# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 2 Phase-Field Method  
As mentioned previously, the phase-field method belongs to the diffusive approaches for describing fractures in deformable bodies. Crack geometry is described by a diffusive zone, where zone width is controlled by the length scale parameter \(l\) in the domain \(\Omega\) (Figure 1). The diffusive zone provides a smooth transition from intact and sound material to fully damaged material, governed by the phase-field scalar parameter \(\phi[0,1]\), where \(\phi=0\) denotes intact material while \(\phi=1\) represents fully damaged material. The basic idea of the phase-field method for fracture modelling is based on the variational approach to the fracture process [17]. The governing model is formulated as a minimization problem of the internal energy functional, which consists of the bulk energy \(\Psi^{\rm b}\) stored in the body and the fracture dissipated energy \(\Psi^{\rm s}\), respectively:  
\[\Psi=\Psi^{\rm b}+\Psi^{\rm s}=\int_{\Omega/\Gamma}\psi(\varepsilon){\rm d} \Omega+\int_{\Gamma}G_{\rm c}{\rm d}\Gamma. \tag{1}\]  
Figure 1: Fracture surface \(\Gamma\) approximated by a phase-field diffusive approach. Following [PERSON]'s theory [PERSON] (1964), material failure occurs when the material-dependant critical value of fracture energy density \(G_{\text{c}}\) is reached. In the phase-field method, the usual problem of tracking discrete fracture surfaces \(\Gamma(t)\) is omitted through their approximation by the crack density function \(\gamma(\phi,\
abla\phi)\). As mentioned above, crack density function is governed by the incorporation of the length scale parameter \(l\) and the phase-field parameter \(\phi\)[PERSON] (2016). In general, the regularised internal energy potential for elastic bodies can be stated as follows:  
\[\Psi(\varepsilon,\phi)=\int_{\Omega}\big{[}g(\phi)\psi_{\text{e}}^{+}( \varepsilon)+\psi_{\text{e}}^{-}(\varepsilon)\big{]}\mathrm{d}\Omega+G_{ \text{c}}\!\int_{\Omega}\gamma(\phi,\
abla\phi)\mathrm{d}\Omega. \tag{2}\]  
In Equation (2), the degradation function \(g(\phi)\) appears, which degrades material stiffness. In the literature, one of the most popular forms of the degradation function is the quadratic form:  
\[g(\phi)=(1-\phi)^{2}. \tag{3}\]  
As can be seen in Equation (2), only the positive part of the strain energy density \(\psi_{\text{e}}^{+}\) is degraded according to the physically consistent assumption that material loses its stiffness only in tensile state, while in compressive state (\(\psi_{\text{e}}^{-}\)), crack closure occurs. Hence, in this research, the elastic strain energy density \(\psi_{\text{e}}\) function is divided into positive and negative parts through a volumetric-deviatoric decomposition ([PERSON], 2016). Aside from the volumetric-deviatoric decomposition, other approaches to decomposition are also possible ([PERSON], 2016). In this paper, 2 standard forms of the crack density function will be used:  
\[\gamma_{\text{AT}-2}(\phi,\
abla\phi)=\frac{1}{2}\bigg{[}\frac{1}{l}\phi^{2}+l |\
abla\phi|^{2}\bigg{]}, \tag{4}\]  
proposed in ([PERSON], 2016; [PERSON] (2016)), referred as AT-2 model, and  
\[\gamma_{\text{TH}}(\phi,\
abla\phi)=\frac{3}{8\sqrt{2}}\bigg{[}\frac{1}{l}2 \phi+l|\
abla\phi|^{2}\bigg{]}, \tag{5}\]  
proposed by [PERSON] et al. (2016), often referred to as the TH model. Furthermore, from the regularised functional in the TH model, a specific fracture energy \(\psi_{\text{c}}=\frac{3}{8\sqrt{2}}\frac{G_{\text{c}}}{I}\), acting as an energetic barrier preventing the damage evolution, can be derived. The scheme presented in this paper also considers elastoplastic constitutive behaviour of materials prior to fracture.