Asymptotic Behavior of $3D$ Unstable Structures Made of Beams

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