PSI - Issue 5

Behzad V. Farahani et al. / Procedia Structural Integrity 5 (2017) 1237–1244 Behzad V. Farahani et al./ Structural Integrity Procedia 00 (2017) 000 – 000

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been used to support the development of several meshless techniques, see e.g. (Belinha 2014)(Wang & Liu 2002a)(Nguyen et al. 2008)(Atluri & Zhu 1998). The literature suggests that each 2D influence-domain should possess approximately = [9,16] nodes, see i.e. (Belinha 2014)(Wang & Liu 2002a)(Nguyen et al. 2008)(Atluri & Zhu 1998). Since the Galerkin weak-formulation is adopted to construct the system of equations, a background integration mesh is obligatory to numerically integrate the integro-differential equations governing the physical phenomenon (Farahani et al. 2016)(Vasheghani Farahani et al. 2015). In this work, a 2D plane stress deformation theory is assumed. The standard FEM and RPIM formulations for 2D plane stress are extensively described in the literature (Belinha 2014), leading to a linear system of equations presented as = . Being the stiffness matrix, f the external 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) In this work, all integration cells are quadrilateral consisting of approximately 9 nodes and × integration points inside, respecting the Gauss-Legendre quadrature scheme. Previous works (Belinha 2014) (Wang & Liu 2002b) reported that this integration scheme maximizes its efficiency if = 3 . 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 (2) 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. In fact, the negative value of the yield function relates to the purely elastic state of the material while the value of zero characterizes the border of the yield surface. 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. This criterion is based on the distortion energy determination in a given material, i.e., of the energy associated with shape changes in that material (as opposed to the energy associated with the changes in volume in the same material). According to this criterion, named after German-American applied mathematician Richard von-Mises (1883-1953), a given structural material is safe as long as the maximum value of the distortion energy per unit volume in that material remains smaller than the distortion energy per unit volume required to cause yield in a tensile test specified of the same material (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 + ( − ) 2 + ( − ) 2 + 6( 2 + 2 + 2 )] 0.5 (3)