Geodesic
Numerical estimation of effective properties of periodic cellular structures using beam and shell finite elements with CAE Fidesys
Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 523-536.

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The development of additive technologies (3D printing) made it possible to manufacture parts and products of a regular porous and cellular structure (in order to reduce the weight of the structure). In this case, the characteristic cell size is much smaller than the scale of the whole product. Numerical strength and related calculations of such structures require a preliminary estimation of the effective properties of such a cellular structure. In this article, a method for the numerical estimation of the effective elastic properties of regular cellular structures is presented, which is based on the numerical solution of boundary value problems of the theory of elasticity on a periodicity cell. Periodic boundary conditions in the form of restraints on the displacements of opposite edges of the cell are successively applied to the cell. The boundary value problem of the theory of elasticity is solved for each type of boundary conditions, and the resulting stress field is averaged over the volume. The effective properties of the cellular material are estimated as a generalized Hooke's law. Composite materials based on a rigid lattice skeleton filled with softer material are considered in the paper. The calculations are carried out using the finite element method with the domestic Fidesys CAE system. Beam finite elements are used in some calculations for the modeling of a lattice skeleton. In some other calculations, a thin layer of a binder between the skeleton and the matrix is taken into account. This layer is modeled using shell finite elements. Graphs of comparing the results of calculations of composite materials with a lattice skeleton modeled by beam elements and the results of similar calculations in which the skeleton is modeled by three-dimensional finite elements are given in the article. In addition, graphs of comparing the results of calculations in which the binder layer is modeled by shell elements and the results of similar calculations in which the binder is modeled by three-dimensional elements are given. The graphs show that with sufficiently thin framework elements (or with a sufficiently thin layer of the binder), the results are quite close. It confirms the applicability of beam and shell elements for the numerical solution of such problems.
Keywords: additive manufacturing, effective properties, solid mechanics, porous and cellular structures, CAE Fidesys. 
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V. A. Levin; K. M. Zingerman; M. Ya. Yakovlev; E. O. Kurdenkova; D. V. Nemtinova. Numerical estimation of effective properties of periodic cellular structures using beam and shell finite elements with CAE Fidesys. Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 523-536. http://geodesic.mathdoc.fr/item/CHEB_2019_20_2_a39/

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