13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- matrix-inclusion interfacial fatigue crack, matrix fatigue crack and all possible contact on the crack edge, fatigue crack initiation and propagation is simulated by a new remeshing method. All possible contact on the crack edge when the crack closed under all possible fatigue loads are sought along every crack edge. The fatigue crack initiates when the fatigue damage exceeds certain critical damage value. The fatigue crack propagation is simulated by gradually seeking crack propagating directions and new crack tips, using a remeshing method that a damaged node at the crack tip is replaced by a pair of nodes, a new crack tip node is assigned at the crack propagating directions and a more pair of nodes are needed to be added on the crack edge near the new crack tip in order to better facilitate the free-traction boundary condition. The first example has been given for Particle-reinforced metal-matrix composites with 20 elliptical inclusions to simulate the fatigue crack initiation and propagation under plane stress conditions. It appears that this method is a more efficient way to deal with the interfacial damage of composite materials. The simulation results are compared with those of general fine finite element model and a good agreement is obtained. In the second example, the results show that the mechanical properties of functionally gradient materials are influenced by the particles’ size, topological structure, and interfacial deboning strength. With the interface cracking the overall stiffness of functionally gradient materials is gradually reduced. 2. The Voronoi Cell Finite Element Model 2.1. Hybrid Element Assumptions and Weak Forms In the Voronoi cell element method, each cell represents a basic structural element of the microstructure, which includes a particulate with its matrix neighborhood. A new cell element, including an interfacial crack and a matrix crack, is shown in Fig. 1. The matrix and inclusion phase in each Voronoi cell eΩ are denoted by mΩ and cΩ , respectively, i.e., e m c Ω =Ω ΩU . The bonded inclusion-matrix interface is indicated by b ∂Ω and the debonded interface is indicated by c ∂Ω on the inclusion side and by m ∂Ω on the matrix side. The element boundary e ∂Ω is assumed to be composed of prescribed displacement boundary d e ∂Ω , prescribed traction boundary t e ∂Ω , inter-element boundary i e ∂Ω and free boundary f e ∂Ω , i.e., e e e e d t i f e ∂Ω =∂Ω ∂Ω ∂Ω ∂Ω U U U . The matrix crack edges 1m ∂Ω , 2m ∂Ω , 3m ∂Ω and inclusion crack edge c ∂Ω have outward normals 1mn , 2mn , 3mn and cn , respectively, while en is the outward normal to the element boundary e ∂Ω . Figure 1. A Voronoi cell finite element with part interfacial crack and matrix crack In an incremental formulation to account for the onset and growth of the fatigue crack, σ is an equilibrated stress field corresponding a strain fieldε; u is a compatible displacement field on the element boundary at the beginning of an increment; Δσ is the equilibrated stress increment in eΩ ; Tractions Prescribed t e ∂Ω Inter-element Boundary i e ∂Ω Displacements Prescribed d e ∂Ω Free Boundary f e ∂Ω Ωm Ωc ∂Ωb ∂Ωc ∂Ωm1 ∂Ωm3 ∂Ωm2
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