# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 1 Introduction  
In recent years, modern engineering structures have experienced increased demand for their application, especially in the automotive, aerospace, and energy industries. The most essential requirements in the mechanical design of structures are the prevention of critical fractures and endurance in repeated loading cycles during operation. Hence, the key role of engineers considering the problem is determination of the maximal load capacity of the structure and the number of loading cycles in operation until the fracture in the material occurs. Fracture is the failure mode which denotes splitting of material at two or more parts. This phenomenon can be divided into brittle and ductile fractures based on the constitutive material response. During the fracture, the material undergoes the usual steps, such as crack nucleation, propagation, branching, and total failure. Unlike the brittle fracture, affected by material elasticity, in ductile fractures, a major influence on the material softening is material plasticity. In practice, solving fracture problems represents a cumbersome task, including prediction of complicated crack topology depending on various factors, such as boundary and loading conditions, material properties, and the microstructure. During repeated loading with amplitudes below the load capacity of a material, fractures are frequently a consequence of material fatigue. Material fatigue causes different material failure mechanisms depending on the working conditions and loading. The process of the fatigue fracture can be divided into low- and high-cyclic fatigue regimes. In the former, cyclically loaded mechanical components exhibit fatigue failure after up to 10,000 cycles, while in the latter, fatigue failure occurs after more than 10,000 cycles. Numerical modelling of the fracture process regarding the fracture description can be divided into discrete and diffusive approaches. In discrete methodologies, a crack is presented as a geometrical discontinuity with a sharp interface. Cohesive zone modelling (CZM), which was first developed by [PERSON] [1] and [PERSON] [2] to solve the stress singularity and nonlinear processes ahead of an existing crack, is one of the most used methods for dealing with discrete crack propagation. It is carried out by generalisedcontact elements that form a narrow band before the crack front and is referred to as the cohesive zone [3]. In the standard approach of CZM finite element implementation as shown in [PERSON] et al. [4], the results received demonstrate a substantial dependence of the findings on mesh size and orientation during crack formation because the crack propagates along the element edges. [PERSON] et al. [5] conducted a thorough literature analysis of CZMs, including both their benefits and drawbacks. Moreover, remeshing algorithms given in [6; 7] and enriched FEM discretization approaches, especially with the extended finite element method (XFEM) [8], were developed to accurately represent various crack topologies. This makes it possible to anticipate the growth of cracks without relying on finite element discretisation. According to [PERSON] et al. [9], XFEM resolves crack tip stress singularities and genuine stress behaviour near the fracture tip with outstanding success; however, a major issue with numerical tracking of the crack discontinuities arises. In contrast to modelling real crack geometry, diffuse crack modelling approaches include a damage parameter that regulates the material stiffness into the constitutive model of the stress release brought on by the crack formation. [PERSON] [10] presented a detailed overview. Many continuum damage models preserve the concept of a local continuum theory which does not take into account the material points in the surroundings. According to [PERSON] [11], a powerful softening behaviour as a consequence of increasing the damage parameter is followed by an intense strain and damage localisation.