Representation of Nye’s Lattice Curvature Tensor by Log Angles

Abstract

The log angles of a rotation matrix are three independent elements of the logarithm of the rotation matrix. Nye’s lattice curvature tensor κ_ij is discussed by using the log angles. For the change in a crystal orientation ΔR with the change in a position Δx_i, it is shown that the elements of κ_ij are written as κ_ij = Δω_i/Δx_j using the log angles Δω_i of ΔR. The log angles for the crystal rotation given by the axis/angle pair are also discussed.

 

This Paper was Originally Published in Japanese in J. Japan Inst. Met. Mater. 82 (2018) 415–418.

Fig. 2 The change in a crystal orientation as much as ΔR with the change in a position from (x₁, x₂, x₃) to (x₁ + Δx₁, x₂ + Δx₂, x₃ + Δx₃). δR is a small-angle rotation which satisfies ΔR ≈ (δR)^N where N is a sufficiently large positive integer. The relationship ΔR ≈ (δR)^N means that the N-times successive rotations of δR with an interval of δx = (Δx₁/N, Δx₂/N, Δx₃/N) is equivalent to ΔR. The angles δϕ_i are the small rotation angles of δR around the x_i axes.

1. Introduction

To obtain an essential understanding of the strength of materials, it is necessary to know development and non-uniformity of plastic deformation in crystals, which are caused by motion and arrangement of dislocations. Since orientations of crystals change as a function of position when dislocation structures are formed in the crystals, efforts have been made to show dislocation structures after plastic deformation by considering such orientation changes. Pioneering work of the efforts is the theoretical study by Nye.¹⁾ Nye has shown that the dislocation density in a crystal can be quantitatively treated when orientation changes in the crystal are evaluated by using the lattice curvature tensor $\boldsymbol{{\kappa}}$. Using the x₁ − x₂ − x₃ orthogonal coordinate system, this lattice curvature tensor $\boldsymbol{{\kappa}}$ describes the orientation change δϕ_i around the x_i axis with the change in a position δx_j in the crystal as κ_ij = δϕ_i/δx_j.

As shown by Nye,¹⁾ the theoretical framework to discuss dislocation structures using orientation changes in crystal grains have been presented about half a century ago. On the other hand, experimental measurements of changes in crystal orientations for a wide area, with a high accuracy and in a short time have been performed recently only after a rapid development of SEM/EBSD (scanning electron microscopy/electron back scattering diffraction) method. However, the procedure to determine values of κ_ij = δϕ_i/δx_j from data obtained by SEM/EBSD method has not been shown clearly in previous studies. When the orientation change in a crystal is given by the rotation angle ΔΦ around the unit vector n_i, Pantleon²⁾ and He et al.³⁾ have shown that the product ΔΦn_i of n_i and ΔΦ is also a vector and the components ΔΦn_i are the components of the rotation angle around the x_i axes. However, it should be noted that, in successive rotations around different axes such as the Euler angles, resultant rotations generally depend on the order of the rotations even if the rotation angle around each axis is the same.⁴⁾ This means that it is not obvious that the components of ΔΦn_i are the components of rotation around the x_i axes, but the reason is not shown in their papers.^2,3) It is hence significant to show the reason if the components of ΔΦn_i can be treated as the components of the rotation angle around the x_i axes.

A crystal orientation can be described by using a rotation matrix R with respect to a reference coordinate system. The matrix R is a 3 × 3 orthogonal matrix whose determinant is 1. The logarithm ln R of R is a 3 × 3 skew-symmetric matrix with three independent elements of real numbers.^4–7) The logarithm ln R of R has been discussed in previous studies to discuss crystal orientations.^4,6–10) The three elements of ln R are called the log angles ω_i by Hayashi et al.⁴⁾ and Onaka and Hayashi,^9,10) which are utilized as the characteristic angles of R. They have discussed geometrical meanings of the log angles ω_i and applied these to the analysis of crystal rotations.^4,9,10)

When a crystal orientation changes as much as ΔR with the change in a position Δx_i, the elements of the lattice curvature tensor $\boldsymbol{{\kappa}}$ are written as κ_ij = Δω_i/Δx_j using the log angles Δω_i of ΔR. We will show this in the present paper. This knowledge enables us to understand the procedure to determine the lattice curvature tensor $\boldsymbol{{\kappa}}$ from data of crystal orientations as those given by the SEM/EBSD method. Measurements of the lattice curvature tensor $\boldsymbol{{\kappa}}$ for plastically deformed metals and evaluation of densities or structures of dislocations from an experimental point of view will be shown in our future work based on the present study.

2. Log Angles

2.1 Logarithmic function

Here we explain characteristics of the logarithmic function before summarizing the log angles. For a real number x, the relationship between x and the exponential function exp x is well-known as¹¹⁾   

\begin{equation} \exp x = 1 + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \ldots = \lim_{p \to \infty}\left(1 + \frac{x}{p}\right)^{p}. \end{equation} (1)

Using y = exp x (y > 0), this relationship is rewritten by the logarithmic function as   

\begin{equation} y = \lim\limits_{p \to \infty}\left(1 + \frac{\ln y}{p}\right)^{p}. \end{equation} (2)

When N is a sufficiently large positive integer, we have   

\begin{equation} y \approx \left(1 + \frac{\ln y}{N}\right)^{N}. \end{equation} (3)

This equation gives us better understanding on variables describing various changes.

Figure 1 shows the schematic illustration of deformation of a rod-like material along the longitudinal direction. The upper figure shows the state before deformation with the initial length L₀ and the state after uniform deformation with the length L = L₀ + ΔL is shown in the lower one. The nominal strain e is defined as e = ΔL/L₀ and the definition of the stretch λ showing the length ratio before and after deformation is   

\begin{equation} \lambda = L/L_{0} = (L_{0} + \Delta L)/L_{0}. \end{equation} (4)

This λ of λ > 0 is written as λ = 1 + e.

Fig. 1

Schematic illustration showing one-dimensional deformation of a bar. The lengths L₀ and L = L₀ + ΔL are those before and after the deformation.

From (3), we have   

\begin{equation} \lambda\approx \left(1 + \frac{\ln\lambda}{N}\right)^{N} \end{equation} (5a)

and this equation is rewritten from (4) as   

\begin{equation} L \approx L_{0}\left(1 + \frac{\ln\lambda}{N}\right)^{N}. \end{equation} (5b)

Since the logarithmic strain ε is   

\begin{equation} \varepsilon = \ln(1 + e) = \ln\lambda, \end{equation} (6)

using a sufficiently large integer N, from (6) and (5b), the relationship among L₀, L and ε is given by   

\begin{equation} L \approx L_{0}\left(1 + \frac{\varepsilon}{N}\right)^{N}. \end{equation} (7)

This eq. (7) shows a characteristic of the logarithmic strain ε as an appropriate measure to describe large deformation. Even for large deformation where |ε| ≪ 1 is not satisfied, if we divide the large deformation into small deformation of N-times, we can reasonably treat the large deformation as an extension of the concept of infinitesimal deformation. That is to say, ε = (ε/N) × N and this ε is understood as the sum of N-times repeated small strain ε/N. This is an interpretation of the logarithmic strain ε given by (7). A similar interpretation can be put on (5a): the stretch λ is the quantity describing the deformation as it occurred, but its logarithm ln λ is a measure to show the extent of the deformation.

2.2 Outline of log angles

The relationship between the 3 × 3 orthogonal matrix R and its logarithm ln R is written as¹²⁾   

\begin{equation} \mathbf{R} = \lim\limits_{p \to \infty}\left(\mathbf{E} + \frac{\ln\mathbf{R}}{p}\right)^{p}, \end{equation} (8)

where E is the 3 × 3 unit matrix. This is given from (2) by expanding the concept of numbers to matrices. The logarithm ln R is the skew-symmetric matrix with elements of real numbers.^5–7,12) The log angles ω_i of R are the elements of ln R and are written as^4,9,10)   

\begin{equation} \ln\mathbf{R} = \begin{pmatrix} 0 & -\omega_{3} & \omega_{2}\\ \omega_{3} & 0 & -\omega_{1}\\ -\omega_{2} & \omega_{1} & 0 \end{pmatrix} . \end{equation} (9)

A calculation method to obtain ln R from R will be shown later in this paper.

As well as the discussion made in 2.1, using a sufficiently large positive integer N, from (8) we have   

\begin{equation} \mathbf{R} \approx \left(\mathbf{E} + \frac{\ln\mathbf{R}}{N}\right)^{N}. \end{equation} (10)

This equation can be rewritten from (9) as   

\begin{equation} \mathbf{R} \approx (\delta\mathbf{R})^{N}, \end{equation} (11a)

where   

\begin{equation} \delta\mathbf{R} = \mathbf{E} + \left(\frac{\ln\mathbf{R}}{N}\right) = \begin{pmatrix} 1 & -(\omega_{3}/N) & (\omega_{2}/N)\\ (\omega_{3}/N) & 1 & -(\omega_{1}/N)\\ -(\omega_{2}/N) & (\omega_{1}/N) & 1 \end{pmatrix} . \end{equation} (11b)

From (11a) and (11b), the meanings of the log angles ω_i are understood as well as the case of the logarithmic strain ε. The third side of (11b) shows that δR is a small-angle rotation matrix since |ω_i/N| ≪ 1 is satisfied for the sufficiently large N.⁴⁾ This δR is interpreted as the result of three successive rotations composed of the angle ω₁/N around the x₁ axis, the angle ω₂/N around the x₂ axis and the angle ω₃/N around the x₃ axis.⁴⁾ Since the rotation angles around the axes are small, any orders of the successive rotations give the same result given by the third side of (11b) by neglecting the second and higher terms of the elements.⁴⁾ This shows that the log angles ω_i are treated as the characteristic angles and the components of R in the sense that the angles ω_i are the sum of the divided rotation angles around the x_i axes.⁴⁾

3. Change of Orientation with Change of Position in Crystal

As shown in Fig. 2, we assume that a crystal orientation changes as much as ΔR with the change of a position from x_i to x_i + Δx_i. Moreover we assume that the change ΔR is written as ΔR ≈ (δR)^N and ΔR is given by the N-time successive rotations of δR. This treatment means that the change of orientation between the positions x_i and x_i + Δx_i is assumed to be uniform and δR corresponds to the orientation change with the position change as much as δx_i = Δx_i/N between x_i and x_i + Δx_i.

Fig. 2

The change in a crystal orientation as much as ΔR with the change in a position from (x₁, x₂, x₃) to (x₁ + Δx₁, x₂ + Δx₂, x₃ + Δx₃). δR is a small-angle rotation which satisfies ΔR ≈ (δR)^N where N is a sufficiently large positive integer. The relationship ΔR ≈ (δR)^N means that the N-times successive rotations of δR with an interval of δx = (Δx₁/N, Δx₂/N, Δx₃/N) is equivalent to ΔR. The angles δϕ_i are the small rotation angles of δR around the x_i axes.

When the small-angle rotation δR is composed of the rotation angles δϕ_i around the x_i axes as shown in Fig. 2, using the log angles Δω_i of ΔR, from (9) and (11b) we have   

\begin{equation} \delta\phi_{i} = \Delta\omega_{i}/N. \end{equation} (12)

Hence the relationship between the changes δϕ_i and δx_i is given by   

\begin{equation} \delta\phi_{i}/\delta x_{j} = \Delta\omega_{i}/\Delta x_{j}. \end{equation} (13)

This relationship means that using Δω_i of ΔR with the position change from x_i to x_i + Δx_i, the average lattice curvature tensor $\boldsymbol{{\kappa}}$ for the region is given by   

\begin{equation} \kappa_{ij} = \Delta\omega_{i}/\Delta x_{j}, \end{equation} (14a)

which is rewritten as   

\begin{equation} \boldsymbol{{\kappa}} = \begin{pmatrix} \Delta\omega_{1}/\Delta x_{1} & \Delta\omega_{1}/\Delta x_{2} & \Delta\omega_{1}/\Delta x_{3}\\ \Delta\omega_{2}/\Delta x_{1} & \Delta\omega_{2}/\Delta x_{2} & \Delta\omega_{2}/\Delta x_{3}\\ \Delta\omega_{3}/\Delta x_{1} & \Delta \omega_{3}/\Delta x_{2} & \Delta\omega_{3}/\Delta x_{3} \end{pmatrix} . \end{equation} (14b)

4. Log Angles for Rotation Matrix Given by the Axis/Angle Pair

The matrix R for the rotation given by the rotation angle Φ (0 ≤ Φ ≤ π) around the unit vector n_i is written as^9,13)   

\begin{equation} \mathbf{R} = \begin{pmatrix} (1 - n_{1}{}^{2})\cos\Phi + n_{1}{}^{2} & n_{1}n_{2}(1 - \cos\Phi) - n_{3}\sin\Phi & n_{3}n_{1}(1 - \cos\Phi) + n_{2}\sin\Phi\\ n_{1}n_{2}(1 - \cos\Phi) + n_{3}\sin\Phi & (1 - n_{2}{}^{2})\cos\Phi + n_{2}{}^{2} & n_{2}n_{3}(1 - \cos\Phi) - n_{1}\sin\Phi\\ n_{3}n_{1}(1 - \cos\Phi) - n_{2}\sin\Phi & n_{2}n_{3}(1 - \cos\Phi) + n_{1}\sin\Phi & (1 - n_{3}{}^{2})\cos\Phi + n_{3}{}^{2} \end{pmatrix} . \end{equation} (15)

This is the representation of R by using the axis/angle pair, n_i and Φ. When the rotation angle is small and satisfies |δΦ| ≪ 1, we have   

\begin{equation*} \sin\delta\Phi \approx \delta\Phi\ \text{and}\ \cos\delta\Phi \approx 1. \end{equation*}

Using these approximations, the rotation matrix δR of δΦ around n_i is written as   

\begin{equation} \delta\mathbf{R} \approx \begin{pmatrix} 1 & -\delta\Phi n_{3} & \delta\Phi n_{2}\\ \delta\Phi n_{3} & 1 & -\delta\Phi n_{1}\\ -\delta\Phi n_{2} & \delta\Phi n_{1} & 1 \end{pmatrix} . \end{equation} (16)

When the relationship between Φ and δΦ is given by   

\begin{equation} \delta\Phi = \Phi/N, \end{equation} (17)

R of Φ around n_i is the same with the N-times successive rotations of δR. Then, using the relationship R ≈ (δR)^N and (9), (11b) and (16), we have the logarithm ln R of R given by (15) is written as^5,6)   

\begin{equation} \ln \mathbf{R} = \begin{pmatrix} 0 & -\Phi n_{3} & \Phi n_{2}\\ \Phi n_{3} & 0 & -\Phi n_{1}\\ -\Phi n_{2} & \Phi n_{1} & 0 \end{pmatrix} . \end{equation} (18)

This shows that the log angles ω_i of R are   

\begin{equation} \omega_{i} = \Phi n_{i}. \end{equation} (19)

Equation (18) is satisfied even if the rotation angle Φ of R is not small.

The above discussion shows that the log angles ω_i of R with Φ around n_i are given by Φn_i. Hence, as shown in the papers by Pantleon²⁾ and He et al.,³⁾ when the rotation angle is ΔΦ, the product ΔΦn_i can be treated as the components of the rotation angles around the x_i axes in the sense that these are the sum of the divided rotation angles around the axes.

From (15) and (18), the relationship between the rotation matrix R and its logarithm ln R is given by   

\begin{equation} \ln\mathbf{R} = \frac{\Phi}{2\sin\Phi}(\mathbf{R} - {}^{t}\mathbf{R}), \end{equation} (20)

where ^tR is the transposed matrix of R. The proof of (2) is shown in Ref. 14). The angle Φ is given by the tarce Tr R of R as¹³⁾   

\begin{equation} \cos\Phi = (\text{Tr}\mathbf{R} - 1)/2. \end{equation} (21)

The logarithm of matrix is generally obtained by the procedure starting from the diagonalization of the matrix. Recent computation programs such as Mathematica have a command to calculate the logarithm of matrix. However, for the logarithm ln R of the rotation matrix R, (20) and (21) are convenient to calculate the values of ln R.

5. Conclusions

The log angles ω_i of a rotation matrix R are three elements of the logarithm ln R of R. Using the log angles ω_i, we have discussed the operation to obtain the lattice curvature tensor $\boldsymbol{{\kappa}}$ from position-dependent changes of crystal orientations in a grain. When the crystal orientation changes as much as ΔR with the change in the position Δx_i, using the log angles Δω_i of ΔR, the elements of the lattice curvature tensor changes $\boldsymbol{{\kappa}}$ are given by κ_ij = Δω_i/Δx_j. When crystal-orientation change is given by the rotation angle ΔΦ around the unit vector n_i, the log angles Δω_i of this orientation change satisfy Δω_i = ΔΦn_i. Both of Δω_i and ΔΦn_i are considered to be the components of the rotation angle around the x_i axes.

Acknowledgment

Funding from the JSPS KAKENHI (Grant Number JP16K06703) is gratefully acknowledged.

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