PSI - Issue 79

Domenico Ammendolea et al. / Procedia Structural Integrity 79 (2026) 467–474

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1. Introduction Fracture in brittle materials is inherently dynamic and challenging to manage due to the material's poor capacity for plastic energy dissipation (Fineberg et al. (1991), Luciano and Willis (2005), Bruno et al. (2013), Tocci Monaco et al. (2021), De Maio et al. (2023)). Once a crack velocity exceeds a critical threshold, the process becomes unstable, leading to rapid crack branching (Yoffe (1951)). This phenomenon, in which a single crack splits into multiple diverging paths, is highly detrimental to structural integrity, as it accelerates fragmentation and increases the risk of sudden, catastrophic failure in common engineering materials such as glass and ceramics. Consequently, a deep understanding of the mechanics and conditions governing crack branching is essential for engineering safety (Luciano and Willis (2003), De Maio et al. (2024a)). Early investigations relied heavily on experimental methods, such as high-speed photomicrography, which provided fundamental insights into factors such as crack velocity and near-tip stress distributions (Fayyad and Lees (2017)). However, these techniques are constrained by high costs, the need for specialized equipment. This reliance on costly experiments has driven researchers toward numerical methods, which offer a robust, systematic, and cost-effective means to model fracture under controlled conditions (Bruno et al. (2009), Caporale et al. (2012), Greco et al. (2015)). These numerical strategies can be broadly classified according to the way the crack is represented, so that it is possible to define the class of discrete crack models and that of smeared crack models. Discrete crack models (such as FEM, XFEM) explicitly represent cracks as discontinuities, while smeared crack models (such as Phase-Field and Peridynamics) account for fracture via diffuse damage zones. Each class has inherent limitations. Traditional FEM models are generally unsuitable for dynamic cracking because they require computationally expensive, potentially unstable remeshing (Trädegård et al. (1998), Song et al. (2008), Ammendolea et al. (2025b)). Although methods like XFEM enhance FEM by allowing cracks to propagate independently of the mesh, they become excessively demanding when dealing with numerous interacting cracks (Belytschko et al. (2003)). Smeared models, while naturally capturing branching as an energy-minimization process, often require extremely fine mesh resolutions (Phase-Field) or face difficulties in accurately imposing boundary conditions (Peridynamics), thereby incurring high computational overhead (Agwai et al. (2011), Borden et al. (2012), Hofacker and Miehe (2012), Ammendolea et al. (2025c)). Despite the versatility of the standard FEM, its application to dynamic fracture remains problematic, as accurately tracking an evolving path—particularly during a branching event—requires frequent, complex tracking and remeshing, which increases both cost and the potential for solution instability. To overcome these inherent FEM limitations, the authors previously developed an advanced approach utilizing the Moving Mesh (MM) technique based on the Arbitrary Lagrangian-Eulerian (ALE) formulation (Greco et al. (2022)). This prior MM/ALE model successfully reduced the need for computationally intensive remeshing by dynamically moving mesh nodes around a single propagating crack tip, demonstrating its efficiency for single-crack dynamics. The present work directly extends this foundation, expanding the MM/ALE framework to simulate the full dynamic crack-branching event. A significant advantage is that the method uses the MM technique to eliminate the necessity of external criteria to define the branching angle after the split, allowing the branching direction to emerge naturally from the calculated stress field. The reliability and efficiency of this enhanced MM/ALE strategy are subsequently assessed through simulations of widely adopted benchmark cases. 2. Theoretical background 2.1. Essential of the Arbitrarily Lagrangian Eulerian formulation The ALE formulation necessitates that the governing equations, usually expressed in the material frame R X , be transformed into a third frame, the Referential coordinate system R χ , which identifies the points of the computational mesh through the referential coordinate ( χ ) (Funari et al. (2016)). The transformation rules for gradient operators and time derivatives are essential for this purpose. The material gradient of a vector field v is mapped into the referential frame using the following expression: R M ∇ =∇ Ψ v vJ (1)