13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- , (11) where denotes the partial derivative with respect to . Clearly, the second variation can only become negative if the factor in front of the last integral in Eq. (11) becomes negative. If the displacement load is smaller than the load (Eq. 8) with the maximal stress response, the second variation (Eq. 11) is positive. Thus, the homogeneous solution is a local minimizer of the total energy and therefore is considered as stable. If the displacement load becomes larger than the load , the factor in front of the last integral becomes negative and thus, the second variation can become negative, too. The homogeneous solution is then no longer a local minimizer and becomes unstable. Consequently, the load with the maximal stress response represents a lower bound for the stability of the homogeneous solution. A further analysis of the stability of homogeneous solutions of different gradient damage formulations, is carried out in [16] and, for a broader class of gradient damage models, in [13–15]. Concerning the specific phase field fracture model under consideration, the main conclusion from these publications is, that for small values of , the actual stability load lies slightly above the lower bound . Only for rather large , the actual stability load is significantly larger than . However, regarding the regularizing character of , this case is of minor interest. 3.1.3 Non-homogeneous solution Figure 2. Function and roots (circles) and (triangles) for different values of (left) and the respective inhomogeneous crack fields (right) In this section, a semi-analytical approach to the computation of the non-homogeneous crack field at supercritical loading is outlined. More details are reported in [10, 11] for similar phase field models. A finite element study of the non-homogeneous solution stages can be found in [12]. The static one dimensional evolution equation (Eq. 4) and the material law (Eq. 2) can be recast in the format . (12) Assuming a differentiable, symmetric solution with a minimum value at , an integration of Eq. (12) with respect to over the interval yields , (13) which can be interpreted as a balance law for the phase field variable with the potential and
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