Numerical Analysis of Disclinations in Connecting Kink Bands Formed by Multiple Basal Shear

Ryutaro Matsumura; Yuri Shinohara; Tomonari Inamura

doi:10.2320/matertrans.MT-MD2022020

Abstract

We analyzed the kinematic (geometric) aspects of kink bands with multiple basal shear using rank-1 connection to investigate the type of disclination and annihilation of disclinations in kink microstructures. We found that wedge disclinations occur in connecting kink bands for realistic magnitude of shear that can occur, regardless of the shear direction. The normal vector of the junction plane between kink bands and Frank vector of the resulting wedge-disclination varied continuously with respect to the changes in the magnitude and direction of the basal shear. Annihilation of disclinations is possible even between the kinks which are formed by multiple basal shears. Kinematical model of the three-dimensional wavy kinks are proposed using the obtained results.

Fig. 16 (a) Schematic of the three-dimensional morphology of a wavy ridge kink. The wavy kinks have a network of wedge disclinations. (b) Schematic of three-dimensional morphology of kink microstructure with annihilation of disclinations.

1. Introduction

Mg–Zn–Y alloys with long-period stacking ordered (LPSO) phase have high specific strength, ductility, and fire resistance than conventional Mg alloys, and are expected to be applied to lightweight transportation equipment.¹^–⁷⁾ Mg–Zn–Y alloys have high yield stress and excellent ductility. Although the origin of the high yield stress and excellent ductility has not been clarified, it has recently been shown that the microstructure formed by kink deformation is the source of the excellent mechanical properties. Therefore, clarifying the geometric aspects of the “kink microstructure” formed by kink deformation is essential to understanding the strength of LPSO-Mg alloys.

Kink deformation generally occurs in materials with strong plastic anisotropy and has been observed in rocks,⁸⁾ wood,⁹⁾ micas,¹⁰⁾ graphite,¹¹⁾ and polymers.¹²⁾ Kink microstructures are formed when a compressive load is applied parallel to a limited slip plane. Typical kink morphologies are kink band, ridge kink, and ortho kink. Kink deformation in metallic materials was first discovered in Cd single crystals by Orowan.¹³⁾ Later, Hess and Barrett¹⁴⁾ discovered kink deformation in Zn single crystals as well. The crystal structure of Zn and Cd is hexagonal close-packed (hcp) with $\text{c}/\text{a} > \sqrt{3} $, and as in other materials where kink deformation has been observed, slip deformation due to dislocation is strongly restricted to the basal plane.

The LPSO phase has a structure of periodically soft layers and hard layers.²^,¹⁵^,¹⁶⁾ The soft layer is composed of α-Mg, the most densely packed phase, and the hard layer is a 4-atom-thick layer composed of clusters with an L1₂-type structure in which Zn and Y atoms are enriched. The structures with the hard layer for every 5, 6, 7, 8 soft layers are called 10H, 18R, 14H and 24R structure, respectively.²^,¹⁷^–²¹⁾ The easy-slip system in this layered structure is ⟨a⟩ basal, which is the closest packed plane of the hexagonal crystal. Therefore, kink deformation is the dominant mode of plastic deformation in LPSO-Mg alloys and, a kink microstructure consisting of kink band, ortho kink, and ridge kink is produced upon plastic deformation.³^,¹³^,¹⁴^,²²^–²⁸⁾ Since the kink microstructure inhibits the activity of basal dislocations, strengthening by the kink microstructure has gained attention as “kink strengthening”.²⁹^,³⁰⁾

The relation between the kink strengthening and the kink microstructure is currently open-question. Some experimental studies show that kink boundary becomes a stronger obstacle to dislocations as the misorientation angle at the kink interface increases.³¹⁾

Previously, Hess and Barrett¹⁴⁾ have proposed that the kink band is formed by the cooperative movement of dislocations on the basal plane. Previously, Inamura³²⁾ revealed the geometric aspects of kink band, ridge kink and ortho kink using rank-1 connection in which comprehensive geometric nonlinearities are taken into account. The results revealed that when two kink bands connect each other, the wedge disclination is inevitably formed at the junction of kink bands. Inamura also showed that the power of the wedge disclination is uniquely determined for a given magnitude of shear. Although the geometrical analysis of the kink has been performed by Nakatani et al.,³³⁾ the existence of the disclination is not explicitly shown in their analysis. Since the disclinations should interact with basal dislocation, the disclinations may play important roles in the strengthening by the kink microstructure.

Furthermore, a disclination in connecting kink bands can be annihilated by the additional connecting with a specific kink band.³²⁾ Since the disclinations has large elastic energy,³⁴^,³⁵⁾ it is physically natural to assume that the kink microstructure is formed in such a way as to reduce the elastic energy of the disclinations. In a kink microstructure with annihilation of disclinations, multiple kinks are connected. When an external force is applied to such kink microstructure, the slip deformation occurs in the kink band with a high Schmid factor. In such a case, an adjacent kink band that connects with the deformed kink band must also be deformed in a cooperative manner while maintaining the state of annihilation of disclinations. Since the slip system of the material is restricted to the basal plane, the cooperative deformation required for the adjacent kink band may not be possible under the external force. This is thought to be one of the factors causing the increase in yield stress. Therefore, the kink microstructure with annihilation of disclinations is also considered to be an important microstructure in the kink strengthening.

However, the kink deformation in Inamura’s analysis is limited to the plane strain state, where a slip system parallel to the compressive direction occurs; the analysis of the kink microstructure is essentially two-dimensional. Some experiments have confirmed that more than two basal slip occurs within a single kink band.³⁶^,³⁷⁾ Yamasaki et al.³⁶^,³⁷⁾ have confirmed that two or more basal slips occur within a single kink band. Hagihara et al.²⁶⁾ have experimentally shown that misorientation and its rotation axis change within a kink band due to the inhomogeneous activation of multiple basal slips. Furthermore, Hagihara et al.²⁶⁾ proposed a three-dimensional morphology model of the kink microstructure in which kinks with different misorientations and misorientation axes are connected to each other, resulting in a wavy kink interface on (0001) side of the ridge kink. It is very important to clarify the geometrical aspect of such a three-dimensional kink microstructure with multiple basal slips in order to clarify the kink strengthening.

Therefore, we extended the kink model to three dimensions by assuming the case where multiple basal slips are active and considering the slip in any direction on the basal plane as a composite of multiple basal slips.³⁸⁾ The analysis shows that wedge disclination or mixed disclination, which is a mixture of wedge and twist disclinations, occurs when kink bands connect with each other, even in the case of three-dimensional deformation. However, the previous report only showed the existence of the disclination without clarification of the type and the power of the disclination. The possibility of annihilation of disclinations due to the kinks formed by multiple basal slips has also not been discussed. Furthermore, it has not been clarified whether the three-dimensional morphology of curved kink as proposed by Hagihara²⁶⁾ or kink microstructure with annihilation of disclinations as proposed by Inamura³²⁾ can be described kinematically in the case where multiple basal slips are activated.

In this study, we investigate the effect of magnitude and direction of the basal shear on the interface normals of the interfaces in connecting kinks, Frank vector and the power of the disclinations by numerical analysis. Based on the obtained results, we clarify the types of disclinations formed by the connecting kink bands with multiple basal slips. The possibility of annihilation of disclinations and three-dimensionally curved kinks composed of a sequence of kink bands are discussed.

2. Analysis Method

Based on the results of previous studies,³⁹⁾ we assume that a piecewise uniform deformation gradient occurs in the kink deformation, and that the continuity of deformation is maintained at the interfaces. Gilman observed kink bands in Zn single crystals.³⁹⁾ The kink interface between the matrix and the kink band is a sharp planar interface, and the crystal orientation changes discontinuously through the interfaces. In addition, the lines on the sample surface continued as uninterrupted lines after passing through the kink interface. These observations clearly indicate that the deformation gradient within the kink band is uniform and the deformation is continuous through the kink interface, whereas the deformation gradient is discontinuous at the kink interface.

These features indicate that the rank-1 connection is maintained at the kink interfaces. Suppose objects are subjected to different uniform deformation gradients F, G. The condition (rank-1 connection) $\boldsymbol{F} - \boldsymbol{G} = \boldsymbol{a} \otimes \hat{\boldsymbol{n}}$ is required for maintaining the continuity of the deformation at the interface $\hat{\boldsymbol{n}}$. a is a vector representing the discontinuity of the deformation gradients.⁴⁰⁾ The material in this study is assumed to be the LPSO phase in LPSO-Mg alloy. However, it should be noted that the results can be applied to any materials which have only one slip plane and undergo kink deformation.

The basal shear is assumed to occur on the basal plane of a single crystal with a hexagonal lattice. Figure 1 shows the orthogonal coordinates of the reference configuration, which are spanned by the mutually perpendicular unit vectors e_x, e_y and e_z. The relationship between the orthogonal coordinates and the hexagonal lattice vectors are the same as in the previous report $\mathbf{a}_{1} = 1/3[2\bar{1}\bar{1}0]//\mathbf{e}_{\text{y}}$, $\mathbf{a}_{2} = 1/3[\bar{1}2\bar{1}0]$, $\mathbf{a}_{3} = 1/3[\bar{1}\bar{1}20]$, a₄ = [0001]//e_z.³⁸⁾ The shear direction s on the basal plane is assumed to form an angle α with e_y. The magnitude of the shear is expressed as |s| = s. The body treated in this analysis is elastoplastic body. The deformations to form kinks are treated as plastic deformation (eigenstrain) and the compatibility among these deformations is analyzed by the rank-1 connection. There is rotational misfit among the plastic deformations and we need additional deformation to keep the continuity of the body. This additional deformation is provided by elastic deformation and corresponding to the elastic strain field by the disclinations. When α = 0, the deformation gradient of the kink band and the interface normal are obtained as follows.³²⁾ The rank-1 connection between the kink band and the matrix is $\mathbf{Q}_{0}\mathbf{S}_{0} - \mathbf{I} = \mathbf{a}_{0} \otimes \hat{\mathbf{n}}_{0}$ and

\begin{equation} \mathbf{S}_{0} = \mathbf{I} + s\mathbf{e}_{\text{y}} \otimes \mathbf{e}_{\text{z}} \end{equation}

(1)

\begin{equation} \mathbf{a}_{0} = \frac{1}{\sqrt{4 + s^{2}}} \begin{pmatrix} 0\\ -s^{2}\\ 2s \end{pmatrix} \end{equation}

(2)

\begin{equation} \hat{\mathbf{n}}_{0} = \frac{1}{\sqrt{4 + s^{2}}} \begin{pmatrix} 0\\ 2\\ s \end{pmatrix} \end{equation}

(3)

\begin{equation} \mathbf{Q}_{0} = \begin{pmatrix} 1 & 0 & 0\\ 0 & \dfrac{4 - s^{2}}{4 + s^{2}} & \dfrac{-4s}{4 + s^{2}}\\ 0 & \dfrac{4s}{4 + s^{2}} & \dfrac{4 - s^{2}}{4 + s^{2}} \end{pmatrix} \end{equation}

(4)

\begin{equation} \mathbf{B}_{0} = \mathbf{Q}_{0}\mathbf{S}_{0} \end{equation}

(5)

S₀ is the deformation gradient of the shear, a₀ is a vector related to the discontinuity of deformation at the kink interface, $\hat{\mathbf{n}}_{0}$ is the normal vector of the kink interface, B₀ is the total deformation gradient of the kink band, Q₀ is the rigid body rotation of the kink band, and q₀ represents the axis of rotation by Q₀.

Fig. 1

Relationship between the orthogonal coordinates and hexagonal lattice. The shear direction (s) on the basal plane is assumed to form an angle α with e_y. q_α = (cos α, sin α, 0) is the rotation axis of the rigid body rotation Q_α.

When α ≠ 0, where the shear direction on the basal plane is arbitrary, as in the schematic in Fig. 2, eqs. (2)–(5) are given by using the rotation matrix R_α around the e_z axis.³⁸⁾

\begin{equation} \mathbf{R}_{\alpha} = \begin{pmatrix} \cos \alpha & - \sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0 & 0 & 1 \end{pmatrix} \end{equation}

(6)

\begin{equation} \mathbf{q}_{\alpha} = (\cos \alpha,\sin \alpha,0) \end{equation}

(7)

\begin{equation} \mathbf{a}_{\alpha} = \mathbf{R}_{\alpha}\mathbf{a}_{0} \end{equation}

(8)

\begin{equation} \hat{\mathbf{n}}_{\alpha} = \mathbf{R}_{\alpha}\hat{\mathbf{n}}_{0} \end{equation}

(9)

\begin{equation} \mathbf{Q}_{\alpha} = \mathbf{R}_{\alpha}\mathbf{Q}_{0}\mathbf{R}_{\alpha}^{\text{T}} \end{equation}

(10)

\begin{equation} \mathbf{B}_{\alpha} = \mathbf{R}_{\alpha}\mathbf{Q}_{0}\mathbf{S}_{0}\mathbf{R}_{\alpha}^{\text{T}} = \mathbf{R}_{\alpha}\mathbf{B}_{0}\mathbf{R}_{\alpha}^{\text{T}} \end{equation}

(11)

Fig. 2

Schematic of a kink band formed by the operation of shear deformation with α ≠ 0. S_α is the deformation gradient of the shear, B_α is the total deformation gradient of the kink band, q_α is the rotation axis of the rigid body rotation Q_α and $\hat{\mathbf{n}}_{\alpha }$ is the normal vector of the kink interface.

The rank-1 connection due to the connecting between a kink band with α = 0 and a kink band with α ≠ 0 is given by

\begin{equation} \mathbf{WB}_{\alpha}(s,\alpha) - \mathbf{B}_{0}(s_{0}) = \mathbf{b} \otimes \hat{\mathbf{m}} \end{equation}

(12)

where W is a three-dimensional rotation matrix, b and $\hat{\mathbf{m}}$ are a three-dimensional column vector. $\hat{\mathbf{m}}$ denotes the junction plane of the normal vector. According to Ball and James,⁴⁰⁾ if eq. (12) satisfies $\mathbf{C} = \mathbf{B}_{0}^{ - \text{T}}\mathbf{B}_{\alpha }^{\text{T}}\mathbf{B}_{\alpha }\mathbf{B}_{0}^{ - 1} \ne \mathbf{I}$, $\lambda_{1}\leq 1$, $\lambda_{2} = 1$ and $\lambda_{3} \geq 1$, there exist solutions W, b and $\hat{\mathbf{m}}$, where $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ are the eigenvalues of C. b and $\hat{\mathbf{m}}$ are obtained by

\begin{equation} \mathbf{b} = \rho\left(\sqrt{\frac{\lambda_{3}(1 - \lambda_{1})}{\lambda_{3} - \lambda_{1}}}\hat{\mathbf{e}}_{1} + k\sqrt{\frac{\lambda_{1}(\lambda_{3} - 1)}{\lambda_{3} - \lambda_{1}}} \hat{\mathbf{e}}_{3}\right) \end{equation}

(13)

\begin{equation} \hat{\mathbf{m}} = \frac{\sqrt{\lambda_{3}} - \sqrt{\lambda_{1}}}{\rho \sqrt{\lambda_{3} - \lambda_{1}}}(-\sqrt{1 - \lambda_{1}} \mathbf{B}_{0}^{\text{T}}\hat{\mathbf{e}}_{1} + k\sqrt{\lambda_{3} - 1} \mathbf{B}_{0}^{\text{T}}\hat{\mathbf{e}}_{3}) \end{equation}

(14)

\begin{align} \mathbf{W} & = (\mathbf{B}_{0} + \mathbf{b} \otimes \hat{\mathbf{m}})\mathbf{B}_{\alpha}^{-\text{T}},\quad |\omega| = \cos^{-1}\left(\frac{\mathop{\text{Tr}}\nolimits\mathbf{W} - 1}{2}\right),\\ {\boldsymbol{\omega}} & = |\omega| \begin{pmatrix} \text{W}_{32} - \text{W}_{23}\\ \text{W}_{13} - \text{W}_{31}\\ \text{W}_{21} - \text{W}_{12} \end{pmatrix} \Bigg/ \begin{vmatrix} \text{W}_{32} - \text{W}_{23}\\ \text{W}_{13} - \text{W}_{31}\\ \text{W}_{21} - \text{W}_{12} \end{vmatrix} \end{align}

(15)

where k = ±1, $\rho \neq 0$ is the normalization constant for $|\hat{\mathbf{m}}| = 1$, and $\hat{\mathbf{e}}_{i}$ are the eigenvectors of C corresponding to the eigenvalues $\lambda_{i}$. Taking k = +1 gives one solution and taking k = −1 gives the other. In the present study, since it is difficult to obtain analytical solutions for W, b, $\hat{\mathbf{m}}$, $\boldsymbol{\omega}$ and $\omega$, numerical solutions are obtained. The software used for the calculations was Mathematica. Numerical calculations were performed by giving real values for s₀, s and α. The eigenvalues and eigenvectors of C were obtained by solving the characteristic equations of C directly. The obtained eigenvalues and eigenvectors were substituted into eqs. (13)∼(15).

3. Result

3.1 The disclination caused by the connecting kink bands

3.1.1 The power of the disclination and Frank vector

As mentioned in the previous section, there are two solutions for eqs. (13) and (14). Always $\hat{\mathbf{m}} = \hat{\mathbf{e}}_{z}$ in one solution, independent of s₀, s and α. In the other solution, $\hat{\mathbf{m}}(s_{0},s,\alpha )$ depends on s₀, s and α (Fig. 3). We call the former solution as trivial connection and the latter as non-trivial connection. The trivial connection represents a state in which different sheared regions are connected through the shear plane (basal plane), and does not represent kink. On the other hand, the non-trivial connection represents a state in which sheared regions are connected on a plane that is not the shear plane, and one of the sheared regions suffers non-zero rotation and thus represents a kink. W is the rigid body rotation of the kink band to satisfy the rank-1 connection when the kink bands connected with each other. $\boldsymbol{\omega}$ and $\omega$ are the rotation axis and rotation angle of the rigid body rotation, respectively, and can be regarded as the Frank vector and the power of the disclination, respectively. Figure 4 shows a schematic of positive and negative wedge disclinations according to Volterra.⁴¹⁾ The disclination line l is parallel to the axis of the hollow cylinder. In the case of the wedge disclination, Frank vector is parallel to the disclination line. The sign of the disclination follows the definition given by Romanov.³⁴^,³⁵⁾ The positive disclination is formed by removing the wedge, and the negative one is formed by inserting a wedge. The direction of Frank vector is defined using right-handed screw as indicated by the red arrow in Fig. 4. The misorientation of the basal plane at the interface ($\theta$) is $\theta = \cos^{-1}((4-s^{2})/(4+s^{2}))$.³²⁾ The maximum value of $\theta$ in experiment is about 60° for LPSO-Mg alloys²⁶^,⁴²⁾ and Cu/Nb laminates.⁴³⁾ Cu/Nb laminate can be regarded as materials with only one slip plane because the slip deformation is strongly constrained on the plane of lamination. The plane of laminate in Cu/Nb laminate is, therefore, equivalent to the basal plane in Fig. 1, whereas hexagonal coordinate system is not necessary in this case. This indicates s ≈ 1.15 in experiments. When considering the kink deformation that actually occurs, it is sufficient to consider s at most 2. The situation where the shear direction of 180° < α < 360° is the same as the situation where the sign of the shear is inverted as α − 180°: it is sufficient to consider 0° ≤ α ≤ 180°. Let $\omega_{\text{t}}$ and $\omega_{\text{nt}}$ be the power of the disclination for trivial and non-trivial connections, respectively. Since the elastic energy of the disclination is proportional to $\omega^{2}$, it is reasonable to assume that the connecting with smaller $\omega$ is preferred.³⁴^,³⁵⁾ Therefore, first of all, we analyze the difference between $\omega_{\text{t}}$ and $\omega_{\text{nt}}$. Figure 5 shows $\Delta\omega = \omega_{\text{t}} - \omega_{\text{nt}}$. The regions with $\Delta\omega > 0$ are shown in red and the regions with $\Delta\omega \leqq 0$ is shown in blue in Fig. 5. For |s₀|, |s| ≦ 2 as assumed, $\omega_{\text{nt}}$ is always smaller than $\omega_{\text{t}}$, independent of α. The elastic energy of disclination is, therefore, lower for the non-trivial connection for realistic situation. For |s₀|, |s| > 2, as |s₀| and |s| increase from 2 and α increases from 0°, $\omega_{\text{t}} < \omega_{\text{nt}}$. Furthermore, for |s₀|, |s| > 4, as |s₀| and |s| increases from 4 and α decreases from 180°, $\omega_{\text{t}} < \omega_{\text{nt}}$. Finally, for s₀, s = 1000, where the magnitude of shear is sufficiently large, $\omega_{\text{nt}} < \omega_{\text{t}}$ for α ≈ 90°, but $\omega_{\text{t}} < \omega_{\text{nt}}$ for the other values of α. In other words, for very large values of the magnitude of shear, the elastic energy of the disclination is lower for the trivial connection. However, since it is not realistic to give such a large deformation, we only need to consider the non-trivial connections. In the following, we consider only the non-trivial connection with |s| ≦ 2.

Fig. 3

Schematic of the junction plane of (a) trivial connection and (b) non-trivial connection. The normal vector of the junction plane of the trivial connection is always $\hat{\mathbf{m}} = \hat{\mathbf{e}}_{z}$ independent of the magnitude and direction of the shear. The normal vector of the junction plane of the non-trivial connection changes depending on the magnitude and direction of the shear.

Fig. 4

Schematic of (a) positive and (b) negative wedge disclination.

Fig. 5

Difference of the power of disclination between trivial and non-trivial connections. In the red region, the non-trivial connection has a smaller power of disclination.

3.1.2 Types of disclination caused by connecting kink bands

We consider the relationship between the junction plane and Frank vector for the non-trivial connection with |s| ≦ 2. As shown in Fig. 3, kink interface $(\hat{\mathbf{m}})$ depends on s₀, s and α. Figure 6 shows Frank vector $(\boldsymbol{\omega})$, the normals of the matrix/kink interfaces $(\hat{\textbf{n}}_{0},\hat{\textbf{n}}_{\alpha })$ and the line of intersection of the two kink interfaces $(\hat{\textbf{n}}_{0} \times \hat{\textbf{n}}_{\alpha })$. $\hat{\textbf{m}}$ and $\boldsymbol{\omega}$ are obtained from eqs. (14) and (15), respectively. $\hat{\textbf{n}}_{0}$ and $\hat{\textbf{n}}_{\alpha }$ are obtained from eqs. (3) and (9), respectively. Numerical analysis showed that $\hat{\textbf{m}}$ is always perpendicular to $\boldsymbol{\omega}$ independent of s₀, s and α. Furthermore, the intersection vector $\hat{\textbf{n}}_{0} \times \hat{\textbf{n}}_{\alpha }$ is always parallel to $\boldsymbol{\omega}$ independent of s₀, s and α. In other words,

\begin{equation} {\boldsymbol{\omega}} \perp \hat{\mathbf{m}}\ \text{and}\ {\boldsymbol{\omega}}\mathrel{/\!/}\hat{\mathbf{n}}_{0} \times \hat{\mathbf{n}}_{\alpha}. \end{equation}

(16)

Fig. 6

Schematic diagram of the junction plane of connecting kink bands in the non-trivial connection. l is the line vector of the disclination, $\boldsymbol{\omega}$ is Frank vector, $\hat{\mathbf{m}}$ is the normal vector of the junction plane, $\hat{\mathbf{n}}_{0} \times \hat{\mathbf{n}}_{\alpha }$ is the intersection vector of the kink interfaces. Frank vector is always perpendicular to $\hat{\mathbf{m}}$ and parallel to $\hat{\mathbf{n}}_{0} \times \hat{\mathbf{n}}_{\alpha }$.

Figure 7 shows the geometric classification of wedge disclination and twist disclination by Volterra’s model.³⁴^,³⁵^,⁴¹⁾ l is the line of the disclination. When l is parallel to $\boldsymbol{\omega}$, the disclination is wedge-type. When l is perpendicular to $\boldsymbol{\omega}$, the disclination is twist I or twist II. $\boldsymbol{\omega} \perp \hat{\textbf{m}}$ and $\boldsymbol{\omega}\mathrel{/\!/}\hat{\textbf{n}}_{0} \times \hat{\textbf{n}}_{\alpha }$ correspond to wedge, $\boldsymbol{\omega}\mathrel{/\!/}\hat{\textbf{m}}$ and $\boldsymbol{\omega} \perp \hat{\textbf{n}}_{0} \times \hat{\textbf{n}}_{\alpha }$ correspond to twist I, and $\boldsymbol{\omega} \perp \hat{\textbf{m}}$ and $\boldsymbol{\omega} \perp \hat{\textbf{n}}_{0} \times \hat{\textbf{n}}_{\alpha }$ correspond to twist II. For the non-trivial connection with |s₀|, |s| ≦ 2, eq. (16) is always satisfied regardless of α, so the only type of disclination formed is the wedge disclination. Twist-type disclination appears in the trivial connection but is not considered in this study.³⁸⁾

Fig. 7

Schematics of wedge disclination and twist disclinations. For the non-trivial connection with |s₀|, |s| ≦ 2, the type of disclination formed by the connecting kink bands is always wedge disclination.

3.1.3 s and α dependence of the normal vector and Frank vector

The α dependence of $\boldsymbol{\omega}$ and $\hat{\textbf{m}}$ is shown in this section. s, s₀ are fixed to s = 0.3, s₀ = −0.3,³²⁾ which corresponds to an average misorientation angle $\theta \sim 17 {}^{\circ}$, as is often confirmed for LPSO-Mg alloys.²⁶^,⁴²⁾ However, the generality of the essential result is not lost. Figure 8(a) and (b) shows the components of $\boldsymbol{\omega}$ and $\hat{\textbf{m}}$ as functions of α. Each component of $\boldsymbol{\omega}$ and $\hat{\textbf{m}}$ is continuous and smooth without discontinuity. This means that as α changes continuously, each component of $\boldsymbol{\omega}$ and $\hat{\textbf{m}}$ also changes continuously. Figure 9 shows schematically the variation of $\boldsymbol{\omega}$ and $\hat{\textbf{m}}$ as α is varied from 0° to 180° for s = 0.3, s₀ = −0.3 basing on the results in Fig. 8(a) and (b). When α = 0°, $\hat{\textbf{m}}\text{ = (0,1,0)}$. As α increases, the x and y components decrease, but z component is always zero. Finally, when α = 180°, $\hat{\textbf{m}} = ( - 1,0,0)$. In this case, $\hat{\textbf{m}}$ rotates in the x-y plane around the z-axis for increase in α, $\hat{\textbf{m}} = ( - \sin \alpha /2,\cos \alpha /2,0)$.

Fig. 8

(a) Each component of $\hat{\mathbf{m}} = ( - \sin \alpha /2,\cos \alpha /2,0)$ for the case of s = 0.3, s₀ = −0.3. (b) Each component of $\boldsymbol{\omega}$ for the case of s = 0.3, s₀ = −0.3. $\boldsymbol{\omega}$ and $\hat{\mathbf{m}}$ vary continuously with respect to α.

Fig. 9

Variation of the normal vector and Frank vector of the junction plane. $\boldsymbol{\omega}$ and $\hat{\mathbf{m}}$ vary continuously for α, so the form of kink changes continuously.

Next, we consider $\boldsymbol{\omega}$ with respect to the change in α. When α = 0°, $\boldsymbol{\omega} = ( - 0.01265,0,0)$. The x component increases monotonically with increasing α, reaching zero at α = 180°. The y and z components are convex curves with a minimum value at α = 91.2°. In this case, $\boldsymbol{\omega}$ can be approximated as $\boldsymbol{\omega} \approx ( - \frac{1}{2}\boldsymbol{\omega}_{\text{x}}(\alpha = 0)\cos \alpha + \boldsymbol{\omega}_{\text{x}}(\alpha = 91.2^{\circ} ),\boldsymbol{\omega}_{\text{y}}(\alpha = 91.2^{\circ} )\sin \alpha ,\boldsymbol{\omega}_{\text{z}}(\alpha = 91.2^{\circ} )\sin \alpha )$ and depends only on α. Thus, $\boldsymbol{\omega}$ and $\hat{\mathbf{m}}$ varies continuously with respect to α.

Furthermore, let us consider a situation where the magnitude of shear changes in addition to the shear direction. Figure 10 shows each component of $\boldsymbol{\omega}$ and $\hat{\mathbf{m}}$ with s₀ = −0.3; α and s are variables. Each component of $\boldsymbol{\omega}$ is continuous and smooth curve without discontinuity for the change in α and s. In contrast, $\hat{\mathbf{m}}$ has discontinuity on the line connecting (s, α) = (−0.3, 0°) and (s, α) = (0.3, 180°). On this singular line, the two kink bands are identical, so there is no solution for the rank-1 connection and the power of disclination is zero. However, except for the singular line, $\hat{\mathbf{m}}$ is continuous and smooth for the change in α and s. The range of $\hat{\mathbf{m}}$ is a unit sphere. It is concluded that except for the singular line, $\boldsymbol{\omega}$ and $\hat{\mathbf{m}}$ change continuously with respect to the continuous change in α and s. This means that the junction plane of kink bands changes smoothly when the shear direction and magnitude of the shear inside the kink change continuously.

Fig. 10

(a) $\boldsymbol{\omega}(s,\alpha )$ for the case of s₀ = −0.3. $\boldsymbol{\omega}(s,\alpha )$ varies continuously for α and s. (b) m(s, α) for the case of s₀ = −0.3. $\boldsymbol{\omega}(s,\alpha )$ varies continuously for α and s except at line connecting (s, α) = (0.3, 180°) and (s, α) = (−0.3, 0°). At the boundary, power of disclination is zero.

3.2 The possibility of annihilation of disclinations

As mentioned in the previous section, when kink bands with multiple basal shears connect, wedge disclination inevitably forms at the junction plane. We show that annihilation of disclination in the present case is possible as previously reported.³²⁾ Annihilation of disclinations occurs when the Frank vectors of disclinations formed by kinks are antiparallel each other and parallel to the intersecting line of the two kink bands, and the powers of the disclinations are equal to each other. Such Frank vectors and the power of disclinations are obtained by analyzing the rank-1 connections for each kink. Figure 11(a) shows connecting kink bands with B₁(s₁, α) and B₂(s₂, 0) (hereafter denoted as B₂(s₂)). Figure 11(b) shows the connecting kink bands with B₂(s₂) and B₃(s₃, β). Ridge and ortho kink can be distinguished by the inner product of the shear directions acting in kink bands. Positive inner product gives ortho kink, and the negative inner product gives ridge kink. The kink in Fig. 11(a) is the ridge-type because s₁s₂ cos α < 0. The kink in Fig. 11(b) is the ortho-type because s₂s₃ cos β > 0. The rank-1 connections are $\mathbf{W}_{\text{R}}\mathbf{B}_{1}(s_{1},\alpha ) - \mathbf{B}_{2}(s_{2}) = \mathbf{b}_{\text{R}} \otimes \hat{\mathbf{m}}_{\text{R}}$ for the ridge kink in Fig. 11(a) and $\mathbf{W}_{\text{O}}\mathbf{B}_{3}(s_{3},\beta ) - \mathbf{B}_{2}(s_{2}) = \mathbf{b}_{\text{O}} \otimes \hat{\mathbf{m}}_{\text{O}}$ for the ortho kink in Fig. 11(b), where W_R and W_O are the rigid body rotations of the kink bands, $\boldsymbol{\omega}_{\text{R}}$ and $\boldsymbol{\omega}_{\text{O}}$ are Frank vectors, $\omega_{\text{R}}$ and $\omega_{\text{O}}$ are the power of the disclinations, b_R and b_O are the vector representing the discontinuities in the deformation gradients, $\hat{\mathbf{m}}_{\text{R}}$ and $\hat{\mathbf{m}}_{\text{O}}$ are the normal vector of the junction planes. W_R, $\boldsymbol{\omega}_{\text{R}}$ and $\omega_{\text{R}}$ are functions of s₁, s₂ and α. On the other hand, W_O, $\boldsymbol{\omega}_{\text{O}}$ and $\omega_{\text{O}}$ are functions of s₂, s₃ and β. The normal vectors of the junction planes in the current configuration are $\mathbf{W}_{\text{R}}^{ - \text{T}}\hat{\mathbf{n}}_{1}$, $\hat{\mathbf{n}}_{2}$ and $\mathbf{W}_{\text{O}}^{ - \text{T}}\hat{\mathbf{n}}_{3} $ as described in Fig. 11(a) and (b). The intersection vector of two kink interfaces is obtained by the outer product of the normal vectors of the current configuration of each interface; $\hat{\mathbf{n}}_{\text{R}}(s_{1},s_{2},\alpha ) = \mathbf{W}_{\text{R}}^{ - \text{T}}\hat{\mathbf{n}}_{1} \times \hat{\mathbf{n}}_{2}$ for the ridge kink and $\hat{\mathbf{n}}_{\text{O}}(s_{2},s_{3},\beta ) = \hat{\mathbf{n}}_{2} \times \mathbf{W}_{\text{O}}^{ - \text{T}}\hat{\mathbf{n}}_{3}$ for the ortho kink. These are summarized in Table 1.

Fig. 11

(a) Schematic diagram of the ridge kink for s₁s₂ cos α < 0. B₁(s₁, α) and B₂(s₂) are the deformation gradients, $\boldsymbol{\omega}_{\text{R}}$ is Frank vector, $\mathbf{W}_{\text{R}}^{ - \text{T}}\hat{\mathbf{n}}_{1}$, $\mathbf{B}_{2}^{ - \text{T}}\hat{\mathbf{m}}_{\text{R}}$ and $\hat{\mathbf{n}}_{2}$ are the normal vector of the junction planes, q_α and q₀ are the rotation axes of the rigid body rotations of the kink bands. (b) Schematic diagram of the ortho kink for s₂s₃ cos β > 0. B₁(s₁, α) and B₂(s₂) are the deformation gradients, $\boldsymbol{\omega}_{\text{R}}$ is Frank vector, $\mathbf{W}_{\text{R}}^{ - \text{T}}\hat{\mathbf{n}}_{1}$, $\mathbf{B}_{2}^{ - \text{T}}\hat{\mathbf{m}}_{\text{R}}$ and $\hat{\mathbf{n}}_{2}$ are the normal vectors of the junction planes, q_α and q₀ are the rotation axes of the rigid body rotations of the kink bands.

Table 1 Rank-1 connections for the ridge kink and ortho kink.

When $\boldsymbol{\omega}_{\text{R}}$ and $\boldsymbol{\omega}_{\text{O}}$ are antiparallel and $\omega_{\text{R}} + \omega_{\text{O}} = 0$, annihilation of disclinations occurs. According to eq. (16) $\boldsymbol{\omega}_{\text{R}}$ and $\boldsymbol{\omega}_{\text{O}}$ are parallel to n_R and n_O, respectively. Therefore $\boldsymbol{\omega}_{\text{R}}$ is antiparallel to n_O. $\boldsymbol{\omega}_{\text{O}}$ is antiparallel to n_R. n_R and n_O are antiparallel to each other for the annihilation. In other words, the conditions under which annihilation of disclinations occurs are as follows.

\begin{equation} \omega_{\text{R}} + \omega_{\text{O}} = 0 \end{equation}

(17)

\begin{equation} \frac{{\boldsymbol{\omega}}_{\text{R}} \cdot {\boldsymbol{\omega}}_{\text{O}}}{|{\boldsymbol{\omega}}_{\text{R}}||{\boldsymbol{\omega}}_{\text{O}}|} = -1 \end{equation}

(18)

\begin{equation} \frac{\mathbf{n}_{\text{R}} \cdot \mathbf{n}_{\text{O}}}{|\mathbf{n}_{\text{R}}||\mathbf{n}_{\text{O}}|} = -1 \end{equation}

(19)

\begin{equation} \frac{\mathbf{n}_{i} \cdot {\boldsymbol{\omega}}_{i}}{|\mathbf{n}_{i}||{\boldsymbol{\omega}}_{i}|} = 1\quad i = \text{R}\ \text{or}\ \text{O}\quad j = \text{R}\ \text{or}\ \text{O} \end{equation}

(20)

\begin{equation} \frac{\mathbf{n}_{i} \cdot {\boldsymbol{\omega}}_{j}}{|\mathbf{n}_{i}||{\boldsymbol{\omega}}_{j}|} = -1\quad i,j = \text{R}\ \text{or}\ \text{O},\quad i \neq j \end{equation}

(21)

The magnitude and direction of the shears that satisfy the annihilation condition can be obtained by solving the simultaneous equations of eqs. (17)–(21) numerically.

As an example, the magnitude and direction of shears for the ortho kink in Fig. 11(b) were set to be s₂ = 0.4, s₃ = 0.3 and β = 5.0° and then s₁ and α for annihilation was numerically solved. In this case, the power of disclination is $|\omega_{\text{O}}| = 0.33827^{\circ}$. Figure 12 shows an example for s₂ = 0.4, s₃ = 0.3 and β = 5.0°. When $\omega_{\text{R}} + \omega_{\text{O}}$, $\boldsymbol{\omega}_{\text{R}}\cdot \boldsymbol{\omega}_{\text{O}}/|\boldsymbol{\omega}_{\text{R}}||\boldsymbol{\omega}_{\text{O}}| + 1$, n_R · n_O/|n_R||n_O| + 1 and $\mathbf{n}_{\text{O}}\cdot \boldsymbol{\omega}_{\text{R}}/|\mathbf{n}_{\text{O}}||\boldsymbol{\omega}_{\text{R}}| + 1$ are simultaneously zero at a certain point, such point gives (s₁, α) for the annihilation. The solution that satisfies the annihilation of disclinations condition is s₁ = −0.06852, α = 22.772° in Fig. 12. Figure 13 depicts the kink microstructure corresponding to the condition of annihilation in Fig. 12.

Fig. 12

$\omega_{\text{R}} + \omega_{\text{O}}$, $\boldsymbol{\omega}_{\text{R}}\cdot \boldsymbol{\omega}_{\text{O}}/|\boldsymbol{\omega}_{\text{R}}||\boldsymbol{\omega}_{\text{O}}| + 1$, n_R · n_O/|n_R||n_O| + 1 and $\mathbf{n}_{\text{O}}\cdot \boldsymbol{\omega}_{\text{R}}/|\mathbf{n}_{\text{O}}||\boldsymbol{\omega}_{\text{R}}| + 1$ with s₂ = 0.4, s₃ = 0.3, β = 5.0°. Inset is the enlargement around the intersection point for the annihilation. When all surfaces intersect at a point with value of vertical axis of 0, the intersection point gives the annihilation of disclinations condition. In this case s₁ = −0.06852 and α = 22.772°.

Fig. 13

Schematic diagram of a kink microstructure with annihilation of disclination. When s₂ = 0.4, s₃ = 0.3, β = 5.0°, s₁ = −0.06852 and α = 22.772°, annihilation of disclinations conditions (eqs. (17)–(21)) are simultaneously satisfied.

Further examples are summarized in Table 2 which shows s₁, s₂, s₃ and α satisfying the annihilation condition for a given value of β. Thanks to the six-fold symmetry of the hexagonal lattice, it is sufficient to consider β of 0∼60°. Since no analytical solution has been obtained, we are not confident that the obtained solutions are the only solutions. However, the annihilation of wedge disclinations is possible even between kink bands which are formed by basal shears with different shear directions.

Table 2 Combination of s₁, s₁, s₃ and α satisfying the annihilation of disclinations condition for given β. There is at least one combination of s₁, s₂, s₃ and α that satisfies the annihilation condition for 0° < β < 60°.

4. Discussion

The results show that the interface normals and Frank vector change continuously with respect to the magnitude and direction of the basal shears in the case where two kink bands formed by multiple basal shears connects. Furthermore, we found that annihilation of disclinations is possible even between kinks formed by multiple basal shears. Hagihara et al.²⁶⁾ observed wavy ridge kink on the (0001) cross section. The crystallographic orientation within such kink band was not constant. In the light of the present study, actual kink is not formed by a single kink, but is formed by connecting fine kink bands with slightly different shear magnitudes. Such connection of fine kink bands requires disclinations at the boundaries and the aggregate of such kinks should takes curved morphology with a network of surrounding disclinations. Three-dimensional morphology of such curved kink is discussed in the light of the present results.

Firstly, the three-dimensional morphology of kink is discussed based on the power of the disclination. Figure 14 shows α dependence of the power of disclination for s = 0.3, s₀ = −0.3. The vertical axis shows the power of disclination normalized by α = 0°. The elastic energy of a wedge disclination is proportional to the square of the power of disclination.³⁴^,³⁵⁾ The square of the power of disclination is, therefore, a measure of the elastic energy of the connecting kink bands. Figure 15 shows morphology of connecting kink bands for α = 0°, 10°, 30°, 90°, 150° and 180°. The normal vector of the junction plane $\hat{\mathbf{m}} = ( - \sin \alpha /2,\cos \alpha /2,0)$. When α = 0°, 10° and 30°, the connecting kink bands is a ridge kink, and the power of disclination is 0.72°, 0.84° and 1.40°, respectively. The elastic energy is maximized at α = 90°, and the power of disclination is 2.19°. The connection becomes ortho-type when α > 90° as mentioned in the previous section. The power of disclination is 1.27° at α = 150° and is 0° at α = 180°. Figure 15 clearly demonstrates that the change in the morphology of the connecting kink bands is sensitive to the shear direction α. On the other hand, as shown in Fig. 14, the elastic energy of the disclination is insensitive to α around α = 0° and 180°. In other words, the three-dimensional morphology of kink can change drastically from a single kink band and ideal ridge kink with a small increase in elastic energy. This point is extremely important when we speculate the three-dimensional morphologies of kinks using the two-dimensional microscopy image.

Fig. 14

α dependence $\omega^{2}/\omega_{0^{\circ}}^{2}$ in case of s = 0.3, s₀ = −0.3.

Fig. 15

Shape of connecting kink at α = 0°, 10°, 30°, 90°, 150° and 180°.

One of the shear direction in one kink is fixed in the above discussion. However, the result has generality because the difference of the two shears in the kink bands is the essential factor to determine the morphology and energy of the kink. Applying coordinate transformation to the system gives general cases with the same results.

Hagihara et al.²⁶⁾ have shown that the kink interface on the (0001) cross section of the ridge kink is wavy with the continuous change in crystallographic orientation within the kink band. According to this study, it is considered to be due to the continuous change in the kink interface caused by the continuous change in the direction and magnitude of the shears inside the kink band. The magnitude and direction of the shears of the wavy kink interface on (0001) cross section is schematically shown in Fig. 16(a). The kink band whose magnitude of shear is s_χ and shear direction is χ is described as (s_χ, χ). As shown in Fig. 16(a), we can consider sequences of kinks which are rank-1 connected: (s_α + nΔs_α, α + nΔα) and (s_β + nΔs_β, β + nΔβ) where n is non-integer. On the other hand, the three-dimensional morphology of kink microstructure with annihilation of disclination is schematically shown in Fig. 16(b). As shown in Fig. 16(b), we can construct rank-1 connected sequences of kink bands: (s_ξ + mΔs_ξ, ξ + mΔξ), (s_ζ + mΔs_ζ, ζ + mΔζ) and (s_η + mΔs_η, η + mΔη) with non-negative integer m. This allows us to represent a deformation gradient with a slight change in magnitude and direction of the shears. In addition to the rank-1 connection between kink bands to form a ridge kink (blue plane in Fig. 16(a)), the connection between ridge kinks (green plane in Fig. 16(a)) can also be expressed by rank-1 connection between kinks to form a wavy ridge kink. The rank-1 connection between ridge kinks also requires disclinations. As shown in Fig. 16(a), wedge disclination dipoles are formed at the junctions because the junction plane is embedded in the body.³²⁾ These wedge disclination dipoles further contribute to reduce the elastic energy. Wavy ridge kinks can, therefore, be formed with such small power of disclinations. Similar situation is also possible for the ridge kink with annihilating disclinations as depicted in Fig. 16(b). This construction of ridge kinks with annihilating disclination is possible when there are solutions for the annihilation in the neighborhood of a solution of annihilation.

Fig. 16

(a) Schematic of the three-dimensional morphology of a wavy ridge kink. The wavy kinks have a network of wedge disclinations. (b) Schematic of three-dimensional morphology of kink microstructure with annihilation of disclinations.

5. Conclusion

We analyzed the kinematic aspects of kink bands with multiple basal shear using rank-1 connection to investigate the type of disclination and annihilation of disclinations in kink microstructures. Following conclusions were obtained.

(1) Wedge disclination forms in the connecting kink bands for realistic magnitude of shear that would occur (|s| ≦ 2), regardless of the shear direction in each kink band. Mixed disclination also can form with a larger power of disclination in this case.
(2) The interface normal between kinks and Frank vector of the wedge disclination change continuously with respect to the magnitude and direction of the shears in each kink band.
(3) The annihilation of disclinations is possible even between the kinks which are formed by multiple basal shears.
(4) The three-dimensional morphology of the connecting kinks is sensitive to the direction of the basal shears in the kinks. On the other hand, when the basal shears are close to parallel or antiparallel, the power of disclination in the connecting kinks is insensitive to the direction of the basal shears. This means that single kink band and ridge kink can change their shape with very small increase in elastic energy.
(5) Three-dimensional models of wavy ridge kinks which are constructed by sequences of kink bands were proposed.

Acknowledgements

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

REFERENCES

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