A homogenisation approach for shell structures using scaled boundary isogeometric analysis on RVE‐level

Abstract

AbstractThis contribution deals with a homogenisation approach for shell structures. A coupled multiscale approach also known as FE2 is used, which satisfies the Hill‐Mandel condition. The macroscopic structure is modelled with shell elements. Assuming periodicity of the mesostructure, which fulfils the scale separation requirement, allows the definition of a representative volume element (RVE). Using the RVE, equivalent homogeneous material parameters are obtained in order to define the overall structure. The shell strains obtained on the macroscopic scale are applied to the RVE. To model the mesostructure as 3D solids, the boundary representation from CAD is adopted. Next, Isogeometric analysis (IGA) is used for the solution on the boundary. In order to analyse the interior of the solid, IGA is combined with the scaled boundary finite element method. While the boundary is approximated using NURBS functions, the interior is approximated by B‐Splines, contrary to the use of an analytical solution in classical SBFEM. As a result, nonlinear problems can be covered. Furthermore, this approach facilitates the direct use of CAD models for analysis, which is especially useful for complex geometries on the mesostructure, e.g. for modelling laminar textile structures, such as woven fabrics or laid webs. Star‐shaped domains needed for the analysis are obtained by subdivision of the overall domain. Using NURBS basis functions for the approximation of the boundary can be problematic because the position of the control points, in contrast to the nodes in FEM, does not necessarily equal the physical coordinates. To enforce prescribed boundary conditions in the framework of homogenisation, this leads to difficulties. To overcome this problem a transition element is utilized to couple the shell kinematics on the macrostructure with the 3D solid on the mesostructure. This results in a method able to use complex geometries from CAD models as representative volume elements for shell structures.