Multiscale computational homogenization of heterogeneous shells at small strains with extensions to finite displacements and buckling

In this paper, a framework for computational homogenization of shell structures is proposed in the context of small strain elastostatics, with extensions to large displacements and large rotations. At the macroscopic scale, heterogeneous thin structures are modeled using a homogenized shell model, based on a versatile three-dimensional 7-parameter shell formulation, incorporating a through-thickness and pre-integrated constitutive relationship. In the context of small strains, we show that the local solution on the elementary cell can be decomposed into 6 strain and 6 strain gradient modes, associated with corresponding boundary conditions. The heterogeneities can have arbitrary morphology, but are assumed to be periodically distributed in the tangential direction of the shell. We then propose an extension of the small strain framework to geometrical nonlinearities. The procedure is purely sequential and does not involve coupling between scales. The homogenization method is validated and illustrated through examples involving large displacements and buckling of heterogeneous plates and shells.

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