13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- The discretization is made by using dual Voronoi {} and Delaunay tessellations {}, and their characteristic functions { } and {} are used as basis functions. Note that, the displacement is discretized by Voronoi tessellations, while strain and stress are discretized by Delaunay tessellations: ( ) ( ), , , , . u x,t u t x x t t x . (2) The following discretized wave equation is derived from L and the discretized functions: ' ' ' 0 M t u t u K , (3) where M is the mass of the th Voronoi tessellation, the superscript ' stands for the 'th adjacent Voronoi tessellation. Note that K ' is the stiffness matrix of PDS-FEM. This K ' coincides with an element stiffness matrix of FEM with linear tetrahedron elements [8]. Also, it should be noted that Eqn. 3 automatically leads to a lumped mass matrix. No approximation is needed to derive the lumped mass matrix, unlike ordinary FEM. This is the advantage of PDS-FEM, since, as shown in Eqn. 3, displacement is discretized as a set of rigid body displacement, or a continuum is regarded as an assembly of rigid body particles. From this discretized wave equation, a discretized Hamiltonian of the following form is defined: ' 1 1 2 2 H q K q p p M , (4) where L p u and q u are the momentum and displacement of the th Voronoi tessellation. We take advantage of the bilateral symplectic algorithm [12] as a robust algorithm of the time integration of Eqn. 4. The main advantage of this algorithm is that in order to achieve the accuracy of the order of tN with t and N being time increment and an integer, it needs 2N times iteration for the interval of 2t. Until now the highest order derived for this algorithm are four. In this paper, the fourth order is used. 3. Modeling of weakly heterogeneity for cracking For brittle materials, it is usually observed that a crack propagates in an unpredictable manner, when subjected to dynamic loading. For instance, kinking and brunching are induced during the process of crack growth, or shattering due to multiple cracking is observed at high loading rate. The key task of this paper is a numerical experiment that uses a set of weakly heterogeneous bodies. In the numerical model, several parameters, such as failure criteria, material properties, flaws’ positions, can hardly be obtained accurately. However, it is not necessary at all to make all these parameters to be the stochastic variables, we can assign only a few to be stochastic variables, and others can be assigned as constants according to experience for simplicity. In this way, all the variability of unknown parameters can be represented by the designed stochastic variables. Generally speaking, two methods can be used to model heterogeneity, adding perturbation to material properties of either deformation or fracture. PDS-FEM takes a simple treatment of weak heterogeneity, as: material properties are uniform except for a parameter for fracture, and cracking are allowed only on some of predetermined weak plane segments.
RkJQdWJsaXNoZXIy MjM0NDE=