PSI - Issue 80

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D.C. Gonçalves et al. / Procedia Structural Integrity 80 (2026) 443–450 Author name / Structural Integrity Procedia 00 (2019) 000–000

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1. Introduction Leading construction, automotive, and aircraft industries rely on structural adhesive joints to provide strong lightweight structures. Adhesive joints allow the joining of dissimilar materials, and uniformly distribute stress, thus reducing stress concentrations, and providing high fatigue durability within dynamic conditions and corrosive environments. However, being the numerical simulation of adhesively bonded components compromised by the intricate stress fields within the adhesive layer and its interfaces, requiring dense nodal discretization in such delicate regions, the development of high convergent techniques is highly motivated. In recent years, meshless methods (Liu, 2011; Liu & Gu, 2005) emerged as an innovative technique aiming to surpass several limitations of classic mesh-based methods, such as the Finite Element Method (FEM). In advanced applications including large deformation, impact, and fracture problems, FEM accuracy is highly dependent on mesh distortion and stress concentrations, usually requiring costly remeshing procedures throughout the analysis. Meshless techniques permit discretizing the problem domain with a flexible and unstructured nodal set. Then, the intricate discretization of the crack tip region can be easily adjusted without the concern of distorted elements, and remeshing/refinement techniques are more efficient. Meshless collocation methods settle the foundation for the early meshless method development in the 1930s (Lanczos, 1938; Liu & Gu, 2005; Slater, 1934). Nonetheless, the Smoothed Particle Hydrodynamics (SPH) method (Gingold & Monaghan, 1977) introduced in 1977 is considered the first fully developed meshless technique (GU, 2005). The SPH is based on the strong formulation, i.e., the partial differential equations are addressed directly allowing to obtain exact solutions. Later, meshless methods based on the weak form were developed to obtain approximate solutions considering variational principles. The Diffuse Element Method (DEM) (Nayroles et al., 1992) uses the Moving Least Square (MLS) approximation (Lancaster & Salkauskas, 1981) to construct the shape functions, being one of the first approximant meshless methods to be developed, and the basis of the Element Free Galerkin Method (EFGM) (Belytschko et al., 1994). Today, the EFGM is well established in the computational mechanics field and widely used in several fields including solid and fracture mechanics. Approximation shape functions lack the delta Kronecker property, requiring additional procedures and care to enforce boundary conditions. On the contrary, interpolating meshless methods allow the direct imposition of the boundary conditions in the global system of equations. Interpolation methods can be traced back to the Natural Element Method (NEM) (Braun & Sambridge, 1995; Sukumar et al., 1998), which uses the Sibson functions to interpolate the field variables. The RPIM (Wang & Liu, 2002) is the combination of the original Point Interpolation Method (PIM) (Liu & Gu, 2001) with radial basis functions (RBF). The PIM uniquely uses polynomial functions to interpolate the field variables, potentially leading to singular solutions in certain nodal arrangements. Adding RBF to the PIM shape functions stabilizes the procedure and produces smoother solutions (Wang & Liu, 2002). Further combination of the RPIM with the NEM led to the development of the NNRPIM, a meshless method based on the Voronoï diagram and RPI shape functions. The Voronoï diagram (Voronoi, 1908) and Delaunay triangulation (Delauney, 1934) are used to determine the influence domains and construct an integration mesh completely dependent on the initial nodal discretization. Seminal works on fracture propagation can be found in the literature, mainly regarding the EFGM, including dynamic problem (Belytschko et al., 1995), 3D applications (Bordas et al., 2008; Rabczuk, Bordas, et al., 2010; Rabczuk, Zi, et al., 2010) and enriched formulations (Fleming et al., 1997). In reference (Khosravifard et al., 2017), the background decomposition method (BDM) is considered to numerically integrate the domain integrals of the enriched EFGM formulation, developed for crack tip problems. Additionally, the RPIM was also considered but considering ordinary RBF. Benchmark analyses and the high accuracy results further demonstrate the potential of meshless methods in linear elastic fracture mechanics (Khosravifard et al., 2017). Regarding the application of meshless methods to model fracture in adhesively bonded joints, the SPH method (Gingold & Monaghan, 1977) was used by Mubashar and Ashcroft (Mubashar & Ashcroft, 2017) to simulate fracture propagation in SLJ. The Symmetric Smoothed Particle Hydrodynamics (SSPH) method was used combined with CZM to simulate fracture propagation in DCB specimens by Tsai et al. (Tsai et al., 2014a). A continuum mechanics formulation was implemented, providing accurate load-displacement predictions. The same authors also implemented a SSPH method combined with CZM to model the fracture propagation in DCB joints (Tsai et al., 2014b). The RPIM has been used to analyse hybrid composite SLJ using a fracture mechanics approach (Bodjona & Lessard, 2015). Accurate mid-thickness stresses within the adhesive layer were obtained using the RPIM. Despite the inherent advantages of meshless methods in fracture