A New Proof That the Number of Linear Elastic Symmetries in Two Dimensions Is Four

Abstract

We present an elementary and self-contained proof that there are exactly four symmetry classes of the elasticity tensor in two dimensions: oblique, rectangular, square, and isotropic. In two dimensions, orthogonal transformations are either reflections or rotations. The proof is based on identification of constraints imposed by reflections and rotations on the elasticity tensor, and it simply employs elementary tools from trigonometry, making the proof accessible to a broad audience. For completeness, we identify the sets of transformations (rotations and reflections) for each symmetry class and report the corresponding equations of motions in classical linear elasticity.

Appendices

Appendix A: Equations of Motion in Two-Dimensional Classical Linear Elasticity

In this section, we consider the equation of motion (8) for representative members of each of the four symmetry classes. The corresponding elasticity tensors are given in Theorem 1.

Oblique: There are no restrictions on the elasticity tensor ℂ under oblique symmetry and thus the oblique equation of motion (in component form) is given by (8), which we write out explicitly:

$$\begin{aligned} \begin{aligned} \rho \ddot{u}_{1} ={}& C_{1111} \frac{\partial ^{2} u_{1}}{\partial x^{2}} +2C_{1112} \frac{\partial ^{2} u_{1}}{\partial x \partial y} + C_{1212} \frac{\partial ^{2} u_{1}}{\partial y^{2}} + C_{1112} \frac{\partial ^{2} u_{2}}{\partial x^{2}} \\ &+ \left ( C_{1122} + C_{1212} \right ) \frac{\partial ^{2} u_{2}}{\partial x \partial y} + C_{2212} \frac{\partial ^{2} u_{2}}{\partial y^{2}} + b_{1}, \end{aligned} \end{aligned}$$

(46a)

$$\begin{aligned} \begin{aligned} \rho \ddot{u}_{2} ={}& C_{1112} \frac{\partial ^{2} u_{1}}{\partial x^{2}} + \left ( C_{1122} + C_{1212} \right ) \frac{\partial ^{2} u_{1}}{\partial x \partial y} + C_{2212} \frac{\partial ^{2} u_{1}}{\partial y^{2}} + C_{1212} \frac{\partial ^{2} u_{2}}{\partial x^{2}} \\ &+ 2 C_{2212} \frac{\partial ^{2} u_{2}}{\partial x \partial y} + C_{2222} \frac{\partial ^{2} u_{2}}{\partial y^{2}} + b_{2}. \end{aligned} \end{aligned}$$

(46b)

Rectangular: Imposing the rectangular symmetry restrictions (20) on (46a) and (46b) produces the rectangular equation of motion (in component form):

$$\begin{aligned} \rho \ddot{u}_{1} ={}& C_{1111} \frac{\partial ^{2} u_{1}}{\partial x^{2}} + C_{1212} \frac{\partial ^{2} u_{1}}{\partial y^{2}} + \left ( C_{1122} + C_{1212} \right ) \frac{\partial ^{2} u_{2}}{\partial x \partial y} +b_{1}, \end{aligned}$$

(47a)

$$\begin{aligned} \rho \ddot{u}_{2} ={}& \left ( C_{1122} + C_{1212} \right ) \frac{\partial ^{2} u_{1}}{\partial x \partial y} + C_{1212} \frac{\partial ^{2} u_{2}}{\partial x^{2}} + C_{2222} \frac{\partial ^{2} u_{2}}{\partial y^{2}} + b_{2}. \end{aligned}$$

(47b)

Square: Imposing the square symmetry restrictions (27) on (46a) and (46b) produces the square equation of motion (in component form):

$$\begin{aligned} \rho \ddot{u}_{1} ={}& C_{1111} \frac{\partial ^{2} u_{1}}{\partial x^{2}} + C_{1212} \frac{\partial ^{2} u_{1}}{\partial y^{2}} + \left ( C_{1122} + C_{1212} \right ) \frac{\partial ^{2} u_{2}}{\partial x \partial y} + b_{1}, \end{aligned}$$

(48a)

$$\begin{aligned} \rho \ddot{u}_{2} ={}& \left ( C_{1122} + C_{1212} \right ) \frac{\partial ^{2} u_{1}}{\partial x \partial y} + C_{1212} \frac{\partial ^{2} u_{2}}{\partial x^{2}} + C_{1111} \frac{\partial ^{2} u_{2}}{\partial y^{2}} + b_{2}. \end{aligned}$$

(48b)

Isotropic: Imposing the isotropic symmetry restrictions (31) on (46a) and (46b) produces the isotropic equation of motion (in component form):

$$\begin{aligned} \rho \ddot{u}_{1} ={}& C_{1111} \frac{\partial ^{2} u_{1}}{\partial x^{2}} + \frac{1}{2} \left ( C_{1111} - C_{1122} \right ) \frac{\partial ^{2} u_{1}}{\partial y^{2}} + \frac{1}{2} \left ( C_{1111} + C_{1122} \right ) \frac{\partial ^{2} u_{2}}{\partial x \partial y} + b_{1}, \end{aligned}$$

(49a)

$$\begin{aligned} \rho \ddot{u}_{2} ={}& \frac{1}{2} \left ( C_{1111} + C_{1122} \right ) \frac{\partial ^{2} u_{1}}{\partial x \partial y} + \frac{1}{2} \left ( C_{1111} - C_{1122} \right ) \frac{\partial ^{2} u_{2}}{\partial x^{2}} + C_{1111} \frac{\partial ^{2} u_{2}}{\partial y^{2}} + b_{2}. \end{aligned}$$

(49b)

Appendix B: Linearization of the Components of the Elasticity Tensor Under Reflections and Rotations

In this section, we linearize the expressions of the transformed components of the elasticity tensor in order to identify interesting properties of the two-dimensional elasticity tensor. Under the transformation $\mathbf{Q} = \mathbf{Ref}(\alpha )$, the components of the elasticity tensor ℂ are (see (11))

$$\begin{aligned} C_{1111}' ={}& \left ( {\frac {C_{{1111}}}{8}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{\frac { C_{{2222}}}{8}} \right ) \cos ( 8\alpha ) + \left ( {\frac {C_{{1112}}}{2}} -{\frac {C_{{2212}}}{2}} \right ) \sin ( 8\alpha ) \\ &+ \left ( {\frac {C_{{1111}}}{2}}-{\frac {C_{ {2222}}}{2}} \right ) \cos ( 4\alpha ) + \left ( C_{{1112}} + C_{{2212}} \right ) \sin ( 4\alpha ) \\ &+ \left ({\frac {3 C_{{ 1111}}}{8}}+{\frac {C_{{1122}}}{4}}+{\frac {C_{{1212}}}{2}}+{\frac {3 C_{{2222}}}{8}}\right ), \\ C_{1112}' ={}& -\left ( {{\frac {C_{{1112}}}{2}} - \frac {C_{{2212}}}{2}} \right ) \cos ( 8 \alpha ) + \left ( {\frac {C_{{1111}}}{8}}-{\frac {C_{ {1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{\frac {C_{{2222}}}{8}} \right ) \sin ( 8\alpha ) \\ &- \left ( { \frac {C_{{1112}}}{2}} + {\frac {C_{{2212}}}{2}} \right ) \cos ( 4\alpha ) + \left ( {\frac {C_{{1111}}}{4}}-{\frac {C_{{2222}}}{4}} \right ) \sin ( 4\alpha ), \\ C_{1122}' ={}& -\left ( {\frac {C_{{1111}}}{8}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{\frac {C_{{2222}}}{8}} \right ) \cos ( 8\alpha ) - \left ( {\frac {C_{{1112}}}{2}} - {\frac {C_{{2212}}}{2}} \right ) \sin ( 8\alpha ) \\ &+ \left ( {\frac {C_{{1111}}}{8}}+{\frac {3 C_{{ 1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{\frac {C_{{2222}}}{8}} \right ), \\ \\ C_{1212}' ={}&-\left ( {\frac {C_{{1111}}}{8}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{\frac {C_{{2222}}}{8}} \right ) \cos ( 8\alpha ) - \left ( {\frac {C_{{1112}}}{2}} - {\frac {C_{{2212}}}{2}} \right ) \sin ( 8\alpha ) \\ &+ \left ( {\frac {C_{{1111}}}{8}}-{\frac {C_{{ 1122}}}{4}}+{\frac {C_{{1212}}}{2}}+{\frac {C_{{2222}}}{8}} \right ), \\ C_{2212}' ={}& \left ( {\frac {C_{{1112}}}{2}}-{\frac {C_{{2212}}}{2}} \right ) \cos ( 8 \alpha ) - \left ( {\frac {C_{{1111}}}{8}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{\frac {C_{{2222}}}{8}} \right ) \sin ( 8\alpha ) \\ &- \left ( {{ \frac {C_{{1112}}}{2}} + \frac {C_{{2212}}}{2}} \right ) \cos ( 4\alpha ) + \left ( {\frac {C_{{1111}}}{4}} -{\frac {C_{{2222}}}{4}} \right ) \sin ( 4\alpha ), \\ C_{2222}' ={}& \left ( {\frac {C_{{1111}}}{8}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{\frac { C_{{2222}}}{8}} \right ) \cos ( 8\alpha ) + \left ( {\frac {C_{{1112}}}{2}} -{\frac {C_{{2212}}}{2}} \right ) \sin ( 8\alpha ) \\ &- \left ( {\frac {C_{ {1111}}}{2}} - {\frac {C_{{2222}}}{2}} \right ) \cos ( 4\alpha ) -\left ( C_{{1112}} + C_{{2212}} \right ) \sin ( 4\alpha ) \\ &+ \left ( {\frac {3 C_{{ 1111}}}{8}}+{\frac {C_{{1122}}}{4}}+{\frac {C_{{1212}}}{2}}+{\frac {3 C_{{2222}}}{8}} \right ). \end{aligned}$$

(50)

Similarly, under the transformation $\mathbf{Q} = \mathbf{Rot}(\alpha )$, the components of the elasticity tensor ℂ are

$$\begin{aligned} C_{1111}' ={}& \left ( {\frac {C_{{1111}}}{8}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{\frac { C_{{2222}}}{8}} \right ) \cos ( 4\alpha ) - \left ( {\frac {C_{ {1112}}}{2}} - {\frac {C_{{2212}}}{2}} \right ) \sin ( 4\alpha ) \\ &+ \left ( {\frac {C_{{1111}}}{2}}-{\frac {C_{{2222}}}{2}} \right ) \cos ( 2\alpha ) - \left (C_{{1112}} + C_{{2212}} \right ) \sin ( 2\alpha ) \\ & + \left ( {\frac {3 C_{{1111}}}{8}}+{\frac {C_{{1122}}}{4}}+{\frac {C_{{1212}}}{2}} + {\frac {3 C_{{ 2222}}}{8}}\right ), \\ C_{1112}' ={}& \left ( {\frac {C_{{1112}}}{2}} -{ \frac {C_{{2212}}}{2}} \right ) \cos ( 4 \alpha ) + \left ( {\frac {C _{{1111}}}{8}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{\frac {C_{{2222}}}{8}} \right ) \sin ( 4\alpha ) \\ &+\left ( {\frac {C_{{1112}}}{2}} + {\frac {C_{{2212}}}{2}} \right ) \cos ( 2\alpha ) + \left ( {\frac {C_{{1111}}}{4}} -{\frac {C_{{2222}}}{4}} \right ) \sin ( 2\alpha ), \\ C_{1122}' ={}& -\left ( {\frac {C_{{1111}}}{8}} -{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} +{\frac {C_{{2222}}}{8}} \right ) \cos ( 4\alpha ) + \left ( {\frac {C_{{1112}}}{2}}-{\frac {C_{{2212}}}{2}} \right ) \sin ( 4\alpha ) \\ &+ \left ( {\frac {C_{{1111}}}{8}} +{\frac {3 C_{{ 1122}}}{4}}-{\frac {C_{{1212}}}{2}}+ {\frac {C_{{2222}}}{8}} \right ), \\ \\ C_{1212}' ={}& -\left ( {\frac {C_{{1111}}}{8}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{\frac {C_{{2222}}}{8}} \right ) \cos ( 4\alpha ) + \left ( {\frac {C_{{1112}}}{2}} -{\frac {C_{{2212}}}{2}} \right ) \sin ( 4\alpha ) \\ &+ \left ( {\frac {C_{{1111}}}{8}}-{\frac {C_{{ 1122}}}{4}}+{\frac {C_{{1212}}}{2}}+{\frac {C_{{2222}}}{8}} \right ), \\ C_{2212}' ={}& -\left ( { {\frac {C_{{1112}}}{2}} - \frac {C_{{2212}}}{2}} \right ) \cos ( 4 \alpha ) - \left ( {\frac { C_{{1111}}}{8}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{ \frac {C_{{2222}}}{8}} \right ) \sin ( 4\alpha ) \\ &+ \left ( {{\frac {C_{{1112}}}{2}} + \frac {C_{{2212}}}{2}} \right ) \cos ( 2\alpha ) + \left ( {\frac {C_{{1111}}}{4}} -{\frac {C_{{2222}}}{4}} \right ) \sin ( 2\alpha ), \\ C_{2222}' ={}& \left ( {\frac {C_{{1111}}}{8}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}}+{\frac { C_{{2222}}}{8}} \right ) \cos ( 4\alpha ) - \left ( {\frac {C_{ {1112}}}{2}} - {\frac {C_{{2212}}}{2}} \right ) \sin ( 4\alpha ) \\ &- \left ( {\frac {C_{{1111}}}{2}} - {\frac {C_{{2222}}}{2}} \right ) \cos ( 2\alpha ) + \left ( C_{{1112}} + C_{{2212}} \right ) \sin ( 2\alpha ) \\ &+ \left ( {\frac {3 C_{{ 1111}}}{8}}+{\frac {C_{{1122}}}{4}}+{\frac {C_{{1212}}}{2}}+{\frac {3 C_{{2222}}}{8}} \right ). \end{aligned}$$

(51)

One immediate observation from (50) and (51) is that the expressions do not contain sine and cosine of odd multiples of $\alpha$. This is not entirely unexpected as the transformation (see (11) with $\mathbf{Q}$ given by (12) or (13)) employs terms of the form $(\cos (2 \alpha ))^{n} (\sin (2 \alpha ))^{4-n}$ in (50) and $(\cos (\alpha ))^{n} (\sin (\alpha ))^{4-n}$ in (51) for $n \in \left \{ 0,1,2,3,4 \right \}$, which when linearized, produce constant terms and terms of the form $\cos (2m \alpha )$ and $\sin (2m \alpha )$ for $m \in \left \{1,2,3,4 \right \}$. For rotations, this was observed in [12] wherein the so-called parameters of Tsai and Pagano were introduced. Additionally, there are deeper connections between the harmonic decomposition of the elasticity tensor and this linearized form of the transformed components of ℂ under rotations which were explored in [7].

Appendix C: Orthogonal Transformations in Two Dimensions

For the sake of completeness, this section presents a proof that, in two dimensions, orthogonal transformations are either rotations or reflections. The proof presented here is straightforward and we make no claims of originality.

Proposition 1

In two dimensions, orthogonal transformations are either rotations or reflections.

Proof

Let

$$ \mathbf{A} := \left [ \textstyle\begin{array}{c@{\quad}c} a & b \\ c & d \end{array}\displaystyle \right ] $$

(52)

be an orthogonal transformation in two dimensions. Consequently, $\mathbf{A} \mathbf{A}^{T} = \mathbf{I}$ with $\mathbf{I}$ the identity transformation, which implies

$$ \begin{aligned} 1 = a^{2} + b^{2}, \, 0 = ac +bd, \text{ and } 1 = c^{2} + d^{2}. \end{aligned} $$

(53)

From (53), we see that the points $(a,b)$ and $(c,d)$ lie on the unit circle and without loss of generality we may introduce angles $\theta , \phi \in [0,2 \pi )$ such that

$$ \begin{aligned} a = \cos (\theta ), \, b = \sin (\theta ), \, c = \cos ( \phi ), \text{ and } d = \sin (\phi ). \end{aligned} $$

(54)

From (53) and (54), we have

$$ 0 = ac +bd = \cos (\theta ) \cos (\phi ) + \sin (\theta ) \sin (\phi ) = \cos (\theta -\phi ) \Rightarrow \theta -\phi = \frac{\pi}{2} + k \pi , $$

(55)

for some $k \in \mathbb{Z}$.

If $k$ is even, then from the periodicity of the sine and cosine functions, as well as the fact that the sine function is simply a quarter period shift of the cosine function, we have

$$ \mathbf{A} = \left [ \textstyle\begin{array}{c@{\quad}c} \cos (\theta ) & \sin (\theta ) \\ \cos ( \theta - \frac{\pi}{2} ) & \sin ( \theta - \frac{\pi}{2} ) \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{c@{\quad}c} \cos (\theta ) & \sin (\theta ) \\ \sin ( \theta ) & -\cos ( \theta ) \end{array}\displaystyle \right ]. $$

(56)

From (13), we see that (56) corresponds to a reflection about the line through the origin making an angle of $\frac{\theta}{2}$ with the $x$-axis.

Alternatively, if $k$ is odd, then from the periodicity of the sine and cosine functions, the fact that the sine function is simply a quarter period shift of the cosine function, and the odd and even properties of the sine and cosine functions, respectively, we have

$$ \mathbf{A} = \left [ \textstyle\begin{array}{c@{\quad}c} \cos (\theta ) & \sin (\theta ) \\ \cos ( \theta + \frac{\pi}{2} ) & \sin ( \theta + \frac{\pi}{2} ) \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{c@{\quad}c} \cos (\theta ) & \sin (\theta ) \\ -\sin ( \theta ) & \cos ( \theta ) \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{c@{\quad}c} \cos (-\theta ) & -\sin (-\theta ) \\ \sin ( -\theta ) & \cos ( -\theta ) \end{array}\displaystyle \right ]. $$

(57)

From (12), we see that (57) corresponds to a clockwise rotation by an angle $\theta$ about the origin, or equivalently a counterclockwise rotation by an angle $-\theta$ about the origin. □

Appendix D: Invariance of the Elasticity Tensor with Respect to $\mathbf{Ref}(\alpha )$

This section presents a proof that, under particular conditions on its components, the elasticity tensor ℂ is invariant with respect to a specific reflection transformation.

Lemma 3

Let $C_{ijkl}$, for $i,j,k,l \in \left \{1,2 \right \}$, be the components of the elasticity tensor ℂ. If $C_{1111} = C_{2222}$, $C_{1112} = -C_{2212}$, and there exists an $\alpha \in \left ( 0, \frac{\pi}{4} \right )$ such that

$$ C_{1111} - C_{1122} - 2 C_{1212} = 2C_{1112} \left ( \cot (2\alpha ) - \tan (2\alpha ) \right ), $$

(58)

then ℂ is invariant under the transformation $\mathbf{Ref}(\alpha )$.

Proof

Recall ℂ is invariant under the transformation $\mathbf{Ref}(\alpha )$ if (11) holds with $\mathbf{Q} = \mathbf{Ref}(\alpha )$. Conveniently, the components of the elasticity tensor under the transformation $\mathbf{Ref}(\alpha )$ are presented in (50). Demonstrating $C_{ijkl}' = C_{ijkl}$ for $i,j,k,l \in \left \{1,2 \right \}$ in (50) proves the invariance of the elasticity tensor under the transformation $\mathbf{Ref}(\alpha )$. We start by substituting the assumptions $C_{2222} = C_{1111}$ and $C_{2212} = -C_{1112}$ into (50) to obtain

$$\begin{aligned} C_{1111}' ={}& \left ( {\frac {C_{{1111}}}{4}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \cos ( 8\alpha ) + C_{{1112}} \sin ( 8\alpha ) + \left ({ \frac {3 C_{{ 1111}}}{4}}+{\frac {C_{{1122}}}{4}}+{\frac {C_{{1212}}}{2}} \right ), \\ C_{1112}' ={}& - C_{{1112}} \cos ( 8 \alpha ) + \left ( { \frac {C_{{1111}}}{4}}-{\frac {C_{ {1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \sin ( 8\alpha ), \\ C_{1122}' ={}& -\left ( {\frac {C_{{1111}}}{4}}-{ \frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \cos ( 8\alpha ) - C_{{1112}} \sin ( 8\alpha ) \\ &+ \left ( {\frac {C_{{1111}}}{4}}+{ \frac {3 C_{{ 1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ), \\ C_{1212}' ={}&-\left ( {\frac {C_{{1111}}}{4}}-{ \frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \cos ( 8\alpha ) - C_{{1112}} \sin ( 8\alpha ) \\ &+ \left ( {\frac {C_{{1111}}}{4}}-{ \frac {C_{{ 1122}}}{4}}+{\frac {C_{{1212}}}{2}} \right ), \\ C_{2212}' ={}& C_{{1112}} \cos ( 8 \alpha ) - \left ( { \frac {C_{{1111}}}{4}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \sin ( 8\alpha ), \\ C_{2222}' ={}& \left ( {\frac {C_{{1111}}}{4}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \cos ( 8\alpha ) + C_{{1112}} \sin ( 8\alpha ) + \left ( { \frac {3 C_{{ 1111}}}{4}}+{\frac {C_{{1122}}}{4}}+{\frac {C_{{1212}}}{2}} \right ). \end{aligned}$$

(59)

Next, we subtract $C_{ijkl}$ from both sides of each equation, where the indices correspond to the indices of $C_{ijkl}'$ in each equation. This is followed by combining like terms to find

$$ \begin{aligned} C_{1111}' -C_{1111} ={}& -\left ( { \frac {C_{{1111}}}{4}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \left (1-\cos ( 8\alpha ) \right ) + C_{{1112}} \sin ( 8\alpha ), \\ C_{1112}' -C_{1112}={}& - C_{{1112}} \left ( 1+\cos ( 8 \alpha ) \right ) + \left ( {\frac {C_{{1111}}}{4}}-{\frac {C_{ {1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \sin ( 8\alpha ), \\ C_{1122}' -C_{1122}={}& \left ( {\frac {C_{{1111}}}{4}}-{ \frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \left (1- \cos ( 8 \alpha ) \right ) - C_{{1112}} \sin ( 8\alpha ), \\ C_{1212}' -C_{1212}={}&\left ( {\frac {C_{{1111}}}{4}}-{ \frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \left (1 - \cos ( 8 \alpha ) \right ) - C_{{1112}} \sin ( 8\alpha ), \\ C_{2212}' -C_{2212}={}& C_{{1112}} \left (1 + \cos ( 8 \alpha ) \right ) - \left ( {\frac {C_{{1111}}}{4}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \sin ( 8\alpha ), \\ C_{2222}' -C_{2222}={}& -\left ( {\frac {C_{{1111}}}{4}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \left (1 - \cos ( 8\alpha ) \right ) + C_{{1112}} \sin ( 8\alpha ). \end{aligned} $$

(60)

Note that in the equations for $C_{2212}'$ and $C_{2222}'$ in (60), we employed the assumptions $C_{2212} = -C_{1112}$ and $C_{2222} = C_{1111}$, respectively. To complete the proof, we demonstrate the right-hand side of each equation in (60) is null. Up to a change in sign, there are only two unique expressions on the right-hand sides of the equations in (60):

$$ \begin{aligned} &\left ( {\frac {C_{{1111}}}{4}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \left (1- \cos ( 8 \alpha ) \right ) - C_{{1112}} \sin ( 8\alpha ), \\ &C_{{1112}} \left ( 1+\cos ( 8 \alpha )\right ) - \left ( { \frac {C_{{1111}}}{4}}-{\frac {C_{ {1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) \sin ( 8\alpha ). \end{aligned} $$

(61)

Based on the final assumption (58), we substitute $\left ( {\frac {C_{{1111}}}{4}}-{\frac {C_{{1122}}}{4}}-{\frac {C_{{1212}}}{2}} \right ) = \frac{1}{2} C_{1112} ( \cot (2\alpha ) - \tan (2\alpha ) )$ into (61) to obtain

$$ \begin{aligned} &\left ( \frac{1}{2} (\cot (2 \alpha ) - \tan (2 \alpha )) (1-\cos (8\alpha )) - \sin (8 \alpha ) \right ) C_{1112}, \\ &\left (1+\cos (8\alpha ) - \frac{1}{2} ( \cot (2\alpha )-\tan (2 \alpha ))\sin (8\alpha ) \right ) C_{1112}. \end{aligned} $$

(62)

Employing the trigonometric identities $\cot (2\alpha )-\tan (2\alpha ) = 2 \cot (4\alpha ), 1-\cos (8 \alpha ) = 2 \sin ^{2}(4\alpha ), 1+\cos (8\alpha ) = 2 \cos ^{2}(4 \alpha )$, and $\sin (8\alpha ) = 2 \sin (4 \alpha ) \cos (4 \alpha )$, we may transform (62) into

$$ \begin{aligned} &\left ( 2 \cot (4\alpha ) \sin ^{2}(4\alpha ) -2 \sin (4 \alpha ) \cos (4\alpha )\right )C_{1112}=0, \\ &\left (2 \cos ^{2}(4\alpha ) - 2 \cot (4 \alpha ) \sin (4 \alpha ) \cos (4\alpha ) \right ) C_{1112} = 0. \end{aligned} $$

(63)

The equalities in (63) follow by recalling the identity $\cot (4\alpha ) = \frac{\cos (4 \alpha )}{\sin (4\alpha )}$. Consequently, the right-hand sides of (60) are null and thus $C_{ijkl}' = C_{ijkl}$ for $i,j,k,l \in \left \{1,2 \right \}$. We conclude ℂ is invariant under the transformation $\mathbf{Ref}(\alpha )$. □