# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 2 Phase-Field Method
### Extension of Phase-Field Method to Fatigue  
Material fatigue is a phenomenon caused by cyclic loading. The cyclic loading maximum value is usually below the material yield strength and slowly deteriorates the material during loading cycles. After the corresponding number of loading cycles, a fatigue crack initiates. After the initiation stage, stable propagation of the fatigue crack prevails until the critical crack size, when unstable and rapid fatigue crack growth emerges. According to the literature overview, there are many solutions offering extension of the phase-field method towards fatigue problems. In this paper, the approach derived in [48] is adopted. Accordingly, the regularised internal energy potential consisting of (2) and (6) is extended to fatigue, assuming the TH model:  
\[\Psi\big{(}\mathrm{e}^{\mathrm{e}},\mathrm{e}^{\mathrm{P}},\phi,\overline{ \psi}\big{)}=\int_{\Omega}\big{\{}g(\phi)\big{[}\psi_{\mathrm{e}}^{+}(\mathrm{ e}^{\mathrm{e}})+\psi_{\mathrm{p}}(\mathrm{e}^{\mathrm{P}})\big{]}+\psi_{ \mathrm{e}}^{-}(\mathrm{e}^{\mathrm{e}})\big{\}}\mathrm{d}\Omega\ +\ \int_{\Omega} \hat{F}\big{(}\overline{\psi}\big{)}\psi_{\mathrm{c}}\big{[}2\phi+l|\
abla\phi |^{2}\big{]}\mathrm{d}\Omega, \tag{15}\]  
introducing the fatigue degradation function \(\hat{F}\). As can be seen from (15), the fatigue degradation function directly affects the energetic threshold and the crack driving state function:  
\[\widetilde{D}_{\mathrm{TH}}=\frac{\psi_{\mathrm{e}}^{+}+\psi_{\mathrm{p}}}{ \hat{F}\big{(}\overline{\psi}\big{)}\psi_{\mathrm{c}}}-1. \tag{16}\]  
The value of the fatigue degradation function is computed by means of the energy density accumulation variable \(\overline{\psi}\), which can be formulated as:  
\[\overline{\psi}(t)=\int_{0}^{t}\psi_{\mathrm{e}}(t)H\Big{(}-\psi_{\mathrm{e} }\Big{)}\mathrm{d}t, \tag{17}\]  
where \(H\big{(}-\dot{\psi}_{\mathrm{e}}\big{)}\) denotes the Heaviside function, equal to 1 if \(\dot{\psi}_{\mathrm{e}}>0\) and equal to 0 if \(\dot{\psi}_{\mathrm{e}}\leq 0\). In this way, the accumulation variable obtains an increase only during unloading, avoiding the influence on the proportional loading cases. Equation (17) can be recast into incremental form for \(N\)-th loading cycle:  
\[\begin{array}{l}\overline{\psi}_{N}=\overline{\psi}_{N-1}+\Delta\overline{ \psi},\\ \Delta\overline{\psi}=\overline{\psi}_{N-1}-\overline{\psi}_{N}.\end{array} \tag{18}\]The choice of appropriate fatigue degradation functions can be found in [PERSON] et al. [48], along with necessary properties of the fatigue degradation function. In this paper, 2 forms of fatigue degradation functions have been utilised:  
\[\begin{split}\hat{F}_{\text{LCF}}\big{(}\overline{\psi}\big{)}= \big{(}1-\frac{\overline{\psi}}{\overline{\psi}_{\infty}}\big{)}^{2}\text{ for }\overline{\psi}\in\big{[}0,\overline{\psi}_{\infty}\big{]},\\ \hat{F}_{\text{HCF}}\big{(}\overline{\psi}\big{)}=\Big{(}\log \frac{\overline{\psi}_{\infty}}{\overline{\psi}}\Big{)}^{2}\text{ for }\overline{\psi}\in\big{[}\overline{\psi}_{\infty},10\overline{\psi}_{\infty} \big{]}.\end{split} \tag{19}\]  
In Equation (19), \(\overline{\psi}_{\infty}\) represents a material-dependant fatigue parameter.