PSI - Issue 66

Domenico Ammendolea et al. / Procedia Structural Integrity 66 (2024) 396–405 Author name / Structural Integrity Procedia 00 (2025) 000–000

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Keywords: Phase field method, adaptive approach, crack propagation, finite element, heterogeneous materials.

1. Introduction Accurately predicting crack propagation remains a significant challenge in assessing the structural integrity of cracked heterogeneous structures, where failure could result in unsafe conditions (Matouš et al., 2017; Pindera et al., 2009). These structures, composed of multiple phases with distinct material properties, often exhibit complex failure patterns due to their heterogeneity. In the last decades, several numerical approaches have been developed to simulate crack propagation, each with its strength and limitations. One of the earliest methods for fracture simulation is the Finite Element Method (FEM), which still remains the one most widely employed because of its flexibility in replicating complex systems (Bessa et al., 2013; Hillerborg et al., 1976). In general, classical FEM-based models represent defects as discontinuity surfaces embedded in the computational domain, requiring remeshing operations as the crack propagates. This feature makes FEM computationally expensive and cumbersome, especially when dealing with complex crack paths and branching into heterogeneous materials. Among all these methods, Cohesive Zone Models (CMZs) are undoubtedly the most used approaches (Barenblatt, 1962; Pascuzzo et al., 2020; Tvergaard and Hutchinson, 1992). To avoid remeshing events, several strategies have been developed, such as the Extended Finite Element Method (XFEM) (Belytschko and Black, 1999; Vellwock and Libonati, 2024), the Diffuse Interface (DI) method (De Maio et al., 2022; Gaetano et al., 2022a, 2022b; Greco et al., 2020), and the Moving Mesh (MM) approach (Ammendolea et al., 2024, 2023, 2020; De Maio et al., 2024). The XFEM extends the traditional FEM approach by incorporating discontinuous enrichment functions in the finite element approximation, allowing it to model cracks more effectively. Despite this improvement, complexities arise in formulation finite elements, material law, performing numerical integrations and accurately capturing highly localized damaged zones in heterogeneous structures (Sukumar et al., 2001). In DI approaches, which are based on the insertion interface elements with softening constitutive behavior along all internal boundaries of a fixed bulk mesh, the computational burdens increase significantly, making their adoption impractical in the case of large computational domains. The MM technique, consistent with the Arbitrary Lagrangian Formulation (ALE), moves the nodes of the given mesh with respect to the material frame to replicate the crack trajectory evolution, resulting in a significant reduction in the remeshing events as the main advantage. Although MM is a highly efficient strategy, the Phase-Field Method (PFM) has recently gained attention as a robust approach for simulating crack growth process in both homogeneous and heterogeneous material (Dinachandra and Alankar, 2020; Kuhn and Müller, 2010; Miehe et al., 2010). The PFM replaces the sharp crack interface with a smeared one by utilizing a continuous scalar field, where the transition from intact to fully fractured material is described by a smooth function. This makes it extremely effective for predicting crack initiation, propagation, branching, coalescence and failure, and handling complex crack patterns without remeshing. However, the standard phase-field formulation presents some issues when applied to large-scale problems, particularly due to the need for fine mesh resolution in areas where cracks are expected to occur. To overcome this, adaptive methods have been introduced that refine the mesh only in necessary regions, thereby significantly reducing computational costs while preserving accuracy (Noii et al., 2020; Plewa et al., 2005). The aim of the present work is to propose an efficient FE adaptive phase-field approach for reproducing crack propagation in both heterogeneous and homogeneous structures under general loading conditions. In particular, the proposed model combines the advantages of the traditional PFM for handling intricate fracture trajectories, with an adaptive meshing strategy to efficiently manage the computational demands. After the introduction, the rest of the manuscript is organized as follows. Section 2 reports a brief overview of the theoretical formulation of the Cohesive Phase-Field Model proposed for the first time by Wu (Wu, 2017). Section 0 describes the numerical implementation of the proposed strategy, showing a detailed description of both the adaptive mesh technique and the algorithm developed to manage the propagation process. Section 4 presents two numerical applications to assess the reliability of the proposed methodology through comparisons with experimental data and numerical results obtained by other approaches reported in the literature. Finally, in Section 5 some final remarks and future perspectives of the proposed model are discussed.