Issue 29

L. Cabras et alii, Frattura ed Integrità Strutturale, 29 (2014) 9-18; DOI: 10.3221/IGF-ESIS.29.02 11 In analyzing the trajectory of the central point B, we also follow the more general formulation given in [15]. In Fig. 1, p is the length of the arms, θ the internal angle between them and α is the angle between the two straight lines along which the points A, D and C, E are constrained to move. B is the “coupler” point of the linkage. The equation for the one- parameter trajectory followed by the point B is obtained fixing the values of the geometric variables p, θ, α; then, the position of B is determined by the angle γ. When we couple the movement of the linkage ABC with the linkage EBD, we obtain a relation between angles θ and α. The common point B follow the radial line OB: x cosθ 1 sinθ y   (1) The two linkages are assembled in order to create a radially foldable structure, as depicted in Fig. 2 and to avoid crossover with other pairs in a polar arrangement of the fully radially foldable structure the angle γ has to satisfy the bound η π γ ηα  , (2) where EDBCAB      . Different configurations are shown in Fig. 2b; the point B for each pair of linkages moves radially and the corresponding Poisson's ratios is equal to -1. We consider the triangular geometries with α=2π/3. (a) (b) (c) Figure 2 : (a) Radially foldable structure with geometric parameters. (b) Configurations of the single degree of freedom lattices at different values of the geometrical parameter. (c) The radial distance OB as a function of γ is also given for p=1. Construction of periodic lattice The kinematically compatible periodic structures shown in Fig. 3 is obtained by a periodic distribution of the single cell elements shown in Fig. 2 as in [16]. Figure 3 : Periodic microstructure. Three different configurations, for different values of α are shown. The grey dashed region is the unit cell of the Bravais lattice where t 1 and t 2 are the primitive vectors. The microstructure is composed of shaped elements with 12 arms of the same length. A system of cross couple is built where two elements are disposed in two different planes. Each cross couple is mutually constrained to have the same displacement at the central point where a hinge is introduced. Different couples of crosses are then constrained each other by internal hinges at the external end of each arm. The periodic structures have a Bravais periodic lattice [17] consisting of points: R = n 1 t 1 +n 2 t 2 (3)