## 5 Phase Field Methods
### PFM Variational Formulation  
Griffith [86] postulated that the total potential energy \(\Pi\) of an elastic body undergoing elastic fracture comprises the contributions of the elastic strain energy \(\Pi_{e}\) and the fracture energy \(\Pi_{f}\)  
\[\Pi\left(\mathbf{u},\Gamma\right)=\Pi_{e}+\Pi_{f}+W_{ext}=\int_{\Omega}\psi_{ e}d\Omega+\int_{\Gamma}\mathcal{G}_{c}d\Gamma+W_{ext} \tag{52}\]  
where \(\psi_{e}\) is the elastic energy density and \(\mathcal{G}_{c}\) is the critical fracture energy density, and \(W_{ext}\) is the work done by the external forces. The elastic energy density for the case of an isotropic medium is defined as  
\[\psi_{e}\left(\mathbf{\varepsilon}\right)=\frac{1}{2}\lambda\left[\mathrm{Tr} \left(\mathbf{\varepsilon}\right)\right]^{2}+\mu\left[\mathrm{Tr}\left(\mathbf{ \varepsilon}^{2}\right)\right] \tag{53}\]  
where \(\lambda\) and \(\mu\) are the Lame constants. Phase field modelling of fracture approximates the fracture surface integral expression introduced in Equation (52) with a volume integral defined over the entire deformable domain \(\Omega\) according to Equation (54) below. \[\int_{\Gamma}\mathcal{G}_{c}d\Gamma\approx\int_{\Omega}\mathcal{G}_{c}\Gamma_{ \Gamma}\left(c,\
abla c\right)d\Omega \tag{54}\]  
where \(c=c\left(\mathbf{x}\right)\in\left[0,1\right]\)\(\forall\mathbf{x}\in\Omega\) is the scalar phase field representing crack. Using Equation (54), the expression of the potential energy of the elastic deformable body introduced in Equation (52) can be modified into the following form  
\[\Pi\approx\int\limits_{\Omega}\psi_{e}d\Omega+\qquad\overbrace{\int\limits_{ \Omega}\mathcal{G}_{c}F_{\Gamma}d\Omega}^{Fracture\ Energy\ Approximation}-\left(\int\limits_{\Omega}u_{i}b_{i}d\Omega+\int\limits_{ \partial\Omega_{\mathbf{i}}}u_{i}\bar{t}_{i}d\Omega_{\mathbf{i}}\right) \tag{55}\]  
The functional \(F_{\Gamma}\) assumes the following generic form  
\[F_{\Gamma}=\frac{1}{c_{w}}\left(\frac{1}{2l_{0}}\omega(c)+2l_{0}|\
abla c|^{2} \right), \tag{56}\]  
where \(l_{0}\in\mathbb{R}\) is a length scale parameter and \(\omega(c)\) and \(c_{w}\) are the generic crack geometric function and associated constant; these assume different expressions based on the type of fracture surface energy approximation used. With the introduction of the crack surface density function in Equation (56), the discrete description of a sharp crack \(\Gamma_{c}\) in Figure 1 is transformed onto a diffused crack description as shown in Figure 6 via the regularized crack functional \(\Gamma_{l_{0}}(c)\) which is scaled by the length-scale parameter \(l_{0}\) (57). \[\Gamma_{l_{0}}(c)=\int\limits_{\Omega}F_{\Gamma}\left(c,\
abla c\right)d\Omega \tag{57}\]  
The length scale parameter \(l_{0}\) is the regularisation length over which damage diffuses as shown in Figure 6. In the conventional phase field formulation, originally presented in Bourdin et al. [69], the peak force reached before the onset of fracture depends on the value of length-scale parameter \(l_{0}\). Higher values of the length-scale parameter lead to lower peak forces and vice versa. In recent formulations, see, e.g., Wu and Nguyen [251] and Geelen et al. [250] this is alleviated, hence providing a significant advantage in enhancing the critical-stress predicting capabilities of the phase field method. In Miehe et al. [256], generalized crack-driving forces with a failure criteria based on the maximum principal stress were introduced which also succeeded in predicting critical fracture loads unaffected by the length-scale parameter. However in notched structures, a crack nucleation principle based purely on the maximum principal stress criteria suffers from the curse of stress singularity at the notch-tip as also highlighted in [257]. Providing different expressions for \(\omega(c)\) and \(c_{w}\) results in variants of the phase field approximation; key variants, i.e., the second and fourth order quadratic approximations and the second order linear phase field approximation are discussed in Sections 5.2.1, 5.2.2, and 5.2.3, respectively.