13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Crack Nucleation in Phase Field Fracture Models Charlotte Kuhn1,*, Ralf Müller1 1 Institute of Applied Mechanics, TU Kaiserslautern, P.O.B. 3049, 67653 Kaiserslautern, Germany * Corresponding author: chakuhn@rhrk.uni-kl.de Abstract Phase field fracture models are able to reproduce a wide range of phenomena, which are observed in fracture experiments. These phenomena include the nucleation of new cracks in initially undamaged material. However, none of the material parameters of a phase field fracture model is directly connected to the fracture strength of the material. Thus, the critical stress for the nucleation of new cracks is not a priori clear. Crack nucleation in a phase field fracture model is preceded by a localization of the initially homogeneous crack field in an area surrounding the nucleating crack. For homogeneous problems, it can be shown analytically that the onset of the localization is caused by the loss of stability of the crack-free homogeneous solution of the phase field equations at a certain load level. This critical stability load provides a definition of the fracture stress in the phase field model depending on the stiffness of the material, the cracking resistance and the internal length of the phase field model. The analytical findings are illustrated in finite element simulations of the phase field fracture model. Further numerical investigations analyze the crack nucleation behavior of the phase field model in more complicated scenarios, where analytical stability results are not available. Keywords Phase field model, Fracture, Finite element method, Crack nucleation, Stability 1. Introduction Conclusions drawn from numerical simulations often play a crucial role in the design process of structural components. In order to obtain a reliable prediction of the integrity of a structure, a fracture model must be able to reproduce a wide range of phenomena which are observed at fracture events. On the one hand, this requires criteria for the stability of pre-existing cracks as well as criteria for the nucleation of new cracks in originally undamaged material. On the other hand, the fracture model must also predict the geometry of the crack path, including possible kinking of a crack or bifurcation into several crack branches. Unlike many other continuum fracture models, which are equipped with a whole toolbox of different criteria in order to meet these requirements, the phase field approach provides a unified framework for the simulation of the entire fracture process. Different phase field fracture models have been introduced and discussed e.g. in references [1–6]. More recently, phase field fracture models based on Bourdin's regularization of the variational formulation of brittle fracture [7] have been formulated in [8–10]. All these models differ in detail, but in all formulations cracks are represented by a scalar phase field order parameter, which indicates the condition of the material and interpolates smoothly between broken and undamaged material. Cracking is addressed as a phase transition problem, and the crack evolution, obtained implicitly through the solution of the coupled field equations, covers the whole range of phenomena which need to be considered. Concerning a finite element implementation of the fracture model, the phase field approach is advantageous because the diffuse phase field cracks do not lead to discontinuous jumps in the displacement field. Thus, the discretization can be done with standard finite element shape functions, and no remeshing is required in order to simulate crack propagation. The fracture behavior of phase field fracture models is mainly adjusted by two parameters. The cracking resistance is a material parameter, which is a measure for the surface or fracture energy, needed to create new fracture surfaces. By means of configurational forces, it can be illustrated, that the propagation of pre-existing cracks in the phase field model agrees with the energetic considerations of classical Griffith theory, see e.g. [8, 11, 12]. The second parameter is a length
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