Several numerical results substantiate our developments. On the element technologies different mesh types at the Global and Local domains were considered for efficient large scale problems. The modeling of crack formation at the Local scale is achieved in a convenient way by continuum phase-field formulations to fracture, which are based on the regularization of sharp crack discontinuities. This work is concerned with the extensions of the Global-Local method to problems of 3D brittle fracture. The successful application of the method to non-linear problems such as finite strain hydraulic and ductile fracture in 2D leads naturally to the question of its effectiveness and robustness in the third dimension. In this regard, failure is analyzed on a lower (Local) scale, while dealing with a purely linear problem on an upper (Global) scale. This method has the potential to tackle practical field problems in which a large-structure might be considered and fracture propagation is a localized phenomenon. This work addresses a robust and efficient Global-Local approach for numerically solving three dimensional (3D) fracture-mechanics problems. The efficiency and accuracy of this new method is shown in 2D simulations. In contrast to classical XFEM / GFEM, the presented method does not require level set techniques or explicit representations of crack geometries, considerably simplifying the simulation of crack initiation, propagation, and coalescence. This combination allows for the application of significantly coarser meshes than it is possible in PFM while still obtaining accurate solutions. The method is based on the combination of a phase-field approach with an ansatz transformation for the simulation of fracture processes and an enrichment technique for the displacement field as it is used in the extended finite element method (XFEM) or generalised finite element method (GFEM). It has the potential to drastically reduce computational cost compared to the classical phase-field method (PFM). In this contribution, an enriched phase-field method for the simulation of 2D fracture processes is presented. The efficient simulation of complex fracture processes is still a challenging task. Furthermore, by an assessment of crack growth rates obtained from several numerical simulations by a conventional approach for the description of fatigue crack growth, it is shown that the presented model is able to predict realistic behavior. The proposed model is able to predict nucleation as well as growth of a fatigue crack. This new contribution is governed by the evolution of fatigue damage, which can be approximated by a linear law, namely the Miner’s rule, for damage accumulation. While in first phase field models the material’s fracture toughness becomes degraded to simulate fatigue crack growth, we present an alternative method within this work, where the driving force for the fatigue mechanism increases due to cyclic loading. However, fracture due to cyclic mechanical fatigue, which is a very important phenomenon regarding a safe, durable and also economical design of structures, is considered only recently in terms of phase field modeling. The framework was so far applied to quasi static and dynamic fracture for brittle as well as for ductile materials with isotropic and also with anisotropic fracture resistance. The field has gained attention properly due to its benefiting features for the numerical simulations even for complex crack problems. Phase field modeling of fracture has been in the focus of research for over a decade now.
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