$\begin{split}&\frac{1}{2}\Big{[}c_{1111}\varepsilon_{11}^{2}+4c_{1112}\varepsilon_{11}\varepsilon_{12}+2c_{1113}\varepsilon_{11}\nabla
A_{13}^{*}\cdot\bar{\mathbold{\varepsilon}}+2c_{1122}\varepsilon_{11}\varepsilon_{22}+2c_{1123}\varepsilon_{11}\nabla
A_{23}^{*}\cdot\bar{\mathbold{\varepsilon}}\\\
&+2c_{1133}\varepsilon_{11}\nabla
A_{33}^{*}\cdot\bar{\mathbold{\varepsilon}}+4c_{1212}\varepsilon_{12}^{2}+4c_{1213}\varepsilon_{12}\nabla
A_{13}^{*}\cdot\bar{\mathbold{\varepsilon}}+4c_{1222}\varepsilon_{12}\varepsilon_{22}+4c_{1223}\varepsilon_{12}\nabla
A_{23}^{*}\cdot\bar{\mathbold{\varepsilon}}\\\
&+4c_{1233}\varepsilon_{12}\nabla
A_{33}^{*}\cdot\bar{\mathbold{\varepsilon}}+c_{1313}\left(\nabla
A_{13}^{*}\cdot\bar{\mathbold{\varepsilon}}\right)^{2}+2c_{1322}\nabla
A_{13}^{*}\cdot\bar{\mathbold{\varepsilon}}\varepsilon_{22}+2c_{1323}\nabla
A_{13}^{*}\cdot\bar{\mathbold{\varepsilon}}\nabla
A_{23}^{*}\cdot\bar{\mathbold{\varepsilon}}\\\ &+2c_{1333}\nabla
A_{13}^{*}\cdot\bar{\mathbold{\varepsilon}}\nabla
A_{33}^{*}\cdot\bar{\mathbold{\varepsilon}}+c_{2222}\varepsilon_{22}^{2}+2c_{2223}\varepsilon_{22}\nabla
A_{23}^{*}\cdot\bar{\mathbold{\varepsilon}}+2c_{2233}\varepsilon_{22}\nabla
A_{33}^{*}\cdot\bar{\mathbold{\varepsilon}}+c_{2323}\left(\nabla
A_{23}^{*}\cdot\bar{\mathbold{\varepsilon}}\right)^{2}\\\ &+2c_{2333}\nabla
A_{23}^{*}\cdot\bar{\mathbold{\varepsilon}}\nabla
A_{33}^{*}\cdot\bar{\mathbold{\varepsilon}}+c_{3333}\left(\nabla
A_{33}^{*}\cdot\bar{\mathbold{\varepsilon}}\right)^{2}\Big{]},\end{split}$
(47)

if $\displaystyle\varepsilon_{33}\geq\nabla
A_{33}^{*}\cdot\bar{\mathbold{\varepsilon}}$, while it equals

$\begin{split}&\frac{1}{2}\Big{[}c_{1111}\varepsilon_{11}^{2}+4c_{1112}\varepsilon_{11}\varepsilon_{12}+2c_{1113}\varepsilon_{11}\nabla
A_{13}^{**}\cdot\hat{\mathbold{\varepsilon}}+2c_{1122}\varepsilon_{11}\varepsilon_{22}+2c_{1123}\varepsilon_{11}\nabla
A_{23}^{**}\cdot\hat{\mathbold{\varepsilon}}\\\
&+2c_{1133}\varepsilon_{11}\varepsilon_{33}+4c_{1212}\varepsilon_{12}^{2}+4c_{1213}\varepsilon_{12}\nabla
A_{13}^{**}\cdot\hat{\mathbold{\varepsilon}}+4c_{1222}\varepsilon_{12}\varepsilon_{22}+4c_{1223}\varepsilon_{12}\nabla
A_{23}^{**}\cdot\hat{\mathbold{\varepsilon}}\\\
&+4c_{1233}\varepsilon_{12}\varepsilon_{33}+c_{1313}\left(\nabla
A_{13}^{**}\cdot\hat{\mathbold{\varepsilon}}\right)^{2}+2c_{1322}\nabla
A_{13}^{**}\cdot\hat{\mathbold{\varepsilon}}\varepsilon_{22}+2c_{1323}\nabla
A_{13}^{**}\cdot\hat{\mathbold{\varepsilon}}\nabla
A_{23}^{**}\cdot\hat{\mathbold{\varepsilon}}\\\ &+2c_{1333}\nabla
A_{13}^{**}\cdot\hat{\mathbold{\varepsilon}}\varepsilon_{33}+c_{2222}\varepsilon_{22}^{2}+2c_{2223}\varepsilon_{22}\nabla
A_{23}^{**}\cdot\hat{\mathbold{\varepsilon}}+2c_{2233}\varepsilon_{22}\varepsilon_{33}+c_{2323}\left(\nabla
A_{23}^{**}\cdot\hat{\mathbold{\varepsilon}}\right)^{2}\\\ &+2c_{2333}\nabla
A_{23}^{**}\cdot\hat{\mathbold{\varepsilon}}\varepsilon_{33}+c_{3333}\varepsilon_{33}^{2}\Big{]},\end{split}$
(48)

if $\displaystyle\varepsilon_{33}<\nabla
A_{33}^{*}\cdot\bar{\mathbold{\varepsilon}}$.

In the presence of isotropy, the final formula is

$\boxed{\begin{aligned}
&W_{\mathrm{d},\mathrm{lin}}\left(\mathbold{\varepsilon},{\mathbold
e}_{3}\right)=\\\
&\begin{cases}\textstyle\frac{1}{2}\left(\lambda+2\mu+\frac{3\lambda^{2}}{\lambda+2\mu}\right)(\varepsilon_{11}^{2}+\varepsilon_{22}^{2})+\left(\lambda+\frac{3\lambda^{2}}{\lambda+2\mu}\right)\varepsilon_{11}\varepsilon_{22}+2\mu\varepsilon_{12}^{2}&\text{if
}\varepsilon_{33}\geq\frac{\lambda}{\lambda+2\mu}(\varepsilon_{11}+\varepsilon_{22}),\\\
\textstyle\frac{\lambda+2\mu}{2}(\varepsilon_{11}^{2}+\varepsilon_{22}^{2}+\varepsilon_{33}^{2})+\lambda(\varepsilon_{11}\varepsilon_{22}+\varepsilon_{11}\varepsilon_{33}+\varepsilon_{22}\varepsilon_{33})+2\mu\varepsilon_{12}^{2}&\text{if
}\varepsilon_{33}<\frac{\lambda}{\lambda+2\mu}(\varepsilon_{11}+\varepsilon_{22}),\end{cases}\end{aligned}}$
(49)

and the formula for a general unit vector $\displaystyle{\mathbold n}$ is
calculated as follows: we consider any $\displaystyle{\mathbold Q}\in SO(3)$
such that $\displaystyle{\mathbold Q}{\mathbold n}={\mathbold e}_{3}$, as in
(34), and use that

$W_{\mathrm{d},\mathrm{lin}}(\mathbold{\varepsilon},{\mathbold
n})=W_{\mathrm{d},\mathrm{lin}}({\mathbold Q}\mathbold{\varepsilon}{\mathbold
Q}^{T},{\mathbold e}_{3}).$

If $\displaystyle W$ is not isotropic, then $\displaystyle
W_{\mathrm{d},\mathrm{lin}}(\mathbold{\varepsilon},{\mathbold n})$ is
calculated as in (47)–(48), but replacing $\displaystyle c_{ijkl}$ with
$\displaystyle\tilde{c}_{ijkl}$ and $\displaystyle\varepsilon_{ij}$ with
$\displaystyle\tilde{\varepsilon}_{ij}$, similarly to Subsection 7.A.

##  8 Numerical Implementation

We use the finite element method to compute numerical solutions, implemented
in the open source FEniCS library, which is now used widely for problems in
mechanics, e.g., [46, 11]; our implementation adapts the code from [39]. We
use $\displaystyle{\mathbold y}({\mathbold x})$ and $\displaystyle{\mathbold
d}({\mathbold x})$ as the primary unknown functions, and approximate them
using standard piecewise linear elements. We consider two classes of problems.

First, we compute the energy-minimizing deformation $\displaystyle{\mathbold
y}$, keeping $\displaystyle{\mathbold d}$ fixed at a prescribed configuration
that corresponds roughly to a crack; we refer to this type of problem as a
“frozen crack”. For these problems, the energy is minimized under load using
the FEniCS Dolfin Adjoint library [50].

Second, we compute the fully-coupled problem of crack growth, wherein both
$\displaystyle{\mathbold y}$ and $\displaystyle{\mathbold d}$ evolve to
minimize the energy under a time-dependent external loading. To compute the
minimization at each load step, we find that alternate minimization over
$\displaystyle{\mathbold y}$ and $\displaystyle{\mathbold d}$ works well,
following [39]. Specifically, at each load step, we perform first a
minimization over $\displaystyle{\mathbold y}$ with $\displaystyle{\mathbold
d}$ fixed, and then minimize over $\displaystyle{\mathbold d}$ with
$\displaystyle{\mathbold y}$ fixed, and alternate this until we converge. That
is, given the iterates $\displaystyle{\mathbold y}^{i}$ and
$\displaystyle{\mathbold d}^{i}$, we obtain the next iterate using:

$\begin{split}&{\mathbold y}^{i+1}=\operatorname*{argmin}_{{\mathbold
y}}E[{\mathbold y},{\mathbold d}^{i}],\\\ &{\mathbold
d}^{i+1}=\operatorname*{argmin}_{{\mathbold d}}E[{\mathbold
y}^{i+1},{\mathbold d}]\end{split}$ (50)

and we repeat until $\displaystyle\max\\{\lVert{\mathbold y}^{i+1}-{\mathbold
y}^{i}\rVert,\lVert{\mathbold d}^{i+1}-{\mathbold d}^{i}\rVert\\}<10^{-3}$.
Once we reach a converged solution, we update the time-dependent load.

An important aspect of crack growth is that the minimization over
$\displaystyle{\mathbold d}$ must respect the physics that cracks do not heal
even when the load is reversed. Following standard approaches, e.g. [12], we
implement this irreversibility as follows. We first choose a critical value
$\displaystyle d_{c}$, taken to be $\displaystyle 0.95$ in this paper. If the
converged vector field $\displaystyle{\mathbold d}_{0}$ in a loading step
satisfies $\displaystyle|{\mathbold d}_{0}|\geq d_{c}$, we impose the
pointwise constraint $\displaystyle{\mathbold d}=\frac{{\mathbold
d}_{0}}{|{\mathbold d}_{0}|}$ for all future times, and minimize over
$\displaystyle{\mathbold d}$ that respect this constraint.

In addition to preventing the magnitude of $\displaystyle{\mathbold d}$ from
decreasing once it satisfies $\displaystyle|{\mathbold d}|\geq d_{c}$, the
constraint also prevents the crack normal from changing orientation. This is
important in the setting considered in this paper. Specifically, if we did not
prevent the orientation of the crack from changing, we could concieve of a
sequence of loading steps that would lead to the unphysical result that a
crack normal changes orientation even if the crack – defined by the magnitude
of $\displaystyle{\mathbold d}$ – is fixed.

We use $\displaystyle\epsilon=0.015$ in a nondimensionalized length scale for
the phase-field regularization parameter introduced in (5). Given this value
of $\displaystyle\epsilon$, we use a mesh that is sufficiently refined as to
resolve the interfaces.

##  9 Numerical Calculations

We solve the model numerically for three settings: first, we look at the
elastic response of frozen cracks in which $\displaystyle{\mathbold d}$ is
held fixed to a given configuration, as a means of testing the efficacy of the
model; second, we look at a configuration with cyclic shear loading that leads
to crack closure as well as crack branching; and, third, we look at fracture
in a cavity under cyclic loading that leads to crack closure and the growth of
multiple cracks, motivated by recent experiments on similar settings.

All of the settings considered above use the large-deformation neo-Hookean
model in 2D (Section 4.A). We choose material parameters corresponding roughly
to PMMA, which is a common material for experimental research [61]. We use
$\displaystyle\lambda=$\displaystyle 2.576\text{\,}\mathrm{GPa}$$,
$\displaystyle\mu=$\displaystyle 1.104\text{\,}\mathrm{GPa}$$, and
$\displaystyle G_{c}=$\displaystyle
285\text{\,}\mathrm{N}\text{\,}{\mathrm{m}}^{-1}$$ for the frozen cracks. For
growing cracks, we set $\displaystyle\lambda=0$ to avoid Poisson’s ratio
effects. All plots of energy density and stress are normalized by
$\displaystyle\mu$.

###  9.A Mechanical Response of a frozen crack

Figure 4 shows the specimen the chosen configuration for
$\displaystyle{\mathbold d}$. We use a circular crack to ensure that the model
works even in configurations that are far from a classical sharp crack. We
compute the elastic response of the specimen for the fundamental deformation
modes from Figure 2. The deformation modes are applied by imposing the
expressions in Section 2.B as affine boundary conditions on the specimen. For
instance, considering mode (a) and using that $\displaystyle{\mathbold
n}={\mathbold e}_{2}$ in this specimen, we define $\displaystyle{\mathbold
F}^{0}={\mathbold e}_{1}\otimes{\mathbold e}_{1}+F^{0}_{22}{\mathbold
e}_{2}\otimes{\mathbold e}_{2}$ with $\displaystyle F^{0}_{22}>1$; and then
minimize the elastic energy subject to $\displaystyle{\mathbold y}={\mathbold
F}^{0}{\mathbold x}$ on the boundary of the specimen.

Figure 4: The chosen configuration for the $\displaystyle{\mathbold d}$ field
for the frozen crack calculations. We use a circular configuration to test the
effective energy in configurations that are far from an ideal sharp crack.

Figures 5 and 6 show the elastic energy density, deformation, and the traction
on the plane with normal $\displaystyle{\mathbold e}_{2}$ for the fundamental
deformation modes. Specifically, Figure 5 shows modes (a) and (b) for which
our idealized crack should have no elastic response. We see that the elastic
energy in the crack is much smaller than the intact material; the deformations
in the crack are much larger than in the intact material; and there are no
shear or tensile normal tractions in the crack.

Figure 6 plots the same quantities for modes (c), (d), (e). For modes (c) and
(e), we notice that the elastic energy density, deformation, and traction are
uniform to the precision of the numerical approximations; the specimen behaves
as expected for a crack that is closing, and the traction across the crack
faces are indeed compressive. For mode (d), on the other hand, we find that
the specimen has the clear signature of the crack in the elastic energy
density, the deformation, and the traction. This can be understood as arising
due to an interplay between the imposed affine boundary conditions and the
fact that the crack faces pull apart due to lateral shrinkage driven by the
load – i.e., a Poisson’s ratio effect – because the crack faces cannot support
tension. To show this, we use affine boundary conditions that are set up to
give the normal stress in the vertical direction to be $\displaystyle 0$, and
find that the deformation and elastic energy density are indeed uniform in
this setting (Fig. 7); i.e., when allowed to relax the stress, the specimen
behaves like an idealized sharp crack.

(a) Elastic energy, overlaid on a mesh showing deformation, for deformation
mode (a) (b) Traction for deformation mode (a)

(c) Elastic energy, overlaid on a mesh showing deformation, for deformation
mode (b) (d) Traction for deformation mode (b)

Figure 5: Left column: elastic energy density (overlaid on a mesh that shows
the deformation); right column: traction on the plane with normal
$\displaystyle{\mathbold e}_{2}$ for the deformation modes that should have
zero effective crack energy.

(a) Elastic energy, overlaid on a mesh showing deformation, for deformation
mode (c) (b) Traction for deformation mode (c)

(c) Elastic energy, overlaid on a mesh showing deformation, for deformation
mode (d) (d) Traction for deformation mode (d)

(e) Elastic energy, overlaid on a mesh showing deformation, for deformation
mode (e) (f) Traction for deformation mode (e)

Figure 6: Left column: elastic energy density (overlaid on a mesh that shows
the deformation); right column: traction on the plane with normal
$\displaystyle{\mathbold e}_{2}$ for the deformation modes that should have
nonzero effective crack energy. Figure 7: Mode (d), with the affine boundary
conditions set up to effectively allow relaxation in the vertical direction.
The elastic energy density and the deformation are both essentially uniform.

###  9.B Crack Growth under Cyclic Shear Loading with Crack Face Contact

We now consider an example in which cracks change direction, close, and branch
due to a cyclic shear loading. We use a specimen as in Figure 8(a) that
contains an initial crack. Our boundary conditions are as follows: on the
bottom face, we fix the displacement to zero; on the left and right faces, we
fix the traction to zero; and on the top face, we fix the vertical
displacement to zero and the horizontal displacement to cause shearing.

First, we shear the specimen to the right, and that causes the initial crack
to kink, i.e., change direction, and grow towards the bottom-right (Figs.
8(bc)). This is consistent with the crack normal being aligned with the
direction of maximum tension. We notice 2 stress concentrations: one is from
the crack tip, and the other is at the point that the crack kinks. The stress
concentration at the kink can be understood as, roughly, due to the re-entrant
corner.

Next, we reverse the shear on the specimen to the left, and that causes a
branch crack to nucleate at the kink – driven by the high stresses there – and
grow towards the right (Figs. 8(de)). The direction of maximum tension would
suggest that the crack grow towards the top-right, but we impose constraints –
requiring that both components of $\displaystyle{\mathbold d}$ are non-
negative – to drive the crack to grow horizontally after branching. We do this
to set up a situation without reentrant-corner-like geometries, enabling us to
better understand the stresses at the branch point.

We highlight that the crack that grew under the shearing to the right now
closes and the crack faces contact. From Figure 8(e), we see that the elastic
energy shows only a very small signature of the presence of the closed crack,
suggesting that the model works well in capturing the response of closed
cracks.

We also highlight that are higher stresses near the branch point, which would
not occur in an idealized Y-configuration crack pattern with a straight upper
crack branch. These stresses occur because $\displaystyle{\mathbold d}$ is a
smooth field and it rotates as it transitions between crack branches. That is,
the second argument of the effective crack energy density $\displaystyle
W_{\mathrm{d}}(\nabla{\mathbold y},{\mathbold n})$ takes a range of values,
and hence the response deviates from the idealized sharp crack response.

(a) The initial crack configuration before cyclic shear loading.

(b) Crack changing direction under shear to the right. (c) Elastic energy
density under shear to the right.

(d) Crack branching under subsequent shear to the left. (e) Elastic energy
density under subsequent shear to the left.

Figure 8: Crack configuration and elastic energy density under cyclic shear
loading.

###  9.C Cracks in a Cavity under Cyclic Loading

We consider the growth of fracture patterns in a cavity that is subject to a
remote load. While our work is motivated by recent carefully controlled
experiments that show a range of interesting behavior [48, 49, 59, 42], these
experiments have additional physics such as inertia and cavitation that is not
considered here.

We consider a square specimen with a circular cavity as in Figure 9(a). We
begin with 8 initial short pre-cracks. We apply the remote loading by imposing
affine boundary conditions, i.e., imposing $\displaystyle{\mathbold
y}={\mathbold F}^{0}{\mathbold x}$ on the outer boundary, where
$\displaystyle{\mathbold F}^{0}$ is proportional to the identity. The surface
of the cavity is traction free.

Figure 9(b) shows the elastic energy density of the specimen initially under
compression as a baseline. That is, the elastic energy density is what we
would have in an intact uncracked specimen.

Figures 9(c,d) show the crack configuration and elastic energy density,
respectively, when the specimen is then loaded under tension. We find that 4
of the 8 cracks grow while the others do not, and the elastic energy shows the
expected regularized stress concentration at the crack tips. We highlight that
the loading had to be imposed in small increments to capture the crack
configuration shown in the figure, because the cracks would grow rapidly and
reach the boundary soon after the state shown in the figures.

Figures 9(e,f) show the crack configuration and elastic energy density,
respectively, when the specimen is subsequently loaded under compression. The
crack configuration is largely the same, though the cracks have become
visually narrower; this is due to our approach to healing, wherein we allow
healing when $\displaystyle{\mathbold d}$ has magnitude below a critical value
(Section 8). An important highlight is that the elastic energy density in
Figure 9(f) is identical to the uncracked case in Figure 9(b), despite the
completely different crack configurations in these settings. The elastic
response of the cracked configuration under compression is identical to the
uncracked configuration under compression because all the cracks have closed,
suggesting that the model is working well in capturing the desired response.

(a) Initial unloaded configuration with pre-cracks. (b) Elastic energy density
of the initial configuration under compression.

(c) Crack growth under tensile load. (d) Elastic energy density of the cracked
configuration under tensile load.

(e) Crack configuration under a second cycle of compressive load; crack
patterns do not evolve under compression. (f) Elastic energy density of the
cracked configuration under compressive load; response is identical to the
intact initial configuration.

Figure 9: Crack growth in a circular cavity under cyclic remote loading.

##  10 Discussion

In this paper, we have presented an effective energy for phase-field cracks
that provides the correct crack response when the crack is subject to complex
loadings that cause contact across the crack faces. Our approach is valid in
the setting of finite deformations, which enables application to soft
materials in complex configurations (e.g., to complement theoretical studies
such as [51, 33, 43]), as well as to stiff materials where strains are small
but fractured pieces can have large rotations or where compression driving
crack closure is significant (e.g., [40]).

While the numerical results presented above are promising, there remain
several important directions for future research:

  * •

Quasiconvexity is a central property for well-posed problems in finite
elasticity, e.g., [16, 10, 25], and it remains to check that $\displaystyle
W_{\mathrm{d}}$ is quasiconvex in the $\displaystyle{\mathbold F}$ variable.
While lack of convexity typically leads to instabilities or microstructure,
our numerical computations have not shown that behavior. In this context, we
notice that an alternative definition of $\displaystyle W_{\mathrm{d}}$ could
have been

$W_{\mathrm{d}}({\mathbold F},{\mathbold n})=\min_{{\mathbold y}={\mathbold
F}{\mathbold x}\text{ on }\partial Q_{{\mathbold n}}}\int_{Q_{{\mathbold
n}}}W(\nabla{\mathbold y})\ \mathrm{d}V_{{\mathbold x}},$

where: $\displaystyle Q_{{\mathbold n}}$ is the cube of volume one centered at
the origin with edges parallel to the orthonormal basis
$\displaystyle\\{{\mathbold t}_{1},{\mathbold t}_{2},{\mathbold n}\\}$;
$\displaystyle{\mathbold y}$ is Sobolev in the two half-cubes $\displaystyle
Q_{{\mathbold n}}^{\pm}=\\{{\mathbold x}\in Q_{{\mathbold n}}:\pm{\mathbold
x}\cdot{\mathbold n}>0\\}$, but can have a jump in the interface
$\displaystyle\\{{\mathbold x}\in Q_{{\mathbold n}}:{\mathbold
x}\cdot{\mathbold n}=0\\}$; $\displaystyle{\mathbold y}$ does not
interpenetrate. This definition renders a quasiconvex $\displaystyle
W_{\mathrm{d}}$ and, in addition, simulates the effective response of two
blocks under the affine deformation $\displaystyle{\mathbold F}$ applied on
the boundary (Fig. 10).

Figure 10: Obtaining the effective crack energy through the homogenized
effective response of two blocks.

However, explicitly carrying out the minimization is challenging.

  * •

The limiting sharp-interface model that appears as $\displaystyle\epsilon\to
0$, and the corresponding consequences on the crack face tractions, needs to
be clarified.

  * •

The numerical studies on branched cracks show that the crack normal is not
well defined at the branch point, which leads to undesired responses at the
branch point when the crack is subject to complex loads. While it is not
immediately clear to us as to how to eliminate this, possibly it is useful to
borrow from damage theory in using a tensorial parameter [44] that can
potentially represent a richer kinematics such as multiple normals at a
spatial location.

  * •

We have considered the setting of idealized frictionless crack faces. A
natural extension is to consider friction, which is relevant to a large class
of materials. While relaxation of the energy is not applicable to this
setting, the QR decomposition provides an approach to transparently separate
the relative slip of the crack faces from other deformation modes. Therefore,
we expect that the kinematics introduced in this paper will be useful in
considering more realistic boundary conditions.

## Software Availability

A version of the code developed for this work is available at
https://github.com/maryhzd/Phase-field.git

###### Acknowledgements.

We thank George Gazonas, Anthony Rollett, and Noel Walkington for useful
discussions; Pradeep Sharma for pointing us to relevant prior work in
continuum damage mechanics; Agencia Estatal de Investigación of the Spanish
Ministry Research and Innovation (Project MTM2017-85934-C3-2-P), Army Research
Office (MURI W911NF-19-1-0245), Office of Naval Research (N00014-18-1-2528),
and National Science Foundation (CMMI MOMS 1635407, DMREF 2118945, DMS
2108784) for support; Center for Nonlinear Analysis for hosting a visit by
Carlos Mora-Corral; and National Science Foundation for XSEDE resources
provided by Pittsburgh Supercomputing Center. This paper draws from the
doctoral dissertation of Vaibhav Agrawal at Carnegie Mellon University.

##  Appendix A Shortcomings of the Energy Splitting Method

The simplest approach to model the energy density of the damaged volume is to
set the damaged energy to $\displaystyle 0$. However, this leads to unphysical
behavior such as interpenetration of the crack faces and an incorrect
mechanical response of a body with a crack. The papers [47] and [9] proposed
different ways of decomposing the energy, and associating only certain terms
of the energy to the damaged region. While the energy splitting method vastly
improved the difficulties caused by simply using zero energy for the crack, it
does not consider the crack orientation in the energetic decomposition.
Consequently, energy splitting leads to unphysical response in some settings;
further, it is unclear how to extend it beyond the setting of linear isotropic
elasticity. We discuss below some simple examples where the splitting methods
provide unexpected results. We emphasize that the incorrect stress field at
crack tips can affect the crack growth behavior significantly, even if the
stresses are largely correct away from the crack tip.

In all examples, we consider a large specimen with a given far-field stress or
strain, and a finite crack oriented with normal $\displaystyle{\mathbold
e}_{2}$. We use $\displaystyle\mathbold{\sigma}_{split}$ to denote the
response of the cracked material predicted by the energy splitting model.

We define the intact energy and the corresponding stress by

$W_{intact}(\mathbold{\varepsilon})=\frac{1}{2}\lambda(\operatorname{tr}\mathbold{\varepsilon})^{2}+\mu|\mathbold{\varepsilon}|^{2}\quad\Rightarrow\quad\mathbold{\sigma}_{intact}=\lambda\left(\operatorname{tr}{\mathbold{\varepsilon}}\right){\mathbold
I}+2\mu\mathbold{\varepsilon}.$ (51)

A value $\displaystyle\mu>0$ is required for convexity of $\displaystyle
W_{intact}$. For simplicity we consider $\displaystyle\lambda>0$, and
consequently the Poisson ratio $\displaystyle\nu$ is positive.

###  A.1 Splitting Based on the Principal Strain

Following [47], the compressive ($\displaystyle\psi_{0}^{-}$) and tensile
($\displaystyle\psi_{0}^{+}$) energies are defined by

$\psi_{0}^{\pm}:=\frac{1}{2}\lambda\langle{\varepsilon}_{1}+{\varepsilon}_{2}+{\varepsilon}_{3}\rangle_{\pm}^{2}+\mu\langle{\varepsilon}_{1}\rangle_{\pm}^{2}+\langle{\varepsilon}_{2}\rangle_{\pm}^{2}+\langle{\varepsilon}_{3}\rangle_{\pm}^{2},$

where $\displaystyle{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}$
denote the principal strains, and the corresponding principal directions are
$\displaystyle{\mathbold n}_{1},{\mathbold n}_{2},{\mathbold n}_{3}$. Also,
$\displaystyle\langle x\rangle_{+}=\max\\{0,x\\}$ and $\displaystyle\langle
x\rangle_{-}=\min\\{0,x\\}$. The effective energy of the crack is assumed to
consist of only the compressive part $\displaystyle\psi_{0}^{-}$. The
corresponding stress is

$\mathbold{\sigma}_{split}=\sum_{a=1}^{3}\left(\lambda\langle{\varepsilon}_{1}+{\varepsilon}_{2}+{\varepsilon}_{3}\rangle_{-}+2\mu\langle{\varepsilon}_{a}\rangle_{-}\right){\mathbold
n}_{a}\otimes{\mathbold n}_{a}.$ (52)

We notice from this expression that $\displaystyle\mathbold{\sigma}_{split}$
is always non-tensile in every direction.

#### Uniaxial tension parallel to the crack.

Consider the specimen subject to a far-field stress
$\displaystyle\sigma_{0}{\mathbold e}_{1}\otimes{\mathbold e}_{1}$ with
$\displaystyle\sigma_{0}>0$, giving a far-field strain of:

$\mathbold{\varepsilon}=\frac{\sigma_{0}}{\mu(3\lambda+2\mu)}\begin{pmatrix}\lambda+\mu&0&0\\\
0&-\lambda/2&0\\\ 0&0&-\lambda/2\end{pmatrix}.$ (53)

The corresponding stress response in the crack in this model is
$\displaystyle\mathbold{\sigma}_{split}=-\frac{\sigma_{0}}{3\lambda+2\mu}\left({\mathbold
e}_{2}\otimes{\mathbold e}_{2}+{\mathbold e}_{3}\otimes{\mathbold
e}_{3}\right)$.

We emphasize two aspects of this result. First, there is an unexpected (since
$\displaystyle\nu>0$) compressive stress along the $\displaystyle{\mathbold
e}_{2}$ direction. Second, the stress along the $\displaystyle{\mathbold
e}_{1}$ direction is zero, while the expectation is that for far-field stress
parallel to crack, the material should behave as an intact material and
sustain the far-field stress as it is. The incorrect stress field can
potentially cause spurious crack growth: the effective zero stiffness for
tension parallel to cracks can lead to spurious stress concentration around
the tip as the intact material ahead of the tip can sustain far-field stress
in $\displaystyle{\mathbold e}_{1}$ direction but the cracked phase cannot.
Further, this can have a significant spurious impact on the T-stress which is
an important parameter for ductile crack growth [60].

#### Shear traction across the crack.

Consider the specimen subject to a far-field strain given by

$\mathbold{\varepsilon}=\frac{\tau}{2\mu}\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix}$

with $\displaystyle\tau>0$. The corresponding stress response in the crack in
this model is

$\mathbold{\sigma}_{split}=\frac{\tau}{2}\begin{pmatrix}-1&1&0\\\ 1&-1&0\\\
0&0&0\end{pmatrix}$ (54)

which predicts a shear traction across the crack faces, which violates the
classical crack face traction conditions.

###  A.2 The Hydrostatic-Deviatoric Split

[9] proposes a splitting that allows the crack to resist compressive
hydrostatic stress, but not tensile hydrostatic and deviatoric stresses. One
starts by writing the isotropic linear elastic energy as

$W_{intact}(\mathbold{\varepsilon})=\frac{1}{2}\kappa\langle\operatorname{tr}\mathbold{\varepsilon}\rangle_{-}^{2}+\frac{1}{2}\kappa\langle\operatorname{tr}\mathbold{\varepsilon}\rangle_{+}^{2}+\mu|\mathbold{\varepsilon}_{D}|^{2}$

where $\displaystyle\kappa=\lambda+2\mu/3>0$ is the bulk modulus and
$\displaystyle\mathbold{\varepsilon}_{D}:=\mathbold{\varepsilon}-\frac{1}{3}\left(\operatorname{tr}{\mathbold{\varepsilon}}\right){\mathbold
I}$ is the deviatoric component of the strain. The effective energy of the
crack is assumed to consist only of the term
$\displaystyle\frac{1}{2}\kappa\langle\operatorname{tr}\mathbold{\varepsilon}\rangle_{-}^{2}$,
giving the stress reponse

$\mathbold{\sigma}_{split}=\left(\kappa\langle\operatorname{tr}\mathbold{\varepsilon}\rangle_{-}\right){\mathbold
I}.$

We notice from this expression that the crack response is always non-tensile
hydrostatic, regardless of the applied loading.

#### Uniaxial tension parallel to the crack.

Consider the specimen subject to a far-field stress
$\displaystyle\sigma_{0}{\mathbold e}_{1}\otimes{\mathbold e}_{1}$ with
$\displaystyle\sigma_{0}>0$, and the corresponding far-field strain is in
(53).

In this model, we find $\displaystyle\mathbold{\sigma}_{split}={\bf 0}$. As
described in Section A.1, the expectation is that material sustains the far-
field tensile stress parallel to crack faces as it is. Further, the result
that $\displaystyle\mathbold{\sigma}_{split}=\bf 0$ has similar implications:
spurious stress concentrations at the crack tip that affects the crack driving
force and the T-stress.

#### Uniaxial compression normal to the crack.

Consider the specimen subject to a far-field stress
$\displaystyle-\sigma_{0}{\mathbold e}_{2}\otimes{\mathbold e}_{2}$ with
$\displaystyle\sigma_{0}>0$. We expect the crack faces to contact and the
response to be identical to the intact material.

The far-field strain corresponding to this stress is

$\mathbold{\varepsilon}=\frac{\sigma_{0}}{\mu(3\lambda+2\mu)}\begin{pmatrix}\lambda/2&0&0\\\
0&-(\lambda+\mu)&0\\\ 0&0&\lambda/2\end{pmatrix},$

which should also be response of the crack. However, this model predicts the
crack response
$\displaystyle\mathbold{\sigma}_{split}=-\frac{\sigma_{0}}{3}{\mathbold I}$.
Similar results can be obtained in the general case of multiaxial compressive
loading.

##  Appendix B Proofs

In this appendix, we provide the proofs of various statements in the main body
of the work.

###### Proof of Proposition 2.1.

This is actually a restatement of the well-known QR decomposition ([38, Th.
2.6.1] or [68, Thms. 7.1 and 7.2]), according to which given any
$\displaystyle{\mathbold F}\in\mathbb{R}^{3\times 3}_{+}$ there exist unique
$\displaystyle{\mathbold R}\in SO(3)$ and $\displaystyle{\mathbold A}$ upper
triangular (with respect to the basis $\displaystyle\\{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}\\}$) with positive diagonal elements
such that $\displaystyle{\mathbold F}={\mathbold R}{\mathbold A}$. This
$\displaystyle{\mathbold A}$ must be of the form $\displaystyle{\mathbold
A}_{{\mathbold t}_{1},{\mathbold t}_{2},{\mathbold n}}(A_{{\mathbold
n}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold t}_{1}},A_{{\mathbold
t}_{2}{\mathbold t}_{2}},A_{{\mathbold t}_{1}{\mathbold n}},A_{{\mathbold
t}_{2}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold t}_{2}})$ for some
$\displaystyle A_{{\mathbold n}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold
t}_{1}},A_{{\mathbold t}_{2}{\mathbold t}_{2}}>0$ and $\displaystyle
A_{{\mathbold t}_{1}{\mathbold n}},A_{{\mathbold t}_{2}{\mathbold
n}},A_{{\mathbold t}_{1}{\mathbold t}_{2}}\in\mathbb{R}$. ∎

###### Proof of Lemma 2.1.

We have, succesively,

${\mathbold F}={\mathbold R}{\mathbold A},\qquad{\mathbold F}^{-T}={\mathbold
R}{\mathbold A}^{-T},\qquad{\mathbold F}^{-T}{\mathbold n}={\mathbold
R}{\mathbold A}^{-T}{\mathbold n},\qquad\left|{\mathbold F}^{-T}{\mathbold
n}\right|=\left|{\mathbold A}^{-T}{\mathbold n}\right|.$

Now, since

${\mathbold A}=A_{{\mathbold n}{\mathbold n}}{\mathbold n}\otimes{\mathbold
n}+A_{{\mathbold t}_{1}{\mathbold t}_{1}}{\mathbold t}_{1}\otimes{\mathbold
t}_{1}+A_{{\mathbold t}_{2}{\mathbold t}_{2}}{\mathbold
t}_{2}\otimes{\mathbold t}_{2}+A_{{\mathbold t}_{1}{\mathbold n}}{\mathbold
t}_{1}\otimes{\mathbold n}+A_{{\mathbold t}_{2}{\mathbold n}}{\mathbold
t}_{2}\otimes{\mathbold n}+A_{{\mathbold t}_{1}{\mathbold t}_{2}}{\mathbold
t}_{1}\otimes{\mathbold t}_{2},$

we can easily calculate

${\mathbold A}^{T}=A_{{\mathbold n}{\mathbold n}}{\mathbold
n}\otimes{\mathbold n}+A_{{\mathbold t}_{1}{\mathbold t}_{1}}{\mathbold
t}_{1}\otimes{\mathbold t}_{1}+A_{{\mathbold t}_{2}{\mathbold
t}_{2}}{\mathbold t}_{2}\otimes{\mathbold t}_{2}+A_{{\mathbold
t}_{1}{\mathbold n}}{\mathbold n}\otimes{\mathbold t}_{1}+A_{{\mathbold
t}_{2}{\mathbold n}}{\mathbold n}\otimes{\mathbold t}_{2}+A_{{\mathbold
t}_{1}{\mathbold t}_{2}}{\mathbold t}_{2}\otimes{\mathbold t}_{1},$

as well as

$\displaystyle\displaystyle{\mathbold A}^{-T}=$
$\displaystyle\displaystyle\frac{1}{A_{{\mathbold n}{\mathbold n}}}{\mathbold
n}\otimes{\mathbold n}+\frac{1}{A_{{\mathbold t}_{1}{\mathbold
t}_{1}}}{\mathbold t}_{1}\otimes{\mathbold t}_{1}+\frac{1}{A_{{\mathbold
t}_{2}{\mathbold t}_{2}}}{\mathbold t}_{2}\otimes{\mathbold
t}_{2}+\left(-\frac{A_{{\mathbold t}_{1}{\mathbold n}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}A_{{\mathbold n}{\mathbold n}}}+\frac{A_{{\mathbold
t}_{1}{\mathbold t}_{2}}A_{{\mathbold t}_{2}{\mathbold n}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}A_{{\mathbold t}_{2}{\mathbold t}_{2}}A_{{\mathbold
n}{\mathbold n}}}\right){\mathbold n}\otimes{\mathbold t}_{1}$
$\displaystyle\displaystyle-\frac{A_{{\mathbold t}_{2}{\mathbold
n}}}{A_{{\mathbold t}_{2}{\mathbold t}_{2}}A_{{\mathbold n}{\mathbold
n}}}{\mathbold n}\otimes{\mathbold t}_{2}-\frac{A_{{\mathbold t}_{1}{\mathbold
t}_{2}}}{A_{{\mathbold t}_{1}{\mathbold t}_{1}}A_{{\mathbold t}_{2}{\mathbold
t}_{2}}}{\mathbold t}_{2}\otimes{\mathbold t}_{1},$

so

${\mathbold A}^{-T}{\mathbold n}=\frac{1}{A_{{\mathbold n}{\mathbold
n}}}{\mathbold n}\quad\text{and}\quad\left|{\mathbold A}^{-T}{\mathbold
n}\right|=\frac{1}{A_{{\mathbold n}{\mathbold n}}},$

which concludes the proof. ∎

###### Proof of Proposition 2.2.

We will only prove a), the rest of the claims being analogous. Let
$\displaystyle(A_{{\mathbold n}{\mathbold n}}^{(j)},A_{{\mathbold
t}_{1}{\mathbold n}}^{(j)},A_{{\mathbold t}_{2}{\mathbold n}}^{(j)})$ be a
sequence (indexed by $\displaystyle j\in\mathbb{N}$) in
$\displaystyle(0,\infty)\times\mathbb{R}\times\mathbb{R}$ satisfying

$\displaystyle\displaystyle\inf_{\begin{subarray}{c}A_{{\mathbold n}{\mathbold
n}}>0\\\ A_{{\mathbold t}_{1}{\mathbold n}},A_{{\mathbold t}_{2}{\mathbold
n}}\in\mathbb{R}\end{subarray}}W\left({\mathbold A}_{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}}(A_{{\mathbold n}{\mathbold
n}},A_{{\mathbold t}_{1}{\mathbold t}_{1}},A_{{\mathbold t}_{2}{\mathbold
t}_{2}},A_{{\mathbold t}_{1}{\mathbold n}},A_{{\mathbold t}_{2}{\mathbold
n}},A_{{\mathbold t}_{1}{\mathbold t}_{2}})\right)$
$\displaystyle\displaystyle=\lim_{j\to\infty}W\left({\mathbold A}_{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}}(A_{{\mathbold n}{\mathbold
n}}^{(j)},A_{{\mathbold t}_{1}{\mathbold t}_{1}},A_{{\mathbold
t}_{2}{\mathbold t}_{2}},A_{{\mathbold t}_{1}{\mathbold
n}}^{(j)},A_{{\mathbold t}_{2}{\mathbold n}}^{(j)},A_{{\mathbold
t}_{1}{\mathbold t}_{2}})\right).$

We have that

$\det{\mathbold A}_{{\mathbold t}_{1},{\mathbold t}_{2},{\mathbold
n}}(A_{{\mathbold n}{\mathbold n}}^{(j)},A_{{\mathbold t}_{1}{\mathbold
t}_{1}},A_{{\mathbold t}_{2}{\mathbold t}_{2}},A_{{\mathbold t}_{1}{\mathbold
n}}^{(j)},A_{{\mathbold t}_{2}{\mathbold n}}^{(j)},A_{{\mathbold
t}_{1}{\mathbold t}_{2}})=A_{{\mathbold t}_{1}{\mathbold t}_{1}}A_{{\mathbold
t}_{2}{\mathbold t}_{2}}A_{{\mathbold n}{\mathbold n}}^{(j)}$

and

$\displaystyle\displaystyle\left|{\mathbold A}_{{\mathbold t}_{1},{\mathbold
t}_{2},{\mathbold n}}(A_{{\mathbold n}{\mathbold n}}^{(j)},A_{{\mathbold
t}_{1}{\mathbold t}_{1}},A_{{\mathbold t}_{2}{\mathbold t}_{2}},A_{{\mathbold
t}_{1}{\mathbold n}}^{(j)},A_{{\mathbold t}_{2}{\mathbold
n}}^{(j)},A_{{\mathbold t}_{1}{\mathbold t}_{2}})\right|$
$\displaystyle\displaystyle=\sqrt{A_{{\mathbold t}_{1}{\mathbold
t}_{1}}^{2}+A_{{\mathbold t}_{1}{\mathbold t}_{2}}^{2}+(A_{{\mathbold
t}_{1}{\mathbold n}}^{(j)})^{2}+A_{{\mathbold t}_{2}{\mathbold
t}_{2}}^{2}+(A_{{\mathbold t}_{2}{\mathbold n}}^{(j)})^{2}+(A_{{\mathbold
n}{\mathbold n}}^{(j)})^{2}}.$

Condition (11) implies that there exist $\displaystyle m,M>0$ such that

$m\leq A_{{\mathbold n}{\mathbold n}}^{(j)}\leq M,\quad-M\leq A_{{\mathbold
t}_{1}{\mathbold n}}^{(j)}\leq M,\quad-M\leq A_{{\mathbold t}_{2}{\mathbold
n}}^{(j)}\leq M,\qquad j\in\mathbb{N}.$

Therefore, there exist $\displaystyle A_{{\mathbold n}{\mathbold
n}}^{*},A_{{\mathbold t}_{1}{\mathbold n}}^{*},A_{{\mathbold t}_{2}{\mathbold
n}}^{*}$ with

$m\leq A_{{\mathbold n}{\mathbold n}}^{*}\leq M,\quad-M\leq A_{{\mathbold
t}_{1}{\mathbold n}}^{*}\leq M,\quad-M\leq A_{{\mathbold t}_{2}{\mathbold
n}}^{*}\leq M$

such that, for a subsequence (not relabelled),

$\lim_{j\to\infty}(A_{{\mathbold n}{\mathbold n}}^{(j)},A_{{\mathbold
t}_{1}{\mathbold n}}^{(j)},A_{{\mathbold t}_{2}{\mathbold
n}}^{(j)})=(A_{{\mathbold n}{\mathbold n}}^{*},A_{{\mathbold t}_{1}{\mathbold
n}}^{*},A_{{\mathbold t}_{2}{\mathbold n}}^{*}).$

As $\displaystyle W$ is continuous,

$\displaystyle\displaystyle\lim_{j\to\infty}W\left({\mathbold A}_{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}}(A_{{\mathbold n}{\mathbold
n}}^{(j)},A_{{\mathbold t}_{1}{\mathbold t}_{1}},A_{{\mathbold
t}_{2}{\mathbold t}_{2}},A_{{\mathbold t}_{1}{\mathbold
n}}^{(j)},A_{{\mathbold t}_{2}{\mathbold n}}^{(j)},A_{{\mathbold
t}_{1}{\mathbold t}_{2}})\right)$
$\displaystyle\displaystyle=W\left({\mathbold A}_{{\mathbold t}_{1},{\mathbold
t}_{2},{\mathbold n}}(A_{{\mathbold n}{\mathbold n}}^{*},A_{{\mathbold
t}_{1}{\mathbold t}_{1}},A_{{\mathbold t}_{2}{\mathbold t}_{2}},A_{{\mathbold
t}_{1}{\mathbold n}}^{*},A_{{\mathbold t}_{2}{\mathbold n}}^{*},A_{{\mathbold
t}_{1}{\mathbold t}_{2}})\right)$

and the proof is complete. ∎

###### Proof of Proposition 2.3.

Following the notation of Definition 2.1, we define, additionally,

${\mathbold A}^{\prime}={\mathbold A}_{{\mathbold t}_{1},{\mathbold
t}_{2},{\mathbold n}}(A^{\prime}_{{\mathbold n}{\mathbold n}},A_{{\mathbold
t}_{1}{\mathbold t}_{1}},A_{{\mathbold t}_{2}{\mathbold
t}_{2}},A^{\prime}_{{\mathbold t}_{1}{\mathbold n}},A^{\prime}_{{\mathbold
t}_{2}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold t}_{2}}).$

By frame-indifference, $\displaystyle W({\mathbold A}^{\prime})=W({\mathbold
R}{\mathbold A}^{\prime})$, and $\displaystyle{\mathbold R}{\mathbold
A}^{\prime}={\mathbold R}{\mathbold A}{\mathbold A}^{-1}{\mathbold
A}^{\prime}={\mathbold F}{\mathbold A}^{-1}{\mathbold A}^{\prime}$. We
calculate

${\mathbold A}^{-1}={\mathbold A}_{{\mathbold t}_{1},{\mathbold
t}_{2},{\mathbold n}}\left(\frac{1}{A_{{\mathbold n}{\mathbold
n}}},\frac{1}{A_{{\mathbold t}_{1}{\mathbold t}_{1}}},\frac{1}{A_{{\mathbold
t}_{2}{\mathbold t}_{2}}},\frac{-A_{{\mathbold t}_{1}{\mathbold
n}}}{A_{{\mathbold t}_{1}{\mathbold t}_{1}}A_{{\mathbold n}{\mathbold
n}}}+\frac{A_{{\mathbold t}_{1}{\mathbold t}_{2}}A_{{\mathbold
t}_{2}{\mathbold n}}}{A_{{\mathbold t}_{1}{\mathbold t}_{1}}A_{{\mathbold
t}_{2}{\mathbold t}_{2}}A_{{\mathbold n}{\mathbold n}}},-\frac{A_{{\mathbold
t}_{2}{\mathbold n}}}{A_{{\mathbold t}_{2}{\mathbold t}_{2}}A_{{\mathbold
n}{\mathbold n}}},-\frac{A_{{\mathbold t}_{1}{\mathbold t}_{2}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}A_{{\mathbold t}_{2}{\mathbold t}_{2}}}\right).$

and

${\mathbold A}^{-1}{\mathbold A}^{\prime}=\\\ {\mathbold A}_{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}}\left(\frac{A^{\prime}_{{\mathbold
n}{\mathbold n}}}{A_{{\mathbold n}{\mathbold
n}}},1,1,\frac{A^{\prime}_{{\mathbold t}_{1}{\mathbold n}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}}-\frac{A_{{\mathbold t}_{1}{\mathbold
t}_{2}}A^{\prime}_{{\mathbold t}_{2}{\mathbold n}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}A_{{\mathbold t}_{2}{\mathbold
t}_{2}}}-\frac{A_{{\mathbold t}_{1}{\mathbold n}}A^{\prime}_{{\mathbold
n}{\mathbold n}}}{A_{{\mathbold t}_{1}{\mathbold t}_{1}}A_{{\mathbold
n}{\mathbold n}}}+\frac{A_{{\mathbold t}_{1}{\mathbold t}_{2}}A_{{\mathbold
t}_{2}{\mathbold n}}A^{\prime}_{{\mathbold n}{\mathbold n}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}A_{{\mathbold t}_{2}{\mathbold t}_{2}}A_{{\mathbold
n}{\mathbold n}}},\frac{A^{\prime}_{{\mathbold t}_{2}{\mathbold
n}}}{A_{{\mathbold t}_{2}{\mathbold t}_{2}}}-\frac{A_{{\mathbold
t}_{2}{\mathbold t}_{3}}A^{\prime}_{{\mathbold n}{\mathbold n}}}{A_{{\mathbold
t}_{2}{\mathbold t}_{2}}A_{{\mathbold n}{\mathbold n}}},0\right).$

Performing the changes

$\displaystyle\displaystyle A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}}:=\frac{A^{\prime}_{{\mathbold t}_{1}{\mathbold n}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}}-\frac{A_{{\mathbold t}_{1}{\mathbold
t}_{2}}A^{\prime}_{{\mathbold t}_{2}{\mathbold n}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}A_{{\mathbold t}_{2}{\mathbold
t}_{2}}}-\frac{A_{{\mathbold t}_{1}{\mathbold n}}A^{\prime}_{{\mathbold
n}{\mathbold n}}}{A_{{\mathbold t}_{1}{\mathbold t}_{1}}A_{{\mathbold
n}{\mathbold n}}}+\frac{A_{{\mathbold t}_{1}{\mathbold t}_{2}}A_{{\mathbold
t}_{2}{\mathbold n}}A^{\prime}_{{\mathbold n}{\mathbold n}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}A_{{\mathbold t}_{2}{\mathbold t}_{2}}A_{{\mathbold
n}{\mathbold n}}},\qquad A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}}:=\frac{A^{\prime}_{{\mathbold t}_{2}{\mathbold n}}}{A_{{\mathbold
t}_{2}{\mathbold t}_{2}}}-\frac{A_{{\mathbold t}_{2}{\mathbold
t}_{3}}A^{\prime}_{{\mathbold n}{\mathbold n}}}{A_{{\mathbold t}_{2}{\mathbold
t}_{2}}A_{{\mathbold n}{\mathbold n}}},$ $\displaystyle\displaystyle
A^{\prime\prime}_{{\mathbold n}{\mathbold n}}:=\frac{A^{\prime}_{{\mathbold
n}{\mathbold n}}}{A_{{\mathbold n}{\mathbold n}}},$

it is immediate to see that

$\displaystyle\displaystyle\Big{\\{}{\mathbold A}_{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}}\left(\frac{A^{\prime}_{{\mathbold
n}{\mathbold n}}}{A_{{\mathbold n}{\mathbold
n}}},1,1,\frac{A^{\prime}_{{\mathbold t}_{1}{\mathbold n}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}}-\frac{A_{{\mathbold t}_{1}{\mathbold
t}_{2}}A^{\prime}_{{\mathbold t}_{2}{\mathbold n}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}A_{{\mathbold t}_{2}{\mathbold
t}_{2}}}+\frac{(-A_{{\mathbold t}_{1}{\mathbold n}}A_{{\mathbold
t}_{2}{\mathbold t}_{2}}+A_{{\mathbold t}_{1}{\mathbold t}_{2}}A_{{\mathbold
t}_{2}{\mathbold n}})A^{\prime}_{{\mathbold n}{\mathbold n}}}{A_{{\mathbold
t}_{1}{\mathbold t}_{1}}A_{{\mathbold t}_{2}{\mathbold t}_{2}}A_{{\mathbold
n}{\mathbold n}}},\frac{A^{\prime}_{{\mathbold t}_{2}{\mathbold
n}}}{A_{{\mathbold t}_{2}{\mathbold t}_{2}}}-\frac{A_{{\mathbold
t}_{2}{\mathbold t}_{3}}A^{\prime}_{{\mathbold n}{\mathbold n}}}{A_{{\mathbold
t}_{2}{\mathbold t}_{2}}A_{{\mathbold n}{\mathbold n}}},0\right)$
$\displaystyle\displaystyle\hskip 300.00046pt:\,A^{\prime}_{{\mathbold
n}{\mathbold n}}>0,\,A^{\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime}_{{\mathbold t}_{2}{\mathbold n}}\in\mathbb{R}\Big{\\}}$
$\displaystyle\displaystyle=\left\\{{\mathbold A}_{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}}\left(A^{\prime\prime}_{{\mathbold
n}{\mathbold n}},1,1,A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold
n}},0\right):\,A^{\prime\prime}_{{\mathbold n}{\mathbold
n}}>0,\,A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold n}}\in\mathbb{R}\right\\}.$

This shows that

$\displaystyle\displaystyle\min_{\begin{subarray}{c}A^{\prime}_{{\mathbold
n}{\mathbold n}}>0\\\ A^{\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime}_{{\mathbold t}_{2}{\mathbold
n}}\in\mathbb{R}\end{subarray}}W\left({\mathbold A}_{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}}(A^{\prime}_{{\mathbold n}{\mathbold
n}},A_{{\mathbold t}_{1}{\mathbold t}_{1}},A_{{\mathbold t}_{2}{\mathbold
t}_{2}},A^{\prime}_{{\mathbold t}_{1}{\mathbold n}},A^{\prime}_{{\mathbold
t}_{2}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold t}_{2}})\right)$
$\displaystyle\displaystyle=\min_{\begin{subarray}{c}A^{\prime\prime}_{{\mathbold
n}{\mathbold n}}>0\\\ A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold
n}}\in\mathbb{R}\end{subarray}}W\left({\mathbold F}{\mathbold A}_{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}}\left(A^{\prime\prime}_{{\mathbold
n}{\mathbold n}},1,1,A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold n}},0\right)\right)$

and formula (13) holds. Analogously, one can show that

$\displaystyle\displaystyle\min_{A^{\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime}_{{\mathbold t}_{2}{\mathbold n}}\in\mathbb{R}}W\left({\mathbold
A}_{{\mathbold t}_{1},{\mathbold t}_{2},{\mathbold n}}(A_{{\mathbold
n}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold t}_{1}},A_{{\mathbold
t}_{2}{\mathbold t}_{2}},A^{\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime}_{{\mathbold t}_{2}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold
t}_{2}})\right)$
$\displaystyle\displaystyle=\min_{A^{\prime\prime}_{{\mathbold
t}_{1}{\mathbold n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold
n}}\in\mathbb{R}}W\left({\mathbold F}{\mathbold A}_{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}}\left(A_{{\mathbold n}{\mathbold
n}},1,1,A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold n}},0\right)\right).$

∎

###### Proof of Proposition 2.4.

We will use Proposition 2.3 and trace the dependence of the quantities
involved.

We start with a). According to (7), it is clear that $\displaystyle
A_{{\mathbold n}{\mathbold n}}$ does not depend on $\displaystyle{\mathbold
t}_{1},{\mathbold t}_{2}$. Let us see that $\displaystyle A_{{\mathbold
n}{\mathbold n}}^{*}$ does not depend either, so let
$\displaystyle\\{{\mathbold t}^{\prime}_{1},{\mathbold
t}^{\prime}_{2},{\mathbold n}\\}$ be another orthonormal basis. We first
notice that

${\mathbold t}_{1}\otimes{\mathbold t}_{1}+{\mathbold t}_{2}\otimes{\mathbold
t}_{2}={\mathbold t}^{\prime}_{1}\otimes{\mathbold t}^{\prime}_{1}+{\mathbold
t}^{\prime}_{2}\otimes{\mathbold t}^{\prime}_{2},$

since both terms act as the identity in
$\displaystyle\operatorname{span}\\{{\mathbold t}_{1},{\mathbold
t}_{2}\\}=\operatorname{span}\\{{\mathbold t}^{\prime}_{1},{\mathbold
t}^{\prime}_{2}\\}$ and as zero in
$\displaystyle\operatorname{span}\\{{\mathbold n}\\}$. On the other hand,
there exist an invertible matrix

$\begin{pmatrix}a_{11}&a_{12}\\\ a_{21}&a_{22}\end{pmatrix}$ (55)

such that

$\begin{cases}{\mathbold t}_{1}=a_{11}{\mathbold
t}^{\prime}_{1}+a_{12}{\mathbold t}^{\prime}_{2}\\\ {\mathbold
t}_{2}=a_{21}{\mathbold t}^{\prime}_{1}+a_{22}{\mathbold
t}^{\prime}_{2}.\end{cases}$

With this we find that for any $\displaystyle A^{\prime\prime}_{{\mathbold
n}{\mathbold n}}>0$ and $\displaystyle A^{\prime\prime}_{{\mathbold
t}_{1}{\mathbold n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold
n}}\in\mathbb{R}$ we have

$\displaystyle\displaystyle A^{\prime\prime}_{{\mathbold n}{\mathbold
n}}{\mathbold n}\otimes{\mathbold n}+{\mathbold t}_{1}\otimes{\mathbold
t}_{1}+{\mathbold t}_{2}\otimes{\mathbold t}_{2}+A^{\prime\prime}_{{\mathbold
t}_{1}{\mathbold n}}{\mathbold t}_{1}\otimes{\mathbold
n}+A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold n}}{\mathbold
t}_{1}\otimes{\mathbold n}$
$\displaystyle\displaystyle=A^{\prime\prime}_{{\mathbold n}{\mathbold
n}}{\mathbold n}\otimes{\mathbold n}+{\mathbold
t}^{\prime}_{1}\otimes{\mathbold t}^{\prime}_{1}+{\mathbold
t}^{\prime}_{2}\otimes{\mathbold
t}^{\prime}_{2}+\left(a_{11}A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}}+a_{21}A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold n}}\right){\mathbold
t}_{1}\otimes{\mathbold n}+\left(a_{12}A^{\prime\prime}_{{\mathbold
t}_{1}{\mathbold n}}+a_{22}A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold
n}}\right)A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold n}}{\mathbold
t}_{1}\otimes{\mathbold n}.$

Thus, performing the changes

$\begin{cases}A^{\prime\prime}_{{\mathbold t}^{\prime}_{1}{\mathbold
n}}=a_{11}A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}}+a_{21}A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold n}}\\\
A^{\prime\prime}_{{\mathbold t}^{\prime}_{2}{\mathbold
n}}=a_{12}A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}}+a_{22}A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold n}},\end{cases}$

we have shown that

${\mathbold A}_{{\mathbold t}_{1},{\mathbold t}_{2},{\mathbold
n}}\left(A^{\prime\prime}_{{\mathbold n}{\mathbold
n}},1,1,A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold n}},0\right)={\mathbold
A}_{{\mathbold t}^{\prime}_{1},{\mathbold t}^{\prime}_{2},{\mathbold
n}}\left(A^{\prime\prime}_{{\mathbold n}{\mathbold
n}},1,1,A^{\prime\prime}_{{\mathbold t}^{\prime}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}^{\prime}_{2}{\mathbold n}},0\right).$

Having in mind that the matrix (55) is invertible, this shows that

$\displaystyle\displaystyle\left\\{{\mathbold A}_{{\mathbold t}_{1},{\mathbold
t}_{2},{\mathbold n}}\left(A^{\prime\prime}_{{\mathbold n}{\mathbold
n}},1,1,A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold
n}},0\right):\,A^{\prime\prime}_{{\mathbold n}{\mathbold
n}}>0,\,A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold n}}\in\mathbb{R}\right\\}$
$\displaystyle\displaystyle=\left\\{{\mathbold A}_{{\mathbold
t}^{\prime}_{1},{\mathbold t}^{\prime}_{2},{\mathbold
n}}\left(A^{\prime\prime}_{{\mathbold n}{\mathbold
n}},1,1,A^{\prime\prime}_{{\mathbold t}^{\prime}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}^{\prime}_{2}{\mathbold
n}},0\right):\,A^{\prime\prime}_{{\mathbold n}{\mathbold
n}}>0,\,A^{\prime\prime}_{{\mathbold t}^{\prime}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}^{\prime}_{2}{\mathbold
n}}\in\mathbb{R}\right\\}$

and, analogously,

$\displaystyle\displaystyle\left\\{{\mathbold A}_{{\mathbold t}_{1},{\mathbold
t}_{2},{\mathbold n}}\left(A^{\prime\prime}_{{\mathbold n}{\mathbold
n}},1,1,A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold
n}},0\right):\,A^{\prime\prime}_{{\mathbold t}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}_{2}{\mathbold n}}\in\mathbb{R}\right\\}$
$\displaystyle\displaystyle=\left\\{{\mathbold A}_{{\mathbold
t}^{\prime}_{1},{\mathbold t}^{\prime}_{2},{\mathbold
n}}\left(A^{\prime\prime}_{{\mathbold n}{\mathbold
n}},1,1,A^{\prime\prime}_{{\mathbold t}^{\prime}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}^{\prime}_{2}{\mathbold
n}},0\right):\,A^{\prime\prime}_{{\mathbold t}^{\prime}_{1}{\mathbold
n}},A^{\prime\prime}_{{\mathbold t}^{\prime}_{2}{\mathbold
n}}\in\mathbb{R}\right\\}.$

This shows that $\displaystyle A_{{\mathbold n}{\mathbold n}}^{*}$ and
$\displaystyle W_{\mathrm{d}}$ are the same for both basis, so proving a).

Now we show b). Changing $\displaystyle{\mathbold n}$ with
$\displaystyle-{\mathbold n}$ does not alter $\displaystyle A_{{\mathbold
n}{\mathbold n}}$, as can be shown from (7). In order to show that the rest of
the quantities remain equal under this change, we first notice that
$\displaystyle\\{-{\mathbold t}_{1},-{\mathbold t}_{2},-{\mathbold n}\\}$ is
also an orthonormal basis and, for all $\displaystyle A_{{\mathbold
n}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold t}_{1}},A_{{\mathbold
t}_{2}{\mathbold t}_{2}}>0$ and $\displaystyle A_{{\mathbold t}_{1}{\mathbold
n}},A_{{\mathbold t}_{2}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold
t}_{2}}\in\mathbb{R}$,

${\mathbold A}_{{\mathbold t}_{1},{\mathbold t}_{2},{\mathbold
n}}(A_{{\mathbold n}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold
t}_{1}},A_{{\mathbold t}_{2}{\mathbold t}_{2}},A_{{\mathbold t}_{1}{\mathbold
n}},A_{{\mathbold t}_{2}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold
t}_{2}})={\mathbold A}_{-{\mathbold t}_{1},-{\mathbold t}_{2},-{\mathbold
n}}(A_{{\mathbold n}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold
t}_{1}},A_{{\mathbold t}_{2}{\mathbold t}_{2}},A_{{\mathbold t}_{1}{\mathbold
n}},A_{{\mathbold t}_{2}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold
t}_{2}}).$

This formula, together with (13), show that $\displaystyle A_{{\mathbold
n}{\mathbold n}}^{*}$ is the same for $\displaystyle{\mathbold n}$ and
$\displaystyle-{\mathbold n}$. In fact, with Proposition 2.4 it also implies
the conclusion of b).

We now show c), so let $\displaystyle{\mathbold Q}\in SO(3)$. From (7) we see
immediately that $\displaystyle A_{{\mathbold n}{\mathbold n}}$ does not
change when $\displaystyle{\mathbold F}$ is replaced with
$\displaystyle{\mathbold Q}{\mathbold F}$, since $\displaystyle|({\mathbold
Q}{\mathbold F})^{-T}\,{\mathbold n}|=|{\mathbold Q}^{-T}{\mathbold
F}^{-T}\,{\mathbold n}|=|{\mathbold F}^{-T}\,{\mathbold n}|$. Now, in view of
(13) and Proposition 2.4, the frame-indifference of $\displaystyle W$ readily
implies that $\displaystyle A_{{\mathbold n}{\mathbold n}}^{*}$ and the whole
$\displaystyle W_{\mathrm{d}}$ is the same for $\displaystyle{\mathbold F}$
and $\displaystyle{\mathbold Q}{\mathbold F}$.

In order to show d), so let $\displaystyle{\mathbold Q}\in\mathcal{S}$. When
the pair $\displaystyle({\mathbold F},{\mathbold n})$ is replaced with
$\displaystyle({\mathbold F}{\mathbold Q}^{T},{\mathbold Q}{\mathbold n})$,
the last expression of (7) does not change, since, $\displaystyle|({\mathbold
F}{\mathbold Q}^{T})^{-T}{\mathbold Q}{\mathbold n}|=|{\mathbold
F}^{-T}{\mathbold Q}^{-1}{\mathbold Q}{\mathbold n}|=|{\mathbold
F}^{-T}\,{\mathbold n}|$. This shows that $\displaystyle A_{{\mathbold
Q}{\mathbold n}\,{\mathbold Q}{\mathbold n}}=A_{{\mathbold n}{\mathbold n}}$.
In order to show that the rest of the quantities remain equal under this
change, we first notice that $\displaystyle\\{{\mathbold Q}{\mathbold
t}_{1},{\mathbold Q}{\mathbold t}_{2},{\mathbold Q}{\mathbold n}\\}$ is also
an orthonormal basis and, for all $\displaystyle A_{{\mathbold n}{\mathbold
n}},A_{{\mathbold t}_{1}{\mathbold t}_{1}},A_{{\mathbold t}_{2}{\mathbold
t}_{2}}>0$ and $\displaystyle A_{{\mathbold t}_{1}{\mathbold n}},A_{{\mathbold
t}_{2}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold t}_{2}}\in\mathbb{R}$,

$\displaystyle\displaystyle{\mathbold F}{\mathbold A}_{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}}(A_{{\mathbold n}{\mathbold
n}},A_{{\mathbold t}_{1}{\mathbold t}_{1}},A_{{\mathbold t}_{2}{\mathbold
t}_{2}},A_{{\mathbold t}_{1}{\mathbold n}},A_{{\mathbold t}_{2}{\mathbold
n}},A_{{\mathbold t}_{1}{\mathbold t}_{2}})$
$\displaystyle\displaystyle={\mathbold F}{\mathbold Q}^{T}{\mathbold
A}_{{\mathbold Q}{\mathbold t}_{1},{\mathbold Q}{\mathbold t}_{2},{\mathbold
Q}{\mathbold n}}(A_{{\mathbold n}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold
t}_{1}},A_{{\mathbold t}_{2}{\mathbold t}_{2}},A_{{\mathbold t}_{1}{\mathbold
n}},A_{{\mathbold t}_{2}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold
t}_{2}}){\mathbold Q}$

and, hence, since $\displaystyle{\mathbold Q}$ is a symmetry for
$\displaystyle W$,

$\displaystyle\displaystyle W\left({\mathbold F}{\mathbold A}_{{\mathbold
t}_{1},{\mathbold t}_{2},{\mathbold n}}(A_{{\mathbold n}{\mathbold
n}},A_{{\mathbold t}_{1}{\mathbold t}_{1}},A_{{\mathbold t}_{2}{\mathbold
t}_{2}},A_{{\mathbold t}_{1}{\mathbold n}},A_{{\mathbold t}_{2}{\mathbold
n}},A_{{\mathbold t}_{1}{\mathbold t}_{2}})\right)$
$\displaystyle\displaystyle=W\left({\mathbold F}{\mathbold Q}^{T}{\mathbold
A}_{{\mathbold Q}{\mathbold t}_{1},{\mathbold Q}{\mathbold t}_{2},{\mathbold
Q}{\mathbold n}}(A_{{\mathbold n}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold
t}_{1}},A_{{\mathbold t}_{2}{\mathbold t}_{2}},A_{{\mathbold t}_{1}{\mathbold
n}},A_{{\mathbold t}_{2}{\mathbold n}},A_{{\mathbold t}_{1}{\mathbold
t}_{2}})\right).$

This shows that $\displaystyle A_{{\mathbold Q}{\mathbold n}\,{\mathbold
Q}{\mathbold n}}^{*}=A_{{\mathbold n}{\mathbold n}}^{*}$ and $\displaystyle
W_{\mathrm{d}}({\mathbold F}{\mathbold Q}^{T},{\mathbold Q}{\mathbold
n})=W_{\mathrm{d}}({\mathbold F},{\mathbold n})$, which completes the proof of
d).

Finally, e) is a particular case of d) when $\displaystyle\mathcal{S}=SO(3)$.
∎

###### Proof of Lemma 6.1.

Taking $\displaystyle\mathbold{\varepsilon}$ as

$\varepsilon_{11}=0,\qquad\varepsilon_{12}=\frac{1}{2},\qquad\varepsilon_{22}=0$

and using
$\displaystyle\boldsymbol{\mathsf{C}}\mathbold{\varepsilon}:\mathbold{\varepsilon}>0$,
we find that $\displaystyle c_{1212}>0$. Analogously, taking

$\varepsilon_{11}=0,\qquad\varepsilon_{12}=-\frac{c_{1222}}{2\sqrt{c_{1212}}},\qquad\varepsilon_{22}=\sqrt{c_{1212}},$

we find that $\displaystyle c_{1212}c_{2222}-c_{1222}^{2}>0$. ∎

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