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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3

were comparable to those obtained with commercial software (ANSYS). In other areas of application, hyper-elastic had also been used with MMs, for example, the Ogden model for biological tissue (Joldes et al., 2019), and the Yeoh model for fluid-structure interaction using the Meshless Local Petrov-Galerkin (MLPG) method (Moon et al., 2020). Turning to the hyper-elastic material models, most of the meshless numerical applications are based on Neo Hookean and Mooney-Rivlin. Nevertheless, the methodology described by de Souza Neto et al., (1995) about the Ogden model can be expanded to MMs. In addition, the Neo-Hookean and Mooney-Rivlin material models can be considered as special cases of the Ogden model (de Souza Neto et al., 1995; Ogden, 1982); consequently, the implementation of this model is more versatile for analysing rubber-like behaviour. This work aims to implement the Ogden hyper-elastic model into the NNRPIMmeshless method. The methodology was validated using the geometry and data from the literature, i.e., (Khosrowpour et al., 2019), and against the solutions from commercial software (ABAQUS 6.17, Dassault Sistèmes, Inc.). This methodology would be used as a base for further applications of the NNRPIM to analyse adhesive joints. 2. The natural neighbour radial point interpolation method The NNRPIM method was developed as an evolution of the RPIM and the NEM (Belinha, 2015). In this method, the geometry, or domain  , is discretised as a set of n nodes 1 2 { , , , } d n x x x R    N . This MM has the advantage of establishing nodal connectivity based only on the nodal distribution (Belinha, 2015). The nodal connectivity is established using Voronoï cells and Delaunay tessellation. The Voronoï diagram allows to establish the nodal connectivity, by means of influence-cells ( V i ), whilst the Delaunay tessellation permits to build the background integration mesh. Then, for an interest point inside the domain ( I  Ω x ), all the nodes inside V i have an influence on x I . Subsequently, any variable related to can be described as the sum of two shape functions, one radial-based (RBF) and one polynomial-based (PBF), as shown in Equation (1) (Belinha, 2015; Belinha et al., 2016; Farahani et al., 2019).

n

m 



  I x

    x x

    x x

(1)

,

u

R a

P b





i

I

i

I

j

I

j

I

1

1

i

j





where   i I R x corresponds to the RBF,   I i a x is its non-constant coefficient.   I j P x corresponds to the PBF, whilst   I j b x is its non-constant coefficient, as described in (Belinha, 2015). The RBF used in this work corresponds to the multiquadrics radial basis function (MQ-RBF), which uses the Euclidean norm (Equation (2)).

1

   2          2 I i I i I x x y y z z 



2 2 

r

(2)



Ii

i

   Ii



. p

(3)

2 r c  

2

R r

Ii

From Equation (3), two parameters can be observed, c and p . The values for these parameters had been determined in previous work (Belinha, 2015), being 0.0001 c  , and 0.9999 p  . Subsequently, the values of the non-constant coefficients ( a i and b j ) are calculated by applying the Equation (1) to all nodes inside the domain. This process is simplified by converting Equation (1) into matrix notation s   u Ra Pb . Once these values are determined, it is possible to express Equation (1) as a function of  i ( x I ), which corresponds to the shape functions, as follows: (4) Once the shape functions are defined, it is possible to define its derivatives, which can then be applied to build the   I     x x   I 1 . n i i i u u