels within PSBs

are on the order of 1 μm and is significantly larger than that of the one between dislocation veins. The cyclic plastic deformation therefore localizes at the PSBs [4].

**Figure 1. (a)** A schematic illustration of a persistent slip band (PSB), vein structure, specimen surface, and the surface persistent slip marking (PSM). **(b)** The cyclic stress–strain curve of single-crystal Cu [8].

PSBs typically form within a specific range of resolved plastic shear strain amplitude of Δγpl/2 ≈ 0.0001~0.01, within which the volume fraction of the PSBs linearly varies from 0% to 100%. The precise values of this range are also material-dependent. This behavior for single-crystal Cu is shown by the cyclic stress–strain curve obtained by Mughrabi [8] (see Figure 1b). Accordingly, the plastic strain amplitude within the PSB is constant at around 1% [9]. In addition, the width of the PSBs is slightly material-dependent but generally invariant with respect to the loading amplitude. As a result, the change in loading amplitude only alters the PSB density—i.e., the higher the loading amplitude is, the higher the PSB density becomes [10].

In polycrystalline metals, the location of the PSBs may shift from the surface to subsurface under very low stress or strain amplitudes (i.e., in the very high cycle fatigue regime) owing to the absence of global plasticity and the dominance of the subsurface, localized plastic deformations due to the elastic incompatibility among grains or between grains and defects [10]. Thus, the corresponding PSB–grain boundary interaction may lead to subsurface fatigue crack initiation [11]. Nevertheless, the PSM-induced surface crack initiation is still the governing mechanism in the low cycle fatigue (LCF) and high cycles fatigue (HCF) regimes encountered in many engineering applications [12–14]. Therefore, any surface treatments that act to suppress or interfere with the formation and/or operation of the PSBs may delay the initiation of fatigue cracks and substantially extend fatigue life. Well-known methods in this regard include intermittent surface repolishing, shot/sand surface peening, and surface hard coatings [3,15–19].

Surface repolishing aims to completely remove the PSMs—together with any other roughness created due to cyclic loading—at appropriate intervals mid-service, eliminating any stress concentrations (even crack embryos). Fatigue life had been shown to be extended almost indefinitely using this method, as long as sufficient serviceable material remained [15]. However, this method, due to its subtractive nature, is not suitable for part surfaces that require tight tolerances. Shot/sand peening, on the other hand, is performed prior to service and induces plastic deformation on the surface layer driven by the impact of shot or sand particles, which leaves a compressive residual stress and a refined microstructure on the surface [16,18]. However, this method results in a relatively rough surface finish and is not suitable for mating surfaces. Lastly, surface hard coatings can circumvent some disadvantages of the other methods because they are generally very thin (a few hundred nm to a μm) and create a smooth surface [19].

There are extensive works focused on examining the effect of various types of coatings—including metallic mono-/multi-layers, ceramic, metallic glass, diamond-like carbon—on the fatigue resistance of metallic specimens [17,20–26]. It has been generally observed that the coatings can improve the HCF performance of the test specimens, while they are often detrimental to the LCF performance. In the LCF regime, higher loading amplitudes tend to induce fracture within the coatings and/or lead to delamination at the coating–substrate interfaces. The precise effect of coatings on the fatigue resistance of a material greatly depends on several factors, including the coating–substrate material combination, mechanical properties and thickness of the coating, and the binding and shear strengths of the coating–substrate interface, etc. [17,19,27]. However, due to the time-intensive nature of the fatigue data generation, systemic evaluations of the influence of the aforementioned factors are still lacking.

Using crystal plasticity (CP) simulations, we aim to elucidate the effect of the coating thickness and the applied cyclic plastic strain amplitude on several characteristics of PSB–coating interaction that tend to influence the fatigue performance of coated materials. These characteristics include (1) the suppressive effects of coatings on the operation of PSBs, (2) the normal stress developed in the coating layer along the loading direction, (3) the interfacial normal stress developed perpendicular to the interface, and (4) the interfacial shear stress developed parallel to the interface. The first characteristic retards, while the rest accelerate, the initiation of fatigue cracks in the substrate. This work by no means attempts to perform direct CP simulations of fatigue failure of coated metallic materials. Instead, it focuses on the interaction between the coating and the PSBs under different loading amplitudes and aspires to generate an understanding that may benefit the geometrical design and material selection of the coatings. Indeed, cyclic damage (in the forms of dislocations and vacancy) accumulates during each cycle via cyclic plasticity and, upon reaching a critical level, leads to the initiation of fatigue cracks [28,29]. The surface coatings' suppression of the plastic deformation per cycle is therefore indicative of the coatings' beneficial effects on a part's overall fatigue resistance. As such, instead of simulating the accumulation of cyclic damage over the entire fatigue life, this work only considers the loading portion of a single cyclic loading period after the PSBs has formed.
