Reduced Order Homogenization for Multiscale Analysis of Nonlinear Composites

Heterogeneous materials are nowadays used in several fields of structural engineering. Such materials, regarded as composites, have a heterogeneous microstructure in which two or more constituents are combined in order to reach improved mechanical properties. Most of the composites include constituents characterized by a nonlinear behaviour, hence, it is important to consider the inelastic phenomena arising at the microscale, to accurately predict the macroscopic response of the heterogeneous material. A modeling approach allowing for the heterogeneous nature of the composite to be considered during the design process is provided by the Multiscale Analysis, in which both the macroscopic scale and the microscopic scale are involved. At the microscale, a Unit Cell, being a representaive sample of the heterogeneous nonlinear material, is studied in order to derive the behaviour of an equivalent homogeneous macroscopic material. In the scale transition process, usually regarded as homogenization, efficient numerical tools are needed in order to reduce the computational cost due to the large quantity of internal variables, coming from the evaluation of the elastoplastic material models at the microscopic level. Reduced Order Models (ROM) are introduced with the aim of lowering the number of internal variables of the problem and to provide accurate solutions with reasonable computational cost and time. This thesis is mainly dedicated to the development of a ROM for the homogenization of nonlinear heterogeneous materials; starting from the Hashin-Shtrikman analytical homogenization scheme, a piecewise uniform distribution of the microscopic quantities is assumed, and thus, the proposed ROM is referred as PieceWise Uniform Hashin-Shtrikman (PWUHS) technique. In particular, the PWUHS is developed for the solution of homogenization problems of nonlinear composites and extended to Mises plasticity with linear hardening. Numerical results demonstrate the accuracy of the proposed homogenization scheme, which is compared to the well known PieceWise Uniform Transformation Field Analysis (PWUTFA) in order to investigate the similarities and the advantages of both reduced order models. PWUHS is implemented in the framework of Multiscale Analysis for studying the response of auxetic composites and numerical results are compared to the experimental counterpart to assess the efficiency of the proposed multiscale scheme.