## 5 Phase Field Methods
### Material Degradation  
The expression of the potential energy introduced in Equation (52) implies that in a given conservative system, any increase in the fracture energy due to a unit increase in the fracture surface has to be compensated by a corresponding decrease in the elastic strain energy. Hence, the expression of the elastic energy must be coupled to the evolution of the phase field \(c\) as the latter dictates the value of the fracture energy. In physical terms, the phase field has to account for the gradual degradation of material stiffness as cracks propagate through the medium. Mathematically, this has been expressed through the definition of a degradation function, \(g\left(c\right)\), which is then used to reduce the value material elastic energy density giving rise to the so-called isotropic phase field methods. Driven from the fact that such an approach led to unrealistic and in cases erroneous results, e.g., cracks initiating and propagating due to pure compression later attempts postulated material degradation on the basis of an energy split, i.e.,  
\[\psi_{e}=g\left(c\right)\psi_{e}^{+}+\psi_{e}^{-} \tag{67}\]  
where \(\psi_{e}^{+}\) and \(\psi_{e}^{-}\) are the elastic strain energy densities whose expressions are specific to the type of energy split adopted, see, e.g., Miehe et al. [228] for an energy decomposition based on the spectral decomposition of the strain tensor and Amor et al. [264] for a volumetric/deviatoric decomposition giving rise to the so-called anisotropic degradation models. It is of interest to note that although anisotropic models mitigated the unrealistic crack patterns derived from the isotropic ones for most typical stress states, the problem is not yet fully resolved. The volumetric split defined in [264] may still result in degradation under a pure compressive stress state. The spectral decomposition model defined in [228] leads to a strongly non-linear stress-strain relation that has been shown to be computationally taxing (see e.g. [265] for a detailed comparison of these two models). The expression of the degradation function \(g(c)\) is not unique see, e.g., [250; 255; 266; 267; 268; 269; 270; 269; 271]. A widely used definition for the degradation function that is compatible with the first and second order quadratic approximations provided in Equations (60) and (61), respectively is  
\[g\left(c\right)=\left[\left(1-k\right)c^{2}+k\right] \tag{68}\]  
where \(k\) in Equation (68) is a model parameter utilized in several applications, see, e.g., [264; 272] as a way to avoid ill-posedness. Geelen et al. [250] introduced a quasi-quadratic definition of \(g(c)\) to be employed in conjuction with the linear approximation defined in Equation (65) that is defined as  
\[g(c)=\frac{c^{2}}{c^{2}+m(1-c)[1+p(1-c)]}\ \ \text{with}\ p\geq 1\ \ \text{and}\ l_{0}<\frac{3E\mathcal{G}_{c}}{4(p+2)\sigma_{c}^{2}} \tag{69}\]  
where \(m=\left(3\mathcal{G}_{c}\right)/\left(8l_{0}\psi_{c}\right)=g^{\prime}(c_{0})\) is the initial slope of the degradation function \(g(c)\) and \(p\) provides the initial slope and shape parameters for the softening curve assuming \(c_{0}=1\) as the initial phase-field. Here, \(\psi_{c}=(\sigma_{c}^{2})/(2E)\) is the critical fracture energy per unit volume of the material, in which \(\sigma_{c}\) and \(E\) represent the critical tensile strength and Young's modulus of the material respectively. This definition, however, comes with an additional upper bound restriction on the value of length-scale parameter \(l_{0}\) which is necessary to achieve optimal convergence. The upper bound on the regularization length is related to the characteristic length of the fracture process zone \(l_{FPZ}=\left(E\mathcal{G}_{c}\right)/(\sigma_{c}^{2})\), see [250; 273] for details.