PSI - Issue 33

I.J. Sánchez-Arce et al. / Procedia Structural Integrity 33 (2021) 149–158 Sánchez-Arce et al. / Structural Integrity Procedia 00 (2019) 000–000

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1. Introduction Nowadays, with the advent of more non-metallic materials and modern manufacturing processes (like additive manufacture) to the engineering practice, adhesives and adhesion play an important role. In addition, computer simulation and modelling facilitate the application of such technologies. Therefore, material models are an important part of this process. Flexible adhesives, in particular, present a rubber-like behaviour (Crocker et al., 1999). One of the first models describing the rubber-like behaviour of materials was developed by Mooney, (1940), later known as the Mooney-Rivlin hyper-elastic model. This model considers the effect of the first ( I 1 ) and second ( I 2 ) invariants of the Cauchy-Green tensor on the strain-energy. Later, (Treloar, 1943) developed a model in which the strain energy was only a function of I 1 ; this model is known as Neo-Hookean (Ogden, 1972). The invariants of the Cauchy-Green tensor can be expressed as a function of the principal stretches (  i , i=1,2,3); based on this, Ogden, (1972) proposed a model in which the strain-energy is a function of  i elevated to a power  . This model is more general and has the particularity that if  =1 it describes the Neo-Hookean material, while if  =2 it describes a Mooney Rivlin (de Souza Neto et al., 1995; Ogden, 1982). Subsequently, taking advantage of the original concept proposed by Rivlin about the strain-energy being an infinite series (Ogden, 1972), Yeoh, (1993) proposed the strain-energy as a function of I 1 , for which the Neo-Hookean is a special case; the Yeoh model commonly considers three terms of the infinite series, each term is elevated to a power. Although more hyper-elastic models exist, the aforementioned models would describe the behaviour of materials like adhesives (Crocker et al., 1999). In addition, these hyper-elastic models also depend on other constants, often related to the shear modulus ( G ), which have to be determined from curve fitting experimental data (Baaser, 2010; Rackl, 2015). A detailed description of these and other hyper-elastic models can be found in (Ogden, 1982; Yeoh, 1993). Analyses of adhesive joints involving hyper-elastic materials are often performed with commercial Finite Element Method (FEM) software. For example, hybrid Single-lap joints (SLJ), i.e., aluminium and composite substrates, bonded with a flexible adhesive were analysed by (Lubowiecka et al., 2012). The adhesive was experimentally characterised and modelled as Neo-Hookean, Mooney-Rivlin, and Arruda-Boyce. Numerical results of the joints were close to the experimental data (Lubowiecka et al., 2012). For instances, (Alami and Bilal, 2015) studied SLJs with acrylic adhesive considered as Mooney-Rivlin. Chiminelli et al., (2019) analysed SLJs and Double cantilever beam (DCB) geometries, assuming several hyper-elastic models. However, from the parameter selection, it can be observed that the analysed cases corresponded to Neo-Hookean and Mooney-Rivlin regardless of the model used. For the polyurethane adhesive used, the Mooney-Rivlin provided the best results with respect to the experimental (Chiminelli et al., 2019). Meshless methods (MM) had been developed to perform analyses in the areas where the Finite Element Method (FEM) presents limitations, such as in the cases where mesh distortion is evident (Chen et al., 2017; Liu, 2016). Although these methods are a tool that can be used to analyse adhesive joints, their application in such field is scarce, as reviewed by (Ramalho et al., 2020). Many of the recent applications of MMs to analyse adhesive joints are performed using the Natural Neighbour Radial Point Interpolation Method (NNRPIM) an efficient and robust MM (Ramalho et al., 2019; Sánchez-Arce et al., 2021). The methodology of the mentioned applications already considers the interaction between different material types as well as non-linear material behaviour. Therefore, these methodologies can be used as starting point for hyper-elastic analyses using meshless methods. The hyper-elastic behaviour of rubber-like materials has been studied by means of the FEM, where some element developments had been done (Brink and Stein, 1996; de Souza Neto et al., 1995). Moreover, these implementations were included in commercial FEM software, as mentioned above. On the other hand, the implementation of hyper elastic models into MM is scarce, perhaps because of the variety of MMs available or because these methods are not widely known. Liu and Gu, (2005) analysed a block under compression by using the Radial Point Interpolation Method (RPIM); the material was considered as Neo-Hookean. Although the solution is presented, there is no description of the implementation nor the methodology followed. Khosrowpour et al., (2019) used a strong form meshless approach to analyse two plates under tensile load, one solid, one perforated. The material model implemented was the compressible Mooney-Rivlin. They also used the RPIM and, in addition, three different approaches for imposing boundary conditions were explored. Their solutions were compared against those obtained with commercial software. Later, Bui et al., (2020) implemented the Neo-Hookean hyper-elastic model with the RPIM. This method, as the NNRPIM, possesses the Kronecker delta property easing the process of applying boundary conditions. The results