# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 1 Introduction
[PERSON] and [PERSON] [12] established that the difficulty in the FEM discretisation framework is exhibited by the non-objective results brought on by substantial mesh refinement and alignment dependence. Therefore, non-local [13] and gradient-enhanced continuum techniques [14] have been created to address this issue via the use of length scale parameters, which introduce non-local state variables that typically depend on their local counterparts. Compared to experiments, which are frequently very expensive, impossible, or unattainable, numerical modelling of a fracture represents a more acceptable solution. Recently, the phase-field method has gained significant attention in the scientific community for modelling of the fracture phenomena. In physics, phase-field fracture models have been established in the Ginzburg-Laudau theory shown in [15], while in mechanics, phase-field fracture models use [PERSON]'s theory [16] as a base. [PERSON] proposed the concept of a brittle fracture, introducing an energetic fracture criterion. According to [PERSON]'s theory, material fails locally upon exceeding a specified fracture energy related to a critical energy release rate. [PERSON] and [PERSON] [17] implemented generalisation of the [PERSON] criterion and the next step toward the phase-field modelling of a fracture, which relies on total energy minimization, consisting of fracture and elastic energy. The proposed methodology can forecast both the initiation of cracks and the emergence of complicated crack patterns. The variational formulation is regularised by [PERSON] [18], enabling the determination of the crack evolution and numerical integration. According to [19; 20], the phase-field method can effectively be used for solving all governing fracture processes, such as crack initiation, propagation, merging, kinking, and branching. To accurately capture high gradients of the phase-field variable, a very fine mesh is necessary within the diffuse crack region. Unfortunately, this typically results in a significant increase in computing costs compared to FE simulation using the sharp crack interface model with adaptive remeshing [7; 21]. Therefore, an adaptive mesh refinement strategy within the phase-field method is increasingly being developed as a solution to this issue [22; 23]. The most significant contribution to the phase-field modelling of the quasi-static brittle fracture was achieved by [PERSON] et al. [24]. [PERSON] et al. [25] presented a succinct outline of a phase-field brittle fracture. Recently, many papers have been published on the subject of brittle fractures [23; 26; 27; 28] and extend to ductile damage of solids [29]. The proposed research topics include the phase-field method with small-strain [30] and large-strain assumptions [31]. Due to its generality and applicability to a wide variety of problems, the phase-field method has gained increasing popularity as a numerical method for damage modelling. Except for standard fracture problems dealing with metal structures, the phase-field method can also be used for assess ment of fatigue fractures in fibre-reinforced composites [32], hyperelastic solids [33], and piezo-electric materials [34]. A detailed overview is given by [PERSON] et al. [35]. Recently, researchers studying phase-field frameworks have dedicated great effort towards the derivation of an extension to fatigue. In [36; 37], phase-field fatigue models are presented without physical interpretation or empirical data. A novel approach is presented by [PERSON] et al. [38], coupling the additional scalar parameter (which describes the fracture) with thermal and fatigue behaviour. Moreover, [PERSON] [39] and [PERSON] [40] introduced the Ginzburg-Landau form, considering fatigue potential as a material degradation and softening during cyclic loading. In [41], a contribution to energy density is introduced, which is calculated as the sum of additional driving forces during cyclic loading. [PERSON] et al. [42], [PERSON] et al. [43], [PERSON] et al. [28], and [PERSON] et al.