Abstract
In this paper, we study the asymptotic behavior of an varepsilon -periodic 3D stable structure made of beams of circular cross-section of radius r when the periodicity parameter varepsilon and the ratio {r/varepsilon } simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.
Highlights
The aim of this work is to study the asymptotic behavior of an ε-periodic 3D stable structure made of “thin” beams of circular cross-section of radius r when the periodicity parameter ε
By “thin”, we mean that the radius r of the beams is much smaller than the periodicity parameter ε and that we deal with the case where ε and r/ε simultaneously tend to 0
From the mathematical point of view, this means that the processes of homogenization and dimension reduction do not commute
Summary
The aim of this work is to study the asymptotic behavior of an ε-periodic 3D stable structure made of “thin” beams of circular cross-section of radius r when the periodicity parameter ε. In a forthcoming paper, we will investigate the unstable and auxetic 3D periodic structures made of beams and we will see that all the estimates of Lemma 5 will remain except (4.5)1 These decompositions allow to obtain Korn type inequalities as well as relevant estimates of the centerline displacements. 5 we deal with an ε-periodic stable 3D structure made of r-thin beams, Sε,r For this structure we introduce a linearized elasticity problem and specify the assumptions on the applied forces. 8, in order to obtain the limit unfolded problem we split it into three problems: the first involving the limit warpings (these fields are concentrated in the cross-sections, this step corresponds mainly to the process of dimension reduction), the second involving the local extensional and inextensional limit displacements posed on the skeleton structure and the third involving the macroscopic limit displacement posed in the homogeneous domain.
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