1 Introduction

Light metals, such as magnesium and aluminum alloys, are very attractive structural materials for achieving high performance and energy savings in machines and structures. This is because of their advantages, such as a high strength to weight ratio and high specific stiffness (Mordike and Ebert, 2001). Therefore, the use of light metals (especially magnesium alloys) as structural materials has recently been attracting increasing interest in many applications. In particular, some magnesium alloys are considered to be replacing some aluminum alloys in aerospace and automobile fields with a concomitant saving in weight, and good casting properties in automotive industries (Dahle et al., 2001; Tang et al., 2005; Wang and Fan, 2006). However, the active chemical reactivity and low fatigue strength of magnesium alloys have deterred many potential users from incorporating magnesium into their designs. Recently, anodizing coatings and plasma electrolytic oxidation (PEO) coatings have become effective methods for improving the corrosion resistance of magnesium alloys (Nykyforchyn et al., 1998; Yerokhin et al., 1999; 2004; 2005; Khan et al., 2005; Guo et al., 2007; Rajasekaran et al., 2008). Unfortunately, some results have indicated that anodized or PEO coating-substrate structures have reduced fatigue strength or fatigue life compared with uncoated substrates (Eifert et al., 1999; Yerokhin et al., 2004; 2005; Rajasekaran et al., 2008). Wang et al. (2014; 2015) provided a method for improving the fatigue strength of coated-substrate of Al 2024-T4 alloy by using PEO with an impregnation epoxy resin sealing pore method. One of reasons for the improvement in fatigue strength was that the PEO coating with impregnation epoxy resin not only “welds” the original thermal cracks, but also fills in the micro pores or meso pits previously formed in the PEO coating process or hard anodized process. This so-called synergistic effect between coating and substrate is able significantly to improve the surface fatigue crack initiation resistance and the fatigue crack propagation resistance. However, it is still unclear how the relations of the interface influence the surface deformation and micro cracking behavior of the coating-substrate structure, especially how to evaluate critical stress failure in the coating layer or substrate. For example, little is known about the effect of increasing the coating thickness or adding multiple coating layers, or the effect of the interface on crack initiation and propagation mechanisms in the micro scale. Although Tvergaard (2001; 2004) reported interface crack nucleation and growth under mixed mode loading conditions based on special interface elements in theory, relevant experimental results have rarely been reported. In addition, the suitability of elastic mechanics theories and finite element (FE) methods for thin anodized coatings (hard thickness layers) and relatively thick substrates (relatively soft thickness layers) has been debated. This is because the elastic theories used to evaluate the failure stress are based on the plane strain state which assumes that the effect of width of coating layer on the strain or stress can be ignored. On the other hand, there are difficulties in mesh refinement using the FE method (Hutchinson and Suo, 1991; Fawcett, 1998; Yan and White, 1999; Michler and Blank, 2001; Zhao et al., 2002; Schwaiger and Kraft, 2003; Yang et al., 2014; Dean et al., 2015). Therefore, to investigate failure issues of thin coating layers/substrates, experimental study is the best and directly approach (Wang and Xu, 2003; 2004; Wang et al., 2004; 2009; Li et al., 2008a; 2008b; Dean et al., 2015). In addition, it is necessary to estimate the delivery criterion of stress close to the interface of thin anodized coatings and thick substrates using SEM in-situ technology (Wang et al., 2004; Wang and Fan, 2006; Jia et al., 2012; Ren et al., 2013; Ren and Wang, 2014).

2 Testing sample preparation and method

All specimens consisted of an anodized MgO coating and AZ91D substrate with a rectangular cross-section (30 mm×5 mm×2 mm). Experiments were carried out by SEM in-situ observation under a three-point bending loading (Fig. 1a). The sample subjected to the bending loading was inserted into the vacuum chamber of the SEM system (Fig. 1b). Force diagrams of a sample with a two-layer-laminated beam are shown in Figs. 1a and 1c.

Fig. 1
figure 1

A schematic of the coating-substrate structure based on SEMin-situtechnology

(a) Shape and size of the sample; (b) Experimental methodology; (c) Mechanics model of a two-layer-laminated beam. L is the span length, MZ is the bending moment, hC and hS are the thickness of the coating and substrate, respectively, d is the displacement of neutral axis, and P is the applied loading

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The chemical composition of the AZ91D substrate was 9.50% Al, 0.90% Zn, 0.17% Mn, 0.05% Si in weight, with very low contents of copper, iron, and nickel. The anodized coating on the AZ91D substrate was prepared by a galvanostatic pulse technique in K2SiO3+KOH+KF solutions with a small quantity of proprietary additives for about 15 min. The square pulse current had an average current density im of 0.05–0.1 A/cm2, the frequency ƒ was 1000 Hz and the ratio (α) of switch time on (ton) to switch time off (toff) (α=ton:toff) was about 5%. The solution temperature was about 30 °C and the two end voltages were 330 V and 350 V, respectively. The thickness of the anodized coating layer measured from the cross-section of the coating-substrate using the SEM was about 30–40 µm (Fig. 2a). Some micro voids of about 1 µm in size were present in the cross-section of the coating layer. These micro voids resulted from the release of residual stress during the anodized coating process. Therefore, the residual stress in the anodized coating was ignored. In addition, the interface between the anodized coating and the AZ91D substrate had a better integrality (Fig. 2a), although the interface was not regular, as shown by the arrows in Fig. 2a, where there is neither interface cracking nor debonding in the region of observation. The free surface microstructure of the anodized coating is shown in Fig. 2b. The free surface roughness was estimated to be about Ra=0.3–0.5 µm using an atomic force microscope (AFM) so as not to cause a change in the stress in the coating layer.

Fig. 2
figure 2

Microstructures at the interface and surface of the coating-substrate structure

(a) Cross-section image (scale bar 10 µm); (b) Free surface image (scale bar 5 µm)

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The main mechanical properties of the anodized coating-substrate structure are listed in Table 1. The Young’s modulus of the anodized coating-AZ91D substrate and AZ91 alloy are similar (Konsta-Gdoutos and Gdoutos, 2005). However, the strength and elongation of the anodized coating-AZ91D substrate were lower than those of the AZ91D bulk substrate. One of reasons is that the sample can be defined as composite layers of structural material so that the strength and toughness of the structural material decreases with increasing thickness of the coating (Li et al., 2002).

Table 1 Mechanical properties of the anodized coating-substrate structure
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The micro crack initiation and propagation tests of the anodized coating-AZ91D substrate were carried out using SEM in-situ observation at room temperature in a vacuum chamber (10−4 Pa) (Figs. 1b and 1c). To achieve a good roughness Ra of the observation cross-section (Ra was about 0.8–0.9 µm), each cross-section was carefully polished with P1000–P2000 (grits page) abrasive papers before the bending tests were carried out (Fig. 2a). A signal from the SEM was directly transferred to a computer via a direct memory access type A/D converter, making it possible to sample 960×1280 frames of SEM images successively in in-situ observation during the bending test. All bending tests were controlled by a stroke ratio of 10−4 mm/min (Wang et al., 2004; Wang and Fan, 2006; Huang et al., 2007; Jia et al., 2012; Ren et al., 2013; Ren and Wang, 2014).

3 Theoretical background

This method is based on the neutral axis maintaining its original position achieved through a side anodized specimen (Fig. 1c). The two-layer-laminated beam was subjected to a bending moment M and shear force S. The coating layer and AZ91D substrate were assumed to display elastic-plastic behavior (Stoney, 1909; Ren et al., 2013; Dean et al., 2015). For a relatively thin coating and relatively low substrate stiffness, the inside stress gradient can be ignored. The shear force S and bending moment M, at a distance of x from the edge in a three-point bending test (Fig. 1c), are given by

$$\begin{array}{*{20}c} {{S_x} = P{{/}}2,} & {{M_x} = Px{{/}}2,} & {0 < x < L{{/}}2,} \end{array}$$
(1a)
$$\begin{array}{*{20}c} {{S_x} = - P{{/}}2,} & {{M_x} = PL{{/}}2 - Px{{/}}2,} & {L{{/}}2 < x < L{.}} \end{array}$$
(1b)

The shear stress τs is taken up mainly by the interface of the substrate (Fig. 1c). The tensile stress σc is able to be estimated roughly using the so-called Stoney equation (Stoney, 1909) as follows:

$${\sigma _{\rm{c}}} \cong {{{E_{\rm{s}}}h_{\rm{s}}^2} \over {6R{h_{\rm{c}}}}},$$
(2)

where the subscripts s and c denote the substrate and coating element, respectively, and Es is Young’s modulus of the AZ91D alloy. Therefore, the bended curvature radius R can be determined based on the known span L, h (h=hc+hs), and the measurable parameter d in the SEM in-situ testing process can be approximately expressed by

$${R^2} = {\left({{L \over 2}} \right)^2} + {(R - d)^2},$$
(3a)
$$R = {1 \over {8h}}({L^2} + 4{d^2}),$$
(3b)

where the displacement d, as shown in Fig. 1c, can be indirectly and accurately measured by the SEM in-situ bending process.

Due to strain compatibility at the interface of the coating-substrate, the maximum tensile stress in the substrate σs (Fig. 1) occurs at the interface of the coating-substrate and is given as follows (Fawcett, 1998; Konsta-Gdoutos and Gdoutos, 2005):

$${\sigma _{\rm{s}}} \cong {{{M_x}} \over {b{h_{\rm{c}}}{h_{\rm{s}}}}}{{{E_{\rm{s}}}} \over {{E_{\rm{c}}}}},$$
(4)

where Ec is Young’s modulus of the coating material, and b (2 mm) is the width of the beam. Li et al. (2002) proposed the formula shown in Eq. (3) for estimating Young’s modulus. The benefit of this method is that the E value obtained represents the overall property of the bulk coating. Therefore, Young’s modulus of the anodized coating layer was also calculated as follows, according to Es:

$$\begin{array}{*{20}c} {2{I_{\rm{c}}}{E_{\rm{c}}} + {I_{\rm{s}}}{E_{\rm{s}}} = {{P{L^3}} \over {48d}},\quad \quad \quad \quad \quad \quad }\\ {{I_{\rm{c}}} = \int\nolimits_{{h_{\rm{s}}}/2}^{{h_{\rm{c}}} + {h_{\rm{s}}}/2} {{y^2}b{\rm{d}}y},\;\;{I_{\rm{s}}} = \int\nolimits_{ - {h_{\rm{s}}}/2}^{{h_{\rm{s}}}/2} {{y^2}b{\rm{d}}y},} \end{array}$$
(5)

where Ic and Is are the moments of inertia coating and substrate, respectively.

The shear stress τs in the substrate is given by

$${\tau _{\rm{s}}} = {{{S_x}} \over {b{h_{\rm{s}}}}}{.}$$
(6)

When the normal stress in the substrate is small relative to the shear stress, it can be assumed that substrate failure occurs when the shear stress reaches the critical value of interface bonding strength. Therefore, the critical failure stress either in the coating layer or at the surface of the substrate occurs when the distance x is L/2.

The presence of a stress or deformation gradient on the coating-substrate structure under the bending test is important in the evaluation of the mechanical behavior of the anodized coating-substrate. The stress distribution in the close vicinity of the coating layer needs to be effectively determined, although the stress gradient in the coating layer can be ignored. Hutchinson and Suo (1991) investigated mixed mode cracking in an interface with different materials from essential mechanics analysis, but the inside stress or strain at the coating layer was not determined, and in particular, there were no experimental data related to the thin coating-film-substrate structure. Yan and White (1999) showed how to estimate the residual stress in the coating layer using an experimental method based on the thickness of both the coating and substrate.

In this study, the flexural stress was estimated using Eq. (7) from Wang et al. (2009) and Eq. (8) from Yan and White (1999). These equations adequately consider the effect of transversal strain on the longitudinal stress. Biaxial stress is different from the Stoney equation (axial stress) in that the flexural stress involves the indirectly measurable parameters of the biaxial elastic modulus and radius of flexion for a two-layer beam.

$${\sigma _{\rm{c}}} = {E_{{\rm{equ}}{.}}}{{h_{\rm{s}}^2} \over {6{h_{\rm{c}}}R}},$$
(7)

where Eequ.=[Es/(1−νs)] is the biaxial elastic modulus of the coating-substrate, νs is the Poisson’s ratio of the AZ91D substrate (Table 1), and σc is the biaxial stress in the coating, which is defined as the restrained stress. A measurement of the flexural stress can be estimated based on the thickness of the coating, Young’s modulus, and Poisson’s ratio as follows (Yan and White, 1999):

$$\begin{array}{*{20}c} {{\sigma _{\rm{c}}} = {{{h_{\rm{c}}}{E_{\rm{c}}}} \over {6R(1 - {\upsilon _{\rm{c}}})}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad }\\ { \times {{(1 + 4\alpha \beta + 6{\alpha ^2}\beta + 4{\alpha ^3}\beta + {\alpha ^4}{\beta ^2})} \over {(1 + \alpha)(1 + \alpha \beta)}},} \end{array}$$
(8)

where \(\alpha = {{{h_{\rm{s}}}} \over {{h_{\rm{c}}}}},\quad \beta = {{{E_{\rm{s}}}(1 - {\upsilon _{\rm{c}}})} \over {{E_{\rm{c}}}(1 - {\upsilon _{\rm{s}}})}}\) and νc is Poisson’s ratio of the coating layer. The thinner the coating layer, the more difficult it is to measure Poisson’s ratio. In addition, as hc≪hs, p≈hs/2 (p is the distance of the neutral axial from the coating-substrate interface). z is the size of the interface layer between the coating layer and the substrate thickness, so z is defined as 0 in the anodized coating-substrate because the interface layer was enough small (about 1–5 µm) (Fig. 2a) to be ignored.

According to the parameters in Table 1 and methods for estimating biaxial stress from Eqs. (7) and (8), which differ mainly in their simplicity and convenience, the surface flexural stresses were considered the obvious effects of the thin coating and mechanical properties. Note that the flexural stress in Eq. (7) depends mainly on the elastic properties of the substrate and the geometric relations of the coating-substrate structure. Typically, the thickness of the anodized coating layer is about 30–40 µm, and the ratio of the thickness is about 100. In such cases, the equivalent Young’s modulus is completely dominated by the properties (Es and νs) of the substrate.

4 Results and discussion

4.1 Quantitative analysis of flexural stress based on SEM in-situ observation

Fig. 3 shows the relationship between the flexural stress and the stroke displacement (d), which is defined as the displacement of the indenter, calculated using three different methods based on Eq. (7), Eq. (8) (Yan and White, 1999), and the flexural stress equation for a traditional beam, as follows:

$$\sigma = {{{M_x}(h)y} \over {{I_z}}}{.}$$
(9)

The moment Mx=const. is a function of the stroke displacement or the flexural radius of the two-layer beam. The flexural stresses measured using the three methods are plotted in Fig. 3. These results indicate that the stresses determined by Eq. (7) and the previous method (Yan and White, 1999) were similar, but the results from using the traditional method based on Eq. (9) differed somewhat from those obtained using Eqs. (7) and (8). One of the reasons is that the effect of the coating layer on the estimated stress in elastic-plastic scope is ignored and the method based on Eq. (9) does not satisfy the requirement of plane strain assumed theory.

Fig. 3
figure 3

Relationship between flexural stress and stroke displacement

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To explain clearly how the flexural stress varied with the deformation of the beam, the curves of flexural stress based on the different methods are given in Fig. 4. As the flexural radius changed from 0 to 100 mm and the displacement d changed from 0 to 1.8 mm, the flexural stresses decreased more rapidly based on Eqs. (7) and (8) than it did using the traditional method. Thus, with slight variation in the flexural radius, the flexural stress at the coating layer showed observable variation. The inflexions of curves based on the results from Eqs. (7) and (8) suggest that failure occurs easily during the inflexion process, because the coating layer of the beam might be subjected to plastic deformation. That is, when the flexural radius of the beam varied from 79 mm to 50 mm, the flexural stress at the free surface of the coating layer increased from 45 MPa to 80 MPa. This can be approximately transformed (ε=σ/E) in the elastic-plastic region with the corresponding strain values of 122 µε (45 MPa) and 2160 µε (80 MPa) at the coating surface. As a general materials mechanics concept, when the engineering strain is equal to 2000 µε, the strength of the material is able to define the offset yield strength. Therefore, plastic deformation may have occurred at the coating layer at that moment. This means that the offset yield stress of the anodized coating is about 80 MPa, which is lower than that of the coating-substrate beam (96 MPa) at 350 V (Table 1). Although the critical failure stress of this coating-substrate beam has been estimated by the above analysis method, the result needs to be verified by experimental results based on the SEM in-situ technology.

Fig. 4
figure 4

Relationship between flexural stress and flexural radius

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4.2 Failure types of coating-substrate beam under three-point bending loading

Fig. 5 shows a topically enlarged image of the cross-section of the coating-substrate structure with the interface. When the flexural stress reaches about 65 MPa (εc=1757 µε, which is defined as the elastic range of the coating-substrate structure) based on Eq. (7), obvious cracks of about 3–5 µm were observed at the shrinkage defects in the coating layer and at the coating-substrate interface. This flexural stress was rather lower than the yield strength of the coating-substrate structure (Table 1) and the above mentioned estimated value of about 80 MPa. This reflects the fact that multiple cracks and interface delamination at the coating layer may occur during the elastic deformation process of the beam, especially near the shrinkage voids. This is because stress is concentrated in these micro regions. Another reason for the failure is that the flexural stress at the coating layer is a biaxial stress including the flexural stress and a restraint stress induced by plastic deformation in the width direction of the substrate. With increasing curvature of the beam (such as when the flexure stress reached about 88 MPa (εc=2378 µε) based on Eq. (7)), the coating layer was subjected to a higher strength level so that fragmentation and spalling occurred and more cracks occurred at the substrate and coating layer (Fig. 6). The local fragmentation and spalling at the coating layer confirmed that the coating layer is subjected mainly to biaxial stress and the local coating completely lost the function of surface protection. Therefore, the critical deformation or flexural stress of the anodizing coatings can be obtained based on the failure analysis. That is, the critical flexural stress was estimated to be between 65 and 88 MPa (the relative radius of the beam is between about 70 and 50 mm). With increasing flexural stress (such as 100 MPa, εc=2703 µε based on Eq. (7)), not only did the coating layer show complete spalling or delamination from the substrate at the flexural top of the beam (Fig. 7a), but also multiple macroscopic cracks formed in the substrate (Fig. 7b). Therefore, the coating-substrate beam suffered complete failure when the flexural stress reached 100 MPa. The failure characteristics of the anodized coating-substrate beam indicated that the types of failure were mainly multiple cracks in the coating layer and substrate, spalling of the coating layer, and interface cracks in the shrinkage region. Therefore, the SEM in-situ observation results indicated that the damage characteristics of the anodized coating-substrate beam can be summarized as follows: (1) the initiation and propagation of multiple cracks occurred mainly at the site of shrinkage in the coating layer and the interface. When the flexural stress reached about 88 MPa, multiple cracks propagated in the substrate and appeared discontinuous. (2) The propagation of macro cracks was not apparent either at the interface or inside the coating. This suggests that the adhesion between the anodized coating layer and the substrate is large enough not to induce interface crack propagation in the macro-scale, but some shrinkage occurs on the interface in the micro-scale. Shrinkage voids lead easily to crack initiation. (3) The spalling/delamination of the coating layer occurred mainly in the vicinity of the maximum flexural stress point. One of the causes of failure due to spalling/delamination is that the coating layer is subjected to the action of biaxial stress.

Fig. 5
figure 5

Enlarged microstructure near the interface region (the flexural stress is 65 MPa, scale bar is 5 µm)

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Fig. 6
figure 6

SEM image at flexural stress (88 MPa) based on Eq. (7) (scale bar is 10 µm)

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Fig. 7
figure 7

(a) SEM image at the flexural stress (100 MPa) based on Eq. (7) (scale bar is 50 µm); (b) SEM image at the flexural stress (23 MPa) based on Eq. (7) (scale bar is 500 µm, R=300 mm)

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4.3 Crack initiation at the interface based on digital image correlation method analysis

Although the cracking and debonding behaviors of the anodized coating-substrate beam can be characterized according the SEM in-situ images, the micro displacement and deformation fields on a cross-section of the beam close to the located failure regions during a discretional applied loading process can also be calculated and quantified using a digital image correlation method (DICM) (Vendroux and Knauss, 1998; Wang and Kang, 2002; Ma and Jin, 2003; Zuo et al., 2010; Li et al., 2013). DICM uses patterns of two SEM in-situ images from the same area under different bending curvatures. SEM images with and without deformation are compared using a pixel and its signature. If an initial deformation state in a small region of the SEM image is chosen as the reference subset, any other deformation state in another SEM image that corresponds to the initial deformation state can be defined as the target subset of image patterns. The two subsets share a relationship between the deformation field, and more importantly, the deformed region. Therefore, the measurable mismatched deformation is changed to compare the extent to which subsets move between the two images. A mathematical correlation coefficient U*(u*, v*) for determining deformation parameters of the gray levels was used to obtain the displacement components u(x, y) and v(x, y) between the speckle variation in two SEM images, in the x and y directions, respectively, as follows:

$$U\ast(u\ast,\;v\ast) = 1 - {{\sum {{f_i}} (x,\;y) \times \sum {{g_i}} (x\ast,\;y\ast)} \over {\sqrt {\sum {f_i^2} (x,\;y) \times \sum {g_i^2} (x\ast,\;y\ast)} }},$$
(10)

where u*, v* are the x-displacement and y-displacement, respectively. fi(x, y), gi(x*, y*) are the gray levels at any point in the reference and deformed images, respectively. The searching correlation was carried out among sub-matrices for each pixel with different values of u* and v*. The correlation coefficient U* indicates how closely the two subsets are related. The matching point is obtained when U* reaches a maximum value where the parameters satisfy u*=u and v*=v, from which the displacement components u(x, y) and v(x, y) can be determined by searching the overlapping subsets in the whole image. Therefore, U*=0 corresponds to image correlation. Because there are systematic errors, random errors, and distortion of images, the correlation coefficient U* cannot equal 0. That is, the minimum of U* is considered a coincidence of the assumed displacement and strain values with the actual deformation components. Thus, the deformation or displacement measurement becomes an optimization process of repeated iteration calculation for the minimum of U*. In this section, the SEM and DICM are combined to quantitatively evaluate the micro deformation of the surface of the coating-substrate beam. The key characteristics of the analysis method are that it is more suitable for identifying deformation at a micro scale and is able to make adequate use of the variation in gray levels with the obvious concave-convex characteristic in the SEM in-situ images (Zuo et al., 2010). Based on the above mentioned principles and advantages of DICM, two SEM images each with different magnifications of 500, 1000, and 2000 times were compared. At the 95% confidence interval, there was a resolution that was about 0.007 to 0.048 of the SEM images. Therefore, this indicated that the precision of the results was less than 0.05 pixels. In the SEM in-situ observation images, the actual length was 0.95 pixels/m and the accuracy of the DICM was 0.05 pixels. Therefore, the error range of the analysis result was about 0.01 µm.

Fig. Fig. 8 shows the estimated deformation level according to the displacement change in the Langrage coordinate by using a mathematical computing software package code (Sharpe et al., 2007; Eberl et al., 2010), which can also be used for calculating other parameters. We found that the result produced by this software agreed with the previous results (Figs. 5 and 6). For example, the maximum relative displacement determined using the mathematical computing software package was about 10.5 µm. To reduce the error and to obtain an accurate deformation field or the deformation mismatch points close to the interface between the anodized coating and the AZ91D substrate, the mathematical computing software package can be used to filter some noise in the distortion image. The peak values of strain distribution in the y-direction indicated that the strain variation was not more than 2500 µε and the inferior peak value of the strain was about 1000 µε. This means that the peak value of about 1000 µε is the critical deformation value between the anodized coating and the AZ91D substrate. This result of about 1000–2500 µε strain roughly corresponds to the critical failure stress (biaxial stress) of about 45–100 MPa. As mentioned above, the failure stress of the anodized coating-AZ91D substrate structure was from about 65 MPa to 88 MPa based on SEM in-situ observations and using Eq. (7). Therefore, DICM technology can be used not only to evaluate the failure point qualitatively, but also to estimate roughly the critical failure stress or strain of the anodized coating-AZ91D substrate (Fig. 8). The failure strain ranges of the coating layer estimated using DICM and Eq. (7) based on the SEM in-situ measurement method were in good agreement. This is because the coating layer is subjected to biaxial stress, not uniaxial stress, and because of the material’s tensile modulus effects.

Fig. 8
figure 8

Strain distributions near the interface based on DICM technology

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5 Conclusions

The quantitative evaluation of failure of an anodized coating-substrate beam based on the flexural radius R of a beam with a thin coating layer was investigated using SEM in-situ observations and DICM. The main results obtained from this study were as follows:

  1. 1.

    The adhesion between the anodized coating layer and the substrate was large enough not to cause interface crack propagation even if an interface crack was initiated. The anodized coating layer had good interface strength.

  2. 2.

    Evaluation of flexural stress under three-point bending loading using Eqs. (7) and (8) was more suitable for the coating-substrate structure than evaluation based on Eq. (2). There was a less than 5% difference between estimates obtained using Eqs. (7) and (8), but the method based on Eq. (7) is simpler. The critical stress of failure (65 MPa) was obtained for the anodized coating-substrate structure under three-point bending loading. The stress was biaxial, including the flexural stress and a restraint stress induced by plastic deformation of the substrate.

  3. 3.

    The types of failure of an anodized coating-substrate beam under flexural stress are multiple cracks at the coating layer, interface, or substrate. Cracks at the interface and in the coating layer are formed preferentially. However, as the flexural stress increases, multiple cracks are initiated and propagated in the substrate. These are accompanied by spalling/delamination of the coating layer in the vicinity of the maximum flexural stress point when the flexural stress is more than 88 MPa, which is lower than the yield strength of an anodized coating-AZ91D substrate structure at 350 V.

  4. 4.

    Estimates of the range of the failure strain of the coating layer obtained using DICM were in good agreement with those obtained using Eq. (7) based on SEM in-situ measurement method. This is because the coating layer is subjected to biaxial stress rather than uniaxial stress, and because of the material’s tensile modulus effects.