Analysis of Inhomogeneous Crystal Rotation in a Grain after Plastic Deformation of a Polycrystalline Low Carbon Steel

Abstract

In the case of metals, it is considered that the crystal grain refinement due to plastic deformation is caused by the operation of various slip systems in a grain. In order to understand the initial stage of the grain subdivision, a simple shear test was performed on a low-carbon-steel polycrystal having an average grain size of approximately 150 μm.

After the application of 10% shear, several regions having different slip lines were observed in a grain. The slip lines were explained by $\{110\} \langle 111 \rangle$ slip-system of body-centered cubic iron, and the operated Burgers vectors were estimated. The crystal orientations were different for the subdivided regions, and misorientation angles of the crystal rotation were approximately 10°. The rotation axes of the crystal rotation across the subdivided regions were determined. The rotation axes were almost perpendicular to either of the estimated Burgers vectors in the subdivided regions. The directions of the rotation axes are discussed.

1. Introduction

Grain refinement is an effective strengthening mechanism for combining high strength with good toughness. In recent years, ultrafine grain metals having grain sizes of 1 μm or less have been produced by severe plastic deformation^1–3). It is known that grain refinement occurs due to crystal orientation change within a grain during plastic deformation. The microstructural evolution in metals progresses by grain subdivision from coarse grains to fine grains caused by activation of different slip systems within a grain^4,5). As a result, the regions may rotate towards a stable orientation with introducing geometrically necessary dislocations (GNDs)⁶⁾ to accommodate the incompatibility of different active-slip-system regions.

Crystal rotation due to plastic deformation is essentially related to the existence of GNDs in the local volume. A graphical illustration of the lattice curvature caused by GNDs has been presented by Ashby⁶⁾ based on the continuum dislocation theory that gives the relation between the lattice curvature and distribution of GNDs within a certain volume⁷⁾. The first application of this framework was performed by Sun et al.⁸⁾ to quantify GND densities using electron back-scatter diffraction (EBSD). More recently, in order to determine all the components of the second-order Nye's dislocation tensor, the three-dimensional EBSD technique was employed⁹⁾. Normally, such experimental works are performed using bicrystals or tricrystals focusing on the crystal rotation neighboring existence grain boundaries.

In this study, to investigate the initial stage of grain subdivision during plastic deformation, we analyzed the crystal orientation change across the subdivided regions in a grain of a low-carbon-steel polycrystal. The crystal orientations and slip systems were estimated from observations of the slip line and EBSD measurements on the surface after plastic deformation. Using the logarithm of the rotation matrix^10–12), the axes of the crystal rotation caused by the plastic deformation are divided into the rotation about the coordinate axes of the crystal system. The changes of crystal rotation axes across the subdivided regions are discussed.

2. Experimental Procedure

The chemical composition of the polycrystalline low-carbon steel is shown in Table 1. The polycrystalline low-carbon steel was prepared by vacuum melting followed by hot rolling and acid pickling. The hot rolled sheets were cold rolled with a reduction of 60%. In order to obtain relatively coarse and equiaxed recrystallized grains, the cold rolled sheets were heat treated at 973 K for 100 s followed by die quenching to room temperature. They were perfectly recrystallized using heat treatment, and the average grain size was approximately 150 μm.

Table 1 Chemical composition of the test material, low-carbon steel.

Compositions (mass%)
C	Si	Mn	P	S
0.0016	<0.01	<0.01	0.0003	0.0002

A planar simple shear test technique¹³⁾ was employed for applying plastic deformation. Figure 1 shows the geometry of the simple shear test specimen with coordinate system, the observation direction (OD//$x_{1}$-axis), shear direction (SD//$x_{2}$-axis), and shear plane normal (SN//$x_{3}$-axis). In the simple shear test frame, the components of the deformation gradient tensor ${\bf F}$ by applying a simple shear strain are given by   

\[ {\bf F} = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & \gamma \\ 0 & 0 & 1 \end{array} \right), \](1)

where $\gamma$ is amount of shear calculated by $\gamma = \delta/h$. $\delta$ and $h$ are simple shear displacement and gauge length (3 mm) respectively. After a 10% shear, the slip-line directions and crystal orientations were observed using a field emission scanning electron microscope (FE-SEM) and EBSD respectively. To obtain these observations, the specimens were mechanically polished followed by electrical polishing to eliminate the deformation layer introduced by the mechanical polishing. The EBSD observations were obtained at a step size of 0.2 μm.

Fig. 1

Schematic representation of a simple-shear-test specimen. The coordinate frame of the simple shear test is also indicated.

3. Results

3.1 Slip-line observation

Figures 2(a) and (b) show the FE-SEM micrograph and inverse pole figure (IPF) map for the same area after the application of a 10% simple shear. The original high-angle grain-boundaries that existed before the simple-shear deformation are identified using the IPF map in Fig. 2(b) and are indicated by white dashed lines in Fig. 2(a). In Fig. 2(a), slip lines are clearly identified on the sample surface. As shown in Fig. 2(a), the central grain in the figure was divided into several regions with parallel slip lines. Figure 3 shows a magnified FE-SEM image of the area of the white solid square in Fig. 2(a). Between the two arrows from (i) to (ii) and (iii) to (iv), there exist transition regions where the slip lines of both sides were observed.

Fig. 2

Microstructural observation of a low-carbon steel after 10% simple-shear deformation. (a) FE-SEM image showing original high-angle grain boundaries before deformation denoted by dashed white lines. A white solid square in this figure indicates the magnified area shown in Fig. 3. (b) Inverse pole figure (IPF) map showing the crystallographic orientation of OD on the OD plane.

Fig. 3

Magnified FE-SEM image of the area indicated in Fig. 2. The arrows heading from the points (i) to (ii) and (iii) to (iv) indicate the lines that the EBSD line-scan profiles have analyzed.

3.2 Analysis of crystal rotation

The objective of this study is to consider the initial stage of the grain subdivision within a grain where the crystal orientation was originally the same. As shown later, the changes of crystal orientation due to a 10% simple-shear deformation are up to approximately 10°. We have analyzed such changes of crystal orientation using the following method.

By using conventional software to treat the EBSD results, we can obtain the Euler angles corresponding to the crystal orientations. However, even if the crystal orientations of two neighboring points are near each other and, for example, the angle between $\langle 100 \rangle$ of them is approximately 10°, the Euler angles may not always cause the result that these $\langle 100 \rangle$ are the same $[100]$. Therefore, in this study, in order to discuss the changes of crystal orientations, we have corrected the Euler angles by considering the symmetry of the cubic structure so that near $\langle 100 \rangle$ can be expressed as the same $[100]$.

Using the corrected Euler angles, we obtained the rotation matrices. The rotation matrix indicating the crystal orientation at each position in the observation region is defined as shown in Fig. 4 with respect to the simple-shear test coordinate frame in Fig. 1. This frame is also the same as that used for the EBSD measurements shown in Fig. 2. Using the reference frame, the additional rotation $\boldsymbol{\Delta {\rm R}}$ from position ${\bf a}_{\bf 1}$ to position ${\bf a}_{\bf 2}$ can be written as   

\[ \boldsymbol{\Delta {\rm R}} = \boldsymbol{{\rm R(a_2)R^{-1}(a_{1})}}, \](2)

where ${\bf R} ({\bf a}_{\bf 1})$ and ${\bf R} ({\bf a}_{2})$ are the rotation matrices that give the crystal orientations at ${\bf a}_{\bf 1}$ and ${\bf a}_{\bf 2}$ respectively, and ${\bf R}^{-1} ({\bf a}_{\bf 1})$ is the inverse matrix of ${\bf R} ({\bf a}_{\bf 1})$.

Fig. 4

Definition of rotation matrices ${\bf R}$ and $\boldsymbol{\Delta {\rm R}}$ showing crystal orientations obtained from the EBSD measurements. ${\bf a}_{\bf 1}$ and ${\bf a}_{\bf 2}$ show the locations at which the EBSD measurements are made. The reference frame used is the simple-shear-test coordinate frame shown in Fig. 1.

The rotation matrix $\boldsymbol{\Delta {\rm R}}$ can be expressed by using a rotation angle $\Phi$ around a unit vector ${\bf n} = (h, k, l)$¹⁴⁾ called the axis/angle pair. The result is shown in Appendix A-1. The logarithm $\ln \boldsymbol{\Delta {\rm R}}$ of $\boldsymbol{\Delta {\rm R}}$^10–12) is a skew symmetric matrix having three independent elements $(\omega_{1}, \omega_{2}, \omega_{3})$. On using the axis/angle pair, $\ln \boldsymbol{\Delta {\rm R}}$ is given by^10–12)   

\[ \ln \boldsymbol{\Delta {\rm R}} = \left( \begin{array}{@{}ccc@{}} 0 & - \omega_{3} & \omega_{2} \\ \omega_{3} & 0 & - \omega_{1} \\ - \omega_{2} & \omega_{1} & 0 \end{array} \right) = \left( \begin{array}{@{}ccc@{}} 0 & - l\Phi & k \Phi \\ l\Phi & 0 & - h\Phi \\ - k \Phi & h\Phi & 0 \end{array} \right), \](3)

The elements $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$ are considered to be the sums of the divided rotation angles around the coordinate axes of the reference frame¹⁰⁾. By using $\ln \boldsymbol{\Delta {\rm R}}$, we can divide a three-dimensional rotation into sets of rotations about the reference frame¹¹⁾. For $\boldsymbol{\Delta {\rm R}}$ with elements of numerical values, $\ln \boldsymbol{\Delta {\rm R}}$ can be calculated using recent mathematical computing programs such as Mathematica 9 or MATLAB.

Figures 5(a) and (b) show the variation of the point-to-origin misorientation angle along the two arrows in Fig. 3 from (i) to (ii) and from (iii) to (iv), respectively. In the line scan, the position ${\bf a}_{\bf 1}$ in the right-hand side of the eq. (2) is set as (i) or (iii), which is the origin of the line scan, and the position ${\bf a}_{\bf 2}$ is set as the analysis point on the line scan. As shown in Fig. 5, the scanned lines can be divided into single-slip-line regions and transition regions of [A]–[C] as shown in Fig. 5(a) and [D]–[F] as shown in Fig. 5(b) by comparing them with the slip line observations shown in Fig. 3. In the figures, the locations of the representative orientations¹⁵⁾ in each region are also indicated by open arrows. The representative orientation is defined as the orientation at a certain point where the sum of the misorientation angles between the certain point and those of other points in the region becomes the minimum¹⁵⁾. Orientation changes along the line scans mainly occurred in the transition regions [B] and [E] in Figs. 5(a) and (b) respectively.

Fig. 5

The variations of the point-to-origin misorientation angles along the arrows shown in Fig. 3. (a) From the origin (i) to (ii) and (b) from the origin (iii) to (iv). Open arrows indicate locations of representative orientations in each region from [A] to [F].

3.3 Operated slip system

In the case of body-centered cubic (bcc) iron, the slip direction is well confirmed as the closest packed $\langle 111 \rangle$ direction, and its Burgers vector is $a/2 \langle 111 \rangle$ where $a$ is a lattice constant. Moreover, there are three postulated possible slip planes $\{110\}$, $\{112\}$ and $\{123\}$ for bcc iron. In order to estimate the slip system for each region, the directions of three possible slip plane traces obtained from the EBSD measurement are compared with the slip lines that were experimentally observed using the FE-SEM in Fig. 3. Figure 6 show the stereographic projections of the representative orientation in regions [A], [C], [D] and [F]. The slip plane traces observed using the FE-SEM are indicated by black solid lines with arrows. In all the cases, the slip planes coincided well with the trace of $\{110\}$. On the $\{110\}$ slip plane, there are two possible $\langle 111 \rangle$ slip directions. By considering the direction of the applied simple shear, we estimated that the operated slip systems in the regions [A] and [F] were $(0\bar{1}1)[\bar{1}11]$, and that in the regions [C] and [D] were $(\bar{1}01) [1\bar{1}1]$ respectively. The majority of the slip systems estimated in this study was the $\{110\} \langle 111 \rangle$ system, and thus, we analyzed only the $\{110\} \langle 111 \rangle$ slip system for the analysis of the crystal rotation axis.

Fig. 6

Stereographic projections of the representative orientations in regions [A], [C], [D] and [F] of Fig. 5. Direction of slip lines observed in Fig. 3 are indicated using black solid lines with solid arrows. Possible slip planes are also indicated using a black broken line for {110}, gray dotted line for {112} and black dotted line for {123}.

3.4 Crystal rotation axis

The rotation axes of the crystal rotation in each region from [A] to [F] shown in Fig. 5(a) and (b) are discussed. The changes of crystal rotation axis in each region along the line scans were analyzed using eqs. (2) and (3). Here, $\boldsymbol{\Delta {\rm R}}$ at a certain point in each region was calculated with ${\bf a}_{\bf 1}$ of the right-hand side of eq. (2) as the position giving the representative orientation in each region. The rotation axes in each region were obtained in this manner. In Fig. 7, the rotation axes of $\boldsymbol{\Delta {\rm R}}$, ${\bf n} = (h, k, l)$, are plotted as small black dots in the stereographic projections showing the representative orientation of each region [A], [B] and [C]. In the figures, the operated slip planes are indicated using a broken gray line in region [A] and as a broken black line in region [C]. In addition, the slip plane normal (SPN), Burgers vector (BV), and direction orthogonal to SPN and BV are indicated by a circle, square and triangle respectively. The accuracy of the rotation axis becomes worse as the rotation angle $\Phi$ becomes smaller. Therefore, only the crystal rotation axes satisfying $\Phi > 0.5^{\circ}$ are plotted in Fig. 7. The rotation axes are distributed around $[211]$ in region [A] and around $[121]$ in region [C]. In the case of the transition region [B], the rotation axes are accumulated between those in regions [A] and [C]. Owing to the limitation of $\Phi > 5^{\circ}$ in region [B], the rotation axes are located exactly in the middle between $[211]$ and $[121]$. Another line scan from (iii) to (iv) was analyzed using the same method. Figure 8 shows the distribution of the rotation axes in transition region [E] when $\Phi > 1.5^{\circ}$. As in the case of region [B], the rotation axes were located in the middle between the directions orthogonal to SPN and BV in regions [D] and [F].

Fig. 7

Rotation axes with rotation angles of more than 0.5° are plotted as black dots in the stereographic projections of the representative orientations for regions [A], [B] and [C]. Rotation axes with rotation angles of more than 5° are also plotted for region [B]. ○, □ and △ indicate the slip plane normal (SPN), Burgers vector (BV) and orthogonal to the SPN and BV, respectively. Open gray symbol and solid black symbol correspond to region [A] and [C] respectively. Broken lines indicate slip planes.

Fig. 8

Stereographic projection of the representative orientation in region [E] in Fig. 5 (b). The rotation axes, indicated by black dots, are plotted for the cases in which the misorientation angle is more than 1.5°. Black and gray broken lines indicate the slip planes in regions [D] and [F] respectively.

4. Discussion

In the initial stage of grain subdivision caused by plastic deformation, transition regions were generated by operation of various slip systems in a grain^4,5). As a result, these regions rotate towards different orientations, and the dislocations on the operated slip plane become GNDs around the transition region in order to accommodate the incompatibility of the different orientations. The accumulation of GNDs around the transition region is illustrated in Fig. 9. Analysis of the crystal rotation axis in each region shows that the rotation axes in the transition region were located between the directions orthogonal to the SPN and BV in both sides of the subdivided regions. The GNDs are necessary for accommodating the difference in directions of rotation. Based on Figs. 7 and 8, the generation of the transition region can be explained as the result of the accumulation of GNDs from both sides of the subdivided regions.

Fig. 9

Schematic of a wall of GNDs caused by the motion of dislocations of various slip systems. Cubes labelled [A], [B] and [C] show the variation of crystal orientation along the arrow.

In Figs. 7 and 8, the crystal rotation axes were analyzed by considering ${\bf a}_{\bf 1}$ in Fig. 4 as the location giving the representative orientation in each region. The stereographic projections in Figs. 7 and 8 show the results for various regions in Fig. 5 with reference to the fixed simple-share test frame. However, the crystal orientation and its slip system continuously rotate as a function of the distance along the line scan. To discuss the crystallographic relation between the changes of the crystal rotation axes and their slip systems including the continuous rotation, the crystal coordinate systems corresponding to the crystal orientations at various locations should be considered. When we use the crystal coordinate system at a certain location ${\bf a}_{\bf 1}$ to express the additional rotation from the orientation at ${\bf a}_{\bf 1}$ given by ${\bf R}({\bf a}_{\bf 1})$ to that at ${\bf a}_{2}$ given by ${\bf R}({\bf a}_{2})$, the additional rotation $\boldsymbol{\Delta {\rm R_C}}$ is given by¹²⁾   

\[ \boldsymbol{\Delta {\rm R_C}} = {\bf R}^{\bf -1}{\bf (a_1) R(a_2)}. \](4)

Figure 10 shows the definition of $\boldsymbol{\Delta {\rm R_C}}$ when we consider various crystal orientations expressed by the $x'_{1}$-$x'_{2}$-$x'_{3}$ crystal coordinates at various ${\bf a}_{\bf 1}$. In the following analysis, the $x'_{1}$-$x'_{2}$-$x'_{3}$ coordinate is constructed by the axis orthogonal to BV and SPN (ORD//$x'_{1}$), BV//$x'_{2}$ and SPN//$x'_{3}$ as shown in Fig. 10.

Fig. 10

Schematic of the calculation of the rotation matrices in the region ΔR_c of eq. (4) with respect to the crystal coordinate frame. The crystal coordinate frames at each a₁ are also indicated.

Using eq. (4), we calculated $\boldsymbol{\Delta {\rm R_C}}$ and obtained the changes of crystal rotation axes at various locations along the line scan at 10 μm intervals. Using $\boldsymbol{\Delta {\rm R_C}}$, the crystal rotation axes can be shown in the same crystal coordinate system. Figure 11(a) shows the results along the line scan of (i)–(ii) as a function of the distance from the origin (i). In Fig. 11, subscripts [A] and [C] of ORD, BV and SPN indicated the referenced slip system operated either in region [A] or [C] that is used to determine these axes' directions. Their crystallographic directions are also indicated in the figure. As shown in Fig. 11(a), the referenced crystal coordinate frame is the slip system in region [C] in this case. In the case of the line scan in region [A], the rotation axes are close to a plane that is orthogonal to ${\rm BV}_{[{\rm A}]}$, and the axes then move toward a plane that is orthogonal to ${\rm BV}_{[{\rm C}]}$.

Fig. 11

Analysis of crystal rotation at 10-μm intervals along the line scan from (I) to (II) in Fig. 3. (a) Changes of crystal rotation axes shown in the stereographic projection. Gray and black chain lines indicate the planes orthogonal to the Burgers vector for regions [A] and [C] respectively. The crystal coordinate frame is also indicated. (b) Changes of ratio of ω_ORD, ω_BV and ω_SPN to Φ as a function of the distance from (i).

Fig. A-1

Schematic of the arrays of screw dislocations in the volume when $\alpha_{11} = \alpha$ and the other values of $\alpha_{ij} = 0$.

By using the logarithm $\ln \boldsymbol{\Delta {\rm R_C}}$ of $\boldsymbol{\Delta {\rm R_C}}$, the rotation can be divided into a crystal coordinate system. A set of rotation angles, $(\omega_{\rm ORD}, \omega_{\rm BV}, \omega_{\rm SPN})$, component of $\Phi$ around ${\rm ORD}_{[{\rm C}]}$-axis, ${\rm BV}_{[{\rm C}]}$-axis and ${\rm SPN}_{[{\rm C}]}$-axis respectively, can be obtained by using $\ln \boldsymbol{\Delta {\rm R_C}}$. Figure 11(b) shows the changes of the ratio of $(\omega_{\rm ORD}, \omega_{\rm BV}, \omega_{\rm SPN})$ to the total rotation angle Φ as a function of the distance from the origin (i). The ratio of $\omega_{\rm SPN}/\Phi$ is dominant in region [A], then $\omega_{\rm ORD}/\Phi$ becomes the main component of the rotation in regions [B] and [C].

The relation between the dislocation density tensor and lattice curvature tensor⁷⁾ explains that the crystal rotation axis is determined by using the Burgers vector of the GNDs, direction of dislocation line and arrays of those dislocations. The majority of the axes of crystal rotation caused by arrays of dislocation become orthogonal to the Burgers vector. The details are shown in Appendix A-2. The changes of crystal rotation axis shown in Fig. 11 can be interpreted by crystal rotation caused by the arrays of GNDs.

5. Conclusions

In order to understand the initial stage of the grain subdivision, a simple shear test was performed on a low-carbon-steel polycrystal having an average grain size of approximately 150 μm. After the application of 10% shear, several regions having different slip lines were observed in a grain. The slip lines were explained by the slip-system of bcc iron and the operated Burgers vectors were estimated. The crystal orientations were different for the subdivided regions, and misorientation angles of the crystal rotation were approximately 10°. The crystal rotation axes across the various subdivided regions were determined. On using the logarithm of the rotation matrix, the rotation axes became almost perpendicular to either of the estimated Burgers vectors in the subdivided regions. The experimental results can be interpreted by the crystal rotation caused by the arrays of GNDs.

Acknowledgement

This work was supported by JSPS KAKENHI Grant Number 16K06703.

Appendix

A-1: Relation between rotation matrix and axis/angle pair

Nine elements of the rotation matrix $\boldsymbol{\Delta {\rm R}}$ can be expressed using Rodrigues' formula as follows¹⁶⁾,   

\[ \begin{split} \boldsymbol{\Delta {\rm R}} & = \left( \begin{array}{@{}ccc@{}} \Delta R_{11} & \Delta R_{12} & \Delta R_{13} \\ \Delta R_{21} & \Delta R_{22} & \Delta R_{23} \\ \Delta R_{31} & \Delta R_{32} & \Delta R_{33} \end{array} \right)\\ & = \left( \begin{array}{@{}ccc@{}} \begin{array}{@{}l@{}} (1 {-} h^{2}) \cos \Phi\\ \quad + {h^2} \end{array} & \begin{array}{@{}l@{}} hk(1 {-} \cos \Phi)\\ \quad - l \sin \Phi \end{array} & \begin{array}{@{}l@{}} lh(1 {-} \cos \Phi)\\ \quad + k\sin \Phi \end{array} \\ \begin{array}{@{}l@{}} hk(1 {-} \cos \Phi)\\ \quad + l\sin \Phi \end{array} & \begin{array}{@{}l@{}} (1 {-} k^{2}) \cos \Phi\\ \quad + k^{2} \end{array} & \begin{array}{@{}l@{}} kl(1 {-} \cos \Phi)\\ \quad - h\sin \Phi \end{array} \\ \begin{array}{@{}l@{}} lh(1 {-} \cos \Phi)\\ \quad - k\sin \Phi \end{array} & \begin{array}{@{}l@{}} kl(1 {-} \cos \Phi)\\ \quad + h\sin \Phi \end{array} & \begin{array}{@{}l@{}} (1 {-} l^{2}) \cos \Phi\\ \quad + l^{2} \end{array} \end{array} \right). \end{split} \](A1)

From this relation, $\Phi$ and ${\bf n} = (h, k, l)$ are calculated as follows.   

\[ \Phi = \cos^{-1} \left( \frac{\Delta R_{11} + \Delta R_{22} + \Delta R_{33} - 1}{2} \right), \](A2)

and   

\[ (h,\ k,\ l) = \left( \frac{\Delta R_{32} {-} \Delta R_{23}}{2\sin\Phi},\ \frac{\Delta R_{13} {-} \Delta R_{31}}{2\sin\Phi},\ \frac{\Delta R_{21} {-} \Delta R_{12}}{2\sin\Phi} \right). \](A3)

A-2: Crystal rotation axis caused by array of dislocations

The dislocation density tensor, $\boldsymbol{\alpha}_{ij}$, and lattice curvature tensor, $\boldsymbol{\kappa}_{ij}$, can be written as follows⁷⁾.   

\[ \boldsymbol{\alpha}_{ij} = \boldsymbol{\kappa}_{ji} - \delta_{ij} \boldsymbol{\kappa}_{kk} \quad (i,\ j,\ k = 1,2,3), \](A4)

where $\delta_{ij}$ is the Kronecker delta function ($\delta_{ij} = 1$ for $i=j$, and $\delta_{ij} = 0$ for $i \ne j$). Another expression of $\boldsymbol{\alpha}_{ij}$ can be obtained using the Burgers vector ${\bf b}_{i}$, unit vector parallel to dislocations ${\bf r}_{i}$, and number of dislocation lines per unit area normal to the lines $n$,   

\[ \boldsymbol{\alpha}_{ij} = n{\bf b}_{i}{\bf r}_{j} \quad (i,j = 1,2,3). \](A5)

Furthermore, $\boldsymbol{\kappa}_{ij}$ can be expressed using the lattice rotation, $\phi_{i}$, about the $x_{i}$ axis,   

\[ \boldsymbol{\kappa}_{ij} = \partial \phi_{i}/\partial x_{j} \quad (i,j = 1,2,3). \](A6)

By substituting eqs. (A4)–(A6), the relation between the Burgers vector and crystal rotation axis is given by   

\[ \begin{split} \boldsymbol{\alpha}_{ij} & = \left( \begin{array}{@{}ccc@{}} nb_{1}r_{1} & nb_{1}r_{2} & nb_{1}r_{3} \\ nb_{2}r_{1} & nb_{2}r_{2} & nb_{2}r_{3} \\ nb_{3}r_{1} & nb_{3}r_{2} & nb_{3}r_{3} \end{array} \right)\\ & = \left( \begin{array}{@{}ccc@{}} \displaystyle - \left(\frac{d\phi_{2}}{dx_{2}} + \frac{d\phi_{3}}{dx_{3}} \right) & \displaystyle \frac{d\phi_{2}}{dx_{1}} & \displaystyle \frac{d\phi_{3}}{dx_{1}} \\[2mm] \displaystyle \frac{d\phi_{1}}{dx_{2}} & \displaystyle - \left(\frac{d\phi_{3}}{dx_{3}} + \frac{d\phi_{1}}{dx_{1}} \right) & \displaystyle \frac{d\phi_{1}}{dx_{3}}\\[2mm] \displaystyle \frac{d\phi_{1}}{dx_{3}} & \displaystyle \frac{d\phi_{2}}{dx_{3}} & \displaystyle - \left(\frac{d\phi_{1}}{dx_{1}} + \frac{d\phi_{2}}{dx_{2}} \right) \end{array} \right). \end{split} \](A7)

In case of all of the edge dislocation components, i.e., $i \ne j$, the relation between the Burgers vector ${\bf b}_{i}$ and lattice rotation about the j axis $\phi_{j}$ is orthogonal. However, for the screw dislocation,   

{\[ \begin{split} & \kappa_{ij} {=} \alpha_{ji} - \frac{1}{2} \delta_{ij} \alpha_{kk} = \left( \begin{array}{@{}ccc@{}} \kappa_{11} & \kappa_{12} & \kappa_{13} \\ \kappa_{21} & \kappa_{22} & \kappa_{23} \\ \kappa_{31} & \kappa_{32} & \kappa_{33} \end{array} \right)\\ & {=} \left( \begin{array}{@{}ccc@{}} \displaystyle \frac{1}{2} (\alpha_{11} {-} \alpha_{22} {-} \alpha_{33}) & \alpha_{21} & \alpha_{31}\\ \alpha_{12} & \displaystyle \frac{1}{2} (\alpha_{22} {-} \alpha_{33} {-} \alpha_{11}) & \alpha_{32}\\ \alpha_{13} & \alpha_{23} & \displaystyle \frac{1}{2} (\alpha_{33} {-} \alpha_{11} {-} \alpha _{22}) \end{array} \right). \end{split} \](A8)

An example of an array of parallel screw dislocations in a small volume $(\delta x_{1}, \delta x_{2}, \delta x_{3})$ on assuming $\alpha_{11} = \alpha$ while the other values of $\alpha_{ij} = 0$ is illustrated in Fig. A-1. The lattice curvature tensor in this assumption can be written as   

\[ \left( \begin{array}{ccc} \kappa_{11} & \kappa_{12} & \kappa_{13} \\ \kappa_{21} & \kappa_{22} & \kappa_{23} \\ \kappa_{31} & \kappa_{32} & \kappa_{33} \end{array} \right) = \left( \begin{array}{ccc} \displaystyle \frac{\alpha}{2} & 0 & 0 \\ 0 & \displaystyle - \frac{\alpha}{2} & 0 \\ 0 & 0 & \displaystyle - \frac{\alpha}{2} \end{array} \right). \](A9)

When the line scan direction is parallel to the $x_{1}$-axis, the crystal rotates about the $x_{1}$-axis. However, when the scan direction is parallel to the $x_{2}$-axis and $x_{3}$-axis, the crystal rotation axis becomes the $x_{2}$-axis and $x_{3}$-axis respectively. Consequently, the rotation axis is not orthogonal to the slip direction only when the Burgers vector of the screw dislocation is parallel to the scan direction.

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