13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- the simulation results in Figure 2a, because energy is dissipated during the tensile reloading (path B-C-D). Similar to Lee and Fenves [21], Figure 2 shows that the model can simulate quasi-brittle material behaviour subjected to cyclic loading for which the hysteresis curve in the tensile region is negligible compared with the compressive counterpart. 4. Numerical Implementation The non-local DP model was implemented using two different Gislocarbon test geometry examples, namely a three point bend (3PB) gemetry and two compact tension (CT) geometries. Plane stress conditions were assumed to allow for short computational times and numerical convergence. The computations were implemented in ABAQUS/Standard (Version 6.9). The material properties used for each model are presented in Table 1. The first example is the two CT specimens tested by Fazluddin [8] and Hodgkins [7]. Note, for Hodgkins’ specimen configuration the width of part of the specimen was reduced in order to allow the progression of the crack to be monitored in three dimensions using X- Ray tomography [7]. The numerical models consisted of 1480 and 1320 four noded plane strain elements for Fazluddin’s and Hodgkins’ geometries respectively (which only represent the symmetrical half). The second example, 3PB specimen, consists of 896 four noded linear plane strain elements and similarly only represents the symmetrical half of the specimen. To avoid convergence issues, only DP elements are utilised in the region in which fracture is expected to occur (elsewhere linear elastic conditions are assumed). Some mesh refinement was performed in the vicinity of the notch-tip to achieve mesh-insensitive results. Boundary conditions are applied with an analytically rigid pin with defined hard frictionless contact in displacement control. The failure envelope is well captured in both the CT specimens and the 3PB specimen. The tensile damage localisation at peak load, which manifests itself by a reduction of the load at fracture, shows similarities to the experimental observations reported by Becker et al. [15]. The experimental data from Hodgkins provided a load-unloading response (Figure 3b) and the non-local DP model is capable, to a certain degree, of simulating this behaviour. Discrepencies exist as some load carrying capability remains in fully degraded elements since the stress-displacement behaviour has been defined with a failure stress of 1 MPa for numerical stability [22]. The 3PB geometry exhibits a more brittle response compared to the CT geometry, which is noticeable from the more severe post peak load drop when comparing Figure 3 and Figure 4. It is worth reiterrating that experimentally obtained R-curve data has shown a geometry dependent fracture behavior of Gilsocarbon, where the non-linearities due to the FPZ and the wake effects result in variations of the apparent fracture toughness [15]. Since the DP model is a non-local approach it is size and geometry independent. The degradation and hence softening of the material is confined to the defined yield criteria and flow rule. As damage develops ahead of the crack tip, the yield surface shrinks in the stress space leading to a softening interfacial constitutive law. Fracture has occurred when the work done by the tractions meets the fracture mechanics based criterion. The DP failure model can be viewed as an extension or generalisation of the cohesive zone model proposed by Zou et al. [34], in that it couples together the effects of stress/displacement curves to derive a combined non-local stress and fracture mechanics based failure criterion. Because the DP model is non-local it is not confined to stress singularities and is thus believed to be a more accurate representation of graphite fracture.
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