PSI - Issue 5

Behzad V. Farahani et al. / Procedia Structural Integrity 5 (2017) 468–475 Behzad V. Farahani et al./ Structural Integrity Procedia 00 (2017) 000 – 000

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1.1 NNRPIM meshless approach In meshless methods, in general, the nodal interdependency is imposed with the “influence - domain” geometric construction, which is obtained after the nodal discretization (Belinha 2014). The NNRPIM is the next step in the radial point interpolation (RPI) methods. In the NNRPIM formulation, the classic ‘influence - domain’ concept, which allows to impose the nodal connectivity, is replaced by the ‘influence - cell’ aspect . The ‘influence - cells’ are obtained using Voronoï diagrams (Voronoï 1908b) and the Delaunay tessellation (Delaunay 1934). In NNRPIM, the problem domain is discretized with a nodal distribution. Then, the Voronoï diagram of the nodal distribution is constructed. The Voronoï diagram is also a discrete structure built by numerous Voronoï cells. Each one of the Voronoï cells is unequivocally associated with only one node discretizing the problem domain. The neighbouring cells of each interest point, permit to define the ‘influence - cell’ of that interest point (Belinha 2014). Besides, Voronoï cells can be used to build the background integration mesh, which consequently will be completely node-dependent. Thus, the NNRPIM can be considered a truly meshless method. The NNRPIM formulation uses the RPI technique to construct the shape functions, used as test function in the Galerkin weak form. Nevertheless, there are some relevant differences between the NNRPIM formulation and the classic RPIM formulation, which are sufficient to modify the method performance. For instance, the NNRPIM uses a polynomial constant basis p ( ) and a MQ-RBF with the following shape parameters: = 1 −4 and = 0.9999 (Belinha 2014). However, NNRPIM has been extended to many computational mechanics fields, such as the static analysis of isotropic and orthotropic plates (Dinis et al. 2008). It was also tested for more demanding applications, such as the material nonlinearity (Dinis et al. 2009b) and the large deformation analysis (Dinis et al. 2009a). In this work, a 2D plane stress deformation theory is assumed. The standard FEM and RPIM formulations for 2D plane stress are extensively described (Belinha 2014) leading to a linear system of equations presented as = . Being the initial stiffness matrix, f the force vector and u the displacement field. Using Hooke’s law, it is possible to obtain a relation between the strain field and stress field, = ⟺ { } = (1 + ) (1 − ) [ 1 0 1 0 0 0 1 − 2 ] { } (1) 1.2 Elasto-plastic Formulation To capture the non-linear behaviour of an elasto-plastic material, it is necessary to define the mathematical law for the plastic component of the deformation. Consequently, three aspects should be taken into account: a yield criterion, indicating the stress level in terms of the stress tensor and permitting to analyse the beginning of the plastic regime; a flow rule, defining the relationship between stress and deformation after plastification; and a hardening law, describing if, and how, the yield criterion depends on the plastic deformation; (Owen & Hinton 1980). The yield criterion allows to define the plastic regime initiation. A yield criterion can be generally formulated as ( , ) = ( ) − ( ) = 0 where is the stress tensor and presents the hardening parameter. The yield function is a scalar function ( ) and ( ) is the yield stress, defining the elastic limit of the material. If the stress state at a certain point leads to ( ) < ( ) , it means that this point shows an elastic behaviour, governed by linear equations of the elasticity theory (Timoshenko 1934), otherwise, it means that the point is in the plastic region ( ) = ( ) , under a loading or unloading condition, in which it depends on the flow vector direction. Following the isotropic von Mises plasticity criterion (also known as the 2 plasticity), the plastic yielding takes place if the second invariant of the deviatoric stress reaches the critical value (Mises 1913) . The yield criterion used in this work is von – Mises criterion (Hill 1998) for an isotropic material, which is known as the Hill yield criterion. The Von Mises criterion states that yield occurs when the principal stresses satisfy the following relation; ( ) = ̅ = [ 2 + 2 + 2 + + + + 3 2 + 3 2 + 3 2 ] 0.5 (2)