PSI - Issue 39

Arturo Pascuzzo et al. / Procedia Structural Integrity 39 (2022) 649–662 Author name / Structural Integrity Procedia 00 (2019) 000–000

651

3

Alternatively to hybrid approaches, effective strategies have been developed by augmenting standard FEM models with enhanced numerical techniques to mitigate the need for remeshing actions to trace the evolution of arbitrarily growing cracks. In this framework, the Moving Mesh technique permits changing the computational mesh by moving the mesh nodes according to specific criteria of the problem under investigation. In this context, the use of the Arbitrary Lagrangian-Eulerian (ALE) formulation offers the possibility of handling the mesh movement with great versatility, reducing the occurrence of relevant finite element distortions, which can affect computational accuracy and stability. It is worth noting that, recent research works have used the ALE to reproduce crack propagation in homogeneous materials under thermo-mechanical loading characterized by highly irregular temperature gradients (Greco et al. (2021a)). It is well known that such conditions implicate the development of articulated crack paths, which are difficult to predict in advance. The results have highlighted the strength of the ALE in managing arbitrarily growing cracks, spending limited computational resources. For this reason, the use of the ALE may be effective in reproducing crack propagation phenomena in FGMs as well. In this work, an effective FE modeling approach based on the use of the Arbitrary Lagrangian-Eulerian Formulation (ALE) for simulating crack propagation phenomena in FGMs is proposed. In particular, the ALE is used to reproduce geometry evolution caused by the advancing cracks. Specifically, the computational nodes of the mesh around the crack tip are moved according to standard fracture criteria, which define the condition for crack nucleation and the direction of propagation. To this end, an accurate evaluation of the fracture variables at the crack front ( e.g. , Stress Intensity Factors, T-stress) is necessary. One of the most used methods to extract fracture variables at the crack front is the Interaction Integral (also known as M -integral method) developed by Yau (Yau et al. (1980)). This method requires refined and regular mesh discretization around the crack tip region to provide reliable predictions. To permit the use of the M -integral method in the framework of the proposed strategy, it is implemented using the ALE formulation. In this way, the M -integral integrates on deforming elements without losing accuracy meanwhile providing a continuous extraction of fracture variables. The paper begins with a brief review of the theoretical notions at the base of the proposed procedure (Section 2). Next, numerical implementation details are reported (Section 3). In particular, a description of the propagation steps involved in the proposed scheme is provided. Finally, numerical results (Section 4) are reported. In particular, the reliability of the proposed model is assessed through comparisons with the predictions of other numerical approaches reported in the literature. 2. Theoretical Background In this section, the main theoretical concepts forming the proposed modeling strategy are reported. At first, an overview of the Arbitrary Lagrangian-Eulerian formulation (ALE) is provided. Subsequently, the governing equations of the fracture propagation problem are outlined. Finally, a discussion on the Interaction Integral method ( M -Integral) in the context of functionally graded materials is reported. 2.1. The Arbitrary Lagrangian-Eulerian formulation (ALE) The ALE formulation is an efficient approach to manage the movement of the mesh nodes of the computational mesh with great flexibility (Greco et al. (2020d), Ammendolea et al. (2021)). In the proposed scheme, the ALE is used to adapt the computational mesh to the evolution of the growing cracks. Hence, the mesh nodes are moved according to fracture conditions prescribed by classic fracture criteria ( e.g. , the maximum hoop stress criterion), which define the condition of crack nucleation and the direction of propagation. To describe the movement of the mesh nodes, the ALE requires the definition of twofold coordinates systems (see Figure 1). The first is a fixed or Referential system (R) that identifies the initial (or referential) position of the mesh nodes from which the movement originates. The second is a coordinate system that recognizes the position of the mesh nodes corresponding to the i -th varied or moved configuration ( i.e. , that one describing a step of the propagation process). Generally, this second system is named as moved or current coordinate system (M). The referential and moved coordinate systems are related reciprocally by means of a continuous and bijective mapping function : R M C C Φ → % with [ ] 1 2 Φ = Φ Φ % , which implies: