Emergence of Disclination and Cooperative Deformation at the Intersection of Kink Interface and Slip Deformation

Abstract

In order to clarify a strengthening principle due to the presence of the kink band, the geometry of the intersection of slip deformation and kink interface was analyzed in a situation where there is only one slip system and no cracking or delamination occurs. When the kink interface is penetrated by slip deformation while maintaining the continuity of deformation, disclinations are always formed at the intersection, and the migration of kink interface is required (cooperative deformation). The disclination and the cooperative deformation always increase the stress for deformation, regardless of how small the penetrating slip is. In layered structures such as long-period stacking ordered (LPSO) Mg alloys, there is a mismatch in the number of layers at the intersection of the kink interface and the slip deformation, which necessitates cutting of the layers. This means that the ⟨c⟩-dislocation is necessary for the cooperative deformation in LPSO-Mg. It is suggested that when the Peierls stress of ⟨c⟩-dislocation is much higher than that of basal slip (easy slip), the presence of kink interface leads to a dramatic increase in the stress for plastic deformation.

1. Introduction

Kink deformation occurs in layered materials with strong plastic anisotropy and has been observed in a variety of materials.^1–5) When the slip system is limited to only one direction on the layer plane, kink deformation, in which shear deformation on the layer plane and rigid body rotation of the layer itself occur simultaneously, is likely to occur.^6–8) This slip deformation on the layer plane that induces kink deformation is called basal slip hereafter. In such materials, kink deformation becomes the dominant deformation mode when a compressive load is applied from a direction nearly parallel to the slip plane.^6–8) Kink band and kink microstructure, which is aggregate of connected kink bands, are formed by the kink deformation.⁷⁾ The interface between kink band and matrix, and the interface between kink bands are called kink interface. Recently, a long-period stacking ordered (LPSO) Mg alloy, such as Mg–Zn–Y alloy, has been found to undergo kink deformation.^9,10) This Mg alloy consists of periodically stacked soft layer of α-Mg (hcp) and hard layer (L1₂ order) enriched with Zn and Y atoms.^11–14) Since the easy-slip system is limited to ⟨a⟩ basal slip due to the presence of the hard layer,¹⁵⁾ kink deformation occurs in this alloy.¹⁰⁾ LPSO-Mg alloy with a large amount of kink introduced by hot extrusion has much higher strength than conventional Mg alloy.⁹⁾ This strengthening is called kink strengthening, and clarification of the strengthening mechanism is strongly desired. The kink interface has been shown experimentally to be an obstacle to the basal slip.^16–18) The kink interface yields a Hall-Petch coefficient comparable to that of a high-angle grain boundary, even though it is a low-angle symmetric tilt grain boundary.^17,18) A different strengthening mechanism is expected to occur at kink interfaces than at normal grain boundaries.

Geometric quantities of kink, such as the kink interface orientation and the crystallographic orientation difference between the matrix and the kink band, are closely related to kink strengthening.¹⁹⁾ However, there have been only a few studies on the geometry of kink,^8,20–22) and most of them have not been related to the strengthening mechanism. Recently, Inamura²³⁾ systematically formulated the geometry of the formation of kink bands using the continuity of deformation (rank-1 connection) under the condition that the slip system is limited to the basal slip only and no crack or delamination occurs. The material was treated as a set of coordinate points in the geometrical analysis to extract the universal nature of kink deformation independent of the physical properties of the material. Inamura’s model clearly showed that disclination is always formed at the junction plane between kink bands due to geometric constraints, and suggesting that the disclination acts as a strengthening mechanism. Following Inamura’s model, one would expect a strengthening mechanism due to geometrical constraints to be present even in situations where the basal slip penetrates the pre-existing kink band. In this study, we construct a model in which a kink interface is sheared by a basal slip and analyze the geometry of the intersection of the basal slip and the kink interface to clarify the geometric aspect of the strengthening mechanism by the kink interface.

2. Analysis Method

It has been experimentally shown that the slip trace and the kink interface are not interrupted at the intersection of the slip deformation and the kink interface.^17,24) These results indicate that the deformation is continuous at the intersection, but the deformation gradient is discontinuous across the kink interface. In other words, the continuity of deformation is maintained at the kink interface even after the shearing by the slip.

We treat the slip deformation on the basal plane as “shear band” with a finite thickness and uniform magnitude of shear in this study. In slip deformation of metal, the crystals above and below the slip plane are displaced by Burgers vector. In geometric point of view, this slip deformation is regarded as a uniform shear with thickness of the atomic plane spacing. The thickness of the shear band can be larger than the atomic plane distance if some successive slip planes were active with the same amount of shear.

Figure 1 shows a model in which a shear band with finite thickness (d) penetrates a pre-existing kink interface that was formed by the deformation gradient Q₁S₁, where Q₁ is rotation in kink and S₁ is the basal shear occurred to form the kink.²³⁾ The unit vectors along x, y, and z axes in the orthogonal coordinate system are denoted by e_x, e_y and e_z, respectively. The unit normal vector of the slip plane is e_z and the unit vector in the slip direction is e_y. $\hat{\mathbf{n}}_{1}$ is the normal vector of the pre-existing kink interface, θ is the amount of crystal rotation by Q₁, ϕ is the angle between the kink interface normal and e_y. S₁, Q₁, θ, $\hat{\mathbf{n}}_{1}$ and ϕ are given by the following equations using a real number s₁ representing the magnitude of the shear.²³⁾

  

\begin{equation} \mathbf{S}_{1} = \mathbf{I} + s_{1}\mathbf{e}_{\text{y}} \otimes \mathbf{e}_{\text{z}} \end{equation} (1)

  

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

  

\begin{equation} \hat{\mathbf{n}}_{1} = \frac{1}{\sqrt{4 + s_{1}{}^{2}}} \begin{pmatrix} 0\\ 2\\ s_{1} \end{pmatrix} \end{equation} (3)

  

\begin{equation} \theta = 2\phi = \cos^{-1}\left(\frac{4 - s_{1}{}^{2}}{4 + s_{1}{}^{2}}\right) \end{equation} (4)

Since the slip deformation is constrained on the basal plane, the shear band buckles at the kink interface to penetrate into the kink. The deformation gradient of the incident shear band (S_m) and the deformation gradient of the shear band penetrating into the kink band (S_k) are given by the following equations using a real number s_m which representing the magnitude of shear by S_m.

  

\begin{equation} \mathbf{S}_{\text{m}} = \mathbf{I} + s_{\text{m}}\mathbf{e}_{\text{y}} \otimes \mathbf{e}_{\text{z}} \end{equation} (5)

  

\begin{equation} \mathbf{S}_{\text{k}} = \mathbf{Q}_{1}\mathbf{S}_{\text{m}}\mathbf{Q}_{1}^{\text{T}} = \mathbf{I} + s_{\text{m}} \{(\mathbf{Q}_{1}\mathbf{e}_{\text{y}}) \otimes (\mathbf{Q}_{1}\mathbf{e}_{\text{z}})\} \end{equation} (6)

The rank-1 connection between S_m and S_k, as shown by the bold line in Fig. 1, is given by the following equation using a rigid body rotation matrix (W), a vector representing the discontinuity of the deformation gradient (b) and the normal vector of the junction plane ($\hat{\mathbf{m}}$).

  

\begin{equation} \mathbf{WS}_{\text{k}} - \mathbf{S}_{\text{m}} = \mathbf{b} \otimes \hat{\mathbf{m}} \end{equation} (7)

It has been shown by Ball and James²⁵⁾ that the solutions (W, b, $\hat{\mathbf{m}}$) exists for eq. (7), if and only if $\mathbf{C} = \mathbf{S}_{\text{m}}^{ - \text{T}}\mathbf{S}_{\text{k}}^{\text{T}}\mathbf{S}_{\text{k}}\mathbf{S}_{\text{m}}^{ - 1} \ne \mathbf{I}$ and λ₁ ≤ 1, λ₂ = 1, λ₃ ≥ 1, where λ₁, λ₂, λ₃ are the eigenvalues of C. The solutions are given as,

  

\begin{equation} \mathbf{b} = \rho \left(\sqrt{\frac{\lambda_{3}(1 - \lambda_{1})}{\lambda_{3} - \lambda_{1}}} \frac{\hat{\mathbf{e}}_{1}}{|\hat{\mathbf{e}}_{1}|} + k\sqrt{\frac{\lambda_{1}(\lambda_{3} - 1)}{\lambda_{3} - \lambda_{1}}} \frac{\hat{\mathbf{e}}_{3}}{|\hat{\mathbf{e}}_{3}|} \right), \end{equation} (8)

  

\begin{equation} \hat{\mathbf{m}} = \frac{\sqrt{\lambda_{3}} - \sqrt{\lambda_{1}}}{\rho\sqrt{\lambda_{3} - \lambda_{1}}}\left(-\sqrt{1 - \lambda_{1}}\mathbf{S}_{\text{m}}^{\text{T}}\frac{\hat{\mathbf{e}}_{1}}{|\hat{\mathbf{e}}_{1}|} + k\sqrt{\lambda_{3} - 1} \mathbf{S}_{\text{m}}^{\text{T}}\frac{\hat{\mathbf{e}}_{3}}{|\hat{\mathbf{e}}_{3}|} \right) \end{equation} (9)

  

\begin{align} & \mathbf{W} = (\mathbf{S}_{\text{m}} + \mathbf{b} \otimes \hat{\mathbf{m}}) \mathbf{S}_{\text{k}}^{-\text{T}},\quad |\omega| = \cos^{-1}\left(\frac{\mathop{\text{Tr}}\nolimits \mathbf{W} - 1}{2} \right),\\ &\quad \text{and} \quad {\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} \Biggm/ \begin{vmatrix} \text{W}_{32} - \text{W}_{23}\\ \text{W}_{13} - \text{W}_{31}\\ \text{W}_{21} - \text{W}_{12} \end{vmatrix} , \end{align} (10)

where k = ±1, ρ ≠ 0 is the normalization constant for $|\hat{\mathbf{m}}| = 1$, and $\hat{\mathbf{e}}_{i}$ is the eigenvector of C corresponding to eigenvalue λ_i, ω and |ω| are axis and angle of the rotation W, respectively. The sign of ω is determined by sign(W₃₂ − W₂₃). There are two solutions for eq. (7). One is given by k = +1 and the other is by k = −1. The solution with k = +1 is indicated by subscript +, and the solution with k = −1 is indicated by a subscript −, hereafter. Firstly, the rank-1 connection of S_m and S_k are solved generally, and then the position of kink interface is overlayed to see what happens at the kink interface penetrated by S_m.

Fig. 1

A pre-existing kink interface penetrated by a shear band. A shear band with finite thickness d penetrates the kink interface which was formed by the deformation gradient Q₁S₁.

3. Results

The eigenvalues of $\mathbf{C} = \mathbf{S}_{\text{m}}^{ - \text{T}}\mathbf{S}_{\text{k}}^{\text{T}}\mathbf{S}_{\text{k}}\mathbf{S}_{\text{m}}^{ - 1}$ were obtained as follows by a brute force computation.

  

\begin{align} \lambda_{1} &= 1 + 8\text{D} \left(1 - \sqrt{1 + \frac{1}{4\text{D}}} \right),\quad \lambda_{2} = 1, \\ \lambda_{3}& = 1 + 8\text{D}\left(1 + \sqrt{1 + \frac{1}{4\text{D}}} \right), \end{align} (11)

where $\text{D} = s_{1}^{2}s_{\text{m}}^{2}(4 + s_{\text{m}}^{2})/(4 + s_{1}^{2})^{2}$. Obviously D > 0 when s_m and s₁ are not zero. The term in the brackets in λ₁ are always negative and the term in the brackets in λ₃ are always positive. Therefore, λ₁ < 1 and λ₃ > 1 when s_m and s₁ are not zero; the necessary and sufficient condition for the existence of solution in eq. (7) is satisfied.

Figures 2(a) and (b) are schematics showing the connection of S_m and S_k. As shown in Fig. 2(a), a rotational gap corresponding to the rigid body rotation W appears at the interface between shear bands S_m and S_k. To fill this gap, a pair of disclinations of ω must be formed, as shown in Fig. 2(b).²³⁾ The vector of the rotation axis of W multiplied by ω represents the Frank vector.²⁶⁾ The angle ω represents the power of the disclination and is obtained by eq. (10). Since the Frank vector is parallel to e_x regardless of s₁ and s_m, the disclinations are wedge type.^23,27,28) Figure 2(c) shows the s₁ and s_m dependence of |ω₊| and |ω₋|. k = +1 solution is plotted in orange and k = −1 solution in blue. |ω₊| and |ω₋| increase for increasing s_m and s₁. s₁ ≠ 0 and s_m ≠ 0; the disclinations are always formed if the shear bands are connected.

Fig. 2

Connection of two shear bands. (a) A rotational gap of ω appearing at the interface between the shear bands. (b) A pair of disclination formed at the edge of the junction plane to fill the gap. (c) s₁ and s_m dependence of the power of the disclination. The disclinations are always formed if the shear bands are connected.

Figure 3(a) is an overlay of kink interface on Fig. 2(b) to show the penetration of the kink interface by the shear bands. Since the junction plane of the shear bands in current configuration ($\hat{\mathbf{m}}_{\text{k}}$) makes an angle δ with $\hat{\mathbf{n}}_{1}$, the kink interface is interrupted at the red dotted line in Fig. 3(a) and the continuity of deformation is broken. For the kink interface to be connected without the discontinuity, the pre-existing kink interface below the shear band must migrate as indicated by the red arrow in Fig. 3(a). This migration of the kink interface is called “cooperative deformation” hereafter. The angle δ is given as follow using $\hat{\mathbf{m}}_{\text{k}}$ and $\hat{\mathbf{n}}_{1}$.

  

\begin{equation} \delta = \cos^{-1} \left(\frac{\hat{\mathbf{n}}_{1} \cdot \hat{\mathbf{m}}_{\text{k}}}{|\hat{\mathbf{n}}_{1}| |\hat{\mathbf{m}}_{\text{k}}|}\right) \end{equation} (12)

Since the length of the intersection of the kink band and shear band (d/cos ϕ) shown in Fig. 1 varies with angle δ, the distance of kink interface migration (x) due to the cooperative deformation is given by

  

\begin{equation} \text{x} = \frac{d\tan\delta}{\cos \phi}, \end{equation} (13)

where d is a variable and ϕ is uniquely determined by s₁. The only term that depends on s_m and s₁ is, therefore, δ. The exact position of the plane $\hat{\mathbf{m}}_{\text{k}}$ is not known in this analysis because the elastic field by the disclinations are not treated. $\hat{\mathbf{m}}_{\text{k}}$ is, however, midway of $\mathbf{S}_{\text{k}}^{ - \text{T}}\hat{\mathbf{m}}$ and $\mathbf{S}_{\text{m}}^{ - \text{T}}\hat{\mathbf{m}}$ when the rotational gap between shear bands is filled by the disclination. Figure 3(b) and (c) show the s_m and s₁ dependence of |δ₊| and |δ₋| for $\mathbf{S}_{\text{k}}^{ - \text{T}}\hat{\mathbf{m}}$ and $\mathbf{S}_{\text{m}}^{ - \text{T}}\hat{\mathbf{m}}$, respectively. For both cases of $\mathbf{S}_{\text{k}}^{ - \text{T}}\hat{\mathbf{m}}$ and $\mathbf{S}_{\text{m}}^{ - \text{T}}\hat{\mathbf{m}}$, |δ₊| and |δ₋| approaches 45° when s_m → 0, deviates from 45° gradually and monotonically for increasing s_m and s₁.

Fig. 3

Penetration of kink interface by shear bands. (a) The lower half of the kink interface must migrate to keep the continuity of deformation (cooperative deformation). s_m and s₁ dependence of |δ₊| and |δ₋| for (b) $\mathbf{S}_{\text{k}}^{ - \text{T}}\hat{\mathbf{m}}$ and (c) $\mathbf{S}_{\text{m}}^{ - \text{T}}\hat{\mathbf{m}}$. |δ₊| and |δ₋| approach 45° when s_m → 0, deviates from 45° gradually and monotonically for increasing s_m and s₁.

4. Discussion

It has been shown that the disclination and the cooperative deformation emerge at the intersection of kink interface and shear band for any condition. The work equivalent to the elastic energy of the disclinations must be supplied externally to penetrate the kink interface. The work required for the cooperative deformation must be supplied by the external load. These extra-works due to the disclination and the cooperative deformation are necessary so that contribute to the strengthening of the material.

Figure 4 shows a schematic of the sheared kink band with s_m = 0.002 (0.2%), s₁ = 0.3, corresponding to θ = 17° which is often observed in experiments,²⁹⁾ ϕ = 8.5308°, and $\hat{\mathbf{m}}_{\text{k}} = \mathbf{S}_{\text{k}}^{ - \text{T}}\hat{\mathbf{m}}_{ + }$. |ω₊| = 0.0336° and |δ₊| = 45.012° for this situation. The dashed lines in gray inside the shear band schematically show the layer planes in each shear band. The region where the kink interface migrated by the cooperative deformation are shown in dark gray.

Fig. 4

Schematic of penetrated kink interface when s_m = 0.002, s₁ = 0.3, $\hat{\mathbf{m}}_{\text{k}} = \mathbf{S}_{\text{k}}^{ - \text{T}}\hat{\mathbf{m}}_{ + }$. ⟨c⟩-dislocation is required for the penetration and the cooperative deformation to occur.

The elastic energy of the disclination is proportional to the square of ω.²⁷⁾ The power of disclination increases for increasing s_m and s₁ so that the extra-work for the penetration of the kink interface is higher as deformation progresses and as the kink angle θ increases. This is consistent with the experimental results that the hardness of LPSO-Mg alloy increases for increasing θ.¹⁹⁾

The plastic work required for the cooperative deformation (P_c) is obtained by multiplying the volume swept by the migrating kink interface (V_m) and the stress for migration of the kink interface (τ_m) as

  

\begin{equation} \text{P}_{\text{c}} = \text{x}\tau_{\text{m}} = \frac{d\tan\delta}{\cos \phi}\tau_{\text{m}}, \end{equation} (14)

for unit area of the migrating kink interface. As seen in Fig. 3(b) and (c), δ is almost 45° for very small s_m and is 45° ± 1° for s_m = 0.1 so that we approximate P_c ∼ dτ_m/cos ϕ. P_c is higher for higher d, higher tilt angle of the basal plane at the kink interface ($\because \theta = 2\phi $) and higher τ_m. Similar result is obtained when $\hat{\mathbf{m}}_{\text{k}} = \mathbf{S}_{\text{m}}^{ - \text{T}}\hat{\mathbf{m}}_{ + }$ is used or another k is used for eq. (8) and (9).

τ_m depends on the structure and material property and can be higher than the yield stress of the easy basal slip in a layered structure as discussed below. Since θ = 2ϕ, the basal plane is symmetrically tilted at the kink interface as if it is a symmetric tilt grain boundary.^30,31) However, the symmetricity is broken at the intersected segment ($\hat{\mathbf{m}}_{\text{k}}$) and the introduction of non-basal dislocations (⟨c⟩-dislocations in LPSO-Mg alloy) is required. This non-basal dislocation stems from the mismatch in the number of basal planes across the intersected segment as demonstrated in Fig. 4. The thickness of the shear band penetrating the kink band is d_k = d cos(δ + ϕ)/cos(δ − ϕ). Since d_k and d are not equal, the number of layer planes in each shear band does not match and ⟨c⟩-dislocation is necessary as shown in Fig. 4. In other words, the hard layer must be cut for the introduction of the ⟨c⟩-dislocation. In LPSO-Mg alloy, ⟨c⟩-dislocation requires higher stress compared to the basal slip so that the stress for migration of kink interface becomes higher than that of critical resolved shear stress of basal slip. In fact, DFT calculation have shown that the stress as high as 700 MPa are required for basal dislocation to passage through the kink interface and migrate the kink interface in the LPSO phase.³²⁾ Given that the amount of kink interface migration is almost constant independent of s_m, P_c contributes to the kink strengthening from the early stage of plastic deformation. It is suggested that when the Peierls stress of ⟨c⟩-dislocation is much higher than that of basal slip, the presence of kink interface leads to a dramatic increase in the stress for plastic deformation.

5. Conclusions

Disclinations are always formed at the intersection of kink interface and incident shear band, and the migration of kink interface is required (cooperative deformation). The disclination and the cooperative deformation always increase the stress for deformation, regardless of how small the penetrating slip is. In layered structures, there is a mismatch in the number of layers at the intersection of the kink interface and the incident shear band, which necessitates cutting of the layers. It is suggested that when the Peierls stress of ⟨c⟩-dislocation is much higher than that of basal slip (easy slip), the presence of kink interface leads to a dramatic increase in the stress for deformation.

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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