PSI - Issue 5

924 Behzad V. Farahani et al. / Procedia Structural Integrity 5 (2017) 920–927 Behzad V. Farahani et al./ Structural Integrity Procedia 00 (2017) 000 – 000 5 first reference point located on the upper hole, 1 , was pinned to satisfy the experimental conditions. Concerning the natural boundary condition, as reported on Table 1, a concentrated force possessing a magnitude of ∆ = −603 [ ] was vertically applied on the second reference point coupled with the bottom hole, 2 , see Fig. 2-a. Indeed, the crack was defined in accordance with the crack assignment on the contour integral in ABAQUS.

(a)

(b)

Fig. 2. Geometry and boundary condition for CT specimen; (a) FE model and the interest region near the crack tip and (b) Meshless RPIM model.

3.2. RPIM Meshless method analysis In this study, a 2D plane stress deformation theory is assumed. The standard RPIM formulations for 2D plane stress are extensively described (J. Belinha 2014) leading to a linear system of equations presented as = . Being K the 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 ] { } (6) 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 [(Wang & Liu 2002), (J. Belinha 2014)] reported that this integration scheme maximizes its efficiency if = 3 . So, the RPIM formulation was exerted to the linear elastic fracture mechanics. The 2D problem was modelled within a numerical plane stress routine developed in MATLB by the authors. The cracked specimen was then elasto statically simulated for each crack length, where the boundary conditions and applied force were fully addressed. The present analysis was carried out using a regular nodal discretization consisting of 4861 and 28155 nodes and integration points, respectively. Taking advantage of the symmetry of the CT specimen, only half of the problem was analysed. The nodal distribution on half of the problem domain is shown in Fig. 2-b. The specimen thickness (2 mm ) was assumed in the integrated formula. The material properties and loading conditions were used the same as the experimental test. Other parameters and coefficients governing the RPIM analysis were used based on the previous works conducted by the authors [(J. Belinha 2014),(B. V Farahani et al. 2016),(Vasheghani Farahani et al. 2015)]. 4. Results and Comparison Using the obtained data from DIC and RPIM meshless approaches with respect to a total number of seven terms ( = 7) in Williams series, Equation (3), the SIF range results for seven crack lengths were acquired using the overdeterministic algorithm. Concentring FE analysis, SIF was directly computed through maximum strain energy