13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- scale , which primarily controls the width of the transition zone of the order parameter between broken and undamaged material. From this point of view, the length scale is merely an auxiliary numerical quantity, which recovers the sharp interface limit for . However, some recent publications [13–16] find that, in a one dimensional setting, the length scale is also crucial for the stability of crack-free, spatially homogeneous phase field solutions. At the critical load level, the crack-free solution becomes unstable and the phase field order parameter localizes until finally a crack forms. Particularly interesting is the fact, that this stability point is also related to the maximal stress response of the crack-free phase field solution. This observation allows for the definition of a fracture strength in the phase field model, which – at first sight – does not feature a material parameter, that is directly connected to the strength of the material, but is able to reproduce crack nucleation. In this work, the crack nucleation behavior of the phase field fracture model introduced in [8] is investigated. In a first step, only the one dimensional case is considered. A stability analysis of the spatially homogeneous, crack-free solution is outlined, which yields the definition of an effective fracture strength in the one dimensional phase field model. In a second step, the problem of crack nucleation is considered in the two dimensional setting. However, a rigorous analytical stability analysis is generally not possible in the case of arbitrary inhomogeneous stress states. Therefore, based on the findings from the one dimensional case, strength estimates are derived for the two dimensional setting. These estimates are then compared to the computed stress states at crack nucleation in a finite element simulation of the phase field model. 2. Phase field formulation In phase field fracture models, cracks are approximated by the zero set of the phase field order parameter . This order parameter is a continuous scalar field quantity which resembles a damage variable and is often referred to as crack field in this context. It interpolates smoothly between cracks, where the order parameter takes the value zero, and undamaged material, where the value of the order parameter is one. By means of a degradation function, the crack field is coupled to the elastic stiffness tensor of the material in order to model the change in stiffness between broken and undamaged material. The core of the phase field model considered in this work is the energy density functional , (1) which was introduced by Bourdin [7] as a regularized approximation of the energy density of a linear elastic fractured body. The first part, which is a function of the linearized strain tensor and the crack field , is the elastic stored energy. The small positive parameter in the degradation function is a residual stiffness, which is introduced in order to avoid numerical difficulties, where equals zero. The second part depends only on the crack field and its gradient . An integration of the second bracketed term over the entire domain yields an approximation of the surface measure of the crack set, when the regularization length is sufficiently small. Multiplied with the cracking resistance it approximates the Griffith type surface energy of the crack set. By virtue of thermodynamic reasoning, the definition of the energy density functional yields the material law (2) for the stress tensor which, together with the equilibrium condition (3)
RkJQdWJsaXNoZXIy MjM0NDE=