\frac{\partial\psi}{\partial\bm{\epsilon}} : \delta\bm{\epsilon} {\,\mathrm{d}}\Omega 
    - \int_\Omega \bm{b} \cdot \delta\bm{u} \,{\,\mathrm{d}}\Omega - \int_{\partial\Omega^t} \bm{t} \cdot \delta\bm{u} \,{\,\mathrm{d}}S \\
    + \int_\Omega g^\prime \left( \phi \right) \psi \left( \bm{\epsilon} \right) \delta\phi {\,\mathrm{d}}\Omega + \mathcal{G}_c  \int_\Omega \left( \frac{1}{\ell}\phi \,\delta\phi + \ell \nabla \phi \cdot \delta\nabla\phi \right) {\,\mathrm{d}}\Omega = 0.\end{gathered}$$ The above equality must hold for arbitrary values of $\delta \bm{u}$ and $\delta\phi$, implying that

\[eq:weakForm\] $$\begin{aligned}
    &\int_\Omega g \left( \phi \right) \frac{\partial\psi}{\partial\bm{\epsilon}} : \delta\bm{\epsilon} \,{\,\mathrm{d}}\Omega = \int_\Omega \bm{b} \cdot \delta\bm{u} \,{\,\mathrm{d}}\Omega + \int_{\partial\Omega^t} \bm{t} \cdot \delta\bm{u} \,{\,\mathrm{d}}S 
    \label{eq:weakForm_momentum} \\
    &\int_\Omega g^\prime \left( \phi \right) \psi \left( \bm{\epsilon} \right) \delta\phi {\,\mathrm{d}}\Omega + \mathcal{G}_c  \int_\Omega \left( \frac{1}{\ell} \phi \delta\phi + \ell \nabla\phi \cdot \nabla \delta\phi \right) {\,\mathrm{d}}\Omega = 0.
    \label{eq:weakForm_phaseField}
    \end{aligned}$$

These constitute the weak form of the governing equations. Noting that $\bm{\sigma} = \partial\psi/\partial\bm{\epsilon}$, the equivalent strong formulation may be obtained by applying Gauss’ divergence theorem yielding the following coupled system:

\[eq:govEq\_StrongForm\] $$\begin{aligned}
    \nabla \cdot \left[ g \left( \phi \right) \bm{\sigma} \right] + \bm{b} &= \bm{0} \text{ on } \Omega \label{eq:linearMomentum} \\
    g \left( \phi \right) \bm{\sigma} \cdot \bm{n} &= \bm{t} \text{ on } \partial\Omega^t \label{eq:linearMomentum_Neumann} \\
    \bm{u} &= \bar{\bm{u}} \text{ on } \partial\Omega^u \label{eq:linearMomentum_Dirichlet} \\
    \mathcal{G}_c \ell_0 \nabla^2\phi - \frac{\mathcal{G}_c}{\ell_0} \phi &= g^\prime \left( 