PSI - Issue 33

R.F.P. Resende et al. / Procedia Structural Integrity 33 (2021) 126–137 Resende et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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distribution, are just some of the advantages of adhesive joints over more traditional bonding methods. However, it should be noted that these also have disadvantages, such as the need for high cure temperatures in some cases and the typical irreversibility of the bond. It is necessary to consider that the behavior of the adhesive joints varies with the type of adhesive, the type of substrate and the overlap length ( L O ). In the first studies of adhesive bonds, the method of predicting the strength of these bonds consisted of analytical models, with Volkersen (1938) and Goland and Reissner (1944) being the pioneers in several theoretical studies. Such studies had a significant advantage in obtaining stress states in adhesive structures using simplified hypotheses in terms of joint geometry, loading and boundary conditions. Through these simplifications, it was possible to obtain an analytical solution for the behavior of joints in an elastic domain. The numerical modelling of adhesive joints has been strongly developed over time, particularly in recent decades. Part of this evolution results from damage modelling by combining the finite element method (FEM) (Penado 1998) with cohesive zone models (CZM) (Fernlund and Spelt 1991, Carvalho and Campilho 2016). This practice consists of combining conventional FEM for regions where there is no damage and a fracture mechanics approach, using cohesive elements, to promote the spread of damage. Through the FEM it is possible to simplify complex problems, dividing their domain into small parts. These small parts are called elements, and the set of elements is called a mesh. This contains not only the elements, but also the connection relations between them (Reddy 2005). However, due to the dependence of the mesh, FEM has some limitations in the analysis of problems with large strains due to the distortion of the elements. Meshless methods have emerged to eliminate the problems associated with the FEM. These first appeared in 1977 using the Smooth Particle Hydrodynamics (SPH) principle (Gingold and Monaghan 1977). This method was originally developed to solve astrophysical problems. It was only in 1990 that this method was extended to solid mechanics (Libersky and Petschek 1991). Over the years, meshless methods have evolved with the emergence of various methods (Dai et al. 2006, Qian et al. 2014). In 2002, the Point Interpolation Method (PIM) was further enhanced, and a radial basis function was added to the polynomial basis function allowing to develop the radial point interpolation method (RPIM) (Wang and Liu 2002, Wang and Liu 2002). In general, most meshless methods use the same procedure. This consists, first, in the discretization of the problem domain. Thus, using a nodal set, it is possible to discretize the domain problem. Nodal discretization can be done regularly or irregularly. The type of discretization directly influences the result of the numerical analysis, and the irregular discretization usually has less precision. On the other hand, places where there is a higher concentration of stresses, such as cracks, must have a higher nodal density. In order to integrate the weak form of Galerkin, which rules the physical phenomenon under study, it is necessary to perform a numerical integration. Thus, a background integration mesh is used. This may be dependent or independent on the nodal mesh, with the independent case presenting higher accuracy in the results. Moreover, dependent integration meshes relying in nodal integration required an extra stabilization, which increases the computational cost (Chen et al. 2001, Sze et al. 2004). Nodal connectivity is ensured by overlapping of the influence domains when using the RPIM. Thus, each integration point searches the closest nodes, forming its influence-domain. With the influence-domain, it is possible to construct the interpolation functions. The RPIM uses interpolation form functions based on the combination of radial basis functions (RBF) with polynomial basis functions. Through the addition of RBF, singularities associated with discretization alignments are eliminated. Few works deal with meshless methods applied to adhesive joints. Tsai et al. (2014) used a numerical approach to simulate the beginning and spread of a crack using Symmetric Smoothed Particle Hydrodynamics (SSPH). The work was carried out considering aluminum double-cantilever beam (DCB), which were subjected to mixed loading modes through the combination of pure load modes I (traction) and II (shear). It was concluded that, about mode I, the results obtained through the SSPH method expose a maximum difference between the calculated and experimental peak loads of 3.1%, while through the FEM this difference is 6.7%. Results for a mixed-mode are like the experimental ones only for a mixity angle of modes lower than 50º. Bodjona and Lessard (2015) carried out a study with the objective of performing a static analysis on adhesive joints and adhesive/fastened joints. To predict joint failure, a maximum stress criterion was used. To validate the model's capacity, the results were compared with a FEM analysis and experimental measurements. The results obtained through the meshless method agreed with the results obtained in the MEF. It was concluded that, compared to the FEM although the computational cost is higher, the meshless method (RPIM) presents a higher precision of results without requiring any adjustment of the model parameters. Ramalho et al. (2019) combined the Critical Longitudinal Strain (CLS) criterion with NNRPIM for the purpose of predicting bond strength for single-lap joints (SLJ). The joint strength was predicted using 3 different