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SUMMARYThis paper addresses the multiscale simulation of fibre-reinforced polymers. The considered composite materials exhibit a hierarchical material structure with three distinct length scales-micro, meso and macro. This feature of the morphology allows for the application of homogenization techniques based on a representative volume element (RVE) that is entirely typical for the local, periodic material structure. The effective macroscopic material behaviour of the composite can be predicted from the properties of the individual constituents and the geometric arrangement of the reinforcing fibres based on the simulation of the material behaviour in the RVE.The heterogeneous material structure in an RVE is modelled by the eXtended finite element method (XFEM). To this, two special element types, called X-element and 2X-element, are derived. They can represent one or two material interfaces within the element domain. For an efficient modelling process, an automated model generation procedure that determines the required element types, locates the material interfaces and performs the subdivision of the X-elements into tetrahedral integration domains has been developed. Problems related to a consistent interface approximation and a continuous displacement field are discussed.In the generated RVE models, a viscoplastic material model accounts for the inelastic material behaviour of the polymeric matrix, whereas the glass-fibres are assumed to have a linear elastic stress-strain behaviour. Using periodic displacement boundary conditions, effective stress-strain curves are computed for glass-fibre-reinforced polypropylene with unidirectional and woven arrangements of the reinforcing fibres.

SUMMARYThe FE-simulation of inhomogeneous structures, such as composite materials, biological tissues or foams, requires the generation of respective finite element meshes. With increasing complexity of the inner architecture of such structures, this becomes a time-consuming and laborious task. Additionally, the risk of forming bad-shaped elements that may lead to ill-conditioned numerical problems grows significantly. A solution to this problem provides the extended finite element method (XFEM). Thereby, the interface between different materials is represented by a local enrichment of the displacement approximation. As a consequence of this, the element boundary need not be aligned to the interface.In order to improve the accuracy of the interface approximation, the development of a plane element based on the XFEM and quadratic shape functions will be presented. This element allows for the description of curved material interfaces. The computation of the element stiffness matrix requires a numerical integration process that accounts for discontinuous fields. Regarding a linear element formulation, this can be achieved by an adapted triangulation of the element domain. However, in the case of a curved interface this solution is not applicable. Hence, non-uniform rational B-Spline (NURBS) surfaces are used to evaluate the integrals numerically.Finally, the results of different examples will show the general properties such as the accuracy of the numerical integration procedure and the convergence behavior of this element formulation.