13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- PDS-FEM uses a boundary facet as a set of pre-determined weak plane segments for possible crack extensions. The location of the weak plane is pre-determined by mesh configuration. In order to model the weak heterogeneity, we add disturbance to the location of mesh mother points, from which, a lot of samples with slightly different fracture property can be generated. However, there are two difficulties in this method: generating unbiased distribution may result in ill-shaped Delaunay elements, i.e., some elements with large aspect ratio; a simple solution of forcing the aspect ratio of the Delaunay elements in a certain range leads to biased distribution of some clusters of Voronoi mother points. The unbiased distribution of the Voronoi mother points and the aspect ratio control of the Delaunay elements are in a trade-off relation. As a compromise, we start from one distribution of the Voronoi mother points with majority of the Delaunay elements being well shaped, and modify this distribution randomly to generate other distributions without changing the geometry of the target model. The mesh size is regarded as a parameter which represents the degree of material heterogeneity; the size becomes smaller as the distribution of material parameters is closer to being uniform. For designing the stochastic model of a real experiment, we need to identify the degree of material heterogeneity or the mesh size. However, with limited observation equipment, the degree of heterogeneity of real samples cannot be measured easily. Therefore, the authors try to start from a standard mesh configuration, which takes a balance between the accuracy and computation overload. If the crack path solutions of the numerical experiments show that the variability is not large enough to include the experimental results, then we have to find ways to increase the degree of heterogeneity in the stochastic model: (1), apply larger mesh size, while ensuring the required accuracy; (2), add additional perturbation to material properties of deformation. Thus, in the numerical experiment presented in section 4, we start from a standard mesh configuration as this: finer meshing is used near the crack tip, to allow a wider choice of crack extension, while meshing becomes coarser farther from the crack tip to save computation overload. 4. Monte-Carlo simulation of crack propagation This section carries out a numerical experiment of executing Monte-Carlo simulation of heterogeneous samples, in order to reproduce the real experiments’ phenomena. 4.1. Problem setting We study a thin plate of 5×24.5×140 mm, which includes two anti-symmetric parallel notches of height 0.6 mm; see Fig. 2. It is assumed that the material is linearly elastic; see Table 1. For dynamic analysis of brittle material we need to consider the time effect, since dynamic fracture is a time-dependent phenomenon [13-14] which depends on the stress pulse duration, also from experiment study, it is observed that, the material strength is higher than static strength for high loading rate [15-16]. Concerning about this, a time dependent material strength failure criterion called Tuler Butcher criterion [17] is adopted in this paper: 1 0 0 f f dt K , (5) for ≧≧0, where and are a threshold stress and the maximum stress, f is fracture duration and Kf is the stress impulse for failure. This criterion means that a crack grows if accumulated stress in fracture duration reaches a critical value. It is assumed that and Kf = 10-8, is equal to the static tensile strength, and f is assigned to be the time step used in time integration; these parameters should be calibrated according to experimental data, which are not
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