GPa (see Sec. 4.2.1). The microstructure evolution during the forward and
reverse MTs are shown in Fig. 22 when
$\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=1200$ GPa, and we compare the results
with Fig. 17 for $\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=800$ GPa .

Forward MT. The $\sf M$ nucleates in a jump-like manner on the GBs and TJ at
$\theta_{n}^{f}=323.1$ K, which is $9.8$ K higher than the sample with lower
junction energy ($\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=800$ GPa). We thus
conclude that the combined effect of a slightly lower numerical width and the
higher junction energy promotes $\sf M$ nucleation. With a further decrease in
$\theta$, the volume fraction of $\sf M$ increases, and we get a fully
martensitic sample at $100$ K. In the wholly transformed samples shown in
Figs. 17(a,b) and 22(a,b), the microstructures are qualitatively similar with
some differences in the number of $\sf M$ plates. However, the microstructures
at the intermediate temperatures in these samples were different.

Reverse MT. As depicted in Fig. 22(c,d), a large $\sf A$ region nucleates
within $G_{3}$ at the upper-right corner at $\theta_{n}^{r}=176.3$ K. At $250$
K, a significant volume fraction of $G_{1}$ and $G_{3}$ are transformed to
$\sf A$, while $G_{2}$ is still fully martensitic; see Fig. 22(a,b). At
$\theta=\theta_{n}^{f}=323.1$ K, residual $\sf M$ is observed only in the GB
and junction regions, and the obtained microstructure is almost the same as
that obtained during the forward MT, as shown in Fig. 22(a,b). A residual $\sf
PM$ is observed on $\Gamma_{13}$, $\Gamma_{23}$, and the junction regions at
$367.8$ K, which barrierlessly transforms to $\sf A$ at $367.9$ K leaving
behind the original austenitic tricrystal.

####  4.2.4 Effect of sample size

To show the effect of the sample size, we consider an austenitic tricrystal of
$50$ nm $\times 50$ nm size. All the parameters are identical to the example
shown in Fig. 17. The results for the forward MT is shown in Fig. 23. The
numerical width of the GB is obtained as $1.07$ nm. The $\sf M$ phase
nucleates at $\theta_{n}^{f}=317.5$ K in a jump-like manner. Although the
volume fraction of $\sf M$ at a given $\theta$ is almost the same in both the
samples, the variant plates are longer and more regularly spaced in the larger
sample. The sample is fully martensitic at $100$ K. There is a difference in
the orientation of the plates in both the martensitic samples. In grain
$G_{1}$ of the larger sample, two sets of variant plates oriented at
$30^{\circ}$ and $60^{\circ}$ with respect to the $\boldsymbol{e}_{1}$-axis
are obtained. The variant plates are mainly oriented at $30^{\circ}$ with
respect to $\boldsymbol{e}_{1}$-axis in $G_{2}$ except for a small region of
the bottom surface. The plates are vertical in $G_{3}$. However, a pair of
horizontal plates are seen near the top and bottom surfaces.

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT

Figure 23: Microstructure evolution during (a,b) forward, and (c,d) reverse
MTs in a $50$ nm $\times$ $50$ nm tricrystal having symmetric planar GBs with
$\theta$ variation. We used
$\Delta\gamma_{12}^{gb}=\Delta\gamma_{13}^{gb}=\Delta\gamma_{23}^{gb}=-0.5$
N/m, $\delta_{12}^{gb}=\delta_{13}^{gb}=\delta_{23}^{gb}=1$ nm, and
$\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=800$ GPa. No strain applied.

Remarks: We could not find the relevant experimental or atomistic results for
quantitatively comparing the transformation temperatures influenced by GBs in
the presence of TJs with our numerical results. However, the nucleation at the
GBs and microstructure development about the GBs are in agreement with the
experimental and atomistic results in bicrystals [28, 29, 50], as discussed in
Sec. 4.1.1 in details. Notably, the existing PF models initiate nucleation at
the GBs by placing artificial nuclei or considering noise in the order
parameters or displacement fields as discussed in details in Sec. 1. on the
other hand, the GB parameters $\Delta\gamma_{IJ}^{gb}$ and $\delta_{IJ}^{gb}$
in the bicrystals, and an additional TJ parameter $\rho_{0}K_{IJK}$ play the
central role in determining the nucleation of $\sf M$ and the subsequent
microstructure evolution, which justifies the novely of the present model.

##  5 Concluding remarks

We have developed a thermodynamically consistent novel multiphase PF approach
for studying GB-induced MTs in polycrystalline solids, where $N$ independent
order parameters are used for describing $\sf A$ and $N(>1)$ martensitic
variants, and another $M$ independent order parameters are used for describing
the $M(>1)$ grains of the sample. The transformation stretch tensor for each
of the grains is taken as a linear combination of the Bain tensors of all the
variants (appropriately rotated about a reference frame) multiplied with
nonlinear interpolation functions of the order parameters related to the
phases. The transformation stretch tensor of the sample is then taken as a
linear combination of the transformation strain tensor for each grain
multiplied with a nonlinear interpolation function related to the grains. The
free energy of the system is composed of the elastic energy, barrier energy
between $\sf A$-$\sf M$ and also between all the variants, the thermal (or
chemical) energy of the phases, the energy penalizing the deviation of the
transformation paths between the variant from the prescribed ones, the energy
of the interfaces between $\sf A$-$\sf M$ and the ones between the variants,
and the isotropic energy (independent of GB orientation) of the GB region. The
energy of the GB region is considered to be composed of the barrier energy
between the grains, gradient energies of the GBs, and energies of the triple
junctions, where the barrier energy coefficients and the gradient energy
coefficients are considered to be functions of the order parameter $\eta_{0}$,
which allows the GB energy to vary during the $\sf A\leftrightarrow\sf M$
transformations due to the structural changes. The total stresses are composed
of the elastic and the structural parts within the $\sf A$-$\sf M$ interfaces,
variant-variant boundaries, and the GBs. In order to obtain the correct form
of the structural stresses, all the barrier energies and the gradient energies
are multiplied with the determinant of the deformation gradient tensor. The
coupled system of mechanical equilibrium equations and the Ginzburg-Landau
equations for all the order parameters related to the $\sf A$-$\sf M$ phases,
the variants, and the grains are derived. The governing partial differential
equations are solved using the finite element method and assuming the
isotropic elastic response of the solid under plane stress condition. The
results for the 2D bicrystals with a planar symmetric tilt GB, a planar
asymmetric tilt GB, and a circular symmetric tilt GB are presented, where the
strong effect of the stress-free austenitic GB width, change in GB energy due
to the MTs, GB misorientation, applied strain, and sample size are studied in
details during both the forward ($\sf A\to\sf M$) and reverse ($\sf M\to\sf
A$) transformations. A rich plot for the temperatures of barrierless (jump-
like) transformations between $\sf A$, $\sf PM$, and $\sf M$ in the GB region
within a bicrystal with symmetric planar tilt GB with a variation of the
stress-free austenitic GB width was plotted, keeping all other parameters
identical. All the transformation temperatures increase with a decrease in the
GB width. The effect of the GB width on temperature hysteresis is also
studied. The role of the incompatibility of the transformation strains for
each variant across the GB on microstructures is studied. The nucleation of
$\sf PM$ or $\sf A$ from a fully martensitic sample during the reverse
transformation was seen at the highest energetic sites of the sample. The GB
parameters are shown to play an important role on the microstructures and
austenite finish temperature. The effect of the GB width, change in GB energy
due to MTs, TJ energy, and sample size on the transformation temperatures and
subsequent microstructure evolution during the forward and reverse MTs is
studied for 2D tricrystals consisting of three planar symmetric tilt GBs
meeting in the middle of the sample at $120^{\circ}$ dihedral angles. The
compatibility of the twinned microstructures across the GBs and TJ is also
studied. The elastic and structural stresses within the fully martensitic
bicrystals and tricrystals are shown, which are observed to be concentrated
mainly within the GB and TJ regions, and across the variant-variant
interfaces. Understanding of the stress distribution is important for the
materials design and their failure analysis [33].

The numerical results have been compared with the limited experimental and
atomistic results available in the literature, and there is a good qualitative
agreement between them. This paper motivates the researchers to conduct
controlled experiments and atomistic studies to better understand the role of
the GB parameters on the nucleation of $\sf PM$ and $\sf M$ at the nanoscale.
The present PF results also suggest that TJ energy and size play a significant
role in the temperatures of transformations and microstructures, which is yet
to be confirmed systematically, both experimentally and through atomistic
simulations. Systematic studies are needed for quantification of the change in
energies during to MTs within the GBs and TJ regions, which can be compared
with the present model for the GB energy (see Eqs. (LABEL:MF2)and (2.17)). The
present PF model can further study the GB-induced MTs in a more realistic
polycrystalline sample with numerous grains. The model can also be extended
for studying the role of other heterogeneities, including point defects,
dislocation pileups, external surface, and the role of diffusion in
polycrystalline solids coupled to GB-induced MTs.

Appendix

##  Appendix A Derivation for the dissipation inequality

In this appendix, we derive the dissipation inequality (2.18), which is
finally used in Sec. 2.4 for deriving the total, elastic, and structural
stresses and the Ginzburg-Landau equations. The general thermodynamically
consistent framework at large strains developed for a single grain in [79, 87]
is extended for polycrystalline samples in this paper. We directly start with
the following dissipation inequality, which is derived using the first and
second laws of thermodynamics like that of [79, 87]:

$\rho_{0}{\mathcal{D}}=\boldsymbol{P}:\dot{\boldsymbol{F}}^{T}-\rho_{0}\dot{\psi}-\rho_{0}s\dot{\theta}+\nabla_{0}\cdot(\boldsymbol{q}_{0}^{\eta}\dot{\eta}_{0})+\sum_{i=1}^{N}\nabla_{0}\cdot(\boldsymbol{q}_{i}^{\eta}\dot{\eta}_{i})+\sum_{I=1}^{M}\nabla_{0}\cdot(\boldsymbol{q}_{I}^{\xi}\dot{\xi}_{I})\geq
0\quad\text{in }\Omega_{0},$ (A.1)

where $s$ is the specific entropy, the vectors $\boldsymbol{q}_{0}^{\eta}$,
$\boldsymbol{q}_{i}^{\eta}$, and $\boldsymbol{q}_{I}^{\xi}$ are the
generalized forces which are conjugate to the gradient of the rate of the
associated order parameters in $\Omega_{0}$, and all other symbols are defined
in Sec. 2. Note that $\boldsymbol{q}_{0}^{\eta}$, $\boldsymbol{q}_{i}^{\eta}$,
and $\boldsymbol{q}_{I}^{\xi}$, which are introduced here for balancing some
terms from the inequality (A.1), are yet to be determined (see [79] for
details). In addition to the inequality (A.1), we also get the Fourier’s
inequality $-\frac{1}{\theta}\boldsymbol{h}_{0}\cdot\nabla_{0}\theta\geq 0$
[79], where $\boldsymbol{h}_{0}$ is the heat flux vector. Based on this
inequality, one can assume the Fourier’s law
$\boldsymbol{h}_{0}=-\boldsymbol{K}_{\theta}\cdot\nabla_{0}\theta$, where
$\boldsymbol{K}_{\theta}$ is the heat conductivity tensor which is symmetric
and positive semi-definite. Note that $\boldsymbol{h}_{0}=\boldsymbol{0}$ for
a sample with uniform temperature.

We now work on the inequality (A.1) and rewrite it in a suitable form. The
material time derivative of the free energy give by Eq. (2.13) is obtained as

$\displaystyle\dot{\psi}$ $\displaystyle=$
$\displaystyle\frac{J_{t}}{\rho_{0}}\frac{\partial\psi_{e}}{\partial\boldsymbol{F}_{e}}\cdot\boldsymbol{F}_{t}^{-T}:\dot{\boldsymbol{F}}^{T}-\frac{J_{t}}{\rho_{0}}\boldsymbol{F}_{e}^{T}\cdot\frac{\partial\psi_{e}}{\partial\boldsymbol{F}_{e}}\cdot\boldsymbol{F}_{t}^{-T}:\dot{\boldsymbol{F}}_{t}^{T}+\frac{J_{t}\psi_{e}}{\rho_{0}}\boldsymbol{F}_{t}^{-T}:\dot{\boldsymbol{F}}_{t}^{T}+J(\breve{\psi}^{\theta}+\psi^{\nabla}+\psi^{\xi}){\boldsymbol{F}}^{-T}:\dot{\boldsymbol{F}}^{T}+$
(A.2)
$\displaystyle\sum_{i=0}^{N}\left(J\frac{\partial(\breve{\psi}^{\theta}+\psi^{\nabla})}{\partial\eta_{i}}+\frac{\partial(\tilde{\psi}^{\theta}+\psi^{p})}{\partial\eta_{i}}+\frac{J_{t}}{\rho_{0}}\left.\frac{\partial\psi_{e}}{\partial\eta_{i}}\right|_{\boldsymbol{F}_{e}}\right)\dot{\eta}_{i}+J\frac{\partial{\psi^{\xi}}}{\partial\eta_{0}}\dot{\eta}_{0}+\sum_{I=1}^{M}\left(J\frac{\partial{\psi}^{\xi}_{loc}}{\partial\xi_{I}}+\frac{J_{t}}{\rho_{0}}\left.\frac{\partial\psi_{e}}{\partial\xi_{I}}\right|_{\boldsymbol{F}_{e}}\right)\dot{\xi}_{I}+$
$\displaystyle
J\sum_{i=0}^{N}\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}}\cdot\dot{\overline{\nabla\eta_{i}}}+J\sum_{I=1}^{M}\frac{\partial\psi^{\nabla}}{\partial\nabla\xi_{I}}\cdot\dot{\overline{\nabla\xi_{I}}}+\frac{\partial\psi}{\partial\theta}\dot{\theta},$

where we have used the following time derivatives
$\dot{\overline{det\,\boldsymbol{A}}}=({det\,\boldsymbol{A}})\boldsymbol{A}^{-T}:\dot{\boldsymbol{A}}^{T}$,
and
$\dot{\overline{\boldsymbol{A}^{-1}}}=-\boldsymbol{A}^{-1}\cdot\dot{\boldsymbol{A}}\cdot\boldsymbol{A}^{-1}$
for an arbitrary invertible second order tensor $\boldsymbol{A}(t)$. The
following term from Eq. (A.2), for example, is rewritten as

$\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}}\cdot\dot{\overline{\nabla\eta_{i}}}=\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}}\cdot\dot{\overline{(\boldsymbol{F}^{-T}\cdot{\nabla_{0}\eta_{i}})}}=\left(\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}}\right)\cdot\dot{\overline{\nabla_{0}\eta_{i}}}-\left(\nabla\eta_{i}\otimes\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}}\right):\dot{\boldsymbol{F}}^{T},$
(A.3)

where we have used the relation
$\nabla\eta_{i}=\boldsymbol{F}^{-T}\cdot\nabla_{0}\eta_{i}$ [79]. Using Eq.
(A.3) into Eq. (A.2) and rearranging the terms, $\dot{\psi}$ is rewritten as

$\displaystyle\dot{\psi}$ $\displaystyle=$
$\displaystyle\left(\frac{J_{t}}{\rho_{0}}\frac{\partial\psi_{e}}{\partial\boldsymbol{F}_{e}}\cdot\boldsymbol{F}_{t}^{-T}+J(\breve{\psi}^{\theta}+\psi^{\nabla}+\psi^{\xi}){\boldsymbol{F}}^{-T}-J\sum_{i=0}^{N}\nabla\eta_{i}\otimes\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}}-J\sum_{I=1}^{M}\nabla\xi_{I}\otimes\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\xi}_{\nabla}}{\partial\nabla\xi_{I}}\right):\dot{\boldsymbol{F}}^{T}+$
(A.4)
$\displaystyle\sum_{i=0}^{N}\left(J\frac{\partial(\breve{\psi}^{\theta}+\psi^{\nabla})}{\partial\eta_{i}}+\frac{\partial({\tilde{\psi}}^{\theta}+\psi^{p})}{\partial\eta_{i}}+\frac{J_{t}}{\rho_{0}}\left.\frac{\partial\psi_{e}}{\partial\eta_{i}}\right|_{\boldsymbol{F}_{e}}\right)\dot{\eta}_{i}+J\frac{\partial{\psi^{\xi}}}{\partial\eta_{0}}\dot{\eta}_{0}+\frac{J_{t}}{\rho_{0}}\left(\psi_{e}\boldsymbol{F}_{t}^{-T}-\boldsymbol{F}_{e}^{T}\cdot\frac{\partial\psi_{e}}{\partial\boldsymbol{F}_{e}}\cdot\boldsymbol{F}_{t}^{-T}\right):\dot{\boldsymbol{F}}_{t}^{T}+$
$\displaystyle\sum_{I=1}^{M}\left(J\frac{\partial{\psi}^{\xi}_{loc}}{\partial\xi_{I}}+\frac{J_{t}}{\rho_{0}}\left.\frac{\partial\psi_{e}}{\partial\xi_{I}}\right|_{\boldsymbol{F}_{e}}\right)\dot{\xi}_{I}+\sum_{i=0}^{N}J\left(\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}}\right)\cdot\dot{\overline{\nabla_{0}\eta}}+\sum_{I=1}^{M}J\left(\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\xi_{I}}\right)\cdot\dot{\overline{\nabla_{0}\xi_{I}}}+\frac{\partial\psi}{\partial\theta}\dot{\theta}.$

Since $\boldsymbol{F}_{t}$ is considered to be a function of all the order
parameters $\eta_{0},\eta_{1},\ldots,\eta_{N}$ and $\xi_{1},\ldots,\xi_{M}$
(see Eq. (2.10)), Eq. (A.4) can further be rewritten as

$\displaystyle\dot{\psi}$ $\displaystyle=$
$\displaystyle\left(\frac{J_{t}}{\rho_{0}}\frac{\partial\psi_{e}}{\partial\boldsymbol{F}_{e}}\cdot\boldsymbol{F}_{t}^{-T}+J(\breve{\psi}^{\theta}+\psi^{\nabla}+\psi^{\xi}){\boldsymbol{F}}^{-T}-J\sum_{i=0}^{N}\nabla\eta_{i}\otimes\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}}-J\sum_{I=1}^{M}\nabla\xi_{I}\otimes\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\xi}_{\nabla}}{\partial\nabla\xi_{I}}\right):\dot{\boldsymbol{F}}^{T}+$
(A.5)
$\displaystyle\sum_{i=0}^{N}\left[J\frac{\partial(\breve{\psi}^{\theta}+\psi^{\nabla})}{\partial\eta_{i}}+\frac{\partial({\tilde{\psi}}^{\theta}+\psi^{p})}{\partial\eta_{i}}+\frac{J_{t}}{\rho_{0}}\left.\frac{\partial\psi_{e}}{\partial\eta_{i}}\right|_{\boldsymbol{F}_{e}}+\frac{J_{t}}{\rho_{0}}\left(\psi_{e}\boldsymbol{F}_{t}^{-T}-\boldsymbol{F}_{e}^{T}\cdot\frac{\partial\psi_{e}}{\partial\boldsymbol{F}_{e}}\cdot\boldsymbol{F}_{t}^{-T}\right):\frac{\partial{\boldsymbol{F}}_{t}^{T}}{\partial\eta_{i}}\right]\dot{\eta}_{i}+$
$\displaystyle
J\frac{\partial{\psi^{\xi}}}{\partial\eta_{0}}\dot{\eta}_{0}+\sum_{I=1}^{M}\left[J\frac{\partial{\psi}^{\xi}}{\partial\xi_{I}}+\frac{J_{t}}{\rho_{0}}\left.\frac{\partial\psi_{e}}{\partial\xi_{I}}\right|_{\boldsymbol{F}_{e}}+\frac{J_{t}}{\rho_{0}}\left(\psi_{e}\boldsymbol{F}_{t}^{-T}-\boldsymbol{F}_{e}^{T}\cdot\frac{\partial\psi_{e}}{\partial\boldsymbol{F}_{e}}\cdot\boldsymbol{F}_{t}^{-T}\right):\frac{\partial{\boldsymbol{F}}_{t}^{T}}{\partial\xi_{I}}\right]\dot{\xi}_{I}+$
$\displaystyle\sum_{i=0}^{N}J\left(\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}}\right)\cdot\dot{\overline{\nabla_{0}\eta}}+\sum_{I=1}^{M}J\left(\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\xi_{I}}\right)\cdot\dot{\overline{\nabla_{0}\xi_{I}}}+\frac{\partial\psi}{\partial\theta}\dot{\theta}.$

Using Eq. (A.5) in the inequality (A.1), and assuming that the dissipation
rate is independent of $\dot{\theta}$, $\dot{\overline{\nabla_{0}\eta_{l}}}$
(for $l=0,\ldots,N$) and $\dot{\overline{\nabla_{0}\xi_{I}}}$ (for
$I=0,\ldots,M$) [79], we get the specific entropy as
$s=-{\partial\psi}/{\partial\theta}$, and the generalized force vectors as

$\boldsymbol{q}_{l}^{\eta}=\rho_{0}J\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{l}}\quad\text{for
}l=0,1,\ldots,N;\quad\text{and}\quad\boldsymbol{q}_{I}^{\xi}=\rho_{0}J\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\xi_{I}}\quad\text{for}\quad
I=1,\ldots,M.$ (A.6)

Using Eqs. (A.5) and (A.6), the dissipation inequality (A.1) is finally
expressed in the form given by (2.18), and it is used in Sec. 2.4 for deriving
the governing equations for the stresses and the Ginzburg-Landau equations.

##  Appendix B A list of the governing equations in the general form

  * •

Order parameters: $\eta_{0}$ is the order parameter describing $\sf
A\leftrightarrow\sf M$ transformations; $\eta_{1},\eta_{2},\ldots,\eta_{N}$
are the order parameters describing the variants ${\sf M}_{1},{\sf
M}_{2},\ldots,{\sf M}_{N}$, respectively; $\xi_{1},\xi_{2},\ldots,\xi_{M}$ are
the order parameters describing grains $G_{1},G_{2},\ldots,G_{M}$,
respectively. The order parameters satisfy

$\sum_{i=1}^{N}\eta_{i}=1;\quad\text{and}\quad\sum_{I=1}^{M}\xi_{I}=1.$ (B.1)

  * •

Kinematic relations:

${\boldsymbol{F}}=\boldsymbol{V}_{e}\cdot\boldsymbol{R}\cdot\boldsymbol{U}_{t};\,\,{\boldsymbol{E}}=\frac{1}{2}({\boldsymbol{F}}^{T}\cdot{\boldsymbol{F}}-{\boldsymbol{I}});\,\,{\boldsymbol{E}}_{e}=\frac{1}{2}({\boldsymbol{F}}_{e}^{T}\cdot{\boldsymbol{F}}_{e}-{\boldsymbol{I}});\,\,\boldsymbol{b}=\frac{1}{2}({\boldsymbol{F}}\cdot{\boldsymbol{F}}^{T}-{\boldsymbol{I}});\,\,\boldsymbol{b}_{e}=\frac{1}{2}({\boldsymbol{F}}_{e}\cdot{\boldsymbol{F}}_{e}^{T}-{\boldsymbol{I}}).$
(B.2)

Transformation deformation gradient:

$\displaystyle\text{ }\quad{\boldsymbol{F}}_{t}$ $\displaystyle=$
$\displaystyle{\boldsymbol{U}}_{t}=\boldsymbol{I}+\sum_{I=1}^{M}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{ti}^{(I)}\phi_{i}(\eta_{i})\varphi(a_{\varepsilon},\eta_{0})\phi_{I}^{\xi}(\xi_{I});$
$\displaystyle\varphi(a_{\varepsilon},\eta_{0})$ $\displaystyle=$
$\displaystyle
a_{\varepsilon}\eta_{0}^{2}(1-\eta_{0})^{2}+\eta_{0}^{3}(4-3\eta_{0});\quad\phi_{i}(\eta_{i})=\eta_{i}^{2}(3-2\eta_{i});\quad\phi_{I}^{\xi}(\xi_{I})=\xi_{I}^{2}(3-2\xi_{I}).$
(B.3)

  * •

Helmholtz’s free energy density:

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\frac{J_{t}}{\rho_{0}}\psi_{e}(\boldsymbol{E}_{e},\tilde{\eta},\tilde{\xi},\theta)+J\breve{\psi}^{\theta}(\tilde{\eta},\theta)+\tilde{\psi}^{\theta}(\tilde{\eta},\theta)+{\psi}^{p}(\tilde{\eta})+J\psi^{\nabla}(\eta_{0},\tilde{\eta}^{\nabla})+J\psi^{\xi}(\tilde{\xi},\tilde{\xi}^{\nabla},\eta_{0}),\,\text{where}$
$\displaystyle\psi_{e}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\boldsymbol{E}_{e}:\left({\boldsymbol{\mathbb{C}}}:\boldsymbol{E}_{e}\right);\quad{\boldsymbol{\mathbb{C}}}(\tilde{\eta},\tilde{\xi})=\sum_{I=1}^{M}\left[(1-\varphi(a,\eta_{0})){\boldsymbol{\mathbb{C}}}_{0}^{(I)}+\varphi(a,\eta_{0})\sum_{i=1}^{N}\phi_{i}(\eta_{i}){\boldsymbol{\mathbb{C}}}_{i}^{(I)}\right]\phi_{I}^{\xi}(\xi_{I});$
$\displaystyle\breve{\psi}^{\theta}$ $\displaystyle=$
$\displaystyle[A_{0M}+(a_{\theta}-3)\Delta\psi^{\theta}(\theta)]\eta_{0}^{2}(1-\eta_{0})^{2}+\varphi(a_{b},\eta_{0})\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}{A}^{\eta}_{ij}\eta_{i}^{2}\eta_{j}^{2};$
$\displaystyle\tilde{\psi}^{\theta}$ $\displaystyle=$
$\displaystyle\psi^{\theta}_{0}(\theta)+\eta_{0}^{2}(3-2\eta_{0})\Delta\psi^{\theta}(\theta);$
$\displaystyle\psi^{p}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}K_{ij}^{\eta}(\eta_{i}+\eta_{j}-1)^{2}\eta_{i}^{2}\eta_{j}^{2};$
$\displaystyle\psi^{\nabla}$ $\displaystyle=$
$\displaystyle\frac{1}{2\rho_{0}}\left[\beta_{0M}|\nabla\eta_{0}|^{2}-\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\beta_{ij}\nabla\eta_{i}\cdot\nabla\eta_{j}\right];$
$\displaystyle\psi^{\xi}$ $\displaystyle=$
$\displaystyle\psi^{\xi}_{loc}+\psi^{\xi}_{\nabla},\qquad\text{where}\quad\psi^{\xi}_{\nabla}=-\sum_{I=1}^{M-1}\sum_{J=I+1}^{M}\frac{\beta_{IJ}^{gb}(\eta_{0})}{2\rho_{0}}\nabla\xi_{I}\cdot\nabla\xi_{J},$
$\displaystyle\psi^{\xi}_{loc}$ $\displaystyle=$
$\displaystyle\sum_{I=1}^{M-1}\sum_{J=I+1}^{M}\left[A_{IJ}^{gb}(\eta_{0})\xi_{I}^{2}\xi_{J}^{2}+K_{IJ}^{gb}(\xi_{I}+\xi_{J}-1)^{2}\xi_{I}^{2}\xi_{J}^{2}\right]+\sum_{H=1}^{M-2}\sum_{I=H+1}^{M-1}\sum_{J=I+1}^{M}K_{HIJ}\xi_{H}^{2}\xi_{I}^{2}\xi_{J}^{2};$
$\displaystyle A_{IJ}^{gb}(\eta_{0})$ $\displaystyle=$
$\displaystyle\frac{18\gamma_{IJ}^{gb}(\eta_{0})}{\delta_{IJ}^{gb}},\quad\beta_{IJ}^{gb}(\eta_{0})=\delta_{IJ}^{gb}\gamma_{IJ}^{gb}(\eta_{0});\quad\gamma_{IJ}^{gb}(\eta_{0})=\gamma_{IJ}^{A}+\Delta\gamma_{IJ}^{gb}\varphi(\eta_{0},a);$
$\displaystyle\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})$ $\displaystyle=$
$\displaystyle{a_{\beta}}{\eta_{0}^{2}}-2[{a_{\beta}}-2(1-a_{0})]{\eta_{0}^{3}}+[{a_{\beta}}-3(1-a_{0})]{\eta_{0}^{4}}+a_{0}.$
(B.4)

  * •

Total, elastic, and structural stresses:

$\displaystyle{\boldsymbol{P}}$ $\displaystyle=$
$\displaystyle{\boldsymbol{P}}_{e}+{\boldsymbol{P}}_{st},\quad\text{where
}{\boldsymbol{P}}_{e}=J_{t}\frac{\partial\psi_{e}(\boldsymbol{F}_{e})}{\partial\boldsymbol{F}_{e}}\cdot\boldsymbol{F}_{t}^{-T};$
$\displaystyle{\boldsymbol{P}}_{st}$ $\displaystyle=$ $\displaystyle
J\rho_{0}(\breve{\psi}_{\theta}+\psi_{\nabla}+\psi_{\xi}){\boldsymbol{F}}^{-T}-J\left(\beta_{0M}\nabla\eta_{0}\otimes\nabla\eta_{0}-\frac{\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})}{2}\sum_{i=1}^{N}\sum_{j=1,\neq
i}^{N}\beta_{ij}^{\eta}\nabla\eta_{i}\otimes\nabla\eta_{j}-\right.$ (B.5)
$\displaystyle\left.\sum_{I=1}^{M}\sum_{H=1,\neq
I}^{M}\frac{\beta_{IH}^{gb}(\eta_{0})}{2}\nabla\xi_{I}\otimes\nabla\xi_{H}\right)\cdot{\boldsymbol{F}}^{-T};$
$\displaystyle\boldsymbol{\sigma}$ $\displaystyle=$
$\displaystyle\boldsymbol{\sigma}_{e}+\boldsymbol{\sigma}_{st},\quad\text{where
}\boldsymbol{\sigma}_{e}=J^{-1}_{e}\frac{\partial\psi_{e}(\boldsymbol{F}_{e})}{\partial{\boldsymbol{F}}_{e}}\cdot{\boldsymbol{F}}_{e}^{T}=J_{t}\,{\boldsymbol{F}}_{e}\cdot\frac{\partial\psi_{e}(\boldsymbol{E}_{e})}{\partial{\boldsymbol{E}}_{e}}\cdot{\boldsymbol{F}}_{e}^{T};$
$\displaystyle\boldsymbol{\sigma}_{st}$ $\displaystyle=$ $\displaystyle
J\rho(\breve{\psi}_{\theta}+\psi_{\nabla}+\psi_{\xi}){\boldsymbol{I}}-\beta_{0M}\nabla\eta_{0}\otimes\nabla\eta_{0}+\frac{\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})}{2}\sum_{i=1}^{N}\sum_{j=1,\neq
i}^{N}\beta_{ij}^{\eta}\nabla\eta_{i}\otimes\nabla\eta_{j}+$ (B.6)
$\displaystyle\sum_{I=1}^{M}\sum_{H=1,\neq
I}^{M}\frac{\beta_{IH}^{gb}(\eta_{0})}{2}\nabla\xi_{I}\otimes\nabla\xi_{H}.$

  * •

Mechanical equilibrium equations (neglecting body forces and inertia):

$\nabla_{0}\cdot\boldsymbol{P}=\boldsymbol{0}\quad\text{in
}\Omega_{0}\quad\text{or equivalently,
}\quad\nabla\cdot\boldsymbol{\sigma}=\boldsymbol{0}\quad\text{in }\Omega.$
(B.7)

  * •

Ginzburg-Landau equations

$\dot{\eta}_{0}=L_{0M}X_{0M};\quad\dot{\eta}_{i}=\sum_{j=1\,(\neq
i)}^{N}L_{ij}^{\eta}(X_{i}^{\eta}-X_{j}^{\eta});\quad\dot{\xi}_{I}=\sum_{J=1\,(\neq
I)}^{M}L_{IJ}^{\xi}(X_{I}^{\xi}-X_{J}^{\xi})\quad\text{(no sum on
}i,\,I\text{), where}$ (B.8) $\displaystyle X_{0M}$ $\displaystyle=$
$\displaystyle\left(J\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}_{e}\cdot\boldsymbol{F}-J_{t}\psi_{e}\boldsymbol{I}\right):\boldsymbol{F}_{t}^{-1}\cdot\frac{\partial\boldsymbol{F}_{t}}{\partial\eta_{0}}-\frac{J_{t}}{2}\boldsymbol{E}_{e}:\left(\frac{\partial\boldsymbol{\mathbb{C}}}{\partial\eta_{0}}:\boldsymbol{E}_{e}\right)-J\rho_{0}\frac{\partial\varphi(a_{b},\eta_{0})}{\partial\eta_{0}}\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}A_{ij}^{\eta}\eta_{i}^{2}\eta_{j}^{2}$
(B.9)
$\displaystyle-\rho_{0}J[A(\theta)+(a_{\theta}-3)\Delta\psi^{\theta}(\theta)](2\eta_{0}-6\eta_{0}^{2}+4\eta_{0}^{3})+\frac{J}{2}\frac{\partial\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})}{\partial\eta_{0}}\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\beta_{ij}^{\eta}\nabla\eta_{i}\cdot\nabla\eta_{j}$
$\displaystyle-\frac{\partial\varphi(\eta_{0},a)}{\partial\eta_{0}}\sum_{I=1}^{M-1}\sum_{H=I+1}^{M}J\rho_{0}\Delta\gamma_{IH}^{gb}\left[\frac{18}{\delta_{IH}^{gb}}\xi_{I}^{2}\xi_{H}^{2}-\frac{\delta_{IH}^{gb}}{2\rho_{0}}\nabla\xi_{I}\cdot\nabla\xi_{H}\right]-\rho_{0}(6\eta_{0}-6\eta_{0}^{2})\Delta\psi^{\theta}+J\nabla\cdot\left(\beta_{0M}\nabla\eta_{0}\right);$
$\displaystyle X_{i}^{\eta}$ $\displaystyle=$
$\displaystyle\left(J\boldsymbol{F}^{-1}\boldsymbol{\sigma}_{e}\boldsymbol{F}-J_{t}\psi_{e}\boldsymbol{I}\right):\boldsymbol{F}_{t}^{-1}\cdot\frac{\partial\boldsymbol{F}_{t}}{\partial\eta_{i}}-\frac{J_{t}}{2}\boldsymbol{E}_{e}:\left(\frac{\partial\boldsymbol{\mathbb{C}}}{\partial\eta_{i}}:\boldsymbol{E}_{e}\right)-2\rho_{0}J\sum_{j=1,j\neq
i}^{N}A_{ij}^{\eta}\eta_{i}\eta_{j}^{2}\varphi(a_{b},\eta_{0})-$ (B.10)
$\displaystyle 2\sum_{j=1,j\neq
i}^{N}\rho_{0}K_{ij}(\eta_{i}+\eta_{j}-1)(2\eta_{i}+\eta_{j}-1)\eta_{i}\eta_{j}^{2}-\frac{J}{2}\nabla\cdot\left(\sum_{j=1,j\neq
i}^{N}\tilde{\varphi}\beta_{ij}^{\eta}\nabla\eta_{j}\right),\quad\text{for
}i=1,2,\ldots,N;$ $\displaystyle X_{I}^{\xi}$ $\displaystyle=$
$\displaystyle\left(J\boldsymbol{F}^{-1}\boldsymbol{\sigma}_{e}\boldsymbol{F}-J_{t}\psi_{e}\boldsymbol{I}\right):\boldsymbol{F}_{t}^{-1}\cdot\frac{\partial\boldsymbol{F}_{t}}{\partial\xi_{I}}-\frac{J_{t}}{2}\boldsymbol{E}_{e}:\left(\frac{\partial\boldsymbol{\mathbb{C}}}{\partial\xi_{I}}:\boldsymbol{E}_{e}\right)-\sum_{H=1,H\neq
I}^{M}J\frac{36\gamma_{IH}^{gb}(\eta_{0})}{(\delta_{IH}^{gb})^{2}}\xi_{I}\xi_{H}^{2}-$
(B.11) $\displaystyle\frac{J}{2}\nabla\cdot\left(\sum_{H=1,H\neq
I}^{M}\beta_{IH}^{\xi}\nabla\xi_{H}\right)\quad\text{for }I=1,2,\ldots,M.$

  * •

Boundary conditions for the order parameters

$\nabla\eta_{0}\cdot\boldsymbol{n}=0;\quad\sum_{j=1}^{N}\beta^{\eta}_{ij}\nabla\eta_{j}\cdot\boldsymbol{n}=0\quad\text{for
}i=1,\ldots,N;\quad\sum_{H=1}^{M}\beta^{\xi}_{IH}\nabla\xi_{H}\cdot\boldsymbol{n}=0\quad\text{for
}I=1,\ldots,M\quad\text{on }S.$ (B.12)

##  Appendix C List of equations for MTs in bicrystals and tricrystals with
two variants under plane stress isotropic elasticity

We list down the governing equations for the 2D bicrystals and tricrystals
used for the simulations under plane stress condition and consider
transformations between $\sf A$ and two variants ${\sf M}_{1}$ and ${\sf
M}_{2}$. Furthermore, we assume isotropic elastic response and the Lamé
constants are identical for $\sf A$ and both the variants, which are denoted
by $\lambda$ and $\mu$. Using Eq. (2.1), we conclude that two independent
order parameters $\eta_{0}$ and $\eta_{1}$ are sufficient to describe the
phases. The order parameter $\xi_{1}$ describes the grains in the bicrystals,
and the grains in the tricrystals are described by the order parameters
$\xi_{1}$, $\xi_{2}$, and $\xi_{3}$, which satisfy $\xi_{1}+\xi_{2}+\xi_{3}=1$
(see Eq. (2.2)).

  * •

Elastic strain and total strain tensor components (indices $a,b=1,2$)

$\displaystyle
b_{ab}=\frac{1}{2}\left(\sum_{c=1}^{2}F_{ac}F_{bc}-\delta_{ab}\right);\quad
b_{(e)ab}=\frac{1}{2}\left(\sum_{c=1}^{2}F_{(e)ac}F_{(e)bc}-\delta_{ab}\right);\quad
b_{(e)33}=-\frac{\lambda^{\prime}}{2\mu}(b_{(e)11}+b_{(e)22});$ $\displaystyle
b_{33}=\frac{1}{2}(F_{33}^{2}-1);\quad
F_{(e)33}=V_{(e)33}=\sqrt{1+2b_{(e)33}};\quad
F_{33}=V_{(e)33}U_{(t)33},\quad\text{where
}\lambda^{\prime}=\frac{2\lambda\mu}{\lambda+2\mu}.$ (C.1) $\text{Also,
}b_{a3}=b_{(e)a3}=F_{a3}=F_{(e)a3}=V_{a3}=V_{(e)a3}=U_{(t)a3}=0\quad\text{for
all }a=1,2.$ (C.2) $J=(F_{11}F_{22}-F_{12}F_{21})F_{33};\quad
J_{e}=(F_{(e)11}F_{(e)22}-F_{(e)12}F_{(e)21})F_{(e)33};\quad
J_{t}=(F_{(t)11}F_{(t)22}-F_{(t)12}F_{(t)21})F_{(t)33}.$ (C.3)

  * •

Transformation stretch tensor components (indices $a,b=1,2$)

Bicrystals:

$\displaystyle F_{(t)ab}$ $\displaystyle=$
$\displaystyle\delta_{ab}+\phi_{1}^{\xi}(\xi_{1})\varepsilon_{(t)ab}^{(1)}+[1-\phi_{1}^{\xi}(\xi_{1})]\varepsilon_{(t)ab}^{(2)},\quad\text{using
}\phi_{1}^{\xi}(\xi_{1})+\phi_{2}^{\xi}(\xi_{2})=1;\,\,\xi_{1}+\xi_{2}=1,\text{
where}$ $\displaystyle\varepsilon_{(t)ab}^{(I)}$ $\displaystyle=$
$\displaystyle\left[\varepsilon_{(t2)ab}^{(I)}+(\varepsilon_{(t1)ab}^{(I)}-\boldsymbol{\varepsilon}_{(t2)ab}^{(I)})\phi_{1}(\eta_{1})\right]\varphi(a_{\varepsilon},\eta_{0})\quad\text{for
}I=1,2.$ $\displaystyle F_{(t)33}$ $\displaystyle=$
$\displaystyle\alpha;\,\,F_{(t)13}=F_{(t)23}=F_{(t)31}=F_{(t)32}=0.$ (C.4)

Tricrystals:

$\displaystyle F_{(t)ab}$ $\displaystyle=$
$\displaystyle\delta_{ab}+\phi_{1}^{\xi}(\xi_{1})\varepsilon_{(t)ab}^{(1)}+\phi_{2}^{\xi}(\xi_{2})\varepsilon_{(t)ab}^{(2)}+\phi_{3}^{\xi}(\xi_{3})\varepsilon_{(t)ab}^{(3)},\,\,\,\text{where
}\xi_{1}+\xi_{2}+\xi_{3}=1;$ $\displaystyle\varepsilon_{(t)ab}^{(I)}$
$\displaystyle=$
$\displaystyle\left[\varepsilon_{(t2)ab}^{(I)}+(\varepsilon_{(t1)ab}^{(I)}-\boldsymbol{\varepsilon}_{(t2)ab}^{(I)})\phi_{1}(\eta_{1})\right]\varphi(a_{\varepsilon},\eta_{0})\quad\text{for
}I=1,2,3.$ $\displaystyle F_{(t)33}$ $\displaystyle=$ $\displaystyle
1+(\alpha-1)\left(\phi_{1}^{\xi}(\xi_{1})+\phi_{2}^{\xi}(\xi_{2})+\phi_{3}^{\xi}(\xi_{3})\right);\,\,\,F_{(t)13}=F_{(t)23}=F_{(t)31}=F_{(t)32}=0.$
(C.5)

  * •

Free energy (indices $a,b,c,d=1,2$)

$\displaystyle\psi_{e}=\left[0.5\lambda^{\prime}(b_{(e)11}+b_{(e)22})^{2}+\mu\sum_{a,b=1}^{2}b_{(e)ab}b_{(e)ab}\right]\sum_{I=1}^{M}\phi^{\xi}_{I},\text{
using }$ (C.6)
$\displaystyle({\mathbb{C}}_{0}^{(I)})_{abcd}=({\mathbb{C}}_{1}^{(I)})_{abcd}=({\mathbb{C}}_{2}^{(I)})_{abcd}=\left[\lambda\delta_{ab}\delta_{cd}+\mu(\delta_{ad}\delta_{bc}+\delta_{ac}\delta_{bd})\right]\sum_{I=1}^{M}\phi^{\xi}_{I};$
(C.7)
$\displaystyle\rho_{0}\breve{\psi}^{\theta}=\rho_{0}[A_{0M}(\theta)+(a_{\theta}-3)\Delta\psi^{\theta}(\theta)]\eta_{0}^{2}(1-\eta_{0})^{2}+\rho_{0}A_{12}^{\eta}\eta_{1}^{2}(1-\eta_{1})^{2}\varphi(a_{b},\eta_{0});$
(C.8)
$\displaystyle\rho_{0}\tilde{\psi}^{\theta}=\rho_{0}\psi_{0}^{\theta}(\theta)+\rho_{0}\Delta\psi^{\theta}(\theta)\eta_{0}^{2}(3-2\eta_{0})\quad\text{with
}\rho_{0}\Delta\psi^{\theta}=-\Delta s(\theta-\theta_{e});$ (C.9)
$\displaystyle\rho_{0}\psi^{p}=0,\quad\text{using Eq. }\eqref{constraint11};$
(C.10)
$\displaystyle\rho_{0}\psi^{\nabla}=\frac{1}{2}\sum_{a=1}^{2}\left(\beta_{0M}\frac{\partial\eta_{0}}{\partial
r_{a}}\frac{\partial\eta_{0}}{\partial
r_{a}}+\beta_{12}^{\eta}\frac{\partial\eta_{1}}{\partial
r_{a}}\frac{\partial\eta_{1}}{\partial
r_{a}}\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})\right);$ (C.11)
$\displaystyle\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})={a_{\beta}}{\eta_{0}^{2}}-2[{a_{\beta}}-2(1-a_{0})]{\eta_{0}^{3}}+[{a_{\beta}}-3(1-a_{0})]{\eta_{0}^{4}}+a_{0}.$
(C.12)

Bicrystals:
$\displaystyle\rho_{0}\psi_{\xi}=\rho_{0}A_{12}^{gb}(\eta_{0})\xi_{1}^{2}(1-\xi_{1})^{2}+\frac{\beta_{12}^{gb}(\eta_{0})}{2}\sum_{a=1}^{2}\frac{\partial\xi_{1}}{\partial
r_{a}}\frac{\partial\xi_{1}}{\partial r_{a}};$ (C.13) Tricrystals:
$\displaystyle\rho_{0}\psi_{\xi}=\rho_{0}A_{12}^{gb}(\eta_{0})\xi_{1}^{2}\xi_{2}^{2}+\rho_{0}A_{23}^{gb}(\eta_{0})\xi_{2}^{2}\xi_{3}^{2}+\rho_{0}A_{13}^{gb}(\eta_{0})\xi_{1}^{2}\xi_{3}^{2}-\frac{1}{2}\sum_{a=1}^{2}\left[{\beta_{12}^{gb}(\eta_{0})}\frac{\partial\xi_{1}}{\partial
r_{a}}\frac{\partial\xi_{2}}{\partial r_{a}}+\right.$ (C.15)
$\displaystyle\left.{\beta_{23}^{gb}(\eta_{0})}\frac{\partial\xi_{2}}{\partial
r_{a}}\frac{\partial\xi_{3}}{\partial
r_{a}}+{\beta_{13}^{gb}(\eta_{0})}\frac{\partial\xi_{1}}{\partial
r_{a}}\frac{\partial\xi_{3}}{\partial
r_{a}}\right]+\rho_{0}(K_{12}^{gb}+K_{23}^{gb}+K_{13}^{gb}+K_{123})\xi_{1}^{2}\xi_{2}^{2}\xi_{3}^{2};\quad$
$\displaystyle\rho_{0}A_{IJ}^{gb}(\eta_{0})=\frac{18\gamma_{IJ}^{gb}(\eta_{0})}{\delta_{IJ}^{gb}},\,\,\beta_{IJ}^{gb}(\eta_{0})=\delta_{IJ}^{gb}\gamma_{IJ}^{gb}(\eta_{0}),\,\text{and
}\gamma_{IJ}^{gb}=\gamma_{IJ}^{A}+(\gamma_{IJ}^{M}-\gamma_{IJ}^{A})\varphi(a,\eta_{0}).$

  * •

Elastic and structural stresses (indices $a,b=1,2$)

$\displaystyle\sigma_{ab}=\sigma_{(e)ab}+\sigma_{(st)ab},\quad\text{and}\quad
P_{ab}=P_{(e)ab}+P_{(st)ab};\quad\text{where}$
$\displaystyle\sigma_{(e)ab}=J_{e}^{-1}\sum_{c,d=1}^{2}V_{(e)ac}V_{(e)cd}\left[\lambda^{\prime}(b_{(e)11}+b_{(e)22})\delta_{db}+2\mu
b_{(e)db}\right];$ (C.16) $\displaystyle
P_{(e)ab}=J_{t}\sum_{c,d,f=1}^{2}V_{(e)ac}V_{(e)cd}\left[\lambda^{\prime}(b_{(e)11}+b_{(e)22})\delta_{df}+2\mu
b_{(e)df}\right]F^{-1}_{bf}.$ (C.17) Bicrystals: $\displaystyle P_{({st})ab}$
$\displaystyle=$ $\displaystyle
J\rho_{0}(\breve{\psi}^{\theta}+\psi^{\nabla}+\psi^{\xi})F^{-1}_{ba}-J\sum_{c=1}^{2}\left[\beta_{0M}\frac{\partial\eta_{0}}{\partial
r_{a}}\frac{\partial\eta_{0}}{\partial
r_{c}}+\beta_{12}^{\eta}\frac{\partial\eta_{1}}{\partial
r_{a}}\frac{\partial\eta_{1}}{\partial
r_{c}}\tilde{\varphi}+\beta_{12}^{gb}(\eta_{0})\frac{\partial\xi_{1}}{\partial
r_{a}}\frac{\partial\xi_{1}}{\partial r_{c}}\right]F^{-1}_{bc};$
$\displaystyle\sigma_{({st})ab}$ $\displaystyle=$ $\displaystyle
J\rho(\breve{\psi}^{\theta}+\psi^{\nabla}+\psi^{\xi})\delta_{ab}-\beta_{0M}\frac{\partial\eta_{0}}{\partial
r_{a}}\frac{\partial\eta_{0}}{\partial
r_{b}}-\beta_{12}^{\eta}\frac{\partial\eta_{1}}{\partial
r_{a}}\frac{\partial\eta_{1}}{\partial
r_{b}}\tilde{\varphi}-\beta_{12}^{gb}(\eta_{0})\frac{\partial\xi_{1}}{\partial
r_{a}}\frac{\partial\xi_{1}}{\partial r_{b}}.$ (C.18) Tricrystals:
$\displaystyle P_{({st})ab}$ $\displaystyle=$ $\displaystyle
J\rho_{0}(\breve{\psi}^{\theta}+\psi^{\nabla}+\psi^{\xi})F^{-1}_{ba}-J\sum_{c=1}^{2}\left[\beta_{0M}\frac{\partial\eta_{0}}{\partial
r_{a}}\frac{\partial\eta_{0}}{\partial
r_{c}}+\beta_{12}^{\eta}\frac{\partial\eta_{1}}{\partial
r_{a}}\frac{\partial\eta_{1}}{\partial
r_{c}}\tilde{\varphi}-\frac{\beta_{12}^{gb}(\eta_{0})}{2}\left(\frac{\partial\xi_{1}}{\partial
r_{a}}\frac{\partial\xi_{2}}{\partial r_{c}}+\right.\right.$
$\displaystyle\left.\left.\frac{\partial\xi_{2}}{\partial
r_{a}}\frac{\partial\xi_{1}}{\partial
r_{c}}\right)-\frac{\beta_{13}^{gb}(\eta_{0})}{2}\left(\frac{\partial\xi_{1}}{\partial
r_{a}}\frac{\partial\xi_{3}}{\partial r_{c}}+\frac{\partial\xi_{3}}{\partial
r_{a}}\frac{\partial\xi_{1}}{\partial
r_{c}}\right)-\frac{\beta_{23}^{gb}(\eta_{0})}{2}\left(\frac{\partial\xi_{2}}{\partial
r_{a}}\frac{\partial\xi_{3}}{\partial r_{c}}+\frac{\partial\xi_{3}}{\partial
r_{a}}\frac{\partial\xi_{2}}{\partial r_{c}}\right)\right]F^{-1}_{bc};$
$\displaystyle\sigma_{({st})ab}$ $\displaystyle=$ $\displaystyle
J\rho(\breve{\psi}^{\theta}+\psi^{\nabla}+\psi^{\xi})\delta_{ab}-\beta_{0M}\frac{\partial\eta_{0}}{\partial
r_{a}}\frac{\partial\eta_{0}}{\partial
r_{b}}-\beta_{12}^{\eta}\frac{\partial\eta_{1}}{\partial
r_{a}}\frac{\partial\eta_{1}}{\partial
r_{b}}\tilde{\varphi}+\frac{\beta_{12}^{gb}(\eta_{0})}{2}\left(\frac{\partial\xi_{1}}{\partial
r_{a}}\frac{\partial\xi_{2}}{\partial r_{c}}+\frac{\partial\xi_{2}}{\partial
r_{a}}\frac{\partial\xi_{1}}{\partial r_{c}}\right)+$ (C.19)
$\displaystyle\frac{\beta_{13}^{gb}(\eta_{0})}{2}\left(\frac{\partial\xi_{1}}{\partial
r_{a}}\frac{\partial\xi_{3}}{\partial r_{c}}+\frac{\partial\xi_{3}}{\partial
r_{a}}\frac{\partial\xi_{1}}{\partial
r_{c}}\right)+\frac{\beta_{23}^{gb}(\eta_{0})}{2}\left(\frac{\partial\xi_{2}}{\partial
r_{a}}\frac{\partial\xi_{3}}{\partial r_{c}}+\frac{\partial\xi_{3}}{\partial
r_{a}}\frac{\partial\xi_{2}}{\partial r_{c}}\right).$
$\sigma_{c3}=\sigma_{(e)c3}=\sigma_{({st})c3}=P_{c3}=P_{3c}=P_{(e)c3}=P_{(e)3c}=P_{({st})c3}=P_{({st})3c}=0\,\,\text{for
}c=1,2,3.$ (C.20)

  * •

Mechanical equilibrium equation neglecting body forces and inertia ($a=1,2$)

$\sum_{b=1}^{2}\frac{\partial P_{ab}}{\partial r_{0b}}=0\,\,\text{in
}\Omega_{0},\quad\text{or equivalently,
}\quad\sum_{b=1}^{2}\frac{\partial\sigma_{ab}}{\partial r_{b}}=0\,\,\text{in
}\Omega.$ (C.21)

  * •

Ginzburg-Landau equations

Bicrystals:

$\dot{\eta}_{0}=L_{0M}^{\eta}X_{0M};\quad\dot{\eta}_{1}=L_{12}^{\eta}X_{12}^{\eta};\quad\dot{\xi}_{1}=L_{12}^{\xi}X_{12}^{\xi};\quad\text{
where}$ (C.22) $\displaystyle X_{0M}$ $\displaystyle=$
$\displaystyle\sum_{a,b,c,d,f=1}^{2}\left(J{F}^{-1}_{ac}\sigma_{(e)cf}F_{fb}-J_{t}\psi_{e}\delta_{ab}\right){F}_{(t)bd}^{-1}\frac{\partial{F}_{(t)da}}{\partial\eta_{0}}-\rho_{0}\Delta\psi_{\theta}(6\eta_{0}-6\eta_{0}^{2})-\rho_{0}A_{12}^{\eta}J\frac{\partial\varphi(a_{b},\eta_{0})}{\partial\eta_{0}}\times$
(C.23)
$\displaystyle\eta_{1}^{2}(1-\eta_{1})^{2}-\rho_{0}J[A_{0M}+(a_{\theta}-3)\Delta\psi^{\theta}(\theta)](2\eta_{0}-6\eta_{0}^{2}+4\eta_{0}^{3})-\frac{J\beta_{12}^{\eta}}{2}\frac{\partial\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})}{\partial\eta_{0}}\sum_{a=1}^{2}\frac{\partial\eta_{1}}{\partial
r_{a}}\frac{\partial\eta_{1}}{\partial r_{a}}-$
$\displaystyle\Delta\gamma_{12}^{gb}\frac{\partial\varphi}{\partial\eta_{0}}\left[18\xi_{1}^{2}(1-\xi_{1})^{2}+0.5\sum_{a=1}^{2}\frac{\partial\xi_{1}}{\partial
r_{a}}\frac{\partial\xi_{1}}{\partial
r_{a}}\right]+\sum_{a=1}^{2}\frac{\partial}{\partial
r_{a}}\left(\beta_{0M}\frac{\partial\eta_{0}}{\partial r_{a}}\right);$
$\displaystyle X_{12}^{\eta}$ $\displaystyle=$ $\displaystyle
X_{1}^{\eta}-X_{2}^{\eta}=\sum_{a,b,c,d,f=1}^{2}\left(J{F}^{-1}_{ac}\sigma_{(e)cf}F_{fb}-J_{t}\psi_{e}\delta_{ab}\right){F}_{(t)bd}^{-1}\frac{\partial{F}_{(t)da}}{\partial\eta_{1}}-$
(C.24)
$\displaystyle\rho_{0}JA_{12}^{\eta}\,(2\eta_{1}-6\eta_{1}^{2}+4\eta_{1}^{3})\varphi(a_{b},\eta_{0})+\sum_{a=1}^{2}\frac{\partial}{\partial
r_{a}}\left[\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})\,\beta_{12}^{\eta}\frac{\partial\eta_{1}}{\partial
r_{a}}\right];\quad\text{and}$ $\displaystyle X_{12}^{\xi}$ $\displaystyle=$
$\displaystyle
X_{1}^{\xi}-X_{2}^{\xi}=\sum_{a,b,c,d,f=1}^{2}\left(J{F}^{-1}_{ac}\sigma_{(e)cf}F_{fb}-J_{t}\psi_{e}\delta_{ab}\right){F}_{(t)bd}^{-1}\frac{\partial{F}_{(t)da}}{\partial\xi_{1}}-A_{12}^{gb}\,(2\xi_{1}-6\xi_{1}^{2}+4\xi_{1}^{3})+$
(C.25) $\displaystyle\sum_{a=1}^{2}\frac{\partial}{\partial
r_{a}}\left[\beta_{12}^{gb}\frac{\partial\xi_{1}}{\partial r_{a}}\right].$

Tricrystals:

$\dot{\eta}_{0}=L_{0M}^{\eta}X_{0M};\quad\dot{\eta}_{1}=L_{12}^{\eta}X_{12}^{\eta};\quad\dot{\xi}_{1}=L_{12}^{\xi}X_{12}^{\xi}+L_{13}^{\xi}X_{13}^{\xi};\quad\dot{\xi}_{2}=L_{12}^{\xi}X_{21}^{\xi}+L_{23}^{\xi}X_{23}^{\xi};\quad\dot{\xi}_{3}=-\dot{\xi}_{1}-\dot{\xi}_{2};$
(C.26)

where $X_{12}^{\xi}=X_{1}^{\xi}-X_{2}^{\xi}$;
$X_{13}^{\xi}=X_{1}^{\xi}-X_{3}^{\xi}$;
$X_{23}^{\xi}=X_{2}^{\xi}-X_{3}^{\xi}$;
$X_{21}^{\xi}=X_{2}^{\xi}-X_{1}^{\xi}$;

$\displaystyle X_{0M}$ $\displaystyle=$
$\displaystyle\sum_{a,b,c,d,f=1}^{2}\left(J{F}^{-1}_{ac}\sigma_{(e)cf}F_{fb}-J_{t}\psi_{e}\delta_{ab}\right){F}_{(t)bd}^{-1}\frac{\partial{F}_{(t)da}}{\partial\eta_{0}}-\rho_{0}\Delta\psi^{\theta}(6\eta_{0}-6\eta_{0}^{2})-\rho_{0}A_{12}^{\eta}J\frac{\partial\varphi(a_{b},\eta_{0})}{\partial\eta_{0}}\times$
(C.27)
$\displaystyle\eta_{1}^{2}(1-\eta_{1})^{2}-\rho_{0}J[A_{0M}+(a_{\theta}-3)\Delta\psi_{\theta}(\theta)](2\eta_{0}-6\eta_{0}^{2}+4\eta_{0}^{3})-\frac{J\beta_{12}^{\eta}}{2}\frac{\partial\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})}{\partial\eta_{0}}\sum_{a=1}^{2}\frac{\partial\eta_{1}}{\partial
r_{a}}\frac{\partial\eta_{1}}{\partial r_{a}}-$
$\displaystyle\frac{\partial\varphi(\eta_{0},a)}{\partial\eta_{0}}\sum_{I=1}^{2}\sum_{J=I+1}^{3}J\Delta\gamma_{IJ}^{gb}\left[\frac{18}{\delta_{IJ}^{gb}}\xi_{I}^{2}\xi_{J}^{2}-\frac{\delta_{IJ}^{gb}}{2}\sum_{a=1}^{2}\frac{\partial\xi_{I}}{\partial
r_{a}}\frac{\partial\xi_{J}}{\partial
r_{a}}\right]+\sum_{a=1}^{2}\frac{\partial}{\partial
r_{a}}\left(\beta_{0M}\frac{\partial\eta_{0}}{\partial r_{a}}\right);$
$\displaystyle X_{12}^{\eta}$ $\displaystyle=$ $\displaystyle
X_{1}^{\eta}-X_{2}^{\eta}=\sum_{a,b,c,d,f=1}^{2}\left(J{F}^{-1}_{ac}\sigma_{(e)cf}F_{fb}-J_{t}\psi_{e}\delta_{ab}\right){F}_{(t)bd}^{-1}\frac{\partial{F}_{(t)da}}{\partial\eta_{1}}-$
(C.28)
$\displaystyle\rho_{0}JA_{12}^{\eta}(2\eta_{1}-6\eta_{1}^{2}+4\eta_{1}^{3})\varphi(a_{b},\eta_{0})+\sum_{a=1}^{2}\frac{\partial}{\partial
r_{a}}\left[\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})\,\beta_{12}^{\eta}\frac{\partial\eta_{1}}{\partial
r_{a}}\right];$ $\displaystyle X_{1}^{\xi}$ $\displaystyle=$
$\displaystyle\sum_{a,b,c,d,f=1}^{2}\left(J{F}^{-1}_{ac}\sigma_{(e)cf}F_{fb}-J_{t}\psi_{e}\delta_{ab}\right){F}_{(t)bd}^{-1}\frac{\partial{F}_{(t)da}}{\partial\xi_{1}}-J_{t}\psi^{e}\frac{\partial\phi^{\xi}_{1}}{\partial\xi_{1}}-\sum_{H=2}^{3}J\frac{36\gamma_{1H}^{gb}}{(\delta_{1H}^{gb})^{2}}\xi_{1}\xi_{H}^{2}-$
(C.29) $\displaystyle
2\rho_{0}(K_{12}^{gb}+K_{13}^{gb}+K_{23}^{gb}+K_{123})\xi_{1}\xi_{2}^{2}\xi_{3}^{2}-\frac{J}{2}\sum_{H=2}^{3}\sum_{a=1}^{2}\frac{\partial}{\partial
r_{a}}\left(\beta_{1H}^{\xi}\frac{\partial\xi_{H}}{\partial r_{a}}\right);$
$\displaystyle X_{2}^{\xi}$ $\displaystyle=$
$\displaystyle\sum_{a,b,c,d,f=1}^{2}\left(J{F}^{-1}_{ac}\sigma_{(e)cf}F_{fb}-J_{t}\psi_{e}\delta_{ab}\right){F}_{(t)bd}^{-1}\frac{\partial{F}_{(t)da}}{\partial\xi_{2}}-J_{t}\psi^{e}\frac{\partial\phi^{\xi}_{2}}{\partial\xi_{2}}-\sum_{H=1,\neq
2}^{3}J\frac{36\gamma_{2H}^{gb}}{(\delta_{2H}^{gb})^{2}}\xi_{2}\xi_{H}^{2}-$
(C.30) $\displaystyle
2\rho_{0}(K_{12}^{gb}+K_{13}^{gb}+K_{23}^{gb}+K_{123})\xi_{1}^{2}\xi_{2}\xi_{3}^{2}-\frac{J}{2}\sum_{H=1,\neq
2}^{3}\sum_{a=1}^{2}\frac{\partial}{\partial
r_{a}}\left(\beta_{2H}^{\xi}\frac{\partial\xi_{H}}{\partial r_{a}}\right);$
$\displaystyle X_{3}^{\xi}$ $\displaystyle=$
$\displaystyle\sum_{a,b,c,d,f=1}^{2}\left(J{F}^{-1}_{ac}\sigma_{(e)cf}F_{fb}-J_{t}\psi_{e}\delta_{ab}\right){F}_{(t)bd}^{-1}\frac{\partial{F}_{(t)da}}{\partial\xi_{3}}-J_{t}\psi^{e}\frac{\partial\phi^{\xi}_{3}}{\partial\xi_{3}}-\sum_{H=1}^{2}J\frac{36\gamma_{3H}^{gb}}{(\delta_{3H}^{gb})^{2}}\xi_{3}\xi_{H}^{2}-$
(C.31) $\displaystyle
2\rho_{0}(K_{12}^{gb}+K_{13}^{gb}+K_{23}^{gb}+K_{123})\xi_{1}^{3}\xi_{2}^{2}\xi_{3}-\frac{J}{2}\sum_{H=1}^{2}\sum_{a=1}^{2}\frac{\partial}{\partial
r_{a}}\left(\beta_{3H}^{\xi}\frac{\partial\xi_{H}}{\partial r_{a}}\right).$

  * •

Boundary conditions for the order parameters

$\sum_{a=1}^{2}\frac{\partial\eta_{0}}{\partial
r_{0a}}n_{0a}=0;\quad\sum_{a=1}^{2}\frac{\partial\eta_{1}}{\partial
r_{0a}}n_{0a}=0;\quad\sum_{a=1}^{2}\frac{\partial\xi_{I}}{\partial
r_{0a}}n_{0a}=0\,\,\text{for all }I=1,\ldots,M\quad\text{on }S_{0},$ (C.32)

which are obtained using Eq. (B.1) in Eq. (B.12), and assuming
$\beta^{\eta}_{12}=\beta^{\eta}_{13}=\beta^{\eta}_{23}$ and
$\beta^{gb}_{12}=\beta^{gb}_{13}=\beta^{gb}_{23}$.

##  Appendix D Compatibility of transformation strains across grain boundary

To understand the martensitic microstructures across the symmetric and
asymmetric GBs, we briefly describe the strain compatibility conditions across
the GBs. Although a large strain-based theory is considered in this paper, we
consider the compatibility condition in terms of the linearized strains across
a sharp GB following Chapter 13 of [1], which is sufficient for understanding
the results in this paper. The Bain strains for the grain $G_{1}$ in the
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2}\\}$ basis (see e.g. Fig. 3(a)) are

$\boldsymbol{\varepsilon}_{t1}^{(1)}=\begin{bmatrix}\varepsilon_{\chi}\cos^{2}\vartheta_{1}+\varepsilon_{\alpha}\sin^{2}\vartheta_{1}&(\varepsilon_{\alpha}-\varepsilon_{\chi})\cos\vartheta_{1}\sin\vartheta_{1}\\\
(\varepsilon_{\alpha}-\varepsilon_{\chi})\cos\vartheta_{1}\sin\vartheta_{1}&\varepsilon_{\alpha}\cos^{2}\vartheta_{1}+\varepsilon_{\chi}\sin^{2}\vartheta_{1}\\\
\end{bmatrix},$ (D.1)
$\boldsymbol{\varepsilon}_{t2}^{(1)}=\begin{bmatrix}\varepsilon_{\alpha}\cos^{2}\vartheta_{1}+\varepsilon_{\chi}\sin^{2}\vartheta_{1}&(\varepsilon_{\chi}-\varepsilon_{\alpha})\cos\vartheta_{1}\sin\vartheta_{1}\\\
(\varepsilon_{\chi}-\varepsilon_{\alpha})\cos\vartheta_{1}\sin\vartheta_{1}&\varepsilon_{\chi}\cos^{2}\vartheta_{1}+\varepsilon_{\alpha}\sin^{2}\vartheta_{1}\\\
\end{bmatrix},$ (D.2)

where $\varepsilon_{\alpha}=\alpha-1$ and $\varepsilon_{\chi}=\chi-1$. The
Bain strains $\boldsymbol{\varepsilon}_{t1}^{(2)}$ and
$\boldsymbol{\varepsilon}_{t2}^{(2)}$ for the grain $G_{2}$ is simply obtained
by replacing $\vartheta_{1}$ with $\vartheta_{2}$ in Eqs. (D.1) and (D.2),
respectively. The variant, say $\sf M_{1}$, would be compatible across the
planar GB in the bicrystal considered in Fig. 3(a) if it satisfies (Chapter 13
of [1])

$\boldsymbol{\varepsilon}_{t1}^{(2)}-\boldsymbol{\varepsilon}_{t1}^{(1)}=0.5(\boldsymbol{s}\otimes\boldsymbol{e}_{2}+\boldsymbol{e}_{2}\otimes\boldsymbol{s}),$
(D.3)

where $\boldsymbol{s}$ is a vector related to the deformation which is to be
determined, and we note that the GB normal is parallel to
$\boldsymbol{e}_{2}$-axis. Operating $\boldsymbol{e}_{1}$ and
$\boldsymbol{e}_{2}$ vectors in Eq. (D.3) one can easily show that the
compatibility of $\sf M_{1}$ across the GB requires to satisfy

$(\boldsymbol{\varepsilon}_{t1}^{(2)})_{11}=(\boldsymbol{\varepsilon}_{t1}^{(1)})_{11},\quad(\boldsymbol{\varepsilon}_{t1}^{(2)})_{12}-(\boldsymbol{\varepsilon}_{t1}^{(1)})_{12}=0.5\boldsymbol{s}\cdot\boldsymbol{e}_{1},\quad\text{and}\quad(\boldsymbol{\varepsilon}_{t1}^{(2)})_{22}-(\boldsymbol{\varepsilon}_{t1}^{(1)})_{22}=\boldsymbol{s}\cdot\boldsymbol{e}_{2}.$
(D.4)

Using Eq. (D.1) and the expression for $\boldsymbol{\varepsilon}_{t1}^{(2)}$
in Eq. (D.4)2,3, one can determine $\boldsymbol{s}$ in terms of
$\varepsilon_{\alpha}$, $\varepsilon_{\chi}$, $\vartheta_{1}$, and
$\vartheta_{2}$. Furthermore, the condition Eq. (D.4)1 requires

$\varepsilon_{\chi}\cos^{2}\vartheta_{1}+\varepsilon_{\alpha}\sin^{2}\vartheta_{1}=\varepsilon_{\chi}\cos^{2}\vartheta_{2}+\varepsilon_{\alpha}\sin^{2}\vartheta_{2}.$
(D.5)

For the arbitrary $\varepsilon_{\alpha}$ and $\varepsilon_{\chi}$, and
$\varepsilon_{\alpha}\neq\varepsilon_{\chi}$, the condition given by Eq. (D.5)
requires that $\vartheta_{1}=-\vartheta_{2}$ for a non-trivial GB, i.e. the GB
must be symmetric. An asymmetric GB thus does not generally satisfy the strain
compatibility equation (D.3). Like this we can show that the other variant
$\sf M_{2}$ also satisfies the strain compatibility across the symmetric GBs,
but not across the asymmetric ones.

Acknowledgements

The support from SERB (Grant number SRG/2020/001194) and the helpful comments
and suggestions from Prof. Valery Levitas (Iowa State University) are
gratefully acknowledged.

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