13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- Due to the different boundary conditions, the elastic contribution differs slightly, depending on the ratio of the Lamé constants. (17) The indices in Eq. (17) refer to the left and right setting depicted in Fig. 3. Neglecting for algebraic simplicity, the corresponding stress in -direction is (18) with . The maximal value (19) is attained at the displacement load . (20) The maximal stress response (Eq. 19) and the corresponding displacement load (Eq. 20) exhibit the same asymptotic behavior with respect to as in the one dimensional setting. For both quantities become infinitely large, while for their value approaches zero. However, in the two dimensional setting, both quantities additionally depend on the factor defined in Eq. (17). As in the one dimensional case, the value of the homogeneous crack field at the maximal stress response is 0.75, independent of all of the phase field model's parameters. Figure 4. Simulation setup (left) and crack tip position with respect to the loading (right) 3.2.2 Numerical evaluation In the following, the maximal stress values from Eq. (19) are compared to the actually computed stress states at crack nucleation in the phase field model, in order to evaluate the effective fracture strength of the phase field model. In the finite element simulation, the notched structure, depicted in the left plot of Fig. 4, is loaded by a linearly increasing displacement load acting in -
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