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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1. Introduction Considerable worldwide efforts are being currently driven to the development and application of advanced high strength steels (AHSS) for automotive applications. Important members of the AHSS family includes dual-phase (DP) steels [(Jiang et al. 1995; Sarwar & Priestner 1996; Saeidi et al. 2015)] Tensile measurements often involve an extensometer or electromechanical strain device. However, Digital Image Correlation (DIC) has become one of the most valuable tools for plastic strain measurement of metal alloys since it readily provides plastic strains beyond diffuse necking which cannot be obtained with the use of extensometers. Recent application of DIC to advanced high strength steels such as press hardened boron steel and dual-phase steels may be found in [(Kang et al. 2007; Savic & Hector 2007; Tarigopula et al. 2008; Mohr & Oswald 2008; Ghadbeigi et al. 2010)]. In the recent years, the significant development of advanced discretization meshless techniques permit them to be considered as a valid alternative to the finite element method (FEM) (Nguyen et al. 2008) (Belytschko et al. 1996). The complete freedom and flexibility in the domain discretization of meshless methods is very attractive. Within meshless methods, the solid domain can be discretized with an arbitrarily set of nodes rather with an element mesh (Belinha 2014). In meshless methods the nodal connectivity is imposed with the influence-domain concept, in opposition to the fixed size element used in the FEM. Furthermore, in contrast with the FEM, to enforce the nodal connectivity the influence-domains may and must overlap each other. As the FEM, RPIM meshless method is a discrete computational approach (Ferreira 2009). The major dissimilarity can be found in the discretization procedure. The FE problem domain is discretised in elements and nodes while the meshless methods discretize the problem domain only with points. Regarding the nodal connectivity, in FEM such connectivity is enforced in the pre-processing phase by the establishment of a predefined finite element mesh. Consequently, each element possesses the intrinsic and sufficient data to enforce the nodal connectivity. Since in meshless methods, there is no predefined “connectivity mesh”, it is required to build one (Farahani et al. 2016). Thus, in meshless methods, in general, the n odal interdependency is imposed with the “influence - domain” geometric construction, which is obtained after the nodal discretization (Belinha 2014). It is an overlap of influence domains that permits to establish the nodal connectivity. To obtain the influence-domain of an interest point, a radial search (centred in the interest point) is performed in order to acquire a certain number of nodes inside a fixed area (2D problems) or a fixed volume (3D problems). Since this approach is very easy to understand (and to implement), it has 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). A generalization on elasto-plastic stress FEM analyses for various constitutive relations including strain softening can be found in (Nayak & Zienkiewicz 1972). In the literature, some relevant works could be found concerning the elastoplastic study for different material behaviours using meshless methods formulation see e.g. [(Tavares et al. 2015; Moreira et al. 2017)]. In this work, DIC, FEM and RPIM are used to analyse a two-dimensional problem considering a small strain formulation and assuming isotropic elasto-plastic materials with isotropic hardening. Therefore, in order to extend and validate the RPIM formulation in the elasto-plastic analysis, the non-linear solution algorithm is employed as the modified Newton-Raphson initial stiffness method and the stress state is returned to the yield surface using a backward-Euler scheme (Crisfield 1991). 1.1 RPIM meshless and solid mechanics basis I n meshless methods, in general, the nodal interdependency is imposed with the “influence - domain” geometric construction, which is obtained after the nodal discretization (Belinha 2014). It is an overlap of influence-domains that permits to establish the nodal connectivity. To obtain the influence-domain of an interest point, a radial search (centred in the interest point) is performed in order to acquire a certain number of nodes inside a fixed area (2D problems) or a fixed volume (3D problems). Since this approach is very easy to understand (and to implement), it has