PSI - Issue 54

Luís D.C. Ramalho et al. / Procedia Structural Integrity 54 (2024) 390–397 Lu´ıs D.C. Ramalho et al. / Structural Integrity Procedia 00 (2023) 000–000

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Sa´nchez-Arce et al. (2021), alternatively some use fracture mechanics criteria Barroso et al. (2020), and others make use of damage mechanics criteria Riccio et al. (2017). However, the most common method to predict the strength of adhesive joints under static loads is using Cohesive Zone Models (CZM) , which combine concepts from fracture and continuum mechanics Sane et al. (2018). Adhesive joints have also been studied using numerical methods under dynamic loading conditions, such as impact, free vibrations and fatigue. The reviews by Ramalho et al. (2022a) and Machado et al. (2018) are focused on that. The current work is focused on impact, therefore an overview of this field is important. Regarding strength predictions, under impact it is also common to use CZM. However, some cohesive law properties are strain rate dependent, which is a limitation of CZM. The properties that are strain rate dependent are the cohesive strength in tension ( t n ) and shear ( t s ), while the other properties do not seem to be strain rate dependent. This has been shown in several works including Valente et al. (2019); Peres et al. (2022); Yang et al. (2021). Triangular CZM were used by Arau´jo et al. (2017) to predict the strength of a SLJ, achieving predictions similar to the experiments. A damage model was used by Boling and Dongyun (2018) to predicted the strength of a composite SLJ under impact. This damage model was applied to the composite adherents. Continuum mechanics were used by Xu et al. (2021), to analyse the stress in the adhesive in a SLJ under impact loading. In that work it was shown that in this type of loading the stress in the adhesive propagates like a wave. The current work aims at using meshless methods in the impact analysis of adhesive joints. There are several di ff erent meshless methods available, the first meshless method was the Smoothed Particle Hydrodynamics (SPH), proposed in 1977 Lucy (1977); Gingold and Monaghan (1977). The meshless methods used in this work are the Radial Point Interpolation Method (RPIM) Wang and Liu (2002b) and the Natural Neighbours RPIM (NNRPIM) Dinis et al. (2007). Both methods have been used in the static analysis of adhesive joints Ramalho et al. (2019, 2022b). The RPIM has previously been used in the impact strength prediction of a SLJ using the Intensity of Singular Stress Field (ISSF) criterion Ramalho et al. (2023b), it has also been used to study the free vibration modes of a SLJ Ramalho et al. (2023a). However, the NNRPIM has never been used to analyse impact or free vibrations in adhesive joints. The main goal of the current work is to use two di ff erent meshless methods, the RPIM and the NNRPIM, to analyse the stress in a SLJ under impact, and to perform a strength prediction for this joint using a simple continuum mechanics criterion. This work is divided into four sections. Section 2 describes the RPIM and the NNRPIM formulations. Section 3 describes problem analysed, and shows the stress analysis and the strength predictions. The conclusions of this work are discussed in Section ?? . The two meshless methods used in this work are the RPIM and the NNRPIM. These methods are similar in several aspects, and the main di ff erence between them is the creation of the integration points and the determination of their respective influence domains. Both methods start with the discretization of a given domain Ω into a nodal set N . Then, the integration points are created and their influence domains determined. In these steps the RPIM and the NNRPIM have di ff erent approaches described in Section 2.1 for the RPIM and Section 2.2 for the NNRPIM. Afterwards, the shape functions are calculated, which are the same for both numerical methods, they is described in detail by Belinha (2014). The shape functions of these methods have a polynomial function and a Radial Basis Function (RBF). The remaining implementation of both the RPIM and the NNRPIM is similar to the FEM and can be found in Belinha (2014). 2. Numerical Model

2.1. Radial Point Interpolation Method

In the RPIM a background integration grid is used to aid in the creation of the integration points. In this work, the grid is composed of quadrilaterals and the integration points are created inside them using a 2 × 2 Gauss-Legendre quadrature, this is exemplified in fig. 1a. Then, the influence domain of each integration point is determined. There are several ways to do this, described by Belinha Belinha (2014). In this work the influence domains are composed by the 16 nodes closest to the integration points, just like in the two example influence domains in fig. 1a. This influence domain size is within the recommended range found in literature Belinha (2014). In fig. 1a the two influence domains