We use standard continuum mechanics notation, where Latin indices refer to spatial coordinates. We use Einstein’s summation convention over repeated indices. All tensors are expressed in Cartesian coordinates. The superscripts
\(\text {E}\) and
\(\text {IE}\) stand for elastic (recoverable) and inelastic (irreversible) deformations, respectively. For the description of the twin, an order (phase-field) parameter,
\(\eta \), is introduced, where
\(\eta = 0\) denotes the parent crystal and
\(\eta = 1\) means the twin. This order parameter as well as displacement,
\(\varvec{u}\), are the primitive variables in space and time that we are searching for. The deformation gradient reads
$$\begin{aligned} F_{ij} = u_{i,j} + \delta _{ij}, \end{aligned}$$
(1)
where comma denotes a derivative in space. We use a material frame, where the derivative is taken in the reference configuration that is chosen to be the initial placement of the continuum body. Kronecker delta,
\(\varvec{\delta }\), is the identity. The deformation gradient,
\(\varvec{F}\), in a large-displacement formulation, is decomposed into elastic and inelastic parts,
$$\begin{aligned} F_{ij} = F^\text {E}_{ik} F^\text {IE}_{kj}, \end{aligned}$$
(2)
where for (inelastic) twinning [
77], we use
$$\begin{aligned} F_{ij}^\text {IE}=\delta _{ij}+\phi (\eta )\gamma _{0} s_i m_j . \end{aligned}$$
(3)
The interpolation function,
\(\phi (\eta )=\eta ^{2}(3-2\eta )\), causes a steep change between the twin and parent crystal [
78] as necessary in phase-field approaches.
\(\gamma _{0}\) is the magnitude of maximum twinning shear, and
\(\varvec{s}\) and
\(\varvec{m}\) are the unit vectors along the twinning direction and normal to the twinning plane, respectively. By following [
79], we decompose the Helmholtz free energy per mass into mechanical and interfacial parts,
$$\begin{aligned} \psi (\varvec{F}, \eta , \nabla \eta ) = \psi ^\text {M}(\varvec{F}, \eta ) + \psi ^\nabla (\eta , \nabla \eta ) \ , \end{aligned}$$
(4)
where the kinetics of interface is controlled by twin order parameter and its first-gradient by the latter. As usual, for the mechanical deformation energy density (per volume), we may use the St. Venant model:
$$\begin{aligned} \rho _0 \psi ^\text {M} = \frac{1}{2} E_{ij} C_{ijkl} E_{kl} \ , \end{aligned}$$
(5)
or the neo-Hookean model:
$$\begin{aligned} \rho _0 \psi ^\text {M} = \frac{\mu }{2} \left( I_C - 3 \right) - \mu \ln {J} + \frac{\lambda }{2} \left( \ln {J}\right) ^{2} \ . \end{aligned}$$
(6)
For nonlinear isotropic elasticity, the neo-Hookean model defined in Eq. (
6) is used. We use right Cauchy–Green deformation tensor,
\(C^\text {E}_{ij} = F^\text {E}_{ki} F^\text {E}_{kj}\), and its invariants,
\(I_C = C^\text {E}_{ii}\),
\(J=\det (\varvec{C}^\text {E})\). The Green–Lagrange strain measure,
\(\varvec{E} = \frac{1}{2}(\varvec{C}^\text {E}- \varvec{\delta })\), accommodates geometric nonlinearity necessary for some applications herein. Lamé parameters,
\(\lambda \),
\(\mu \), or the stiffness tensor of rank four,
\(C_{ijkl}\), are given as material coefficients. The elastic constants are the Voigt-averaged shear and bulk modulus [
80], which are listed in Table
1. For anisotropic elasticity, the elastic coefficients are interpolated between the untwinned
\(C^\text {P}_{ijkl}\) and twinned
\(C^\text {T}_{ijkl}\) domains using the interpolation function,
$$\begin{aligned} C_{ijkl} = C^\text {P}_{ijkl} + ( C^\text {T}_{ijkl} - C^\text {P}_{ijkl} ) \phi (\eta ) \ . \end{aligned}$$
(7)
The same interpolation function is used as in the definition of the inelastic part of the deformation gradient. For the twin phase,
\(\eta =1\), we have the stiffness tensor as a rotation of crystal lattice from the parent phase,
\(\eta =0\), as follows:
$$\begin{aligned} C_{ijkl}^\text {T} = {\mathcal {Q}}_{im}{\mathcal {Q}}_{jn}{\mathcal {Q}}_{ko}{\mathcal {Q}}_{lp} C_{mnop}^\text {P} , \end{aligned}$$
(8)
where
\(\varvec{{\mathcal {Q}}}\) is the reorientation matrix associated with twinning. For a centrosymmetric structure [
81], it becomes
$$\begin{aligned} {\mathcal {Q}}_{ij} = {\left\{ \begin{array}{ll} 2 m_{i} m_{j} - \delta _{ij} &{} \text {type I twins},\\ 2 s_{i} s_{j} - \delta _{ij} &{} \text {type II twins}. \end{array}\right. } \end{aligned}$$
(9)
In the case of a steady-state deformation and neglecting inertial terms, the governing equations for displacement read
$$\begin{aligned} \begin{aligned} P_{ji,j} =&0 \ ,\\ P_{ji} =&\frac{\partial \rho _0 \psi }{\partial F_{ij}} = \frac{\partial \rho _0 \psi ^\text {M}}{\partial F_{ij}} = \frac{\partial \rho _0 \psi ^\text {M}}{\partial E_{kl}} \frac{\partial E_{kl}}{\partial F_{ij}} \\ =&\frac{\partial \rho _0 \psi ^\text {M}}{\partial E_{kl}} F^\text {E}_{il} (\varvec{F}^\text {IE})^{-1}_{jk} \ . \end{aligned} \end{aligned}$$
(10)
The Ginzburg–Landau equation is acquired by a thermodynamically-consistent derivation, as follows:
$$\begin{aligned} \begin{aligned} \dot{\eta } =&-{\mathcal {L}} \Bigg ( \frac{\partial \rho _0 \psi ^\text {M}}{\partial \eta } + \frac{\partial \rho _0 \psi ^\nabla }{\partial \eta } - \left( \frac{\partial \rho _0 \psi ^\nabla }{\partial \eta _{,i}}\right) _{,i} \Bigg ) \ , \end{aligned} \end{aligned}$$
(11)
where the mobility parameter,
\({\mathcal {L}}\), is generally not known and challenging to obtain experimentally. The outcome of this work is the methodology on how to set its numerical value.
The first term is formulated by using the product rule
$$\begin{aligned} \frac{\partial \rho _0 \psi ^\text {M}}{\partial \eta }&= \frac{\partial \rho _0 \psi ^\text {M}}{\partial F_{ij}} \frac{\partial F_{ij}}{\partial \eta } = P_{ji} \frac{\partial F^\text {E}_{ik} F^\text {IE}_{kj} }{\partial \eta } \nonumber \\&= P_{ji} F^\text {E}_{ik} \phi '(\eta ) \gamma _0 s_k m_j \ , \end{aligned}$$
(12)
where
\(\phi '(\eta )=6\eta (1-\eta )\). For the interfacial energy,
\(\psi ^{\nabla }\), we use a standard double-well potential as in [
82,
83] such that the energy density reads
$$\begin{aligned} \rho _0 \psi ^{\nabla }(\eta )=A\eta ^{2}\left( 1-\eta \right) ^{2} + \kappa _{ij} \eta _{,i}\eta _{,j} , \end{aligned}$$
(13)
where
\(A=12\frac{\Gamma }{l}\) characterizes the energy barrier between two stable phases (minima), related to the twin boundary surface energy,
\(\Gamma \), and the twin boundary thickness,
l;
\(\kappa _{ij}=\kappa _{0}\delta _{ij}\) with
\(\kappa _{0}\) being the gradient energy parameter, given as [
68],
\(\kappa _{0} = \frac{3}{4} \Gamma l\). By inserting the energy definitions into the Ginzburg–Landau, we obtain the governing equation for twin order parameter,
$$\begin{aligned} \dot{\eta }&= -{\mathcal {L}} \Big ( P_{ji} F^\text {E}_{ik} \phi '(\eta ) \gamma _0 s_k m_j + 2 A \eta \big ( 1-3\eta + 2\eta ^{2} \big )\nonumber \\&\quad - 2\kappa _{0}\eta _{,ii} \Big ) \end{aligned}$$
(14)
By solving Eqs. (
10), (
14), we obtain
\(\varvec{u}\) and
\(\eta \) fields.