As the ratio, w/a, was varied from 8, to 4, and 2.67, the crack path became curved and deviated towards one side. The curvatures of the three crack trajectories were almost the same. For specimens with a=25.4mm and 38.1mm, (w/a 4), the crack curved to one side quickly as soon as propagation started. However, when the initial crack length, a, was 12.7mm (w/a=8), the crack propagated along a straight line for almost 20mm before the deviation started. This suggests that, for small crack lengths, the T-stress is negative and the crack is stable initially. When the crack length increased to some critical value, the T-stress turned positive, destabilizing the crack, and the crack deviated to one side. Additional details will be shown and discussed when test results are compared with those from numerical simulations. Numerical Studies Both the finite element (FE) and meshless methods were employed in the numerical studies; the former used the commercial software, ABAQUS™, and the latter our in-house developed computer code. Finite Element Analysis The X-FEM (extended finite element method) implemented in ABAQUS™ v.6.9 using 4-node plane strain element, CPE4R, was employed to analyze deformations of the asymmetric CT specimen used in the experimental studies [8, 9]. Assigned boundary conditions simulated as closely as possible those likely to occur in the test configurations. With the loading points shown in Fig. 7a, the following boundary conditions were used in the numerical work. Load point 1: fixed in x and y directions: ux= 0, uy= 0; Load point 2: fixed in the x-direction and y-displacement prescribed: ux= 0, uy= -1 mm. From tensile tests on PMMA at a strain rate of 0.00014/s and room temperature, Elices and Guinea [10] obtained the following average values: Young’s modulus ܧ =3000±30 MPa, yield limit stress ߪ 0.2 =43.9±0.7MPa, rupture stress ܴߪ =74.9±0.2MPa, and Poisson’s ratio ߭ =0.4; these values were used in our simulations. The maximum principal stress of the damage initiation was set equal to ߪ 0.2 =44 MPa. Damage evolution was based on fracture energy ( ܩܿܫ = ܩܿܫܫ = ܭܿܫ 2 (ܧ /(1െ߭ 2)) =312.2 ܬ/݉ 2), linear softening, and mixed mode behavior of power law Į [10]. X-FEM results are show in Figure 7b and will be discussed after meshless methods are introduced. Analysis of the Problem by the SSPH Method Meshless methods were introduced in 1970’s, and include the Smooth Particle Hydrodynamics (SPH) method [11], Element-Free Galerkin method (EFGM) [12-14], Reproducing Kernel Particle Method (RKPM) [15], Meshless Local Petrov-Galerkin (MLPG) [16], Modified Smoothed Particle Hydrodynamics (MSPH) [17, 18] and Symmetric Smoothed Particle Hydrodynamics (SSPH) [19, 20]. Like the FE and the boundary element methods, meshless methods are used to find an approximate solution of an initial-boundary-value problem with the difference that no element connectivity is needed in a meshless method. Various meshless methods differ in the construction of basis functions for the trial solution. The EFGM [12-14] has been used to study linear elastic fracture mechanics (LEFM) problems and uses basis functions found by the moving least squares (MLS) method. However, it employs a background mesh to numerically evaluate various integrals appearing in the weak formulation of the problem and thus is not truly meshless. The enriched basis functions [21] are used to capture the stress singularity near a crack-tip without having a very fine distribution of particles (nodes) there. The computational efficiency can be improved by using an appropriate weight or test function [22] and basis functions that better capture singularity of fields near the crack tip [23]. Advantages of the SSPH method are that spatial derivatives of the trial solution are computed without differentiating the basis functions, and the stiffness matrix is symmetric. Our implementation of the SSPH method does not have the enriching terms to capture the stress singularity. We thus use a fine particle distribution around the crack tip. Additional details of the numerical scheme are provided in [24] where results for the CT specimen are also included. 18
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