13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- cracking releases strain energy and changes stiffness matrix drastically. High robustness is required for the time integration, and we adopt Hamiltonian formulation so that most robust algorithm which is proposed in the field of computational quantum mechanics can be employed for the time integration. In real world, there is no ideal isotropic homogeneous body. Various kinds of disturbances exist everywhere. Due to limitation of the observation technology, the material property and boundary condition can hardly measure accurately. The difference between numerical setting and reality is called disturbance. Since crack is sensitive to local heterogeneity, even with the same setting, the crack paths of experimental samples are still somehow different from each other. In this paper, the authors try to reproduce a fracture experiment of a thin epoxy resin plate. In order to model the heterogeneity, a stochastic model is proposed, which introduces certain perturbation to the homogeneous body. Then, a Monte-Carlo simulation is carried out, from which, the crack patterns observed from corresponding experiments are successfully simulated. The content of the present paper is as follows: section 2 briefly explains the extension of PDS-FEM to dynamic state. We formulate the dynamic extension of PDS-FEM by using discretized Hamiltonian, so that a robust algorithm can be applied to the time integration. Section 3 is devoted to discuss modeling of weak heterogeneity. The modeling is made by using different candidates of possible crack extensions, which is realized by using different meshes. Section 4 contains a Monte-Carlo simulation for a thin epoxy resin plate with a pair of anti-symmetric notches located in the middle. Also, the simulation results are compared with corresponding experiments, which are captured by a high frequency image sensor at the rate of 1 million frames per second. Concluding remarks are pointed out in section 5. 2. Extension of PDS-FEM to dynamic state On the viewpoint of the numerical computation, it is not easy to analyze the crack growth, since cracking not only releases strain energy, but also changes the stiffness matrix. A robust algorithm that can handle such a change is required. The algorithm is also required to guarantee symplecticity, i.e., the total energy and momentum should be conserved during the crack growth. A robust algorithm of time integration has been studied in the field of computational quantum mechanics [11]. To implement such an algorithm, we formulate the dynamic extension of PDS-FEM using Hamiltonian. We start from the following Lagrangean of a linearly and isotropically elastic solid, denoted by B, with elasticity c, density displacement u, stress and strain : 1 1 , ; , : : + : 2 2 B L dv u u ε σ ε c ε u u σ u ε . (1) Voronoi tessellation Delaunay tessellation Figure 1. Two dimension decomposition by using particle discretization scheme The candidate crack path is limited to the boundary of the Voronoi blocks
RkJQdWJsaXNoZXIy MjM0NDE=