13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- The plots of Fig. 5 show the stress component (red) and the crack field (blue) along the -axis at the four load stages marked by the red circles in the right plot of Fig. 4. The vertical black dotted line marks the position of the crack tip after the formation of the initial crack. The horizontal black solid and dash-dotted lines indicate the strength estimates from Eq. (19) for the cases and , respectively. At the load level of (Fig. 5a), no crack nucleation is observed yet, and the numerical solution is stationary and stable. However, very close to the notch ground located at , the stress component already exceeds the strength estimates. The value of the crack field at the notch ground is approximately , which is higher than the assumed critical value . Figure 5b) shows the -stress and the crack field at the beginning of the phase of brutal crack nucleation at the load level . At this stage, the numerical solution becomes unstable and can no longer be considered as stationary. The crack field immediately at the notch ground is decreased to . Due to the loss of stiffness caused by the decreasing crack field, the stress at the notch ground decreases, too, and a stress peak develops in front of the notch. Figure 5c) shows the -stress and crack field at , during the phase of brutal crack extension. The crack field has developed its characteristic exponential shape (cf. Eq. 15), where the material is not yet broken and is constantly zero in the fractured area. During this phase, the peak stress in -direction significantly exceeds the maximal stress response (Eq. 19) of the homogeneous test problems. The last plot, Fig. 5d), shows the stress component and the crack field at , during the phase of stable crack extension. The peak stress is now in good agreement with the strength estimates. The crack field maintains its exponential shape and is only shifted in -direction. As similar results are obtained in simulations with different values of the length parameter , the simulation results yield the following conclusions. For inhomogeneous stress states with maximum stresses below the derived strength estimates, no crack nucleation is observed in the phase field model. Also the exceedance of the assumed strength in a small area does not immediately lead to crack nucleation. Instead, an initial crack forms, if the stress becomes supercritical in a sufficiently large area and the crack field drops below the critical value of 0.75. Thus, the derived strength estimates permit to judge the criticality of a computed stress state prior to crack nucleation and may therefore be interpreted as the effective fracture strength of the phase field model. 4. Conclusion and outlook Despite the regularizing character of the length scale , the stability analysis of the one dimensional model and the numerical results obtained for the two dimensional case, yield a more mechanically motivated interpretation of the parameter . The stability analysis, as well as the numerical simulations, legitimate to interpret the maximal stress response obtained in homogeneous loading scenarios as the effective fracture strength of the phase field model. As a consequence of this interpretation, the length scale is no longer just an arbitrary regularization length, but can be derived according to Eq. (19) from experimentally measurable data, i.e. from the cracking resistance , the fracture strength , and the Lamé constants and of isotropic materials. Thus, in conjunction with the other material parameters of the phase field model, the parameter may be regarded as a material parameter itself. For the phase field formulation, the definition of the fracture strength, together with the ability to master the transition from a crack free initial state into a cracked configuration, justifies the conclusion, that the model naturally combines a strength criterion for the nucleation of new cracks with an energetically motivated Griffith type evolution law for stable crack growth.
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