13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- direction. The radius of the circular notch is . The initial crack field is set to one in the entire domain in order to model the undamaged initial state. The material is assumed to be isotropic with Lamé constants , which is equivalent to a Poisson ratio of . The circular areas of diameter around the load application points (gray) are modeled as stiff by increasing the stiffness and the cracking resistance by a factor of 100. The length parameter is , the artificial residual stiffness in broken areas is set to , and the kinetic coefficient is set to , where the time scale refers to the load factor of the displacement load . The size of the time steps is controlled by an adaptive time stepping procedure. More details on the finite element implementation of the phase field fracture model can be found in [8]. The right plot of Fig. 4 shows the position of the crack tip on the -axis with respect to the displacement load (black solid line). For the evaluation the node with the largest -coordinate where s equals zero is defined as the crack tip position. Before the onset of fracture, the crack tip position is replaced by the position of the notch ground located at . After the critical load level is reached, an initial crack of finite length starting at the notch ground forms brutally along the -axis at a quasi constant load level. The formation of the initial crack is followed by a phase of stable crack extension along the -axis, where the crack grows progressively with the loading. A second phase of brutal crack extension is observed when total rupture occurs after the crack tip passes . The dotted lines indicate the beginning and ending of the phase of stable crack growth. Remarkably, the crack nucleation at the non-singular stress concentration at the notch as well as the transition from brutal to stable crack extension and vice versa are mastered without any technical difficulties. a) b) c) d) Figure 5. Stress in -direction (red) and crack field (blue) along the -axis at different load levels
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