13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- , (6) where the contribution of the remaining stiffness has been neglected for the sake of algebraic simplicity. While the homogeneous crack field (Eq. 5) decreases monotonically with the increasing strain , the stress response (Eq. 6) attains a maximal value , (7) at a displacement load of . (8) Interestingly, the value of the homogeneous crack field at the maximal stress load is 0.75, independent of all of the phase field parameters, see also [10]. This is illustrated in the plots of Fig. 1, which show the evolution of the stress response and the crack field with respect to the displacement loading for different values of . sxX Figure 1. Stress response (black) and crack field (blue) under increasing displacement load for (left), (center) and (right) 3.1.2. Stability analysis In order to analyze the stability of the homogeneous solution (Eq. 5), a family of symmetric test functions with and is introduced. Symmetry is postulated for the sake of simplicity, and the restriction ensures differentiability at . Boundary conditions , kinematic relations, the material law and the fact that the stress is constant yield the corresponding strain field . It can be shown that the first variation of the potential (9) vanishes for the homogeneous static solution, i.e. . (10) Thus, the homogeneous solution is always a local extremum or a saddle point of the energy functional (Eq. 9). The second variation of the potential with respect to is given by the expression
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