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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Table 1. Material properties of DP600. Young’s modulus Poisson ratio

Density

Yield stress

Ultimate tensile stress

Total elongation

= 210 3 ( ) = 0.3 = 7.8 −9 ( 3 ⁄ ) 0.2 = 416.1 ( ) = 630.9 ( ) = 27.2 % 3. Numerical analyses In this section, the bi-failure model is numerically analysed. The computational results obtained with the RPIM and FEM solution are compared with the DIC solution. 3.1. FE analysis The problem was modelled based on the elasto-plastic FEM formulation in ABAQUS©. An explicit model was considered where a 3D deformable shell model was used. The geometric characteristics was considered as presented in Fig. 1-a. Regarding the mesh, 3-node linear triangular elements (S3) were selected to build a FE mesh where the element size varies between 0.5 and 1.0 millimeter with a total number of 7046 and 3739 elements and nodes, respectively, see Fig. 1-b. To guarantee the results validity, more refined elements were assigned to the central section of the specimen covering all holes. Concerning the essential boundary condition, the points located on the upper specimen edge was clamped to follow experimental circumstances. As a natural boundary condition, a uniform displacement enforcement with a magnitude of ̅ = −0.15 [ ] was vertically imposed on the points coupled with the bottom edge, as Fig. 1-b demonstrates. The material properties reported on Table 1 were assumed. Considering intended points 1 and 2 (with the same properties as DIC analysis), the force response was correlated with the displacement variation 21 , see Fig. 2-b. 3.2. RPIM study In this study, the RPIM formulation was extended to the elastoplastic theory in order to solve a bi-failure model. The model geometry was used as presented in Fig. 1-c. The material properties were used in accordance with Table 1. In the non-linear solution algorithm the tangent modulus has been defined as 0 = 210 ( ) . A regular nodal discretization consisting of 3739 and 21138 nodes and integration points, respectively, was constructed as Fig. 1-c shows. The essential and natural boundary conditions followed the FE analysis, as represented in Fig. 1-c. The elastoplastic analysis was performed based on the formulation presented in Subsection 1.2. Due to the non-linearity, a non-linear Newton-Raphson algorithm was used to obtain the non-linear converged solution, as seen in Box 1. A tolerance of 1e-2 was considered while a total number of 30 increments was involved in the resolution process. The RPIM analysis was performed and the load response in terms of the corresponding displacement field variation 21 , on 1 and 2 (with the same properties as DIC analysis) was evaluated as shown in Fig. 2-b. 4. Conclusions In this work, a bi-failure specimen made of a dual phase steel DP600 was elasto-plastically analysed. The experimental test was performed by a uniaxial tensile test and Digital Image Correlation was therefore used to acquire the experimental solution. As a result, the reaction force response correlated with the vertical displacement on a pair of points located in the central section of the specimen was obtained. Numerically, the problem was solved using FEM formulation in ABAQUS©. Thus, assuming isotropic hardening, an isotropic elastoplastic behaviour was considered in ABAQUS and the intended force-displacement response was evaluated. To assess the performance of meshless methods formulation on the elastoplastic model, the RPIM formalism was exerted to the non-linear elastoplastic theory. The yield function relied on von-Mises yield criterion with an isotropic hardening and flow rule. Due to the non-linear material behaviour, a return mapping algorithm together with a Newton-Raphson algorithm was used to obtain the non-linear solution. The force response in terms of displacement variation was thereby acquired and