Canonical forms

The expressions for our anisotropic tensors are provided in a canonical reference frame which exploits the symmetry of the material to present the most compact form of the constitutive tensors.

Symmetries in constitutive models (source) from Pamela Burnley, University of Nevada Las Vegas, based on Nye (1984) Physical Properties of Crystals.

This wikipedia article should also be helpful.

In a general orientation, a rotation tensor \({\cal R}\) is required to transform to / from this canonical frame. For isotropic materials, \({\cal R}\) is the identity matrix. For cubic materials, a single angle describes the offset of the two frames, and for transversely isotropic materials, we need to define a rotation matrix that aligns the normal to the symmetry plane with the vertical axis in the canonical frame. These examples have fewer degrees of freedom than a full rotation matrix and this fact can be used to simplify the general (rotated) form of the constitutive tensors.

Define \({\cal R}_{ij}\) as the rotation matrix that maps the coordinates \(x\) onto the \(x'\) coordiate system, i.e.

\[a_i' = {\cal R}_{ij} \, a_j \]

\({\cal R}_{ij}\) also has this property:

\[ {\cal R}_{ki}\,{\cal R}_{kj} = \delta_{ij} \]

Second order tensors transform as follows:

\[a_{ij}' = {\cal R}_{ik}\,{\cal R}_{jl}\,a_{kl},\]

and for higher rank tensors, we just continue …

\[a_{ijk}' = {\cal R}_{il}\,{\cal R}_{jm}\,{\cal R}_{kn}\,a_{lmn}.\]

In underworld tensor rotation is provided for rank 2 and rank 4 tensors by uw.maths.tensor.tensor_rotation(R, tensor_expression)

Incompressible materials

If we apply constraints to the deformation, we expect to reduce the number of independent material constants. Incompressibility should reduce the number of independent material constants. In an isotropic medium (or a medium with cubic symmetry), incompressibility eliminates the one volumetric material modulus (e.g. the bulk modulus). In general anisotropic media, it is not the case that changes in pressure result in uniform expansion or contraction, and an incompressibility constraint reduces the number of independent material constants. In the transversely isotropic case, there are five independent materia constants in general, reducing to four when the material is incompressible.

It is not a given that the stiffness matrix is trivial to construct / meaningful for incompressible anisotropy and there is some discussion here: https://rastgaragah.wordpress.com/2013/03/12/incompressibility-of-linearly-elastic-material/ (identifies the issue) and this is explained in more detail by Destrade et al. (2002)

Example

First we define the rotation for the transversely isotropic case which depends on the normal vector to the symmetry plane.

A rotation matrix for a transversely isotropic medium is defined by specifying the normal of the symmetry plane (\(\hat{\mathbf{n}} = \{ n_0, n_1, n_2\} \)). The other orientations are arbitrary, so we simply derive them from \(\hat{\mathbf{n}}\) - one vector specified to be perpendicular in the horizontal plane and the third vector is then found from their cross product (\(\hat{\mathbf{s}}\) and \(\hat{\mathbf{t}}\) respectively)

Underworld python script

import underworld3 as uw
import sympy

n = sympy.Matrix(sympy.symarray("n",(3,)))

# Or give a specific value (just specify a vector, then normalise)
# n = sympy.Matrix([1,1,1]) 
# n /= sympy.sqrt(n.dot(n))

if n[0] == 0:
    s = sympy.Matrix([1,0,0])
    t = sympy.Matrix([0,1,0])
    R = sympy.eye(3)

else:
    s = sympy.Matrix((n[1] , -n[0], 0 ))
    s /= sympy.sqrt(s.dot(s))
    t = -n.cross(s) # complete the coordinate triad
    R = sympy.BlockMatrix((s,t,n)).as_explicit()

display(R)

\[\begin{split}\cal{R} = \left[\begin{matrix}\frac{n_{1}}{\sqrt{n_{0}^{2} + n_{1}^{2}}} & - \frac{n_{0} n_{2}}{\sqrt{n_{0}^{2} + n_{1}^{2}}} & n_{0}\\- \frac{n_{0}}{\sqrt{n_{0}^{2} + n_{1}^{2}}} & - \frac{n_{1} n_{2}}{\sqrt{n_{0}^{2} + n_{1}^{2}}} & n_{1}\\0 & \frac{n_{0}^{2}}{\sqrt{n_{0}^{2} + n_{1}^{2}}} + \frac{n_{1}^{2}}{\sqrt{n_{0}^{2} + n_{1}^{2}}} & n_{2}\end{matrix}\right]\end{split}\]

Validation

A simple check on this is to rotate the isotropic constitutive tensor and validate that it is invariant under rotation.

\[\begin{split}C_{IJ} = \left[\begin{matrix}2 \eta_{0} & 0 & 0 & 0 & 0 & 0\\0 & 2 \eta_{0} & 0 & 0 & 0 & 0\\0 & 0 & 2 \eta_{0} & 0 & 0 & 0\\0 & 0 & 0 & 2 \eta_{0} & 0 & 0\\0 & 0 & 0 & 0 & 2 \eta_{0} & 0\\0 & 0 & 0 & 0 & 0 & 2 \eta_{0}\end{matrix}\right]\end{split}\]

Noting that the rotation of the Mandel or Voigt constitutive matrices is complicated by the \(\mathbf{P}\) scaling matrices, we compute rotations on the rank 4 tensor, \(c_{ijkl}\) and transform to the matrix forms as required. We denote the rotation from \(\{IJ\}\) coordinates to \(\{I'J'\}\) as

\[C_{I'J'} = {\cal R}[C_{IJ}]\]

The rotated constitutive model has the following form:

\[\begin{split}C_{I'J'} = \left[\begin{matrix}\frac{2 \eta_{0} \left(n_{0}^{2} n_{2}^{2} + n_{0}^{2} \left(n_{0}^{2} + n_{1}^{2}\right) + n_{1}^{2}\right)^{2}}{\left(n_{0}^{2} + n_{1}^{2}\right)^{2}} & \frac{2 \eta_{0} n_{0}^{2} n_{1}^{2} \left(n_{0}^{2} + n_{1}^{2} + n_{2}^{2} - 1\right)^{2}}{\left(n_{0}^{2} + n_{1}^{2}\right)^{2}} & 0 & 0 & 0 & \frac{2 \sqrt{2} \eta_{0} n_{0} n_{1} \left(n_{0}^{6} + 2 n_{0}^{4} n_{1}^{2} + 2 n_{0}^{4} n_{2}^{2} - n_{0}^{4} + n_{0}^{2} n_{1}^{4} + 2 n_{0}^{2} n_{1}^{2} n_{2}^{2} + n_{0}^{2} n_{2}^{4} - n_{0}^{2} n_{2}^{2} + n_{1}^{4} + n_{1}^{2} n_{2}^{2} - n_{1}^{2}\right)}{n_{0}^{4} + 2 n_{0}^{2} n_{1}^{2} + n_{1}^{4}}\\\frac{2 \eta_{0} n_{0}^{2} n_{1}^{2} \left(n_{0}^{2} + n_{1}^{2} + n_{2}^{2} - 1\right)^{2}}{\left(n_{0}^{2} + n_{1}^{2}\right)^{2}} & \frac{2 \eta_{0} \left(n_{0}^{2} + n_{1}^{2} n_{2}^{2} + n_{1}^{2} \left(n_{0}^{2} + n_{1}^{2}\right)\right)^{2}}{\left(n_{0}^{2} + n_{1}^{2}\right)^{2}} & 0 & 0 & 0 & \frac{2 \sqrt{2} \eta_{0} n_{0} n_{1} \left(n_{0}^{4} n_{1}^{2} + n_{0}^{4} + 2 n_{0}^{2} n_{1}^{4} + 2 n_{0}^{2} n_{1}^{2} n_{2}^{2} + n_{0}^{2} n_{2}^{2} - n_{0}^{2} + n_{1}^{6} + 2 n_{1}^{4} n_{2}^{2} - n_{1}^{4} + n_{1}^{2} n_{2}^{4} - n_{1}^{2} n_{2}^{2}\right)}{n_{0}^{4} + 2 n_{0}^{2} n_{1}^{2} + n_{1}^{4}}\\0 & 0 & 2 \eta_{0} \left(n_{0}^{2} + n_{1}^{2} + n_{2}^{2}\right)^{2} & 0 & 0 & 0\\0 & 0 & 0 & \frac{2 \eta_{0} \left(n_{0}^{2} + n_{1}^{2} + n_{2}^{2}\right) \left(n_{0}^{2} + n_{1}^{2} n_{2}^{2} + n_{1}^{2} \left(n_{0}^{2} + n_{1}^{2}\right)\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \eta_{0} n_{0} n_{1} \left(n_{0}^{2} + n_{1}^{2} + n_{2}^{2}\right) \left(n_{0}^{2} + n_{1}^{2} + n_{2}^{2} - 1\right)}{n_{0}^{2} + n_{1}^{2}} & 0\\0 & 0 & 0 & \frac{2 \eta_{0} n_{0} n_{1} \left(n_{0}^{2} + n_{1}^{2} + n_{2}^{2}\right) \left(n_{0}^{2} + n_{1}^{2} + n_{2}^{2} - 1\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \eta_{0} \left(n_{0}^{2} + n_{1}^{2} + n_{2}^{2}\right) \left(n_{0}^{2} n_{2}^{2} + n_{0}^{2} \left(n_{0}^{2} + n_{1}^{2}\right) + n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} & 0\\\frac{2 \sqrt{2} \eta_{0} n_{0} n_{1} \left(n_{0}^{2} n_{2}^{2} + n_{0}^{2} \left(n_{0}^{2} + n_{1}^{2}\right) + n_{1}^{2}\right) \left(n_{0}^{2} + n_{1}^{2} + n_{2}^{2} - 1\right)}{\left(n_{0}^{2} + n_{1}^{2}\right)^{2}} & \frac{2 \sqrt{2} \eta_{0} n_{0} n_{1} \left(n_{0}^{2} + n_{1}^{2} n_{2}^{2} + n_{1}^{2} \left(n_{0}^{2} + n_{1}^{2}\right)\right) \left(n_{0}^{2} + n_{1}^{2} + n_{2}^{2} - 1\right)}{\left(n_{0}^{2} + n_{1}^{2}\right)^{2}} & 0 & 0 & 0 & \frac{2 \eta_{0} \left(n_{0}^{2} n_{2}^{2} \left(n_{0}^{2} + 2 n_{1}^{2} n_{2}^{2} + 2 n_{1}^{2} \left(n_{0}^{2} + n_{1}^{2}\right) - n_{1}^{2}\right) + n_{0}^{2} \left(n_{0}^{2} + n_{1}^{2}\right) \left(n_{0}^{2} + 2 n_{1}^{2} n_{2}^{2} + 2 n_{1}^{2} \left(n_{0}^{2} + n_{1}^{2}\right) - n_{1}^{2}\right) + n_{1}^{2} \cdot \left(2 n_{0}^{2} + n_{2}^{2} \left(- n_{0}^{2} + n_{1}^{2}\right) - \left(n_{0}^{2} - n_{1}^{2}\right) \left(n_{0}^{2} + n_{1}^{2}\right)\right)\right)}{\left(n_{0}^{2} + n_{1}^{2}\right)^{2}}\end{matrix}\right]\end{split}\]

However, \(\{n_0, n_1, n_2\}\) are not independent because \(\hat{\mathbf{n}}\) is a unit vector. If we add this information and simplify, we recover the isotropic form of \(C\)

Underworld python script

# construct a symbolic, isotropic matrix (Mandel form)

eta0 = sympy.symbols("\eta_0")
C_IJm = 2 * sympy.Matrix.diag([eta0]*6)
display(C_IJm)

## Rotate the matrix 

c_ijkl = uw.maths.tensor.mandel_to_rank4(C_IJm, 3)
C_IJv = uw.maths.tensor.rank4_to_voigt(c_ijkl, 3)
c_ijkl_R = sympy.simplify(uw.maths.tensor.tensor_rotation(R, c_ijkl))
C_IJm_R = sympy.simplify(uw.maths.tensor.rank4_to_mandel(c_ijkl_R, 3))
display(C_IJm_R)

## Is this really invariant under rotation ?? 
## Have to do some manipulation to identify the 
## unit-vector component relationships

C_IJm_R_s1 = C_IJm_R.subs(n[0]**2+n[1]**2+n[2]**2,1)
C_IJm_R_s2 = C_IJm_R_s1.subs(n[0]**2+n[1]**2, 1-n[2]**2).applyfunc(sympy.factor)

display(C_IJm_R_s2.subs(n[0]**2+n[1]**2, 1-n[2]**2).simplify())

Muhlhaus / Moresi transversely isotropic tensor

The Muhlhaus / Moresi transversely isotropic consitituve model is designed to model materials that have a single (usually) weak plane (e.g. an embedded fault).

The constitutive model is

\[\begin{split}\left[\begin{matrix}2 \eta_{0} & 0 & 0 & 0 & 0 & 0\\0 & 2 \eta_{0} & 0 & 0 & 0 & 0\\0 & 0 & 2 \eta_{0} & 0 & 0 & 0\\0 & 0 & 0 & - 2 \Delta\eta + 2 \eta_{0} & 0 & 0\\0 & 0 & 0 & 0 & - 2 \Delta\eta + 2 \eta_{0} & 0\\0 & 0 & 0 & 0 & 0 & 2 \eta_{0}\end{matrix}\right]\end{split}\]

where \(\Delta\eta\) represents the change (usually reduction) in viscosity.

Rotation using \(\cal R\) as defined from the normal to the weak plane (above) gives (Mandel form):

\[\begin{split}\left[\begin{matrix}- \frac{4 \Delta\eta n_{0}^{2} \left(n_{0}^{2} n_{2}^{2} + n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} + 2 \eta_{0} & \frac{4 \Delta\eta n_{0}^{2} n_{1}^{2} \cdot \left(1 - n_{2}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} & 4 \Delta\eta n_{0}^{2} n_{2}^{2} & \frac{2 \sqrt{2} \Delta\eta n_{0}^{2} n_{1} n_{2} \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2} + 1\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \sqrt{2} \Delta\eta n_{0} n_{2} \left(n_{0}^{2} \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2}\right) - n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \sqrt{2} \Delta\eta n_{0} n_{1} \left(- 2 n_{0}^{2} n_{2}^{2} + n_{0}^{2} - n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}}\\\frac{4 \Delta\eta n_{0}^{2} n_{1}^{2} \cdot \left(1 - n_{2}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} & - \frac{4 \Delta\eta n_{1}^{2} \left(n_{0}^{2} + n_{1}^{2} n_{2}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} + 2 \eta_{0} & 4 \Delta\eta n_{1}^{2} n_{2}^{2} & - \frac{2 \sqrt{2} \Delta\eta n_{1} n_{2} \left(n_{0}^{2} - n_{1}^{2} \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2}\right)\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \sqrt{2} \Delta\eta n_{0} n_{1}^{2} n_{2} \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2} + 1\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \sqrt{2} \Delta\eta n_{0} n_{1} \left(- n_{0}^{2} - 2 n_{1}^{2} n_{2}^{2} + n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}}\\4 \Delta\eta n_{0}^{2} n_{2}^{2} & 4 \Delta\eta n_{1}^{2} n_{2}^{2} & - 4 \Delta\eta n_{2}^{2} \left(n_{0}^{2} + n_{1}^{2}\right) + 2 \eta_{0} & 2 \sqrt{2} \Delta\eta n_{1} n_{2} \left(- n_{0}^{2} - n_{1}^{2} + n_{2}^{2}\right) & 2 \sqrt{2} \Delta\eta n_{0} n_{2} \left(- n_{0}^{2} - n_{1}^{2} + n_{2}^{2}\right) & 4 \sqrt{2} \Delta\eta n_{0} n_{1} n_{2}^{2}\\\frac{2 \sqrt{2} \Delta\eta n_{0}^{2} n_{1} n_{2} \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2} + 1\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \sqrt{2} \Delta\eta n_{1} n_{2} \left(n_{0}^{2} n_{1}^{2} - n_{0}^{2} + n_{1}^{4} - n_{1}^{2} n_{2}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} & 2 \sqrt{2} \Delta\eta n_{1} n_{2} \left(- n_{0}^{2} - n_{1}^{2} + n_{2}^{2}\right) & - \frac{2 \Delta\eta \left(n_{0}^{2} n_{2}^{2} - n_{1}^{2} n_{2}^{2} \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2}\right) + n_{1}^{2} \left(n_{0}^{2} + n_{1}^{2}\right) \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2}\right)\right)}{n_{0}^{2} + n_{1}^{2}} + 2 \eta_{0} & \frac{2 \Delta\eta n_{0} n_{1} \left(n_{2}^{2} \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2}\right) + n_{2}^{2} - \left(n_{0}^{2} + n_{1}^{2}\right) \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2}\right)\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \Delta\eta n_{0} n_{2} \cdot \left(2 n_{0}^{2} n_{1}^{2} - n_{0}^{2} + 2 n_{1}^{4} - 2 n_{1}^{2} n_{2}^{2} + n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}}\\\frac{2 \sqrt{2} \Delta\eta n_{0} n_{2} \left(n_{0}^{4} + n_{0}^{2} n_{1}^{2} - n_{0}^{2} n_{2}^{2} - n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \sqrt{2} \Delta\eta n_{0} n_{1}^{2} n_{2} \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2} + 1\right)}{n_{0}^{2} + n_{1}^{2}} & 2 \sqrt{2} \Delta\eta n_{0} n_{2} \left(- n_{0}^{2} - n_{1}^{2} + n_{2}^{2}\right) & \frac{2 \Delta\eta n_{0} n_{1} \left(n_{2}^{2} \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2}\right) + n_{2}^{2} - \left(n_{0}^{2} + n_{1}^{2}\right) \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2}\right)\right)}{n_{0}^{2} + n_{1}^{2}} & - \frac{2 \Delta\eta \left(- n_{0}^{2} n_{2}^{2} \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2}\right) + n_{0}^{2} \left(n_{0}^{2} + n_{1}^{2}\right) \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2}\right) + n_{1}^{2} n_{2}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} + 2 \eta_{0} & \frac{2 \Delta\eta n_{1} n_{2} \cdot \left(2 n_{0}^{4} + 2 n_{0}^{2} n_{1}^{2} - 2 n_{0}^{2} n_{2}^{2} + n_{0}^{2} - n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}}\\\frac{2 \sqrt{2} \Delta\eta n_{0} n_{1} \left(- 2 n_{0}^{2} n_{2}^{2} + n_{0}^{2} - n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \sqrt{2} \Delta\eta n_{0} n_{1} \left(- n_{0}^{2} - 2 n_{1}^{2} n_{2}^{2} + n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} & 4 \sqrt{2} \Delta\eta n_{0} n_{1} n_{2}^{2} & \frac{2 \Delta\eta n_{0} n_{2} \left(- n_{0}^{2} + 2 n_{1}^{2} \left(n_{0}^{2} + n_{1}^{2} - n_{2}^{2}\right) + n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \Delta\eta n_{1} n_{2} \cdot \left(2 n_{0}^{4} + 2 n_{0}^{2} n_{1}^{2} - 2 n_{0}^{2} n_{2}^{2} + n_{0}^{2} - n_{1}^{2}\right)}{n_{0}^{2} + n_{1}^{2}} & \frac{2 \Delta\eta \left(- n_{0}^{4} - 4 n_{0}^{2} n_{1}^{2} n_{2}^{2} + 2 n_{0}^{2} n_{1}^{2} - n_{1}^{4}\right)}{n_{0}^{2} + n_{1}^{2}} + 2 \eta_{0}\end{matrix}\right]\end{split}\]

and we can easily demonstrate that this collapses back to the isotropic form if \(\Delta\eta \leftarrow 0\). We can also show this is equivalent to the alternate expression for the tensor provided in the original papers (an exercise for the reader !!).

Underworld python script

# Muhlhaus definition of C_IJ (mandel form)

eta1 = sympy.symbols("\eta_1")
delta_eta = sympy.symbols("\Delta\eta")

C_ijkl_MM = uw.maths.tensor.rank4_identity(3) * 0
C_IJm_MM = uw.maths.tensor.rank4_to_mandel(C_ijkl_MM, 3)
C_IJm_MM[0,0] = 2*eta0
C_IJm_MM[1,1] = 2*eta0
C_IJm_MM[2,2] = 2*eta0
C_IJm_MM[3,3] = 2*(eta0-delta_eta)  # yz
C_IJm_MM[4,4] = 2*(eta0-delta_eta)  # xz
C_IJm_MM[5,5] = 2*eta0  # xy

display(C_IJm_MM)

## We know that the isotropic part is invariant under rotation, so we only need to 
## examine the non-isotropic part.

C_ijkl_MM = uw.maths.tensor.mandel_to_rank4(C_IJm_MM - C_IJm , 3)
C_ijkl_MM_R = sympy.simplify(uw.maths.tensor.tensor_rotation(R, C_ijkl_MM))

C_IJm_MM_R = sympy.simplify(uw.maths.tensor.rank4_to_mandel(C_ijkl_MM_R, 3)) + C_IJm
C_IJv_MM_R = sympy.simplify(uw.maths.tensor.rank4_to_voigt(C_ijkl_MM_R, 3)) + C_IJv

display(C_IJm_MM_R)

# Check what happens if we set delta eta to zero
C_IJm_MM_iso = C_IJm_MM_R.subs(delta_eta,0).applyfunc(sympy.factor).subs(n[0]**2+n[1]**2, 1-n[2]**2).simplify()
display(C_IJm_MM_iso)

Han & Wahr, 1997 (full transverse isotropic tensor)

In the Han & Wahr (1997) paper, the expression for incompressible transverse-isotropy is as follows

\[\begin{split}\left[\begin{matrix}2 \eta_{0} + \mu_{0} & \mu_{0} & 0 & 0 & 0 & 0\\\mu_{0} & 2 \eta_{0} + \mu_{0} & 0 & 0 & 0 & 0\\0 & 0 & - 2 \Delta\eta + 2 \eta_{0} + \mu_{1} & 0 & 0 & 0\\0 & 0 & 0 & - 2 \Delta\eta + 2 \eta_{0} & 0 & 0\\0 & 0 & 0 & 0 & - 2 \Delta\eta + 2 \eta_{0} & 0\\0 & 0 & 0 & 0 & 0 & 2 \eta_{0}\end{matrix}\right]\end{split}\]

Note that the notation differs from their paper. I have replaced their \(\nu1, \nu2\) with \(\eta0, \eta1\) to be consistent with the forms defined above. I have replaced their \(\eta\) with \(\mu\) to avoid the confusion that results from the first change.

Applying the rotation, \(\cal R\) and attempting to coerce sympy to simplify the constitutive matrix:

\[\begin{split}\left[\begin{matrix}\frac{2 \Delta\eta n_{0}^{4} n_{2}^{4} - 2 \Delta\eta n_{0}^{4} + 4 \Delta\eta n_{0}^{2} n_{1}^{2} n_{2}^{2} - 4 \Delta\eta n_{0}^{2} n_{1}^{2} + 2 \eta_{0} n_{2}^{4} - 4 \eta_{0} n_{2}^{2} + 2 \eta_{0} + \mu_{0} n_{0}^{4} n_{2}^{4} + 2 \mu_{0} n_{0}^{2} n_{1}^{2} n_{2}^{2} + \mu_{0} n_{1}^{4} + \mu_{1} n_{0}^{4} n_{2}^{4} - 2 \mu_{1} n_{0}^{4} n_{2}^{2} + \mu_{1} n_{0}^{4}}{\left(n_{2} - 1\right)^{2} \left(n_{2} + 1\right)^{2}} & \frac{2 \Delta\eta n_{0}^{2} n_{1}^{2} n_{2}^{4} - 4 \Delta\eta n_{0}^{2} n_{1}^{2} n_{2}^{2} + 2 \Delta\eta n_{0}^{2} n_{1}^{2} + \mu_{0} n_{0}^{4} n_{2}^{2} + \mu_{0} n_{0}^{2} n_{1}^{2} n_{2}^{4} + \mu_{0} n_{0}^{2} n_{1}^{2} + \mu_{0} n_{1}^{4} n_{2}^{2} + \mu_{1} n_{0}^{2} n_{1}^{2} n_{2}^{4} - 2 \mu_{1} n_{0}^{2} n_{1}^{2} n_{2}^{2} + \mu_{1} n_{0}^{2} n_{1}^{2}}{\left(n_{2} - 1\right)^{2} \left(n_{2} + 1\right)^{2}} & 2 \Delta\eta n_{0}^{2} n_{2}^{2} + \mu_{0} n_{0}^{2} n_{2}^{2} + \mu_{0} n_{1}^{2} + \mu_{1} n_{0}^{2} n_{2}^{2} & \frac{\sqrt{2} n_{1} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} n_{2}^{2} - 2 \Delta\eta n_{0}^{2} + \mu_{0} n_{0}^{2} n_{2}^{2} + \mu_{0} n_{1}^{2} - \mu_{1} n_{0}^{4} - \mu_{1} n_{0}^{2} n_{1}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & \frac{\sqrt{2} n_{0} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} n_{2}^{2} + 2 \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} n_{2}^{2} + \mu_{0} n_{1}^{2} - \mu_{1} n_{0}^{4} - \mu_{1} n_{0}^{2} n_{1}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & \frac{\sqrt{2} n_{0} n_{1} \cdot \left(2 \Delta\eta n_{0}^{2} n_{2}^{2} + 2 \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} n_{2}^{2} + \mu_{0} n_{1}^{2} + \mu_{1} n_{0}^{2} n_{2}^{2} - \mu_{1} n_{0}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)}\\\frac{2 \Delta\eta n_{0}^{2} n_{1}^{2} n_{2}^{4} - 4 \Delta\eta n_{0}^{2} n_{1}^{2} n_{2}^{2} + 2 \Delta\eta n_{0}^{2} n_{1}^{2} + \mu_{0} n_{0}^{4} n_{2}^{2} + \mu_{0} n_{0}^{2} n_{1}^{2} n_{2}^{4} + \mu_{0} n_{0}^{2} n_{1}^{2} + \mu_{0} n_{1}^{4} n_{2}^{2} + \mu_{1} n_{0}^{2} n_{1}^{2} n_{2}^{4} - 2 \mu_{1} n_{0}^{2} n_{1}^{2} n_{2}^{2} + \mu_{1} n_{0}^{2} n_{1}^{2}}{\left(n_{2} - 1\right)^{2} \left(n_{2} + 1\right)^{2}} & \frac{4 \Delta\eta n_{0}^{2} n_{1}^{2} n_{2}^{2} - 4 \Delta\eta n_{0}^{2} n_{1}^{2} + 2 \Delta\eta n_{1}^{4} n_{2}^{4} - 2 \Delta\eta n_{1}^{4} + 2 \eta_{0} n_{2}^{4} - 4 \eta_{0} n_{2}^{2} + 2 \eta_{0} + \mu_{0} n_{0}^{4} + 2 \mu_{0} n_{0}^{2} n_{1}^{2} n_{2}^{2} + \mu_{0} n_{1}^{4} n_{2}^{4} + \mu_{1} n_{1}^{4} n_{2}^{4} - 2 \mu_{1} n_{1}^{4} n_{2}^{2} + \mu_{1} n_{1}^{4}}{\left(n_{2} - 1\right)^{2} \left(n_{2} + 1\right)^{2}} & 2 \Delta\eta n_{1}^{2} n_{2}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} n_{2}^{2} + \mu_{1} n_{1}^{2} n_{2}^{2} & \frac{\sqrt{2} n_{1} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} + 2 \Delta\eta n_{1}^{2} n_{2}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} n_{2}^{2} - \mu_{1} n_{0}^{2} n_{1}^{2} - \mu_{1} n_{1}^{4}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & \frac{\sqrt{2} n_{0} n_{2} \cdot \left(2 \Delta\eta n_{1}^{2} n_{2}^{2} - 2 \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} n_{2}^{2} - \mu_{1} n_{0}^{2} n_{1}^{2} - \mu_{1} n_{1}^{4}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & \frac{\sqrt{2} n_{0} n_{1} \cdot \left(2 \Delta\eta n_{0}^{2} + 2 \Delta\eta n_{1}^{2} n_{2}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} n_{2}^{2} + \mu_{1} n_{1}^{2} n_{2}^{2} - \mu_{1} n_{1}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)}\\2 \Delta\eta n_{0}^{2} n_{2}^{2} + \mu_{0} n_{0}^{2} n_{2}^{2} + \mu_{0} n_{1}^{2} + \mu_{1} n_{0}^{2} n_{2}^{2} & 2 \Delta\eta n_{1}^{2} n_{2}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} n_{2}^{2} + \mu_{1} n_{1}^{2} n_{2}^{2} & 2 \Delta\eta n_{2}^{4} - 4 \Delta\eta n_{2}^{2} + 2 \eta_{0} + \mu_{0} n_{2}^{4} - 2 \mu_{0} n_{2}^{2} + \mu_{0} + \mu_{1} n_{2}^{4} & - \sqrt{2} n_{1} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} + 2 \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} - \mu_{1} n_{2}^{2}\right) & - \sqrt{2} n_{0} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} + 2 \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} - \mu_{1} n_{2}^{2}\right) & \sqrt{2} n_{0} n_{1} \cdot \left(2 \Delta\eta n_{2}^{2} + \mu_{0} n_{2}^{2} - \mu_{0} + \mu_{1} n_{2}^{2}\right)\\\frac{\sqrt{2} n_{1} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} n_{2}^{2} - 2 \Delta\eta n_{0}^{2} + \mu_{0} n_{0}^{2} n_{2}^{2} + \mu_{0} n_{1}^{2} + \mu_{1} n_{0}^{2} n_{2}^{2} - \mu_{1} n_{0}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & \frac{\sqrt{2} n_{1} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} + 2 \Delta\eta n_{1}^{2} n_{2}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} n_{2}^{2} + \mu_{1} n_{1}^{2} n_{2}^{2} - \mu_{1} n_{1}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & - \sqrt{2} n_{1} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} + 2 \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} - \mu_{1} n_{2}^{2}\right) & \frac{2 \left(\Delta\eta n_{0}^{2} n_{2}^{2} + 2 \Delta\eta n_{1}^{2} n_{2}^{4} - 2 \Delta\eta n_{1}^{2} n_{2}^{2} + \Delta\eta n_{1}^{2} + \eta_{0} n_{2}^{2} - \eta_{0} + \mu_{0} n_{1}^{2} n_{2}^{4} - \mu_{0} n_{1}^{2} n_{2}^{2} + \mu_{1} n_{1}^{2} n_{2}^{4} - \mu_{1} n_{1}^{2} n_{2}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & 2 n_{0} n_{1} \cdot \left(2 \Delta\eta n_{2}^{2} - \Delta\eta + \mu_{0} n_{2}^{2} + \mu_{1} n_{2}^{2}\right) & \frac{2 n_{0} n_{2} \left(\Delta\eta n_{0}^{2} + 2 \Delta\eta n_{1}^{2} n_{2}^{2} - \Delta\eta n_{1}^{2} + \mu_{0} n_{1}^{2} n_{2}^{2} - \mu_{0} n_{1}^{2} + \mu_{1} n_{1}^{2} n_{2}^{2} - \mu_{1} n_{1}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)}\\\frac{\sqrt{2} n_{0} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} n_{2}^{2} + 2 \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} n_{2}^{2} + \mu_{0} n_{1}^{2} + \mu_{1} n_{0}^{2} n_{2}^{2} - \mu_{1} n_{0}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & \frac{\sqrt{2} n_{0} n_{2} \cdot \left(2 \Delta\eta n_{1}^{2} n_{2}^{2} - 2 \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} n_{2}^{2} + \mu_{1} n_{1}^{2} n_{2}^{2} - \mu_{1} n_{1}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & - \sqrt{2} n_{0} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} + 2 \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} - \mu_{1} n_{2}^{2}\right) & 2 n_{0} n_{1} \cdot \left(2 \Delta\eta n_{2}^{2} - \Delta\eta + \mu_{0} n_{2}^{2} + \mu_{1} n_{2}^{2}\right) & \frac{2 \cdot \left(2 \Delta\eta n_{0}^{2} n_{2}^{4} - 2 \Delta\eta n_{0}^{2} n_{2}^{2} + \Delta\eta n_{0}^{2} + \Delta\eta n_{1}^{2} n_{2}^{2} + \eta_{0} n_{2}^{2} - \eta_{0} + \mu_{0} n_{0}^{2} n_{2}^{4} - \mu_{0} n_{0}^{2} n_{2}^{2} + \mu_{1} n_{0}^{2} n_{2}^{4} - \mu_{1} n_{0}^{2} n_{2}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & \frac{2 n_{1} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} n_{2}^{2} - \Delta\eta n_{0}^{2} + \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} n_{2}^{2} - \mu_{0} n_{0}^{2} + \mu_{1} n_{0}^{2} n_{2}^{2} - \mu_{1} n_{0}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)}\\\frac{\sqrt{2} n_{0} n_{1} \cdot \left(2 \Delta\eta n_{0}^{2} n_{2}^{2} + 2 \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} n_{2}^{2} + \mu_{0} n_{1}^{2} + \mu_{1} n_{0}^{2} n_{2}^{2} - \mu_{1} n_{0}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & \frac{\sqrt{2} n_{0} n_{1} \cdot \left(2 \Delta\eta n_{0}^{2} + 2 \Delta\eta n_{1}^{2} n_{2}^{2} + \mu_{0} n_{0}^{2} + \mu_{0} n_{1}^{2} n_{2}^{2} + \mu_{1} n_{1}^{2} n_{2}^{2} - \mu_{1} n_{1}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & \sqrt{2} n_{0} n_{1} \cdot \left(2 \Delta\eta n_{2}^{2} + \mu_{0} n_{2}^{2} - \mu_{0} + \mu_{1} n_{2}^{2}\right) & \frac{2 n_{0} n_{2} \left(\Delta\eta n_{0}^{2} + 2 \Delta\eta n_{1}^{2} n_{2}^{2} - \Delta\eta n_{1}^{2} + \mu_{0} n_{1}^{2} n_{2}^{2} - \mu_{0} n_{1}^{2} - \mu_{1} n_{0}^{2} n_{1}^{2} - \mu_{1} n_{1}^{4}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & \frac{2 n_{1} n_{2} \cdot \left(2 \Delta\eta n_{0}^{2} n_{2}^{2} - \Delta\eta n_{0}^{2} + \Delta\eta n_{1}^{2} + \mu_{0} n_{0}^{2} n_{2}^{2} - \mu_{0} n_{0}^{2} - \mu_{1} n_{0}^{4} - \mu_{1} n_{0}^{2} n_{1}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)} & \frac{2 \left(\Delta\eta n_{0}^{4} + 2 \Delta\eta n_{0}^{2} n_{1}^{2} n_{2}^{2} + \Delta\eta n_{1}^{4} + \eta_{0} n_{2}^{2} - \eta_{0} + \mu_{0} n_{0}^{2} n_{1}^{2} n_{2}^{2} - \mu_{0} n_{0}^{2} n_{1}^{2} + \mu_{1} n_{0}^{2} n_{1}^{2} n_{2}^{2} - \mu_{1} n_{0}^{2} n_{1}^{2}\right)}{\left(n_{2} - 1\right) \left(n_{2} + 1\right)}\end{matrix}\right]\end{split}\]

The underworld / sympy implementation follows:

Underworld python script

## The full incompressible, trans-iso model (4 independent unknowns) 
## from Han and Wahr (1997)

# Extra viscosity terms
mu0 = sympy.symbols(r"\mu_0")
mu1 = sympy.symbols(r"\mu_1")

I = uw.maths.tensor.rank4_identity(3) * 0
C_IJm_HW = uw.maths.tensor.rank4_to_mandel(I, 3)
C_IJm_HW[0,0] = mu0 + 2 * eta0 
C_IJm_HW[0,1] = mu0  
C_IJm_HW[1,0] = mu0  
C_IJm_HW[1,1] = mu0 + 2 * eta0
C_IJm_HW[2,2] = mu1 + 2 * (eta0 - delta_eta)
C_IJm_HW[3,3] = 2 * (eta0 - delta_eta) # yz
C_IJm_HW[4,4] = 2 * (eta0 - delta_eta)  # xz
C_IJm_HW[5,5] = 2 * eta0  # xy

display(C_IJm_HW)

C_ijkl_HW = uw.maths.tensor.mandel_to_rank4(C_IJm_HW - C_IJm, 3)
display(uw.maths.tensor.rank4_to_mandel(C_ijkl_HW, 3)) 

C_ijkl_HW_R = sympy.simplify(uw.maths.tensor.tensor_rotation(R, C_ijkl_HW))
sympy.simplify(uw.maths.tensor.rank4_to_mandel(C_ijkl_HW_R, 3)) + C_IJm

display(C_IJm_HW_R)

## Maybe this can be simplified if we use the unit vector relationships among n0,n1,n2

C_IJm_HW_R_s1 = C_IJm_HW_R.subs(n[0]**2+n[1]**2+n[2]**2,1)
C_IJm_HW_R_s2 = C_IJm_HW_R_s1.subs(n[0]**2+n[1]**2, 1-n[2]**2).applyfunc(sympy.factor)
display(C_IJm_HW_R_s2)

## Perhaps that's not so helpful

Orthotropic medium

Note all the caveats above regarding incompressibility. The Browaeys & Chevrot (2004) elastic tensors have a bulk modulus term, so it is not completely obvious how to square the assumptions in the first two implementations with this set.

The full formulation should look like this:

\[\begin{split}\left[\begin{matrix}2 \eta_{00} & 2 \eta_{01} & 2 \eta_{02} & 0 & 0 & 0\\2 \eta_{01} & 2 \eta_{11} & 2 \eta_{12} & 0 & 0 & 0\\2 \eta_{02} & 2 \eta_{12} & 2 \eta_{22} & 0 & 0 & 0\\0 & 0 & 0 & 2 \eta_{33} & 0 & 0\\0 & 0 & 0 & 0 & 2 \eta_{44} & 0\\0 & 0 & 0 & 0 & 0 & 2 \eta_{55}\end{matrix}\right]\end{split}\]

Rotation in this case should be general as it is no longer enough to specify the symmetry plane.

\[\begin{split}\left[\begin{matrix}s_{0} & t_{0} & n_{0}\\s_{1} & t_{1} & n_{1}\\s_{2} & t_{2} & n_{2}\end{matrix}\right]\end{split}\]

\(\hat{\mathbf{n}}\), \(\hat{\mathbf{s}}\) and \(\hat{\mathbf{t}}\) are an arbitrary orthogonal triad of unit vectors (we keep the notation from the Mühlhaus formulation). It is probably not useful to code up this form.

\[\begin{split}\left[\begin{matrix}2 \eta_{00} s_{0}^{4} + 4 \eta_{01} s_{0}^{2} t_{0}^{2} + 4 \eta_{02} n_{0}^{2} s_{0}^{2} + 2 \eta_{11} t_{0}^{4} + 4 \eta_{12} n_{0}^{2} t_{0}^{2} + 2 \eta_{22} n_{0}^{4} + 4 \eta_{33} n_{0}^{2} t_{0}^{2} + 4 \eta_{44} n_{0}^{2} s_{0}^{2} + 4 \eta_{55} s_{0}^{2} t_{0}^{2} & 2 n_{0} \left(\eta_{33} n_{1} t_{0} t_{1} + \eta_{44} n_{1} s_{0} s_{1} + n_{0} \left(\eta_{02} s_{1}^{2} + \eta_{12} t_{1}^{2} + \eta_{22} n_{1}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{0} n_{1} s_{1} + \eta_{55} s_{1} t_{0} t_{1} + s_{0} \left(\eta_{00} s_{1}^{2} + \eta_{01} t_{1}^{2} + \eta_{02} n_{1}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{0} n_{1} t_{1} + \eta_{55} s_{0} s_{1} t_{1} + t_{0} \left(\eta_{01} s_{1}^{2} + \eta_{11} t_{1}^{2} + \eta_{12} n_{1}^{2}\right)\right) & 2 n_{0} \left(\eta_{33} n_{2} t_{0} t_{2} + \eta_{44} n_{2} s_{0} s_{2} + n_{0} \left(\eta_{02} s_{2}^{2} + \eta_{12} t_{2}^{2} + \eta_{22} n_{2}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{0} n_{2} s_{2} + \eta_{55} s_{2} t_{0} t_{2} + s_{0} \left(\eta_{00} s_{2}^{2} + \eta_{01} t_{2}^{2} + \eta_{02} n_{2}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{0} n_{2} t_{2} + \eta_{55} s_{0} s_{2} t_{2} + t_{0} \left(\eta_{01} s_{2}^{2} + \eta_{11} t_{2}^{2} + \eta_{12} n_{2}^{2}\right)\right) & \sqrt{2} \left(n_{0} \left(\eta_{33} t_{0} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{0} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{0} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + s_{0} \left(\eta_{44} n_{0} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{0} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{0} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + t_{0} \left(\eta_{33} n_{0} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{0} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{0} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{0} \left(\eta_{33} t_{0} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{0} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{0} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + s_{0} \left(\eta_{44} n_{0} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{0} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{0} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + t_{0} \left(\eta_{33} n_{0} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{0} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{0} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{0} \left(\eta_{33} t_{0} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{0} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{0} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + s_{0} \left(\eta_{44} n_{0} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{0} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{0} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + t_{0} \left(\eta_{33} n_{0} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{0} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{0} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\right)\\2 n_{1} \left(\eta_{33} n_{0} t_{0} t_{1} + \eta_{44} n_{0} s_{0} s_{1} + n_{1} \left(\eta_{02} s_{0}^{2} + \eta_{12} t_{0}^{2} + \eta_{22} n_{0}^{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{0} n_{1} s_{0} + \eta_{55} s_{0} t_{0} t_{1} + s_{1} \left(\eta_{00} s_{0}^{2} + \eta_{01} t_{0}^{2} + \eta_{02} n_{0}^{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{0} n_{1} t_{0} + \eta_{55} s_{0} s_{1} t_{0} + t_{1} \left(\eta_{01} s_{0}^{2} + \eta_{11} t_{0}^{2} + \eta_{12} n_{0}^{2}\right)\right) & 2 \eta_{00} s_{1}^{4} + 4 \eta_{01} s_{1}^{2} t_{1}^{2} + 4 \eta_{02} n_{1}^{2} s_{1}^{2} + 2 \eta_{11} t_{1}^{4} + 4 \eta_{12} n_{1}^{2} t_{1}^{2} + 2 \eta_{22} n_{1}^{4} + 4 \eta_{33} n_{1}^{2} t_{1}^{2} + 4 \eta_{44} n_{1}^{2} s_{1}^{2} + 4 \eta_{55} s_{1}^{2} t_{1}^{2} & 2 n_{1} \left(\eta_{33} n_{2} t_{1} t_{2} + \eta_{44} n_{2} s_{1} s_{2} + n_{1} \left(\eta_{02} s_{2}^{2} + \eta_{12} t_{2}^{2} + \eta_{22} n_{2}^{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{1} n_{2} s_{2} + \eta_{55} s_{2} t_{1} t_{2} + s_{1} \left(\eta_{00} s_{2}^{2} + \eta_{01} t_{2}^{2} + \eta_{02} n_{2}^{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{1} n_{2} t_{2} + \eta_{55} s_{1} s_{2} t_{2} + t_{1} \left(\eta_{01} s_{2}^{2} + \eta_{11} t_{2}^{2} + \eta_{12} n_{2}^{2}\right)\right) & \sqrt{2} \left(n_{1} \left(\eta_{33} t_{1} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{1} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{1} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + s_{1} \left(\eta_{44} n_{1} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{1} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{1} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + t_{1} \left(\eta_{33} n_{1} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{1} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{1} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{1} \left(\eta_{33} t_{1} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{1} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{1} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + s_{1} \left(\eta_{44} n_{1} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{1} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{1} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + t_{1} \left(\eta_{33} n_{1} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{1} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{1} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{1} \left(\eta_{33} t_{1} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{1} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{1} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + s_{1} \left(\eta_{44} n_{1} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{1} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{1} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + t_{1} \left(\eta_{33} n_{1} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{1} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{1} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\right)\\2 n_{2} \left(\eta_{33} n_{0} t_{0} t_{2} + \eta_{44} n_{0} s_{0} s_{2} + n_{2} \left(\eta_{02} s_{0}^{2} + \eta_{12} t_{0}^{2} + \eta_{22} n_{0}^{2}\right)\right) + 2 s_{2} \left(\eta_{44} n_{0} n_{2} s_{0} + \eta_{55} s_{0} t_{0} t_{2} + s_{2} \left(\eta_{00} s_{0}^{2} + \eta_{01} t_{0}^{2} + \eta_{02} n_{0}^{2}\right)\right) + 2 t_{2} \left(\eta_{33} n_{0} n_{2} t_{0} + \eta_{55} s_{0} s_{2} t_{0} + t_{2} \left(\eta_{01} s_{0}^{2} + \eta_{11} t_{0}^{2} + \eta_{12} n_{0}^{2}\right)\right) & 2 n_{2} \left(\eta_{33} n_{1} t_{1} t_{2} + \eta_{44} n_{1} s_{1} s_{2} + n_{2} \left(\eta_{02} s_{1}^{2} + \eta_{12} t_{1}^{2} + \eta_{22} n_{1}^{2}\right)\right) + 2 s_{2} \left(\eta_{44} n_{1} n_{2} s_{1} + \eta_{55} s_{1} t_{1} t_{2} + s_{2} \left(\eta_{00} s_{1}^{2} + \eta_{01} t_{1}^{2} + \eta_{02} n_{1}^{2}\right)\right) + 2 t_{2} \left(\eta_{33} n_{1} n_{2} t_{1} + \eta_{55} s_{1} s_{2} t_{1} + t_{2} \left(\eta_{01} s_{1}^{2} + \eta_{11} t_{1}^{2} + \eta_{12} n_{1}^{2}\right)\right) & 2 \eta_{00} s_{2}^{4} + 4 \eta_{01} s_{2}^{2} t_{2}^{2} + 4 \eta_{02} n_{2}^{2} s_{2}^{2} + 2 \eta_{11} t_{2}^{4} + 4 \eta_{12} n_{2}^{2} t_{2}^{2} + 2 \eta_{22} n_{2}^{4} + 4 \eta_{33} n_{2}^{2} t_{2}^{2} + 4 \eta_{44} n_{2}^{2} s_{2}^{2} + 4 \eta_{55} s_{2}^{2} t_{2}^{2} & \sqrt{2} \left(n_{2} \left(\eta_{33} t_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{2} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + s_{2} \left(\eta_{44} n_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{2} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + t_{2} \left(\eta_{33} n_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{2} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{2} \left(\eta_{33} t_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + s_{2} \left(\eta_{44} n_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + t_{2} \left(\eta_{33} n_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{2} \left(\eta_{33} t_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + s_{2} \left(\eta_{44} n_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + t_{2} \left(\eta_{33} n_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\right)\\\sqrt{2} \cdot \left(2 n_{1} \left(\eta_{33} n_{0} t_{0} t_{2} + \eta_{44} n_{0} s_{0} s_{2} + n_{2} \left(\eta_{02} s_{0}^{2} + \eta_{12} t_{0}^{2} + \eta_{22} n_{0}^{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{0} n_{2} s_{0} + \eta_{55} s_{0} t_{0} t_{2} + s_{2} \left(\eta_{00} s_{0}^{2} + \eta_{01} t_{0}^{2} + \eta_{02} n_{0}^{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{0} n_{2} t_{0} + \eta_{55} s_{0} s_{2} t_{0} + t_{2} \left(\eta_{01} s_{0}^{2} + \eta_{11} t_{0}^{2} + \eta_{12} n_{0}^{2}\right)\right)\right) & \sqrt{2} \cdot \left(2 n_{1} \left(\eta_{33} n_{1} t_{1} t_{2} + \eta_{44} n_{1} s_{1} s_{2} + n_{2} \left(\eta_{02} s_{1}^{2} + \eta_{12} t_{1}^{2} + \eta_{22} n_{1}^{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{1} n_{2} s_{1} + \eta_{55} s_{1} t_{1} t_{2} + s_{2} \left(\eta_{00} s_{1}^{2} + \eta_{01} t_{1}^{2} + \eta_{02} n_{1}^{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{1} n_{2} t_{1} + \eta_{55} s_{1} s_{2} t_{1} + t_{2} \left(\eta_{01} s_{1}^{2} + \eta_{11} t_{1}^{2} + \eta_{12} n_{1}^{2}\right)\right)\right) & \sqrt{2} \cdot \left(2 n_{1} n_{2} \left(\eta_{02} s_{2}^{2} + \eta_{12} t_{2}^{2} + \eta_{22} n_{2}^{2} + \eta_{33} t_{2}^{2} + \eta_{44} s_{2}^{2}\right) + 2 s_{1} s_{2} \left(\eta_{00} s_{2}^{2} + \eta_{01} t_{2}^{2} + \eta_{02} n_{2}^{2} + \eta_{44} n_{2}^{2} + \eta_{55} t_{2}^{2}\right) + 2 t_{1} t_{2} \left(\eta_{01} s_{2}^{2} + \eta_{11} t_{2}^{2} + \eta_{12} n_{2}^{2} + \eta_{33} n_{2}^{2} + \eta_{55} s_{2}^{2}\right)\right) & 2 n_{1} \left(\eta_{33} t_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{2} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{2} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{2} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right) & 2 n_{1} \left(\eta_{33} t_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right) & 2 n_{1} \left(\eta_{33} t_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + 2 s_{1} \left(\eta_{44} n_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + 2 t_{1} \left(\eta_{33} n_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\\\sqrt{2} \cdot \left(2 n_{0} \left(\eta_{33} n_{0} t_{0} t_{2} + \eta_{44} n_{0} s_{0} s_{2} + n_{2} \left(\eta_{02} s_{0}^{2} + \eta_{12} t_{0}^{2} + \eta_{22} n_{0}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{0} n_{2} s_{0} + \eta_{55} s_{0} t_{0} t_{2} + s_{2} \left(\eta_{00} s_{0}^{2} + \eta_{01} t_{0}^{2} + \eta_{02} n_{0}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{0} n_{2} t_{0} + \eta_{55} s_{0} s_{2} t_{0} + t_{2} \left(\eta_{01} s_{0}^{2} + \eta_{11} t_{0}^{2} + \eta_{12} n_{0}^{2}\right)\right)\right) & \sqrt{2} \cdot \left(2 n_{0} \left(\eta_{33} n_{1} t_{1} t_{2} + \eta_{44} n_{1} s_{1} s_{2} + n_{2} \left(\eta_{02} s_{1}^{2} + \eta_{12} t_{1}^{2} + \eta_{22} n_{1}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{1} n_{2} s_{1} + \eta_{55} s_{1} t_{1} t_{2} + s_{2} \left(\eta_{00} s_{1}^{2} + \eta_{01} t_{1}^{2} + \eta_{02} n_{1}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{1} n_{2} t_{1} + \eta_{55} s_{1} s_{2} t_{1} + t_{2} \left(\eta_{01} s_{1}^{2} + \eta_{11} t_{1}^{2} + \eta_{12} n_{1}^{2}\right)\right)\right) & \sqrt{2} \cdot \left(2 n_{0} n_{2} \left(\eta_{02} s_{2}^{2} + \eta_{12} t_{2}^{2} + \eta_{22} n_{2}^{2} + \eta_{33} t_{2}^{2} + \eta_{44} s_{2}^{2}\right) + 2 s_{0} s_{2} \left(\eta_{00} s_{2}^{2} + \eta_{01} t_{2}^{2} + \eta_{02} n_{2}^{2} + \eta_{44} n_{2}^{2} + \eta_{55} t_{2}^{2}\right) + 2 t_{0} t_{2} \left(\eta_{01} s_{2}^{2} + \eta_{11} t_{2}^{2} + \eta_{12} n_{2}^{2} + \eta_{33} n_{2}^{2} + \eta_{55} s_{2}^{2}\right)\right) & 2 n_{0} \left(\eta_{33} t_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{2} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{2} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{2} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right) & 2 n_{0} \left(\eta_{33} t_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right) & 2 n_{0} \left(\eta_{33} t_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + 2 s_{0} \left(\eta_{44} n_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + 2 t_{0} \left(\eta_{33} n_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\\\sqrt{2} \cdot \left(2 n_{0} \left(\eta_{33} n_{0} t_{0} t_{1} + \eta_{44} n_{0} s_{0} s_{1} + n_{1} \left(\eta_{02} s_{0}^{2} + \eta_{12} t_{0}^{2} + \eta_{22} n_{0}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{0} n_{1} s_{0} + \eta_{55} s_{0} t_{0} t_{1} + s_{1} \left(\eta_{00} s_{0}^{2} + \eta_{01} t_{0}^{2} + \eta_{02} n_{0}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{0} n_{1} t_{0} + \eta_{55} s_{0} s_{1} t_{0} + t_{1} \left(\eta_{01} s_{0}^{2} + \eta_{11} t_{0}^{2} + \eta_{12} n_{0}^{2}\right)\right)\right) & \sqrt{2} \cdot \left(2 n_{0} n_{1} \left(\eta_{02} s_{1}^{2} + \eta_{12} t_{1}^{2} + \eta_{22} n_{1}^{2} + \eta_{33} t_{1}^{2} + \eta_{44} s_{1}^{2}\right) + 2 s_{0} s_{1} \left(\eta_{00} s_{1}^{2} + \eta_{01} t_{1}^{2} + \eta_{02} n_{1}^{2} + \eta_{44} n_{1}^{2} + \eta_{55} t_{1}^{2}\right) + 2 t_{0} t_{1} \left(\eta_{01} s_{1}^{2} + \eta_{11} t_{1}^{2} + \eta_{12} n_{1}^{2} + \eta_{33} n_{1}^{2} + \eta_{55} s_{1}^{2}\right)\right) & \sqrt{2} \cdot \left(2 n_{0} \left(\eta_{33} n_{2} t_{1} t_{2} + \eta_{44} n_{2} s_{1} s_{2} + n_{1} \left(\eta_{02} s_{2}^{2} + \eta_{12} t_{2}^{2} + \eta_{22} n_{2}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{1} n_{2} s_{2} + \eta_{55} s_{2} t_{1} t_{2} + s_{1} \left(\eta_{00} s_{2}^{2} + \eta_{01} t_{2}^{2} + \eta_{02} n_{2}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{1} n_{2} t_{2} + \eta_{55} s_{1} s_{2} t_{2} + t_{1} \left(\eta_{01} s_{2}^{2} + \eta_{11} t_{2}^{2} + \eta_{12} n_{2}^{2}\right)\right)\right) & 2 n_{0} \left(\eta_{33} t_{1} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{1} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{1} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{1} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{1} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{1} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{1} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{1} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{1} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right) & 2 n_{0} \left(\eta_{33} t_{1} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{1} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{1} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{1} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{1} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{1} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{1} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{1} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{1} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right) & 2 n_{0} \left(\eta_{33} t_{1} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{1} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{1} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + 2 s_{0} \left(\eta_{44} n_{1} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{1} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{1} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + 2 t_{0} \left(\eta_{33} n_{1} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{1} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{1} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\end{matrix}\right]\end{split}\]

In the Browaeys formulation, the orthorhombic part has two unique values (in the Voigt notation this gives three unique entries in the \(C_{IJ}\) matrix). The canonical (Mandel) form is

\[\begin{split}\left[\begin{matrix}2 \xi_{0} & 0 & 2 \xi_{1} & 0 & 0 & 0\\0 & - 2 \xi_{0} & - 2 \xi_{1} & 0 & 0 & 0\\2 \xi_{1} & - 2 \xi_{1} & 0 & 0 & 0 & 0\\0 & 0 & 0 & - 2 \xi_{1} & 0 & 0\\0 & 0 & 0 & 0 & 2 \xi_{1} & 0\\0 & 0 & 0 & 0 & 0 & 0\end{matrix}\right]\end{split}\]

The rotated form:

\[\begin{split}\left[\begin{matrix}2 \eta_{00} s_{0}^{4} + 4 \eta_{01} s_{0}^{2} t_{0}^{2} + 4 \eta_{02} n_{0}^{2} s_{0}^{2} + 2 \eta_{11} t_{0}^{4} + 4 \eta_{12} n_{0}^{2} t_{0}^{2} + 2 \eta_{22} n_{0}^{4} + 4 \eta_{33} n_{0}^{2} t_{0}^{2} + 4 \eta_{44} n_{0}^{2} s_{0}^{2} + 4 \eta_{55} s_{0}^{2} t_{0}^{2} & 2 n_{0} \left(\eta_{33} n_{1} t_{0} t_{1} + \eta_{44} n_{1} s_{0} s_{1} + n_{0} \left(\eta_{02} s_{1}^{2} + \eta_{12} t_{1}^{2} + \eta_{22} n_{1}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{0} n_{1} s_{1} + \eta_{55} s_{1} t_{0} t_{1} + s_{0} \left(\eta_{00} s_{1}^{2} + \eta_{01} t_{1}^{2} + \eta_{02} n_{1}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{0} n_{1} t_{1} + \eta_{55} s_{0} s_{1} t_{1} + t_{0} \left(\eta_{01} s_{1}^{2} + \eta_{11} t_{1}^{2} + \eta_{12} n_{1}^{2}\right)\right) & 2 n_{0} \left(\eta_{33} n_{2} t_{0} t_{2} + \eta_{44} n_{2} s_{0} s_{2} + n_{0} \left(\eta_{02} s_{2}^{2} + \eta_{12} t_{2}^{2} + \eta_{22} n_{2}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{0} n_{2} s_{2} + \eta_{55} s_{2} t_{0} t_{2} + s_{0} \left(\eta_{00} s_{2}^{2} + \eta_{01} t_{2}^{2} + \eta_{02} n_{2}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{0} n_{2} t_{2} + \eta_{55} s_{0} s_{2} t_{2} + t_{0} \left(\eta_{01} s_{2}^{2} + \eta_{11} t_{2}^{2} + \eta_{12} n_{2}^{2}\right)\right) & \sqrt{2} \left(n_{0} \left(\eta_{33} t_{0} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{0} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{0} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + s_{0} \left(\eta_{44} n_{0} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{0} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{0} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + t_{0} \left(\eta_{33} n_{0} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{0} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{0} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{0} \left(\eta_{33} t_{0} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{0} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{0} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + s_{0} \left(\eta_{44} n_{0} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{0} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{0} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + t_{0} \left(\eta_{33} n_{0} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{0} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{0} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{0} \left(\eta_{33} t_{0} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{0} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{0} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + s_{0} \left(\eta_{44} n_{0} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{0} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{0} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + t_{0} \left(\eta_{33} n_{0} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{0} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{0} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\right)\\2 n_{1} \left(\eta_{33} n_{0} t_{0} t_{1} + \eta_{44} n_{0} s_{0} s_{1} + n_{1} \left(\eta_{02} s_{0}^{2} + \eta_{12} t_{0}^{2} + \eta_{22} n_{0}^{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{0} n_{1} s_{0} + \eta_{55} s_{0} t_{0} t_{1} + s_{1} \left(\eta_{00} s_{0}^{2} + \eta_{01} t_{0}^{2} + \eta_{02} n_{0}^{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{0} n_{1} t_{0} + \eta_{55} s_{0} s_{1} t_{0} + t_{1} \left(\eta_{01} s_{0}^{2} + \eta_{11} t_{0}^{2} + \eta_{12} n_{0}^{2}\right)\right) & 2 \eta_{00} s_{1}^{4} + 4 \eta_{01} s_{1}^{2} t_{1}^{2} + 4 \eta_{02} n_{1}^{2} s_{1}^{2} + 2 \eta_{11} t_{1}^{4} + 4 \eta_{12} n_{1}^{2} t_{1}^{2} + 2 \eta_{22} n_{1}^{4} + 4 \eta_{33} n_{1}^{2} t_{1}^{2} + 4 \eta_{44} n_{1}^{2} s_{1}^{2} + 4 \eta_{55} s_{1}^{2} t_{1}^{2} & 2 n_{1} \left(\eta_{33} n_{2} t_{1} t_{2} + \eta_{44} n_{2} s_{1} s_{2} + n_{1} \left(\eta_{02} s_{2}^{2} + \eta_{12} t_{2}^{2} + \eta_{22} n_{2}^{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{1} n_{2} s_{2} + \eta_{55} s_{2} t_{1} t_{2} + s_{1} \left(\eta_{00} s_{2}^{2} + \eta_{01} t_{2}^{2} + \eta_{02} n_{2}^{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{1} n_{2} t_{2} + \eta_{55} s_{1} s_{2} t_{2} + t_{1} \left(\eta_{01} s_{2}^{2} + \eta_{11} t_{2}^{2} + \eta_{12} n_{2}^{2}\right)\right) & \sqrt{2} \left(n_{1} \left(\eta_{33} t_{1} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{1} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{1} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + s_{1} \left(\eta_{44} n_{1} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{1} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{1} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + t_{1} \left(\eta_{33} n_{1} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{1} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{1} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{1} \left(\eta_{33} t_{1} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{1} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{1} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + s_{1} \left(\eta_{44} n_{1} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{1} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{1} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + t_{1} \left(\eta_{33} n_{1} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{1} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{1} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{1} \left(\eta_{33} t_{1} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{1} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{1} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + s_{1} \left(\eta_{44} n_{1} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{1} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{1} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + t_{1} \left(\eta_{33} n_{1} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{1} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{1} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\right)\\2 n_{2} \left(\eta_{33} n_{0} t_{0} t_{2} + \eta_{44} n_{0} s_{0} s_{2} + n_{2} \left(\eta_{02} s_{0}^{2} + \eta_{12} t_{0}^{2} + \eta_{22} n_{0}^{2}\right)\right) + 2 s_{2} \left(\eta_{44} n_{0} n_{2} s_{0} + \eta_{55} s_{0} t_{0} t_{2} + s_{2} \left(\eta_{00} s_{0}^{2} + \eta_{01} t_{0}^{2} + \eta_{02} n_{0}^{2}\right)\right) + 2 t_{2} \left(\eta_{33} n_{0} n_{2} t_{0} + \eta_{55} s_{0} s_{2} t_{0} + t_{2} \left(\eta_{01} s_{0}^{2} + \eta_{11} t_{0}^{2} + \eta_{12} n_{0}^{2}\right)\right) & 2 n_{2} \left(\eta_{33} n_{1} t_{1} t_{2} + \eta_{44} n_{1} s_{1} s_{2} + n_{2} \left(\eta_{02} s_{1}^{2} + \eta_{12} t_{1}^{2} + \eta_{22} n_{1}^{2}\right)\right) + 2 s_{2} \left(\eta_{44} n_{1} n_{2} s_{1} + \eta_{55} s_{1} t_{1} t_{2} + s_{2} \left(\eta_{00} s_{1}^{2} + \eta_{01} t_{1}^{2} + \eta_{02} n_{1}^{2}\right)\right) + 2 t_{2} \left(\eta_{33} n_{1} n_{2} t_{1} + \eta_{55} s_{1} s_{2} t_{1} + t_{2} \left(\eta_{01} s_{1}^{2} + \eta_{11} t_{1}^{2} + \eta_{12} n_{1}^{2}\right)\right) & 2 \eta_{00} s_{2}^{4} + 4 \eta_{01} s_{2}^{2} t_{2}^{2} + 4 \eta_{02} n_{2}^{2} s_{2}^{2} + 2 \eta_{11} t_{2}^{4} + 4 \eta_{12} n_{2}^{2} t_{2}^{2} + 2 \eta_{22} n_{2}^{4} + 4 \eta_{33} n_{2}^{2} t_{2}^{2} + 4 \eta_{44} n_{2}^{2} s_{2}^{2} + 4 \eta_{55} s_{2}^{2} t_{2}^{2} & \sqrt{2} \left(n_{2} \left(\eta_{33} t_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{2} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + s_{2} \left(\eta_{44} n_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{2} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + t_{2} \left(\eta_{33} n_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{2} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{2} \left(\eta_{33} t_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + s_{2} \left(\eta_{44} n_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + t_{2} \left(\eta_{33} n_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right)\right) & \sqrt{2} \left(n_{2} \left(\eta_{33} t_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + s_{2} \left(\eta_{44} n_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + t_{2} \left(\eta_{33} n_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\right)\\\sqrt{2} \cdot \left(2 n_{1} \left(\eta_{33} n_{0} t_{0} t_{2} + \eta_{44} n_{0} s_{0} s_{2} + n_{2} \left(\eta_{02} s_{0}^{2} + \eta_{12} t_{0}^{2} + \eta_{22} n_{0}^{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{0} n_{2} s_{0} + \eta_{55} s_{0} t_{0} t_{2} + s_{2} \left(\eta_{00} s_{0}^{2} + \eta_{01} t_{0}^{2} + \eta_{02} n_{0}^{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{0} n_{2} t_{0} + \eta_{55} s_{0} s_{2} t_{0} + t_{2} \left(\eta_{01} s_{0}^{2} + \eta_{11} t_{0}^{2} + \eta_{12} n_{0}^{2}\right)\right)\right) & \sqrt{2} \cdot \left(2 n_{1} \left(\eta_{33} n_{1} t_{1} t_{2} + \eta_{44} n_{1} s_{1} s_{2} + n_{2} \left(\eta_{02} s_{1}^{2} + \eta_{12} t_{1}^{2} + \eta_{22} n_{1}^{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{1} n_{2} s_{1} + \eta_{55} s_{1} t_{1} t_{2} + s_{2} \left(\eta_{00} s_{1}^{2} + \eta_{01} t_{1}^{2} + \eta_{02} n_{1}^{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{1} n_{2} t_{1} + \eta_{55} s_{1} s_{2} t_{1} + t_{2} \left(\eta_{01} s_{1}^{2} + \eta_{11} t_{1}^{2} + \eta_{12} n_{1}^{2}\right)\right)\right) & \sqrt{2} \cdot \left(2 n_{1} n_{2} \left(\eta_{02} s_{2}^{2} + \eta_{12} t_{2}^{2} + \eta_{22} n_{2}^{2} + \eta_{33} t_{2}^{2} + \eta_{44} s_{2}^{2}\right) + 2 s_{1} s_{2} \left(\eta_{00} s_{2}^{2} + \eta_{01} t_{2}^{2} + \eta_{02} n_{2}^{2} + \eta_{44} n_{2}^{2} + \eta_{55} t_{2}^{2}\right) + 2 t_{1} t_{2} \left(\eta_{01} s_{2}^{2} + \eta_{11} t_{2}^{2} + \eta_{12} n_{2}^{2} + \eta_{33} n_{2}^{2} + \eta_{55} s_{2}^{2}\right)\right) & 2 n_{1} \left(\eta_{33} t_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{2} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{2} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{2} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right) & 2 n_{1} \left(\eta_{33} t_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + 2 s_{1} \left(\eta_{44} n_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + 2 t_{1} \left(\eta_{33} n_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right) & 2 n_{1} \left(\eta_{33} t_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + 2 s_{1} \left(\eta_{44} n_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + 2 t_{1} \left(\eta_{33} n_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\\\sqrt{2} \cdot \left(2 n_{0} \left(\eta_{33} n_{0} t_{0} t_{2} + \eta_{44} n_{0} s_{0} s_{2} + n_{2} \left(\eta_{02} s_{0}^{2} + \eta_{12} t_{0}^{2} + \eta_{22} n_{0}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{0} n_{2} s_{0} + \eta_{55} s_{0} t_{0} t_{2} + s_{2} \left(\eta_{00} s_{0}^{2} + \eta_{01} t_{0}^{2} + \eta_{02} n_{0}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{0} n_{2} t_{0} + \eta_{55} s_{0} s_{2} t_{0} + t_{2} \left(\eta_{01} s_{0}^{2} + \eta_{11} t_{0}^{2} + \eta_{12} n_{0}^{2}\right)\right)\right) & \sqrt{2} \cdot \left(2 n_{0} \left(\eta_{33} n_{1} t_{1} t_{2} + \eta_{44} n_{1} s_{1} s_{2} + n_{2} \left(\eta_{02} s_{1}^{2} + \eta_{12} t_{1}^{2} + \eta_{22} n_{1}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{1} n_{2} s_{1} + \eta_{55} s_{1} t_{1} t_{2} + s_{2} \left(\eta_{00} s_{1}^{2} + \eta_{01} t_{1}^{2} + \eta_{02} n_{1}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{1} n_{2} t_{1} + \eta_{55} s_{1} s_{2} t_{1} + t_{2} \left(\eta_{01} s_{1}^{2} + \eta_{11} t_{1}^{2} + \eta_{12} n_{1}^{2}\right)\right)\right) & \sqrt{2} \cdot \left(2 n_{0} n_{2} \left(\eta_{02} s_{2}^{2} + \eta_{12} t_{2}^{2} + \eta_{22} n_{2}^{2} + \eta_{33} t_{2}^{2} + \eta_{44} s_{2}^{2}\right) + 2 s_{0} s_{2} \left(\eta_{00} s_{2}^{2} + \eta_{01} t_{2}^{2} + \eta_{02} n_{2}^{2} + \eta_{44} n_{2}^{2} + \eta_{55} t_{2}^{2}\right) + 2 t_{0} t_{2} \left(\eta_{01} s_{2}^{2} + \eta_{11} t_{2}^{2} + \eta_{12} n_{2}^{2} + \eta_{33} n_{2}^{2} + \eta_{55} s_{2}^{2}\right)\right) & 2 n_{0} \left(\eta_{33} t_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{2} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{2} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{2} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{2} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{2} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{2} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right) & 2 n_{0} \left(\eta_{33} t_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{2} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{2} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right) & 2 n_{0} \left(\eta_{33} t_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{2} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + 2 s_{0} \left(\eta_{44} n_{2} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{2} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + 2 t_{0} \left(\eta_{33} n_{2} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{2} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{2} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\\\sqrt{2} \cdot \left(2 n_{0} \left(\eta_{33} n_{0} t_{0} t_{1} + \eta_{44} n_{0} s_{0} s_{1} + n_{1} \left(\eta_{02} s_{0}^{2} + \eta_{12} t_{0}^{2} + \eta_{22} n_{0}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{0} n_{1} s_{0} + \eta_{55} s_{0} t_{0} t_{1} + s_{1} \left(\eta_{00} s_{0}^{2} + \eta_{01} t_{0}^{2} + \eta_{02} n_{0}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{0} n_{1} t_{0} + \eta_{55} s_{0} s_{1} t_{0} + t_{1} \left(\eta_{01} s_{0}^{2} + \eta_{11} t_{0}^{2} + \eta_{12} n_{0}^{2}\right)\right)\right) & \sqrt{2} \cdot \left(2 n_{0} n_{1} \left(\eta_{02} s_{1}^{2} + \eta_{12} t_{1}^{2} + \eta_{22} n_{1}^{2} + \eta_{33} t_{1}^{2} + \eta_{44} s_{1}^{2}\right) + 2 s_{0} s_{1} \left(\eta_{00} s_{1}^{2} + \eta_{01} t_{1}^{2} + \eta_{02} n_{1}^{2} + \eta_{44} n_{1}^{2} + \eta_{55} t_{1}^{2}\right) + 2 t_{0} t_{1} \left(\eta_{01} s_{1}^{2} + \eta_{11} t_{1}^{2} + \eta_{12} n_{1}^{2} + \eta_{33} n_{1}^{2} + \eta_{55} s_{1}^{2}\right)\right) & \sqrt{2} \cdot \left(2 n_{0} \left(\eta_{33} n_{2} t_{1} t_{2} + \eta_{44} n_{2} s_{1} s_{2} + n_{1} \left(\eta_{02} s_{2}^{2} + \eta_{12} t_{2}^{2} + \eta_{22} n_{2}^{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{1} n_{2} s_{2} + \eta_{55} s_{2} t_{1} t_{2} + s_{1} \left(\eta_{00} s_{2}^{2} + \eta_{01} t_{2}^{2} + \eta_{02} n_{2}^{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{1} n_{2} t_{2} + \eta_{55} s_{1} s_{2} t_{2} + t_{1} \left(\eta_{01} s_{2}^{2} + \eta_{11} t_{2}^{2} + \eta_{12} n_{2}^{2}\right)\right)\right) & 2 n_{0} \left(\eta_{33} t_{1} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{44} s_{1} \left(n_{1} s_{2} + n_{2} s_{1}\right) + 2 n_{1} \left(\eta_{02} s_{1} s_{2} + \eta_{12} t_{1} t_{2} + \eta_{22} n_{1} n_{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{1} \left(n_{1} s_{2} + n_{2} s_{1}\right) + \eta_{55} t_{1} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 s_{1} \left(\eta_{00} s_{1} s_{2} + \eta_{01} t_{1} t_{2} + \eta_{02} n_{1} n_{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{1} \left(n_{1} t_{2} + n_{2} t_{1}\right) + \eta_{55} s_{1} \left(s_{1} t_{2} + s_{2} t_{1}\right) + 2 t_{1} \left(\eta_{01} s_{1} s_{2} + \eta_{11} t_{1} t_{2} + \eta_{12} n_{1} n_{2}\right)\right) & 2 n_{0} \left(\eta_{33} t_{1} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{44} s_{1} \left(n_{0} s_{2} + n_{2} s_{0}\right) + 2 n_{1} \left(\eta_{02} s_{0} s_{2} + \eta_{12} t_{0} t_{2} + \eta_{22} n_{0} n_{2}\right)\right) + 2 s_{0} \left(\eta_{44} n_{1} \left(n_{0} s_{2} + n_{2} s_{0}\right) + \eta_{55} t_{1} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 s_{1} \left(\eta_{00} s_{0} s_{2} + \eta_{01} t_{0} t_{2} + \eta_{02} n_{0} n_{2}\right)\right) + 2 t_{0} \left(\eta_{33} n_{1} \left(n_{0} t_{2} + n_{2} t_{0}\right) + \eta_{55} s_{1} \left(s_{0} t_{2} + s_{2} t_{0}\right) + 2 t_{1} \left(\eta_{01} s_{0} s_{2} + \eta_{11} t_{0} t_{2} + \eta_{12} n_{0} n_{2}\right)\right) & 2 n_{0} \left(\eta_{33} t_{1} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{44} s_{1} \left(n_{0} s_{1} + n_{1} s_{0}\right) + 2 n_{1} \left(\eta_{02} s_{0} s_{1} + \eta_{12} t_{0} t_{1} + \eta_{22} n_{0} n_{1}\right)\right) + 2 s_{0} \left(\eta_{44} n_{1} \left(n_{0} s_{1} + n_{1} s_{0}\right) + \eta_{55} t_{1} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 s_{1} \left(\eta_{00} s_{0} s_{1} + \eta_{01} t_{0} t_{1} + \eta_{02} n_{0} n_{1}\right)\right) + 2 t_{0} \left(\eta_{33} n_{1} \left(n_{0} t_{1} + n_{1} t_{0}\right) + \eta_{55} s_{1} \left(s_{0} t_{1} + s_{1} t_{0}\right) + 2 t_{1} \left(\eta_{01} s_{0} s_{1} + \eta_{11} t_{0} t_{1} + \eta_{12} n_{0} n_{1}\right)\right)\end{matrix}\right] \end{split}\]

The python code for this is:

Underworld python script

## I can't see much benefit in expanding the full orthorhombic model
## in this way - keeping the rotations separated makes more sense.

eta_00 = sympy.symbols(r"\eta_00")
eta_11 = sympy.symbols(r"\eta_11")
eta_22 = sympy.symbols(r"\eta_22")
eta_33 = sympy.symbols(r"\eta_33")
eta_44 = sympy.symbols(r"\eta_44")
eta_55 = sympy.symbols(r"\eta_55")
eta_01 = sympy.symbols(r"\eta_01")
eta_02 = sympy.symbols(r"\eta_02")
eta_12 = sympy.symbols(r"\eta_12")

I_ijkl = uw.maths.tensor.rank4_identity(3) * 0
C_IJm_ORTHO = uw.maths.tensor.rank4_to_mandel(I_ijkl, 3)

C_IJm_ORTHO[0,0] = 2*eta_00
C_IJm_ORTHO[1,1] = 2*eta_11
C_IJm_ORTHO[2,2] = 2*eta_22
C_IJm_ORTHO[3,3] = 2*eta_33  # yz
C_IJm_ORTHO[4,4] = 2*eta_44  # xz
C_IJm_ORTHO[5,5] = 2*eta_55  # xy
C_IJm_ORTHO[0,1] = C_IJm_ORTHO[1,0] = 2*eta_01
C_IJm_ORTHO[0,2] = C_IJm_ORTHO[2,0] = 2*eta_02
C_IJm_ORTHO[1,2] = C_IJm_ORTHO[2,1] = 2*eta_12

C_ijkl_ORTHO = uw.maths.tensor.mandel_to_rank4(C_IJm_ORTHO, 3)
C_IJv_ORTHO = sympy.simplify(uw.maths.tensor.rank4_to_voigt(C_ijkl_ORTHO, 3))

display(C_IJv_ORTHO)

# Rotation: Use 3 orthogonal unit vectors to define cannonical orientation

n = sympy.Matrix(sympy.symarray("n",(3,)))
s = sympy.Matrix(sympy.symarray("s",(3,)))
t = sympy.Matrix(sympy.symarray("t",(3,)))

# This would work but is less clear in terms of notation
# t = -mesh3.vector.cross(n,s).T # complete the coordinate triad

Rx = sympy.BlockMatrix((s,t,n)).as_explicit()

display(Rx)

C_ijkl_ORTHO = uw.maths.tensor.mandel_to_rank4(C_IJm_ORTHO, 3)
C_ijkl_ORTHO_R = sympy.simplify(uw.maths.tensor.tensor_rotation(Rx, C_ijkl_ORTHO))
uw.maths.tensor.rank4_to_mandel(C_ijkl_ORTHO_R, 3)

# print(sympy.latex(uw.maths.tensor.rank4_to_mandel(C_ijkl_ORTHO_R, 3)))

xi_0 = sympy.symbols(r"\xi_0")
xi_1 = sympy.symbols(r"\xi_0")

I_ijkl = uw.maths.tensor.rank4_identity(3) * 0
C_IJm_BC = uw.maths.tensor.rank4_to_mandel(I_ijkl, 3)

C_IJm_BC[0,0] =  2*xi_0
C_IJm_BC[1,1] = -2*xi_0
C_IJm_BC[3,3] = -2*xi_1  # yz
C_IJm_BC[4,4] = 2*xi_1  # xz
C_IJm_BC[5,5] = 0
C_IJm_BC[0,2] = C_IJm_BC[2,0] = 2*xi_1
C_IJm_BC[1,2] = C_IJm_BC[2,1] = -2*xi_1

C_ijkl_BC = uw.maths.tensor.mandel_to_rank4(C_IJm_BC, 3)
C_IJv_BC = sympy.simplify(uw.maths.tensor.rank4_to_voigt(C_ijkl_BC, 3))

# print(sympy.latex(C_IJm_BC))

C_IJv_BC

# Rotation by Rx

C_ijkl_BC = uw.maths.tensor.mandel_to_rank4(C_IJm_BC, 3)
C_ijkl_BC_R = sympy.simplify(uw.maths.tensor.tensor_rotation(Rx, C_ijkl_ORTHO))
uw.maths.tensor.rank4_to_mandel(C_ijkl_BC_R, 3)

# print(sympy.latex(uw.maths.tensor.rank4_to_mandel(C_ijkl_BC_R, 3)))