13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- and the respective boundary conditions, forms the mechanical part of the problem. The evolution of the crack field is assumed to follow a Ginzburg-Landau type evolution equation, where the rate is proportional to the negative variational derivative of the energy density functional with respect to , i.e. . (4) The symbol denotes the Laplace operator. The positive scalar kinetic coefficient describes the mobility of the process. In the format of Eq. (4), the evolution equation may be regarded as a viscous approximation of the quasi-static case, which is recovered for . In order to model the irreversibility of the fracture process, Dirichlet boundary conditions are applied to the crack field in the subsequent load steps, wherever a crack forms, i.e. the crack field becomes zero. At crack free boundaries with outer normal vector , homogeneous Neumann boundary conditions apply. The coupling of the field equations (Eqs. 2-4) implicitly models the mutual interaction between the elastic stress and strain fields and the crack field . Given a prescribed loading history, the successive solution of the coupled system of equations formed by Eqs. (2-4) yields the evolution of the mechanical stress and strain fields as well as the evolution of the crack field. Note, that no further criteria are required in order to capture even complex crack evolutions, such as the coalescence of different cracks, crack branching or the nucleation of new cracks. 3. Crack nucleation The nucleation of new cracks in an originally undamaged structure is a somewhat delicate topic in the context of a phase field formulation. On the one hand, crack nucleation can be observed in numerical simulations of the phase field model. On the other hand, none of the parameters of the phase field fracture model is directly connected to the fracture strength of the material. In the following, the influences of the different phase field parameters on the effective fracture strength of the phase field model are investigated by means of an analytical stability analysis and numerical simulations. 3.1. The one dimensional case 3.1.1. Homogeneous solution In a first step, a one dimensional problem is considered. A homogeneous bar of length and Young's modulus is strained by an increasing displacement load of at both ends. Before any load is applied, the bar is modeled as undamaged, i.e. . In the one dimensional setting, the equilibrium condition (Eq. 3) immediately implies that the stress must be constant along the entire bar. Under the assumption that the crack field remains spatially constant upon loading, it follows from the material law (Eq. 2, where Young's modulus replaces the stiffness tensor ) that the strain must be constant, too. The kinematic relation yields the strain value prescribed by the boundary displacement. The corresponding static solution for the crack field (5) is obtained from the one dimensional evolution equation. Accordingly, the constant stress in the bar is
RkJQdWJsaXNoZXIy MjM0NDE=