# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 1 Introduction
[44] proposed the addition of degradation fracture energy based on the stress or strain history. In [45], [PERSON] et al. include a toughness degradation law depending on the accumulated elastic strain energy density which governs fatigue failure. Furthermore, [PERSON] et al. [46] introduced the influence of hydrogen on fatigue. All previously presented research proposals in the field of phase-field fatigue assume linear elastic material behaviour only. On the other hand, [PERSON] et al. [47] extended the phase-field fatigue model by considering elastoplastic material behaviour. Unlike the previously mentioned research, [PERSON] et al. [48] presented a phase-field fatigue numerical framework applicable to both high-cycle and low-cycle regimes based on the three-layered staggered algorithm. As demonstrated by the framework, the key features of [PERSON] and Paris law curves are constructed. In [44], material elastoplasticity is empirically included through a local strain approach by Neuber's rule, while a thermo-mechanical fatigue formulation is introduced by [PERSON] et al. [49], incorporating the new degradation function and the Voge-type hardening law. [PERSON] et al. [50] considered a comparison of different fatigue degradation functions, asymptotic and logarithmic. One of the greatest problems in the numerical implementation of the currently proposed phase-field models is their robustness. The phase-field numerical framework can be formulated in a monolithic or staggered manner. The monolithic approach calculates phase-field variables and displacements simultaneously. Unfortunately, the disadvantage of a monolithic scheme is instability during fatal crack propagation because of the non-convexity of the regularised free energy functional regarding displacement and phase-field parameters [26]. The non-convexity has been circumvented in several attempts, for example, by the linearization of the elastic part of total energy [22] by restoring the iterative convergence through application of the line-search algorithm with negative and positive parts [51] and by the Newton scheme with Jacobian adjustment [52]. On the other hand, the staggered scheme calculates phase-field variables and displacements separately, restoring, in this way, convexity of the underlying free energy functional. Hence, the weak formulation is decoupled in two equations by the split operator, and it can be solved in an iterative way. However, as demonstrated by numerous researchers, the staggered scheme requires small-scale loading increments to achieve accurate numerical results. [PERSON] et al. proposed a single iteration staggered procedure [24] which is computationally time consuming and very expensive due to small loading increments. Thus, [PERSON] et al. [53] and [PERSON] et al. [54] resolved the problem of small loading increments by incorporating more iterations during one loading increment, constrained by the stopping criterion. In [52], the gradual convergence rate was circumvented by involving over-relaxed alternate minimisation with Newton-type methods [55], while in [25] the proposed solution bypasses this problem by introducing the stopping criterion which normalizes alternation of the strain energy. In contrast, [PERSON] et al. [27] derived a staggered RCTRL algorithm for brittle fractures with a stopping criterion based on the residual norm, which has been proved as accurate and efficient due to its ability to impose large loading increments. Unfortunately, despite its advantages, the algorithm is computationally exhausting since three mesh layers are required, leading to a large number of unknowns. Therefore, despite tremendous efforts and still increasing popularity of the phase-field methodology, there are still unresolved issues in the phase-field framework. Open questions arise seeking a fast and stable solver strategy, as well as derivation of phase-field models for various physical softening mechanisms. In this paper, a novel staggered phase-field framework is proposed.