functionals approximating can be seen as specific cases of the integral of a general state function pertaining to a gradient damage model: $$\label{eq:stateFunction}
    W_\ell \left( \bm{\epsilon} \left( \bm{u} \right), \phi, \nabla\phi \right) = \frac{1}{2} \bm{\epsilon} : \mathbb{C} \left( \phi \right) : \bm{\epsilon} + w \left( \phi \right) + \frac{1}{2} w_1 \ell^2 \nabla\phi \cdot \nabla\phi$$ where $w \left( \phi \right)$ is a monotonically increasing function in the interval $\left[ 0,1 \right]$ with $w \left( 0 \right) = 0$ and $w \left( 1 \right) = w_1$. They suggest using the linear form $w \left( \phi \right) = w_1 \phi$ which leads to models having a real elastic phase with no premature decrease in material stiffness, at the cost of solving a variational inequality problem for the damage evolution. In contrast, the quadratic form of $w \left( \phi \right)$ employed in leads to material behavior having *no* real elastic phase, with damage already occurring at the onset of loading [@Amor2009]. Bulk energy release resulting from evolution of the phase-field is facilitated through a damage-dependent elasticity tensor $\mathbb{C} \left( \phi \right)$ as seen in . The simplest form which leads to isotropic behavior consists of the multiplicative ansatz $$\label{eq:elasticDamage_basic}
    \mathbb{C} \left( \phi \right) = 