Numerical Estimation of Effective Properties of Periodic Cellular Structures using Beam and Shell Finite Elements with CAE Fidesys
https://doi.org/10.22405/2226-8383-2019-20-2-523-537
Abstract
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.
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.
About the Authors
Vladimir Anatolyevich LevinRussian Federation
Konstasntin Moiseevich Zingerman
Russian Federation
Maksim Yakovlevich Yakovlev
Russian Federation
Ekaterina Olegovna Kurdenkova
Russian Federation
Diana Vladimirovna Nemtinova
Russian Federation
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Review
For citations:
Levin V.A., Zingerman K.M., Yakovlev M.Ya., Kurdenkova E.O., Nemtinova D.V. Numerical Estimation of Effective Properties of Periodic Cellular Structures using Beam and Shell Finite Elements with CAE Fidesys. Chebyshevskii Sbornik. 2019;20(2):523-537. (In Russ.) https://doi.org/10.22405/2226-8383-2019-20-2-523-537
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