Asymptotic Behavior of 3D Unstable Structures Made of Beams

Abstract

In our previous papers (Griso et al. in J. Elast. 141:181–225, 2020; J. Elast., 2021, https://doi.org/10.1007/s10659-021-09816-w), we considered thick periodic structures (first paper) and thin stable periodic structures (second paper) made of small cylinders (length of order varepsilon and cross-sections of radius r). In the first paper r=kappa varepsilon with kappa a fixed constant, varepsilon to 0, while in the second varepsilon to 0 and {r/ varepsilon }to 0. In this paper, our aim is to give the asymptotic behavior of thin periodic unstable structures, when varepsilon to 0, {r/ varepsilon }to 0 and varepsilon ^{2}/ rto 0.Our analysis is again based on decompositions of displacements. As for stable periodic structures, Korn type inequalities are proved. Several classes of unstable and auxetic structures are introduced. The unfolding and limit homogenized problems are really different of those obtained for the thin stable periodic structures. The limit homogenized operators are anisotropic, the spaces containing the macroscopic limit displacements depend on the periodicity cells. It was not the case in the two previous studies. Some examples are given.