Homogenisation for macroscopic shell structures with application to textile‐reinforced mesostructures

Abstract In the analysis of textile‐reinforced shell structures full‐scale models are often impractical due to the complex nature of the meso‐structure. The structure is assumed to be globally periodic which allows the definition of a representative volume element (RVE). In this contribution, the RVE is embedded within a first‐order homogenisation framework, where the meso‐to‐macro transition is governed by the Hill‐Mandel condition. In a coupled multiscale approach the macroscopic scale is analysed using shell elements. Each macroscopic integration point is linked to an RVE which accounts for the total thickness of the shell. The macroscopic shell strains are applied on the RVE by means of suitable boundary conditions. On the mesoscopic scale, a boundary value problem is solved in order to obtain equivalent material parameters which are returned to the macroscopic scale. Here, a discrete model of the textile‐reinforcement on the mesostructure is obtained by adapting the approach from the scaled boundary finite element method (SBFEM) in which solids are described by means of their boundary surfaces. These are scaled into one central point, the so‐called scaling center. The required star‐shaped domains are obtained by subdivision. Combination with the isogeometric analysis (IGA) allows to directly facilitate the NURBS (Non‐Uniform Rational B‐Splines) functions describing the boundaries in commonly used CAD software. A key point is the choice of suitable boundary conditions for the RVE. In this contribution a set of periodic boundary conditions will be presented. The proposed method for the analysis of textile‐reinforced shell structures is validated by means of numerical examples.