13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 3. Numerical Example The developed methodology is applied to the simulation of fatigue response of a single notched rectangular steel plate subjected to cyclic load. The geometry and boundary condition are shown in Figure 4. A fully reversed fatigue load at 20Hz ω= is applied to the top face of the plate in a form of uniformly distributed pressure of 70 MPa while the bottom surface is fixed, which replicates the loading conditions during the actual fatigue test where the bottom surface is clamped in grips and top surface is subjected to push-pull loading. The plate thickness is taken as 2mm. The problem is modeled as plane stress problem. Motivation behind this problem is to simulate the complete fatigue loading history using the presented formulation, calculate damage at gauss points during the simulation, and predict the fatigue crack initiation and propagation. Mode I crack growth is expected in this loading situation. 4.6 4.7 4.8 4.9 5 5.1 x 105 0 0.5 1 1.5 Number of Cycles Crack Length (mm) 70MPa Figure 4: (a) Configuration of the fatigue simulation of a single notched plate. (b) Crack growth vs. number of cycles for the case of completely reversed load of 70 MPa The space-time discretization employs a multiplicative form of the shape function so that the spatial discretization is independent of the temporal discretization. Advantage of this treatment is that the existing meshing scheme widely used in the case conventional FEM can be directly adopted. In the present case, the plate is discretized by quadrilateral mesh with non-uniform mesh density. Fine mesh is prescribed close to the notch in order to capture crack initiation and growth. In the temporal dimension, quadratic shape function is employed for each space-time slab. The regular space-time shape functions are enriched with a harmonic function with frequency of 20Hz . With the two-scale damage model introduced here, crack initiation and propagation are governed by the evolution of the damage. Numerically, once the damage-based the crack criteria is met, the corresponding space element is eliminated. The material parameters chosen correspond to 304L steel and are based on the earlier work by Lemaitre and Desmorat [9-12]. Detailed implementations of the model is documented in an upcoming paper [13] and will not be presented here due to page limitation. Crack growth as a function of the load cycles predicted by the simulation is shown in Figure 4. It is worth noting that up to half-million load cycles are successfully simulated with the proposed numerical approach. In addition, the nature of the crack growth data is similar to that obtained from the experiments. This suggests that, with the calibrated material model accurate fatigue simulation for the HCF loading can be performed using the methodology outlined in this paper.
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