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    Hierarchical RVE-based multiscale modelling of non-linear heterogeneous materials using the finite volume method
    This paper describes the development of a hierarchical multiscale procedure within the finite volume OpenFOAM framework for modelling the mechanical response of non-linear heterogeneous solid materials. This is a first development of hierarchical multi-scale model for solid mechanics to use the finite volume discretisation method. In this computational procedure the information is passed between the macro and micro scales using representative volume elements (RVE), allowing for general, non-periodic microstructures to be considered. Each computational point at the macro scale is assigned an RVE with prescribed microstructural features. The overall macro response accounts for the microstructural effects through the coupling of macro and micro scales, i.e., the macro deformation gradient is passed to the RVE and in turn, the homogenised micro stress-strain response is passed back to the macro scale. The incremental total Lagrangian formulation is used to represent the equilibrium state of the solid domain at both scales and its integral equilibrium equation is discretised using the cell-centred finite volume (FV) method in OpenFOAM. The verification of the model is demonstrated using both 2D and 3D simulations of perforated elastic-plastic plates subjected to tensile loading.