Zr-4 alloy was selected as the substrate material, and its chemical composition is provided in Table 1. The original Zr-4 alloy bar was subjected to an annealing treatment at 800°C for 24 h in vacuum, after which it was cut into dog-bone-shaped tensile samples with a gauge length of 20 mm. The typical microstructure of the resulting Zr-4 alloy is shown in Figure 2A, with a grain size of approximately 20 μm. A multi-arc ion plating technique was used to deposit the Cr coating (purity of approximately 99.9%) on the surface of the Zr-4 alloy. Detailed deposition parameters are listed in Jiang et al. (2020). The surface and cross-sectional morphologies of the Cr-coated Zr-4 substrate are shown in Figures 2B,C. The coating had a dense microstructure with small grains that were 1 μm in size. The coating thickness was relatively uniform with an average thickness of 10 μm. The Cr coating bonded tightly to the substrate, and no micro-cracks or micro-voids were found at either the surface or the interface.

As shown in Figure 3, in situ tensile tests of the Cr-coated samples were performed using a mechanical experimental device equipped with a high-magnification optical microscope, a mechanical loading system (for tensile tests, fatigue tests, and three-point bending tests), and a data acquisition system. As shown in Figure 3B, the test sample was fixed to the mechanical loading system by two clamps. The optical microscope (3,000 × maximum) was suspended over the mechanical loading system to observe the deformation and cracking behavior on the surface of the sample during the tensile process. Tensile tests were performed under the displacement control mode at a constant rate of 5 × 10^–3 mm/s, i.e., a strain rate of 2.5 × 10^–4 s^–1. The tensile test was paused at various moments to capture the crack initiation and its evolution on the coating surface. Note that an area of 1,450 μm × 1,100 μm where the first visible surface crack initiated was imaged to capture the crack evolution. The engineering strain was calculated by dividing the tensile displacement by the original gauge length. The tensile test ended when the sample fractured entirely. After the tensile test, the cross section of the sample was examined using scanning electron microscopy (SEM) to analyze the interfacial cracking behavior.

A rate-dependent crystal plasticity theory based on the classical work of Taylor and Hill (Taylor, 1938; Lee, 1969; Hill and Rice, 1972) is applied to describe the anisotropic constitutive relation of the materials. The model is capable of modeling the plasticity deformation of crystals considering the crystal orientation effects. In this work, the flow rule on each slip system is assumed to follow a power–law relationship (Hutchinson, 1976),

where γ.α is the shear rate, τ^α is the resolved shear stress, a.0α and m are the reference strain rate and rate sensitivity coefficient of slip system α, respectively, g^α is the critical resolved shear stress determined by the hardening state of the material on the crystal scale.

In this study, a Voce hardening model (Agnew et al., 2001) was chosen to model the hardening of the Zr-4 alloy substrate, which has been widely used for hexagonal close-packed (HCP) crystal structure. An exponential hardening model (Peirce et al., 1982) was chosen to model the hardening of the Cr coating, which has a body-centered cubic (BCC) crystal structure. The hardening laws are expressed as follows:

(i) Voce hardening model:

where g0α is the critical shear stress of the α-th slip system in the initial state. g1α is the critical shear stress of the α-th slip system at steady state. γ is the sum of the shear strains of all the slip systems in the current step. θ0α and θ1α are the initial and final hardening rates, respectively.

The increment of the critical shear stress of the α-th slip system in this increment step can be solved by the following formula:

where q_αβ represents material self-hardening (α = β) and latent hardening (α ≠ β). Its value is assigned as 1.0 for coplanar slip systems and 1.4 otherwise.

(ii) Exponential hardening model:

The micromechanical interaction between any slip system β and the fixed slip system α is taken into consideration by

where h^αβ represents the hardening matrix and is given as

where h0α, a, g₀, and g₁ are the slip hardening parameters. In the simulation of Zr-4 deformation, twinning is regarded as a special slip system. Due to the polarity of the twinning, the lattice rotation caused by twinning is different from the lattice rotation caused by slip. In this study, the models proposed by Tomé et al. (1991) and Paul (1978) were used.

Figure 8 displays the experimental results, including in situ observations of the surface crack evolution and cross-sectional morphologies of vertical and interfacial cracks. As seen in Figures 8a–c, the first visible surface crack was found in the Cr coating when the tensile strain (ε) reached 0.4%. As the tensile strain increased, new surface cracks were continuously generated between two adjacent cracks, leading to an increase in the crack density. These multi-cracks were parallel to one another and vertical to the tensile direction. Subsequently, almost no new surface cracks were formed, and the crack density tended to be stable with increasing tensile strain; however, the cracks continued to darken, which indicates that the crack opening displacement (COD) became larger.

To understand the initiation and propagation behavior of cracks on the surface of the coating, the cross-section morphologies at various strains were observed, as shown in Figures 8d–i. Vertical surface cracks were found that initiated in different areas. In most areas, vertical cracks were generated from the interface and propagated to the surface (see Figure 8d), which might be driven by the stress concentration at the interface caused by the large differences in the elastic modulus and plastic deformation between the Cr coating and the Zr alloy substrate. Further analysis and explanation will be presented in the following section using the results of the finite element simulation. In some areas, vertical cracks were also found that initiated from the surface (see Figure 8f), which might be due to local stress concentration caused by impurities or defects present on the coating surface.

Furthermore, after the vertical cracks penetrated through the coating thickness, they continued to propagate along different paths under continuous tension. In most areas, vertical cracks penetrated through the coating to the substrate (see Figure 8g), while few vertical cracks propagated along the interface (see Figure 8h), which reflected the weak interfacial adhesion in these local areas. In some areas, cracks not only penetrated to the substrate but also propagated along the interface (see Figure 8i). The competition between the mechanisms causing vertical and interfacial cracking was highly dependent on the magnitude of the fracture toughness of the Cr coating and the interfacial fracture toughness between the Cr coating and the substrate.

The plastic deformation behavior of the coating system without considering cracking is presented first. Figure 9 displays the distributions of stress components, S_xx, S_yy, and S_xy, in the coating-substrate system at ε = 0.4%. Clearly, the stress components in both the coating and the substrate were non-uniformly distributed along the x direction, and the S_yy and S_xy were non-zero in large areas. These results were quite different from those calculated by a macro-scale FEM that considered isotropic material properties. The different mechanical behaviors of grains with different orientations led to this non-uniform distribution. In addition, the stress distribution near the interface showed a remarkable jump from the substrate to the coating, which was highly related to the difference in the mechanical properties of the two crystal structures (BCC structure for the Cr coating and HCP structure for the Zr-4 substrate) and deformation mismatch between the coating and the substrate. The grain-induced local stress concentration would promote earlier crack initiation under tension.

As shown in Figure 9, the stresses were different along the directions vertical and parallel to the tensile direction. To analyze the stress distribution in detail, a reference coordinate system was defined, as shown in Figure 10. The x-axis was parallel to the tensile direction, and the grains in the Cr coating were numbered from left to right as No. 0 to No. 40. The y-axis was parallel to the coating thickness such that the interface was located at y = 0 μm, and the coating surface was located at y = 10 μm. The stress components, S_xx and S_yy, which were the driving forces for vertical and interfacial cracks, respectively, are the primary focus in the following discussion.

Because the stresses in various grains in the coating were non-uniform, a detailed analysis of the S_xx distributions along the y direction for different grains (i.e., different x) was performed, as shown in Figure 11. No obvious decrease in S_xx were found along x direction. The stress presented a different distribution among grains because of the different crystal orientations in the adjacent grains. Their deformation abilities along the tensile direction were different, as derived from the Schmidt law. Figure 12 shows the S_xx distribution at the interface and the corresponding rotation angles (φ₁, which can represent the grain orientations) around the z-axis in a laboratory coordinate for different grains in the Cr coating. For different grains, the interfacial stress varied from a minimum value of 247 MPa to a maximum value of 312 MPa. The significant variation in the stress is quite different from the results calculated by an isotropic constitutive model in which the stress was independent of the position along the x direction. For the CPFEM, the difference in the Schmidt factor for different Cr grains can rationalize the non-uniform stress distribution. The location where S_xx reached the maximum value could be the most preferential site for vertical crack formation.

Because interfacial cracks were also found in the tensile test, the driving force, i.e., S_yy, was also analyzed. Figure 13 shows S_yy at the interface for different grains in the Cr coating. It is worth noting that for an isotropic constitutive model, S_yy should be zero for different x and y in the coating because no external loading is applied along the y direction. However, for the CPFEM, S_yy in the coating varied greatly from -10 to 45 MPa in different positions, which was caused by the different crystal orientations of the Cr grains and the deformation mismatch between the coating and the substrate. As shown in Figure 13, S_yy near the interface showed the largest change due to the deformation of the substrate. Near the coating surface, S_yy in the coating tended to be nearly zero. The stress S_yy showed tensile value at the interface, which may raise the possibility of initiation of interfacial cracks.

During tensile tests, vertical cracks initiated and propagated through the coating thickness with increasing tensile strain, as shown in Figure 7. The cracking behavior in the Cr coating is directly related to the stress evolution with tensile strain. In turn, the formation of the crack redistributes the local stresses. To analyze this phenomenon, the initiation and propagation of vertical cracks in the Cr coating was simulated using a CZM, and the results of stress and crack evolutions under different tensile strains are displayed in Figure 14. As shown in Figures 14A,D, when ε reached 0.25%, although a crack had not yet formed, damage had already initiated and accumulated in the cohesive elements. The degradation in the cohesive zone led to local stress and strain concentration in the substrate near the interface. As shown in Figures 14B,E, the vertical crack began to form in the Cr coating when ε reached 0.25%, which indicates that the damage value in the cohesive elements reached one, and the relevant elements were deleted to form the crack surface. Once initiated, the vertical crack passed through the coating thickness rapidly to form a macro crack. Comparing Figure 9 and Figure 14B, the stress distribution changed considerably with the formation of the vertical crack. In particular, in the substrate, an area of stress concentration formed near the interface, passing through two adjacent Zr grains. In the Cr coating, the stress near the coating surface relaxed greatly in the region adjacent to the crack surface, while the stress near the interface increased considerably because of the crack tip. As the tensile strain increased, the vertical crack broadened, and the stress and strain concentrations increased considerably (see Figures 14C,F). Once the stress in the substrate reached a critical value, the vertical crack penetrated the substrate along the direction marked by the arrows in Figures 14C,F. The predicted crack deflection was consistent with the experimental observation shown in Figure 8.

Crack formation can also affect the stress–strain curve of the coating–substrate system. As shown in Figure 15, the engineering stress–strain curve changed markedly, particularly during the early period of tension. After the vertical cracks formed, the stress declined slightly when the strain increased from 0.4 to 0.65%, which was caused by stress relaxation with crack propagation. Once the vertical crack penetrated through the coating thickness, it was hindered by the substrate and stopped at the interface. When ε > 0.65%, the stress increased slowly and smoothly with increasing tensile strain.

It is noted that the high stress concentration located at the crack tip may result in continuous crack propagation with a large tensile strain. As discussed previously, two propagation paths were possible: along the original crack direction (i.e., perpendicular to the loading direction), and along the interface between the coating and the substrate. The competition between the vertical and interfacial cracking mechanisms is controlled by the fracture parameters, such as the fracture strength and toughness of the coating and the substrate, and the interfacial fracture toughness. Further experiments and finite element calculations of these competing mechanisms will be developed in the future studies to build a strength assessment method for the ATF coating system.