Verification of Rank-1 Connection Model of Kink Band in Mg–Zn–Y Alloy

Abstract

In order to verify the validity of the rank-1 connection model of kink band, a comparison between the kink described by the rank-1 connection and the kink actually formed in LPSO-Mg alloy was made in this study. Crystal rotation and kink interface (v_k) of kink bands formed by almost single basal slip were analyzed by EBSD and double trace analysis. Theoretical kink interface (n_α) was computed using the experimentally acquired crystal rotation of the kink bands using the rank-1 connection model. A comparison was made between n_α and v_k. Theory and experiment showed agreement within 4°, indicating that the geometry of the kink band is well described by the rank-1 connection.

Fig. 3 (a) IPF map, (b) IQ map observed from direction A₁, (c) IPF map and (d) IQ map observed from direction A₃. The double trace analysis was made for the kink interfaces of K-1 and K-2. (e) Stereographic projections of kink interface v_k (double trace analysis) and kink interface n_α (rank-1 connection) and crystal rotation axis q_α for the kink interfaces of K-1 and K-2. v_k was within 7° of n_α.

1. Introduction

Mg–Zn–Y alloy developed by Kawamura et al.¹⁾ has a long-period stacking ordered (LPSO) phase. The LPSO phase has a periodic layered structure of soft and hard layers.^2,3) The soft layer consists of α-Mg, whereas the hard layer is a four-atom thick layer consisting of clusters with an L1₂-type structure enriched with Zn and Y atoms.^2,4,5) Due to the hard layer, the easy-slip system in this layered structure is restricted to the ⟨a⟩ basal plane of the hexagonal lattice.⁶⁾ Therefore, when the compressive load is applied from a direction parallel to the basal plane, kink deformation becomes the dominant plastic deformation mode in the LPSO phase.⁷⁾ Kink deformation is a plastic deformation in which shear deformation and rigid body rotation occur simultaneously on a constrained slip plane.^8,9) Three types of kink, namely, kink band, ridge kink and ortho kink can be formed by the kink deformation.⁹⁾ Ridge and ortho kink are formed by connecting kink bands. In this study, the microstructure in which these kinks are connected is called kink microstructure. The interface between kink band and the matrix and the interface between kink bands are called kink interface.

Mg/LPSO two-phase alloy, in which kink microstructure is introduced into the LPSO phase by hot extrusion, have a higher specific strength and ductility than conventional Mg alloys.¹⁾ Therefore, Mg–Zn–Y alloy is expected to be a light-weight and high-strength alloy surpass Al alloys. Recently, Hagihara et al.¹⁰⁾ revealed that the unusual strengthening is owing to the kink microstructure and is called “kink strengthening”. The mechanism of the kink strengthening is actively investigated.

Geometrical quantities of the kink microstructure such as the kink interface orientation and crystallographic orientation difference between the matrix and the kink band have been shown to affect the kink strengthening.¹¹⁾ However, there are only limited works on the geometry of the kinks after Hess and Barret.¹²⁾ A theory describing crystallography and geometry of kink in a systematic way is necessary to reveal the mechanism of the kink strengthening.

One of the authors¹³⁾ has systematically formulated the geometry of kinks using the continuity of deformation (rank-1 connection) for the material with only one slip plane and slip direction. This model clearly explains the existence of disclination in the kink microstructure.¹⁴⁾ The model has been modified to take into account the kink band formation by multiple basal slip which is often observed in experiments in LPSO-Mg alloy.^15,16) The modified model provides a geometric explanation for the three-dimensional morphology of curved kink.^16,17) The geometry of kinks revealed by the rank-1 connection seems to provide perspective to understand the nature of the kink microstructure and kink strengthening. However, a comparison between the kink described by the rank-1 connection and the kink actually formed in LPSO-Mg alloy has not yet been made even for a simple kink band which is formed by a single basal shear. In order to verify the validity of the rank-1 connection model of kink band, a comparison between the kink interface calculated by the rank-1 connection and the experimental one was made in LPSO-Mg alloy.

2. Experimental Procedure and Analysis Method

2.1 Specimen, compression test and microscopy

Directionally solidified (DS) specimens with a nominal composition of Mg₈₅Zn₆Y₉ (at%), consisting of a single phase of 18R LPSO phase was used.¹⁸⁾ Master ingot was prepared by induction melting and DS specimens were prepared by Bridgeman technique in an Ar atmosphere.

Compression test was made at room temperature to introduce kink microstructures into the DS specimens. Compressive load parallel to the growth direction induces kink microstructures because the basal plane is almost parallel to the growth direction in the DS specimens.¹⁸⁾ A rectangular specimen approximately 2 mm × 2 mm × 5 mm in size was fabricated using an electrical discharge machine so that the longitudinal direction coincided with the direction of growth. The specimen surfaces were carefully finished by buffing to remove the damaged surface layer. A universal testing machine (Shimadzu AG-IS 1kN) was used for the compression tests. Compression tests were carried out at a nominal strain rate of 1.67 × 10⁻⁴ s⁻¹ and the load was stopped when the nominal strain reached approximately 3–4%.

The kink microstructure was analyzed by optical microscopy (OM) and electron back-scattered diffraction (EBSD) measurement. As the smoothness of the specimen surface has a considerable influence on the accuracy of the EBSD measurement, the specimen surface was buffed and finished by ion milling using ArBlade5000 (HITACHI IM5000 System). Field-emission gun-type scanning electron microscopy (FE-SEM, HITACHI SU5000) was used for the EBSD measurement. The EBSD detector was TSL OIM DVC5. Orientation imaging microscopy (OIM) software was used to analyze the EBSD data.

2.2 Determination of kink interface by double trace analysis

The orientation of the kink interface can be strictly obtained from the two traces of the kink interface on two different surfaces of the specimen. As shown in Fig. 1, EBSD measurement was performed on a single kink from two directions to obtain traces on two distinct surfaces. Firstly, a kink of interest was observed from the A₁ direction, where A₁ − A₂ − A₃ system is the specimen coordinate in Fig. 1. Then, specimen was carefully polished using emery paper, buffing and ion-milling from A₃ direction to expose the kink of interest. A zig was used to keep the orthogonality of the two surfaces.

Fig. 1

Schematic diagram of double trace analysis. Two traces of a kink band were obtained by performing EBSD measurement from two directions. The orientation of the kink interface v_k was determined.

The vectors v₁ and v₂ on the kink interface in the specimen coordinate system were obtained by image analysis as shown in Fig. 1. The normal vector of the kink interface is v₁ × v₂. The coordinate system of the obtained kink interface normal vector is referred to the specimen coordinate system. The kink interface normal in the crystal coordinate system was obtained by transforming v₁ × v₂ from the specimen coordinate system to the crystal coordinate system using the crystal orientation measured by the EBSD. v_k is the kink interface orientation obtained by double trace analysis and expressed in the crystal coordinate system. The v_k was compared with the kink interface orientation obtained from the rank-1 connection using the crystal rotation in the observed kink band as described in section 2.3.

2.3 Analysis of kink interface orientation by rank-1 connection

In the present study, the LPSO phase was treated as hexagonal lattice. c/a was approximated using a = 0.323 and c = 4.697 nm.⁷⁾ Inamura et al.¹³⁾ formulated the deformation gradient of kink band by taking into account slip in any direction on the basal plane as a composite of multiple basal slips. Figure 2 shows the orthonormal Cartesian coordinate system of the reference configuration, which is bounded by mutually orthogonal unit vectors e_x, e_y and e_z. The relationship between the orthogonal coordinate and the hexagonal lattice vectors is $\mathbf{a}_{1} = 1/3[2\bar{1}\bar{1}0]{\mathrel{/\!/}}\mathbf{e}_{\text{y}}$, $\mathbf{a}_{2} = 1/3[\bar{1}2\bar{1}0]$, $\mathbf{a}_{3} = 1/3[\bar{1}\bar{1}20]$, a₄ = [0001]//e₄ as reported by Inamura et al.¹³⁾ The slip direction s on the basal plane is assumed to form an angle α with e_y. The deformation gradient of the kink with slip direction s is obtained by a coordinate transformation of a kink band with slip direction a₁//e_y as follows.¹³⁾

  

\begin{equation} \mathbf{R}_{\alpha}\mathbf{Q}_{0}\mathbf{S}_{0}\mathbf{R}_{\alpha}^{\text{T}} - \mathbf{I} = \mathbf{R}_{\alpha}\mathbf{a}_{0} \otimes \mathbf{R}_{\alpha}\hat{\mathbf{n}}_{0} \end{equation} (1)

S₀ is the deformation gradient of the shear along a₁, R_α is a rotation by a around e_z axis, a₀ is related to the discontinuity of the deformation at the kink interface for a kink band with slip direction a₁//e_y, $\hat{\mathbf{n}}_{0}$ is the normal vector of the kink interface and Q₀ is the rigid body rotation in a kink band with slip direction a₁//e_y. Explicit form of these quantities are given in the previous report.¹³⁾

Fig. 2

Relationship between basis vectors and hexagonal lattice. The slip direction s on the basal plane is assumed to be at an angle of α with e_y. q_α is the crystal rotation axis. s can be decomposed into two basal slips as in the main text.

The crystal rotation when multiple basal slips are active (α ≠ 0) is $\mathbf{Q}_{\alpha } = \mathbf{R}_{\alpha }\mathbf{Q}_{0}\mathbf{R}_{\alpha }^{\text{T}}$. The crystal rotation axis of the kink with slip direction s is q_α = (cos α, sin α, 0). Rotation axis q_α away from $\langle 1\bar{1}00\rangle $ means that the kink band is formed by the activation of more than two basal slips.¹⁹⁾ Inamura et al.¹³⁾ formulated the rotation axis as a function of the magnitudes of shear of the basal slips in the kink band. As shown in Fig. 2, s on the basal plane of hexagonal lattice can be decomposed into two basal slips. In this study, s is analyzed without loss of generality by considering it to be a linear combination of two shear directions a₁ and a₃. s₁ > 0 and s₃ ≥ 0 are the magnitude of shear along a₁ and -a₃ respectively. Let γ = s₃/s₁ ≥ 0, then decomposition of s into s₁ and s₃ is given as,¹³⁾

  

\begin{align} s &= \sqrt{\left(s_{1} + \frac{s_{3}}{2}\right)^{2} {}+ \left(\frac{\sqrt{3}}{2}s_{3}\right)^{2}} = \sqrt{s_{1}^{2} + s_{1}s_{3} + s_{3}^{2}} \\ &= s_{1}\sqrt{\gamma^{2} + \gamma + 1}, \end{align} (2)

where Γ is defined as Γ = 1 + γ + γ² > 0. From eq. (2), the normal vector $\hat{\mathbf{n}}_{\alpha }$ of the kink interface, the crystal rotation axis q_α and the amount of rotation θ_α are given by following equations.¹³⁾

  

\begin{align} \hat{\mathbf{n}}_{\alpha}& = \mathbf{R}_{\alpha}\hat{\mathbf{n}}_{0} \\ &= \left(-\frac{\sqrt{3}\gamma}{\sqrt{\Gamma(4 + \Gamma s_{1}^{2})}},\frac{2 + \gamma}{\sqrt{\Gamma(4 + \Gamma s_{1}^{2})}},s_{1}\sqrt{\frac{\Gamma}{4 + \Gamma s_{1}^{2}}}\right) \end{align} (3)

  

\begin{align} \mathbf{q}_{\alpha} &= (\cos\alpha,\sin\alpha,0) \\ &= \left(\frac{2 + \gamma}{2\sqrt{\Gamma}},\frac{\sqrt{3}\gamma}{2\sqrt{\Gamma}},0\right){\mathrel{/\!/}}[\gamma,\overline{1 + \gamma},1,0] \end{align} (4)

  

\begin{equation} \cos \theta_{\alpha} = \frac{4 - s_{1}^{2}\Gamma}{4 + s_{1}^{2}\Gamma} \end{equation} (5)

The third term in eq. (4) is the four-index representation of q_α in the hexagonal lattice. Note that s₁ and s₃ are obtained by substituting the experimental values of the crystal rotation axis q_α and rotation θ_α by EBSD measurement into eqs. (4) and (5). By substituting the obtained s₁ and s₃ into eq. (3), the normal vector $\hat{\mathbf{n}}_{\alpha }$ of the kink interface is calculated theoretically.

As shown in Fig. 2, q_α is perpendicular to [0001] for a kink band formed only by basal slip.²⁰⁾ According to Yamasaki et al.,²⁰⁾ q_α approaches [0001] when non-basal slip is simultaneously active in addition to the basal slip. The geometry of the kink band with such complex deformations including non-basal slip has not yet been formulated by rank-1 connection. Therefore, to compare theoretical and experimental values of the kink interface, kinks formed without activation of non-basal slips, namely the kink with a crystal rotation axis of $\langle 0\bar{1}10\rangle $, must be selected. The angle between q_α and [0001] obtained from EBSD measurement is defined as χ. Detailed analysis was performed on kink bands with χ close to 90° in this study.

3. Results

3.1 Results of microstructure observation

Figure 3 shows (a) inverse Pole-Figure (IPF) map and (b) Image Quality (IQ) map observed from the A₁ direction, (c) IPF map and (d) IQ map from the A₃ direction. The specimen coordinate system is given as shown in Fig. 3(b), (d). On the IQ map in Fig. 3(b), the position where the EBSD measurement was performed from direction A₃ is indicated by a blue line. The angle between the two surfaces of measurement was 90.6°. The interfaces (K-1, K-2) indicated by the red arrows are kink interfaces between the matrix and the kink band. K-1 and K-2 were sharp and linear so that the deformation inside the kink bands was considered to be almost uniform. The double trace analysis of the kink interface of K-1 and K-2 was therefore made. The analysis showed that v_k was $[99\ \overline{43}\ \overline{57}\ 0]$ for K-1 and $[99\ \overline{41}\ \overline{58}\ 1]$ for K-2 in lattice vector. Conversion of v_k into Miller index just complicates the calculation so that v_k is given in lattice vector.

Fig. 3

(a) IPF map, (b) IQ map observed from direction A₁, (c) IPF map and (d) IQ map observed from direction A₃. The double trace analysis was made for the kink interfaces of K-1 and K-2. (e) Stereographic projections of kink interface v_k (double trace analysis) and kink interface n_α (rank-1 connection) and crystal rotation axis q_α for the kink interfaces of K-1 and K-2. v_k was within 7° of n_α.

As shown by the yellow line in the enlarged view in Fig. 3(d), the kink interface is slightly curved and the determination error of v₁ and v₂ is 3°. Therefore, the error in the kink interface normal in the double trace analysis is 3°.

3.2 Kink interface normal in experiment and theory

n_α was calculated as described in section 2.3 using the rotation angle and axis at points indicated by red triangles in Fig. 3(d). Table 1 shows the rotation axis q_α, the angle between the rotation axis and [0001] (χ), experimental kink interface v_k, the magnitude of shear (s), the direction of slip (α), theoretical kink interface n_α, the angular difference between v_k and n_α (Δφ) for each kink interface. Figure 3(e) shows n_α, v_k and q_α in stereographic projection. The red crosses indicate n_α, the green circles are q_α and the blue triangle is v_k. The net slip direction is in the range of α = 0.98–2.71° so that the observed kink bands were regarded to be formed by almost single basal slip along a₁ direction. As shown in Fig. 3(e), χ is not exactly 90° and indicates small amount of non-basal slip occurred in the kink. The effect of such non-basal slip on the accuracy of the analysis is discussed later. The difference between the theoretical and experimental values of the kink interface orientation, Δφ, was within 7°.

Table 1 Summary of the analysis. q_α and χ were obtained from EBSD measurement, whereas s, α and n_α were obtained from analysis using rank-1 connection. v_k was determined by the double trace analysis.

4. Discussion

Yamasaki et al.²⁰⁾ experimentally showed that kink bands formed only the basal slips gives χ = 90°, whereas occurrence of non-basal slip in addition to the basal slips gives χ ≠ 90°. In the kinks observed in this study, χ = 85.03–89.72° so that a small amount of non-basal slip occurred in addition to the basal slips which is responsible for the kink formation. Equations (1)–(5) do not take into account such an occurrence of the non-basal slip. Therefore, the deviation of the theoretical kink interface from the experimental one is mainly due to these non-basal slip activities. Δφ was about 6.63° for χ = 86.25°, whereas it is only 3.72° for χ = 89.72°. Figure 4 shows the χ dependence of Δφ. The line is given by the least square minimization of the data points. The scatter of Δφ is caused by n_α varying within a range of approximately 1.5° at each measurement point, and it is caused by the differences in s and α. It is clearly seen that Δφ decreases as χ approaches 90°. Extrapolating Δφ at χ = 90° gives Δφ = 3.84°. Therefore, it is plausible to regard the data with minimum Δφ (K2e) in Table 1 is the most reliable one. The deviation of theory from the experiment is estimated to be 4°. In addition, the experimental error of v_k determined by the double trace analysis was 3° due to the curvature of the kink interface as explained in section 3.1 and is comparable to Δφ. The theory is, therefore, in good agreement with the experiment. The description of kink by the rank-1 connection well explains the actual kink and can be regarded as a basic principle to describe the geometry of kink microstructure.

Fig. 4

χ dependence of Δφ. At χ = 90°, which corresponds to the case where only basal slip is active, Δφ = 3.84°.

5. Conclusions

The kink interface normal n_α was calculated by the rank-1 connection using experimental data and compared with the kink interface orientation v_k obtained by double trace analysis.

The analyzed kink band was formed by a basal slip approximately along the a₁ direction and small amount of non-basal slip.

The orientation difference Δφ between v_k and n_α obtained by double trace analysis was less than 7°. This deviation of theory and experiment is due to the occurrence of non-basal slip and experimental error in the double trace analysis.

Error analysis showed that the theoretical kink interface is in good agreement with experimental one within 4° which is comparable to the error in the double trace analysis. The description of kink by the rank-1 connection well explains the actual kink and can be regarded as a basic principle to describe the geometry of kink microstructure.

Acknowledgments

This work was supported by JSPS KAKENHI for Scientific Research on Innovative Areas “MFS Materials Science” (Grant Number JP18H05481) and IIR Research Fellow Program.

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