Issue 51

F. Fabbrocino et alii, Frattura ed Integrità Strutturale, 51 (2020) 410-422; DOI: 10.3221/IGF-ESIS.51.30 411 I NTRODUCTION any researchers devoted their attention to investigate dynamic fracture mechanics by means of analytical or/and computational formulations. In the literature, most of the models are developed in the framework of the Finite Element Method (FEM), because of its robustness and reliability to deal with complex geometries. Smeared crack models are utilized with large success, since they are easy to be implemented and reproduce correctly the amount of energy dissipation during the crack growth [1,2]. However, regularization techniques such as nonlocal or gradient based models are required to prevent material instability problems and to capture the internal characteristic length and displacement discontinuity in presence of microcracks. In addition, mesh dependency problems are typically observed, since a large number of FEs should be introduced in the region in which the crack path is expected. Numerical models based on the Cohesive Zone Modelling (CZM) are frequently adopted for simulating crack evolution in both static and dynamic frameworks [3,4]. An important advantage of the CZMs is their ability to predict directly crack onset and propagation, without introducing preexisting material discontinuities [5,6,7]. However, the initial finite stiffness in the constitutive laws may produce, especially in brittle solids, an excess of compliance with spurious interface traction oscillations. Moreover, a refined mesh is required to predict accurately the fracture variables, which in many cases may lead to instability or non- convergent phenomena. Such problems may be circumvented by the use of re-meshing techniques, but numerical complexities still remain due to the need of local mesh modification algorithm [8]. Despite high computational costs, re- meshing based methods have shown rigorous results to predict crack path and related fracture variables. Alternatively, Embedded or Extended Finite Element Methods (EFEM, XFEM) are able to reproduce crack growth without the use of re-meshing procedures by the enrichment of displacement fields inside an element. In particular, material discontinuities propagate within the elements, leaving unaltered the mesh discretization [9]. However, due to lack of proper criteria for crack branching and interaction, reliable predictions of complex fracture patterns remain a key challenge [10]. Recently, several refined formulations are proposed to improve crack onset criterion and growth procedures with respect to conventional models based on CZM or Fracture Mechanics (FM). For instance, methodologies, based on the Scaled Boundary Finite Element Method (SBFEM) for simulating dynamic crack propagation by adopting polygon elements, are developed in [11]. Although, these numerical schemes present good capabilities for computing the fracture variables by means of re-meshing events, they change the global mesh, leading to large computation costs. Differently, efficient numerical models consistent to the Cracking Elements (CE) are proposed, in which strong discontinuity embedded approach, based on the use of disconnected cracking segments, is developed [12]. The main advantage of this approach is the absence of re- meshing or enrichments events, whereas the main disadvantages are the inability to describe accurately crack path and SIF/ERR functions. Phase-Field Methods (PFMs) have rapidly spread in view of their capability to seamlessly deal with complex crack patterns like initiation, branching, merging and even fragmentation [13]. One of the main disadvantage of the PFM is the fact that the method is still computationally intensive. In order to avoid some of the issue previously mentioned, numerical models based on moving mesh methodology are developed in the literature. In [14], a model based on a combined approach developed in the framework of Fracture Mechanics and moving mesh methodology is able to predict dynamic delamination phenomena in layered structures. Similarly, numerical strategies based on a coupled approach between moving mesh and the cohesive zone modelling are presented in [15-18]. In this case, moving mesh methodology based on ALE approach is introduced only to represent process zone region, leaving the governing equations of the structural model, basically, unaltered [19]. The aim of the present work is to generalize the numerical approach developed in [19,20] to describe dynamic crack propagation in 2D structures. In particular, the approach combines concepts arising from structural mechanics and moving mesh methodology, which are implemented in a unified framework to predict crack growth on the basis of Fracture Mechanics variables. Moving computational nodes are modified starting from a fixed referential coordinate system on the basis of a crack growth criterion to predict directionality and displacement of the tip front. This is achieved by introducing both crack propagation in terms of tip speed and angle of propagation, appropriately. Numerical results demonstrating the effectiveness of the method to simulate crack growth in continuum media under dynamic loading conditions or impact phenomena are proposed. F ORMULATION OF THE MODEL he proposed modeling is based on the combination of FM and Moving Mesh Method (MMM). The former predicts crack growth, by the use of the ERR/SIF concepts and an advancing criterion, whereas the latter is introduced to account geometry changes produced by the crack evolution. M T