Synchronized Formation of Kink Bands in Al/Al₂Cu Mille-Feuille Structured Alloy

Abstract

In this study, kink band formation behavior of an Al/Al₂Cu eutectic alloy with a mille-feuille structure consisting of a hard Al₂Cu phase and a soft Al phase was investigated under compression at room temperature. In-situ surface observations during compression tests revealed multiple micro kink-bands occurring simultaneously within a single lamellar colony. To analyze the formation behavior of micro kink-bands, the spatial distribution of the lamellar structure over a wide area of the specimen was quantitatively evaluated by performing image analysis using the Radon transform. The results showed that the micro kink-bands generated synchronously in one direction and rotated at a uniform angle. Such uniform and fine kink-bands are expected to contribute to the improvement of strength. To unravel the linkage between microstructure, kink band formation, and accompanying kink strengthening, an attempt was made to construct a model that predicts the spacing between micro kink-bands and critical stress for kinking from the features of the lamellar structure. The validity of the model to predict the spacing between kink bands was demonstrated by the experimental results.

1. Introduction

In recent years, automobiles and other transportation equipment have been required to be lightweight while ensuring reliability from the perspectives of environmental, energy, economic, and social sustainability. Among light metals, aluminum (Al) is an indispensable material for reducing the weight of the transportation equipment because of its superiority in terms of lightness, strength, workability, corrosion resistance, and manufacturing cost. In automobiles, Al alloy castings are widely used for engine parts,¹⁾ and 6000 series Al–Mg–Si alloys are used for body panels.²⁾ As the need for energy conservation increases, Al alloys with higher strength are still being actively developed. During its long history of research and development of Al alloys, their strength has been discussed based on strengthening mechanisms such as precipitation strengthening, solid solution strengthening, and grain refinement strengthening. On the other hand, recently, a novel strengthening mechanism called the “kink strengthening phenomenon” was first discovered in Mg alloys with long period stacking ordered (LPSO) phase.^3,4) Kink bands are widely observed in lamellar structures consisting of alternating layers of hard and soft phases (mille-feuille structure), and has been actively studied in other alloys (Fe,⁵⁾ Ti,⁶⁾ Al-based alloys⁷⁾), ceramics⁸⁾ and polymers.⁹⁾ In Al alloys, solidification in the eutectic composition of Al–Cu alloys results in a eutectic lamellar structure consisting of a soft Al phase and a hard Al₂Cu phase. Around 1970, Al/Al₂Cu eutectic alloys were recognized as one of the high-strength metal matrix composites (MMCs), and their microstructure, mechanical properties, and heat resistance were investigated.^10–12) They revealed that the lamellar interface is stable due to its simple crystallographic orientation relationship and that the fracture occurs in shear-mode buckling under compressive loading, but few studies have focused on the kink band formation. After the discovery of kink strengthening phenomenon in LPSO-structured Mg alloys, kink-band formation behavior⁷⁾ and the associated strength increase¹³⁾ in Al/Al₂Cu eutectic alloys were systematically investigated by changing the lamellar structure and test temperature. In the previous reports, it was shown that the homogeneous formation of multiple micro kink-bands leads to a significant increase in strength.¹³⁾ Therefore, with the aim of further strengthening of the Al/Al₂Cu eutectic alloys, it is necessary to clarify the mechanism of uniform and micro kink-band formation. In this study, based on the previous research, an Al/Al₂Cu eutectic alloy was prepared under a solidification condition that achieve a good strength-ductility balance. The purpose of this study is to clarify the formation mechanism of micro kink-bands in the Al/Al₂Cu eutectic alloys.

2. Experimental Procedure and Analysis Method

2.1 Material

Ingots of Al/Al₂Cu eutectic alloy with 17.0 at% Cu (32.5 mass% Cu) were prepared by electric furnace melting. The ingot was subjected to directional solidification (DS) by the Bridgman method in an Ar gas atmosphere at a growth rate of 100 mm/h. Previous reports have shown that a good strength-ductility balance can be obtained at this growth rate.¹³⁾ The microstructure was characterized by scanning electron microscopy (SEM) equipped with an electron backscatter diffraction (EBSD) detector.

2.2 Compression test

From the directionally solidified sample, six rectangle specimens of 2 × 2 × 4 mm³ were cut out by electrical discharge machining (Specimen 1–6). These specimens differed in the number of lamellar colonies contained within the specimens, which will be discussed in the Results section. The longitudinal direction of the specimens corresponded to the solidification direction. To investigate the effect of specimen geometry, a rectangular specimen of 2 × 4 × 4 mm³ was also prepared (Specimen 7). One surface of each specimen was mirror polished for in-situ surface observation during the compression test. Compression tests were performed using Deben MT5000 tensile/compression stage at a room temperature with a crosshead speed of 0.1 mm/min. The loading direction was parallel to the longitudinal direction of the specimen (i.e., the solidification direction). The polished surface of the specimen was observed by Keyence VHX-8000 digital microscope during the compression tests. Since the appearance of the kink bands differed from specimen to specimen in the optical micrographs, the lighting method was changed manually during the compression test in order to clearly observe the kink bands. The full coaxial lighting was used for Specimens 5 and 7, and full ring lighting for the other specimens.

2.3 Lamellar structure analysis by Radon transform

Several methods have recently been proposed to quantitatively evaluate the two-phase microstructures. These include spectral analysis,¹⁴⁾ spatial correlation functions,¹⁵⁾ and persistent homology.¹⁶⁾ These methods have the advantage of being universally applicable, but the high dimensionality of the data obtained makes interpretation of physical meaning difficult. In this study, to quantify the lamellar structure of the Al/Al₂Cu eutectic alloy, image analysis using the Radon transform was performed on the optical micrographs before and after the compression test. The analysis procedure consisted of the following four steps. In the first step, the image of the entire polished surface was segmented into subimages of 128 × 128 pixels (1 pixel = 0.1 µm) allowing for overlap (Fig. 1(a)–(b)). The step size between subimages was set to 60 µm. The bright part of the subimage corresponds to the Al phase and the dark part to the Al₂Cu phase. In the second step, all subimages were binarized by the Otsu method¹⁷⁾ to distinguish between the two phases (Fig. 1(c)). In the third step, the Radon transform image of the binarized subimage was computed to quantify the features of the lamellar structure. The Radon transform is a projection of the image intensity along a line oriented at a specified angle.¹⁸⁾ In the computation, the pixels in the binarized image were divided into four subpixels and each subpixel was projected individually. The Radon transform image is the sum of the Radon transforms of the subpixels. In this study, the projection angle (θ) was defined as 0 degrees when the projection direction is parallel to the loading direction, and counterclockwise is positive. Angular resolution was set to 2 degrees. The example of Radon transform image is shown in Fig. 1(d). In the Radon transform image, several spots with high intensity appears at the projection angle where the projection direction is parallel to the Al/Al₂Cu lamellar interface. Each spot corresponds to the position of each Al₂Cu layer. Therefore, the projection angle at the point of maximum intensity, θ^*, was recorded as the angle between the lamellar interface and the load direction (hereinafter referred to as lamellar angle). To quantify the lamellar spacing, peak detection was performed for the intensity distribution at the projected angle θ = θ^*, and the distance between the peaks was recorded as the lamellar thickness, λ (Fig. 1(e)). The maximum peak intensity at θ = θ^* is related to the perfectness of the lamellar structure, since the intensity of the Radon transform image is maximized when the Al₂Cu layer is perfectly parallel to the projection direction. Thus, the maximum intensity of the Radon transformed image was then recorded as the lamellar uniformity, I^*. The above analysis was performed for all subimages to analyze the lamellar structure of the entire specimen surface (2 × 4 mm² or 4 × 4 mm²).

Fig. 1

Lamellar structure analysis by Radon transform: (a) image segmentation, (b) segmented subimage, (c) binarized subimage, (d) Radon transform image, (e) intensity distribution at the projection angle θ = θ^*. The extracted variables θ^*, λ, and I^* correspond to the lamellar angle (the angle between the lamellar interface and the loading direction), the lamellar thickness, and the uniformity of the lamellar structure, respectively.

3. Results

3.1 Initial microstructure

The EBSD maps of the Al/Al₂Cu eutectic alloy are shown in Fig. 2. A lamellar structure with alternating layers of Al and Al₂Cu phases was clearly observed. The lamellar thickness was approximately 1.5 µm. As shown in previous studies,¹⁹⁾ it was confirmed that the adjacent Al and Al₂Cu phases have a crystallographic orientation relationship characterized by $\{ 211\}_{\text{Al}_{2}\text{Cu}}\parallel \{ 111\}_{\text{Al}}$ and $\langle 120\rangle_{\text{Al}_{2}\text{Cu}}\parallel \langle 110\rangle_{\text{Al}}$. The lamellar structure was not completely uniform, with variations in lamellar thickness and waviness at the interface. However, due to the limited observation range, it was difficult to analyze the morphology of lamellar colonies and the lamellar structure of the entire sample by EBSD analysis. To evaluate the lamellar structure over a wider area, image analysis using Radon transform was performed on optical micrographs of the entire surface of the specimen before compression tests. The maps of the lamellar angle are shown in Fig. 3. Lamellar colonies can be clearly distinguished in the lamellar angle maps as regions of aligned lamellar angles. Each sample consisted of two or more lamellar colonies. The lamellar colonies were elongated parallel to the overall solidification direction. Almost colonies had lamellar angles close to 0 degrees showing that the lamellar interface is parallel to the solidification direction, while some lamellar colonies were inclined to nearly 60 degrees. Also, the lamellar angles were not uniform within each lamellar colony. Colonies that were inclined against the solidification direction and those with distorted colony boundaries tended to have noticeable variation in the lamellar angle within the colony.

Fig. 2

EBSD maps of Al/Al₂Cu eutectic alloy: (a) band contrast map, (b) phase map, (c) inverse pole figure map.

Fig. 3

Lamellar angle map obtained by radon transformation of optical micrographs. (a)–(g) correspond to Specimens 1–7 before compression tests.

3.2 Mechanical behavior and kink formation behavior during compression test

The stress-time curve of Specimen 1 is shown in Fig. 4(a). The compressive strength was 483 MPa. The compression test was stopped when the maximum load was passed and the load began to decrease. Just before the slope of the stress-time curve changed, more than ten deformation bands with a length of 200 µm and a width of 10 µm appeared simultaneously at regular intervals (Fig. 4(b)). Upon further loading, each deformation band increased in size in the length and width directions (Fig. 4(c)). After the maximum stress was reached, a large deformation band formed across the entire specimen and the load began to decrease (Fig. 4(d)). The compressive strains estimated from the optical observations were 0.13%, 0.57%, and 1.62% at (b), (c), and (d) in the figure, respectively. Optical microscopic observation with higher magnification confirmed that the lamellar interface is bent and buckling-like deformation occurred in these deformation bands. Therefore, these deformation bands observed here are kink deformation bands, as reported by previous studies.^7,13) In this alloy, the kink band grew in tens of seconds, which is considerably slower than the kink band growth time of a few µs in LPSO-structured Mg alloys.^20–22) Such slow kink growth is also observed in Zn single crystals²²⁾ and Cu/Nb nanolaminates.²³⁾ The difference in growth rate suggests a different mechanism of kink band formation and propagation, which requires further studies. The micro kink-bands appeared only in the lower central part of the specimen, and was not observed in other areas. Comparison with the lamellar angle map (Fig. 3(a)) shows that micro kink-bands occurred in the lamellar colony where the lamellar interface is parallel to the loading direction. The lamellar colonies on the left and right sides of the specimen were tilted from 20 degrees to 45 degrees with respect to the loading direction, and no kink band was observed in these lamellar colonies.

Fig. 4

In-situ surface observation during a compression test of Specimen 1: (a) stress-time curve, (b) 190 s, (c) 225 s, (d) 260 s.

Similarly, compression tests and in-situ observations were performed on other specimens. The compressive strength of all specimens is listed in Table 1. The strength varied from 306 MPa to 483 MPa. Compared to the map of lamellar colonies shown in Fig. 3, the specimens containing fine lamellar colonies tended to have higher strength. This trend is consistent with previous reports that specimens with smaller lamellar colony sizes have higher strength.¹³⁾ Due to the limited number of specimens, no clear effect of specimen shape on compressive strength was observed. In all specimens, micro kink-bands clusters were observed before the maximum load, and large kink bands were observed around the maximum load. The observation results of the micro kink-bands formed in Specimens 1–7 are summarized in Fig. 5. The micro kink-bands were aligned parallel at regular intervals within a lamellar colony. The kink interface was approximately perpendicular to the loading direction. To quantify the formation behavior of these micro kink-bands, the number of kink bands, the size of the deformed region where the micro kink-bands were observed (dashed line in Fig. 5), and the spacing between kink bands were extracted as shown in Table 2. In the table, the critical stress at which the micro kink-bands were generated and the lamellar angle of the lamellar colony where the kink band belongs are also indicated.

Table 1 Dimensions and compressive strength of Specimens 1–7.

Fig. 5

Clusters of micro kink-bands formed in Specimens 1–7. The red dashed rectangle represents the area used in the buckling model calculation. The blue solid rectangle in (g) represents the area analyzed by Radon transform in Fig. 6.

Table 2 Critical stress, kink band morphology, and lamellar angle of lamellar colonies where micro kink-bands generated in Specimens 1–7.

In order to analyze the relationship between the micro kink-bands and lamellar angle in more detail, the lamellar structure of the deformed region (indicated by the solid rectangle in Fig. 5(g)) was analyzed by Radon transform with higher spatial resolution with segmented image size of 60 × 60 pixels, step size of 20 pixels, and angular resolution of 0.2 degrees. The spatial distribution of lamellar variables before and after the formation of kink-bands are shown in Fig. 6. It can be seen that the lamellar angle changes discontinuously at the kink band boundaries and is almost uniform inside the kink bands. Multiple kink bands showed almost the same lamellar angle. In the distribution of uniformity, the kink band shows a slightly lower value of uniformity (I^*), indicating that the lamellar structure is disturbed inside the kink bands. The lamellar thickness map was almost unchanged before and after the kink deformation. From the results of Fig. 6(a-1) and (a-2), histograms of lamellar angles before and after kink deformation were plotted in Fig. 7. Before the kink deformation, the distribution was unimodal with a peak at −10 degrees, but after the deformation, it became bimodal with a peak at −10 and −18 degrees. Each peak corresponds to the original lamellar angle and the lamellar angle of the kink band. Based on the above analysis results, the kink band formation behavior is estimated to be that when the critical stress is reached, multiple micro kink-bands are formed simultaneously at the same angle of rotation, and as the load increases, the kink bands grow in the longitudinal and width directions.

Fig. 6

Spatial distribution of lamellar variables before and after the formation of micro kink bands: (a-1), (a-2) lamella angle, (b-1), (b-2) uniformity, (c-1), (c-2) lamellar thickness. The analyzed region corresponds to the area indicated by the solid blue rectangle of Specimen 7 in Fig. 6.

Fig. 7

Frequency distribution of lamellar angles before and after the formation of micro kink-bands. The analysis region corresponds to Fig. 9(a-1) and (a-2).

4. Discussion

Two types of kink bands were observed: multiple micro kink-bands formed in a single lamellar colony and large kink bands formed throughout the specimen. These two have opposite effects on kink strengthening, and it is the multiple fine kink bands that contribute to the improvement of strength and elongation.¹³⁾ Therefore, the formation of micro kink-bands will be discussed.

4.1 Effect of microstructural features on micro kink-bands formation

Comparing the lamellar angle maps (Fig. 3) and the kink band generation regions (Fig. 5), there was an overall tendency for micro-kink bands to form in lamellar colonies with small lamellar angles. To examine the effect of microstructural features on kink band formation, the critical stress for kink-bands formation and the spacing of the kink bands were compared with the lamellar structure variables obtained from image analysis. Since buckling generally depends on the geometry of the deformed region, in addition to the lamellar angle, the relationship with the height of the deformed region was also investigated. The results are shown in Fig. 8. The critical stress was not significantly related to the lamellar angle or the height of the deformed region (Fig. 8(a), (c)). Relatively higher critical stress was obtained in Specimens 1, 3 and 4. Figure 5 shows that in these specimens, micro kink-bands occurred mainly in the center of the specimen. This is because in these specimens, the lamellar colonies favorable for kink deformation were sandwiched between the less deformable lamellar colonies. Thus, it is suggested that the critical stress for kink-band formation is influenced not only by the lamellar structure of the deformed region, but also by microstructural features in neighboring lamellar colonies. In Fig. 8(b) and (d), the kink band spacing are plotted against the lamellar angle and the deformation region height, respectively. The spacing is negatively correlated with the lamellar angle and positively correlated with the deformed region height, indicating that the spacing is determined by the geometry of the deformed region and the lamellar structure, with little influence of adjacent lamellar colonies.

Fig. 8

Effects of microstructural features on formation behavior of micro kink-bands: lamellar angle vs. critical stress for kinking (a) and kink-bands spacing (b), height of deformed region vs. critical stress for kinking (c) and kink-bands spacing (d).

4.2 Modeling of micro kink-bands formation

Let us consider a model that describes the behavior of micro kink-bands that generate and grow in a synchronized manner as described above. For the large kink bands in this material, it has been reported that it correspond to buckling of the specimen.¹¹⁾ Thus, the application of the buckling theory to micro kink-bands was examined. Based on the discussion in Section 4.1, the mechanism of micro-kink formation requires consideration of the lamellar structure and the shape of the deformed region. To this end, among various buckling theories, we focused on the ripplocation model of layered materials proposed by Barsoum et al.^24–26) They performed confined buckling experiments on various layered materials consisting of plastic card, thin steel, and Al sheets and found that the confined buckling results in instabilities in which multiple opposite sign ripplocation boundaries (RBs) are nucleated. A schematic diagram of the buckling model is shown in Fig. 9(a). The laminated material is sandwiched from both sides by jigs and elastically restrained. In the figure, P is applied load to the layered materials, H and b are height and width of the layered materials, respectively. Following the theory of elastic buckling, the critical load for buckling (P_crit), the half-wavelength of the ripplocation (L), and the number of waves (m) can be expressed by the following equations:   

\begin{equation} P_{\text{crit}} = \frac{n\pi^{2}EI}{L^{2}} + \frac{kL^{2}}{\pi^{2}}, \end{equation} (1)

  

\begin{equation} L = \pi \left( \frac{nEI}{k} \right)^{\frac{1}{4}}, \end{equation} (2)

  

\begin{equation} m = \frac{H}{\lambda} = \frac{H}{2L}, \end{equation} (3)

where n is total number of layers, E is the Young’s modulus, I is the second moment of area of a single layer, k is spring constant per unit length for jigs, and λ is full-wavelength of ripplocation. When the applied stress exceeds the yield stress of the layered materials and causes irreversible deformation, the RBs are replaced by kink boundaries (KBs). Therefore, the number of kink bands is equal to the m value in the above equation. In the current case of Al/Al₂Cu alloy, single pair of thin Al and Al₂Cu phases were considered as single layer. The region where the micro kink-bands occurred was assumed to be rectangular. For seven specimens, the height H and width T of each deformed region were extracted, as shown by the dashed rectangulars in Fig. 5. The thickness perpendicular to the observation plane of the deformed region was assumed to be b because it cannot be measured. The thickness of the single pair of Al and Al₂Cu layers is t = 1.5 µm. The total number of pairs in the deformed region is n = T/t. We can eliminate k from eqs. (1) and (2):   

\begin{equation} L = \sqrt{\frac{2n\pi^{2}EI}{P_{\text{crit}}}}. \end{equation} (4)

Using the critical stress for kinking (σ_crit) and the cross-sectional area of the specimen (A), the critical load for kink formation obtained in a compression test is represented by P_total = σ_critA. The critical stress of the deformed region is then $P_{\text{crit}} = \frac{bT}{A}P_{\text{total}} = \sigma_{\text{crit}}bT$. The second moment of area of each pair is I = bt³/12. Therefore, L is expressed by the following equation:   

\begin{equation} L = \sqrt{\frac{\pi^{2}Et^{2}}{6\sigma_{\text{crit}}}}. \end{equation} (5)

This equation means that the wavelength of the ripplocation (2L) can be determined independently of the thickness of the deformation region (b) and the spring constant of the constraint (k). The Young’s modulus of the single pair (E) was determined to be 90 GPa by a simple Voigt model with Young’s modulus of Al and Al₂Cu as 70 GPa and 110 GPa,²⁷⁾ respectively. The number of kink bands calculated from eqs. (3) and (5) are plotted in Fig. 9(b). The calculated number of kink bands roughly agreed with experimental values. Thus, it was shown that the spacing of the micro kink-bands can be estimated by the Young’s modulus of the material, the thickness of the constituent layers, and the critical stress for kinking, using eq. (5). The discrepancies between the experiment and the model can be attributed to the fact that some kink bands occur at the edges of the specimens and are not evenly constrained on the left and right sides, and that the lamellar structure may be tilted against the observation plane in the depth direction. To build a more accurate model, it is necessary to consider three-dimensional lamellar orientation and more realistic constraint conditions.

Fig. 9

(a) Schematic of buckling model, (b) experimental and calculated values for the number of kink bands. The numbers in (b) represent specimen numbers.

4.3 Challenges for kink strengthening

It has been suggested that the introduction of fine, uniform kink bands is effective in achieving kink strengthening.¹³⁾ To predict the generation of such kink bands, eq. (5) is useful. In the equation, Young’s modulus (E) and layer thickness (t) are controllable in materials design, but critical stress for kinking (σ_crit) is difficult to control. As discussed previously, the critical stress is affected by neighboring lamellar colonies. This is evident from the fact that specimens 1, 3 and 4 exhibit higher compressive strength (Table 1). In order to unravel the linkages between microstructure, kink-band formation and the resultant kink strengthening, it is a challenge to predict the critical stress of micro kink-bands from the microstructure. In the ripplocation model, the critical load is expressed by eq. (1). In this equation the spring constant k in the surrounding constraint was unknown in this work. By relating k to the features of the surrounding lamellar colonies, it might be possible to formulate an equation that predicts critical stress from microstructural features. This would require further experiments with specimens containing multiple lamellar colonies to accumulate data on adjacent lamellar colonies and critical stresses. In addition, residual stress due to the solidification process and heat treatment will change the lateral constraining force on the lamellar colony. In order to control the critical stress for kinking, it may be necessary to consider the residual stress in the mille-feuille structured materials.

5. Conclusions

In this study, kink formation behavior in Al/Al₂Cu eutectic alloy under compressive loading at room temperature was investigated by in-situ surface observation. We also proposed a method for mapping lamellar structural variables (lamellar angle, lamellar thickness, and uniformity) over a wide area of the specimen by applying a Radon transform to the optical microscope image. The following conclusions can be drawn:

1. (1)    Image analysis using Radon transform revealed that the directionally solidified Al/Al₂Cu eutectic alloy is composed of multiple lamellar colonies with different lamellar angles.
2. (2)    When the compressive stress reached a critical value, multiple fine kink bands with identical lamellar angles formed simultaneously, and as the load increased, the kink bands grew in the longitudinal and width directions. Around the maximum load, large kink bands were formed across the entire specimen.
3. (3)    The spacing of the micro-kink bands correlated with the geometry of the deformed region and the lamellar angle. On the other hand, the critical stress for micro kink-bands formation was strongly influenced by adjacent lamellar colonies.
4. (4)    Based on the ripplocation model in multi-layer materials, a model was proposed to predict the synchronized formation of micro kink-bands in mille-feuille structured materials. Comparison with experimental results of kink-bands spacing shows the validity of the proposed model.

Acknowledgments

This work was partially supported by the JSPS KAKENHI for Scientific Research on Innovative Areas “MFS Materials Science” (Grant No. JP18H05478, JP18H05482 and 21H00093).

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