Modeling of failure in composites by X-FEM and level sets within a multiscale framework
Composites or multi-phase materials are characterized by a distinct heterogeneous microstructure. The failure modes of these materials are governed by several micromechanical effects like debonding phenomena and matrix cracks. The overall mechanical behavior of composites in the linear as well as the nonlinear regime is not only governed by the material properties of the components and their bonds but also by the material layout. In the present contribution the material structure is resolved and modeled on a small scale allowing to deal with these effects. For the numerical simulation we apply a combination of the extended finite-element method (X-FEM) [N. Mos, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, Int. J. Numer. Methods Engrg. 46 (1999) 131150] and the level set method (LSM) [S. Osher, J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on HamiltonJacobi formulations, J. Comput. Phys. 79 (1988) 1249]. In the X-FEM the finite-element approximation is enriched by appropriate functions through the concept of partition of unity. The geometry of material interfaces and cracks is described by the LSM. The combination of both, X-FEM and LSM, turns out to be very natural since the enrichment can be described and even constructed in terms of level set functions. In order to project the material behavior modeled on a small scale onto the large or structural scale, we employ the variational multiscale method (VMM) [T. Hughes, G. Feijoo, L. Mazzei, J.-B. Quiney, The variational multiscale method  a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg. 166 (1998) 324]. This concept is based on an additive split of the displacement field into large and small scale parts. For an efficient solution of the discrete problem we postulate that the small scale displacements are locally supported; in order to achieve this objective one has to assume appropriate constraint conditions. It can be shown that the applied numerical model allows a considerable flexibility concerning the variation of the material design and consequently of the mechanical behavior of a composite.
extended finite-element method level set method variational multiscale method cohesive crack