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In this paper, a computational homogenization technique for thin-structured sheets is proposed, based on the computational homogenization concepts for first- and second-order continua. The actual three-dimensional (3D) heterogeneous sheet is represented by a homogenized shell continuum for which the constitutive response is obtained from the nested analysis of a microstructural representative volume element (RVE), incorporating the full thickness of the sheet and an in-plane representative cell of the macroscopic structure. At an in-plane integration point of the macroscopic shell, the generalized strains, i.e. the membrane deformation and the curvature, are used to formulate the boundary conditions for the microscale RVE problem. At the RVE scale, all microstructural constituents are modeled as an ordinary 3D continuum, described by the standard equilibrium and the constitutive equations. Upon proper averaging of the RVE response, the macroscopic generalized stress and the moment resultants are obtained. In this way, an in-plane homogenization is directly combined with a through thickness stress integration. From a macroscopic point of view, a (numerical) generalized stress–strain constitutive response at every macroscopic in-plane integration point is obtained. Additionally, the simultaneously resolved microscale RVE local deformation and stress fields provide valuable information for assessing the reliability of a particular microstructural design. |
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