Issue 29

L. Cabras et alii, Frattura ed Integrità Strutturale, 29 (2014) 9-18; DOI: 10.3221/IGF-ESIS.29.02 10 are firstly formed by connecting straight ligaments to central nodes which may be circles or rectangles or other geometrical forms. The auxetic effects are achieved through wrapping or unwrapping of the ligaments around the nodes in response to an applied force, as shown in [2], the Poisson's ratio ν of a chiral structure for in-plane deformations, with flexible ribs and rigid node, can be tailored to be around -1. Later Ruzzene and Spadoni, in [3], have considered the behavior of structures by introducing the flexibility of the nodes. Other models, as in [4], derive the auxetic behavior by the rotation of rigid or semi-rigid shapes (triangle, squares, rectangles and tetrahedron) when loaded, this type of structures has been developed to reproduce the behavior of foams and hypothetical nanostructure networked polymers. A different approach is followed by Bathurst and Rothenburg in [5], they formulate the incremental response of an assembly of elastic spheres, considering an isotropic distribution of contacts around a particle. Since the negative Poisson's ratio is a scale independent property the auxetic behavior can be achieved at a macroscopic or microstructural level, or even at the mesoscopic and molecular levels, many models were developed to simulate polymeric structure or anisotropic fibrous composites. The first auxetic microporous polymeric material was investigated in [6]. It was an expanded foam of PTFE which has a highly anisotropic negative ν =-12. Several cases of negative Poisson's ratios have been discovered in the analysis of anisotropic fibrous composites. In these composites there is a high degree of anisotropy and the negative Poisson's ratio only occurs in some directions; in some cases only over a narrow range of orientation angle between the applied load and the fibers. In a recent advance, laminate structures have been presented which give rise to intentional negative Poisson's ratios combined with mechanical isotropy in two dimensions or in three dimensions. These laminates have structure on several levels of scale; they are hierarchical. By appropriate choice of constituent properties one can achieve Poisson's ratios approaching the lower limit of -1. Auxetic systems perform better than classical material in a number of applications, due to their superior properties. They have been shown to provide better indentation resistance [7, 8 and 9] for their property of “densification” in the vicinity of an impact. The auxetic materials form dome shaped structures [10, 11] when they are subjected to out of plane bending moments instead the saddle shape adopted by the common materials. Also, they can be useful when we need better acoustic and vibration properties than the conventional materials [12, 13 and 14]. M ODEL OF PERIODIC LATTICE WITH AUXETIC MACROSCOPIC BEHAVIOR e consider radially foldable structure formed by two angulated elements ABC and DBE, shown in Fig. 1, connected together through a hinge in B. Figure 1 : Pair of linkages movable with a single degree of freedom. The two rigid linkages ABC and EBD are shown in grey and black, respectively. They are constraint at the “coupler” point B to have the same displacement components. Points A and D and E and C can only move along straight lines. The coupled angulated elements ABC and DBE can roto-translate with a single degree of freedom and the end point A, C, D and E can only translate along to Ox-axis and the axis inclined by the angle α with respect to the Ox-axis, respectively. W