Issue 29

L. Cabras et alii, Frattura ed Integrità Strutturale, 29 (2014) 9-18; DOI: 10.3221/IGF-ESIS.29.02 12 where n 1,2 are integers and t 1 , 2 the primitive vectors spanning the lattice. E FFECTIVE PROPERTIES OF THE PERIODIC AUXETIC LATTICE acroscopic properties are derived analytically for our lattice. The triangular lattice have three-fold symmetries and basic considerations on the symmetry group of the material lead to the conclusion that its constitutive behavior is isotropic (in the plane of deformation). Therefore, it will be necessary to compute two effective elastic constants. Stability constraints the in-plane Poisson's ratio to range between -1 and 1. Effective properties are denoted as K*(bulk modulus), E*(Young's modulus), μ*(shear modulus) and ν*(Poisson's ratio) and macroscopic stress and strain as σ and ε , respectively. The structure is composed of slender crosses and classical structural theories can be conveniently applied to analyze the response of the elastic system. In particular, each arm of a single cross is modeled as an Euler beam undergoing flexural and extensional deformations. Each beam have Young's modulus E, cross-sectional area A and second moment of inertia J. Additional springs have longitudinal stiffness equal to k L or rotational stiffness equal to k R (see Fig. 4). We also introduce the non-dimensional stiffness ratio parameters α 1 =k L p/(EA), α 2 =k L p 3 /(EJ), α 3 =k R /(pEA) and α 4 =k R p/(EJ). Macroscopic stresses are computed averaging the resultant forces on the boundary of the unit cell. Periodic boundary condition have been applied on the boundary of the unit cell so that displacements are periodic and forces are anti- periodic. Additional constrains are introduced to prevent rigid body motions. To solve the structure we apply the Principle of Virtual Work (PVW). It states that, if a structure is in equilibrium, then, for any arbitrary small virtual displacement satisfying kinematic boundary conditions, the work done by the external forces must equal the work done by the internal forces. In the following, we apply the PVW in two steps: in the first we find the internal actions (bending moments M, axial forces P and spring forces S L or moments M R ) of the structure searching for the kinematic admissible configuration in the set of statically admissible ones (Flexibility Method) and in the second step we compute the macroscopic displacements. (a) (b) Figure 4: Lattice reinforced with elastic springs. (a) Longitudinal springs of stiffness k L . (b) Rotational spring of stiffness k R . The dashed contour indicates a typical unit cell of the periodic elastic system. This procedure has the advantage to maintain the analytical treatment sufficiently simple. We point out that all the results have been also verified numerically implementing a finite element code in Comsol Multiphysics®. We consider the elastic structure as in Fig. 5a, subjected to normal and tangential external forces supposed to be known and corresponding to a macroscopic stress having components 11  and 22  different from zero. We define an equivalent statically determined system disconnecting two springs and introducing the dual static parameter as unknown X, equal for the two springs, as shown in Fig. 5b. Then, the general field of tension Ξ (Ξ=M, N, S L or M R ) in equilibrium with the external loads is: Ξ= Ξ 0 + XΞ 1 (7) where Ξ 0 is the solution of the static scheme in equilibrium with the external loads and X=0; while the field Ξ 1 is the solution of the static scheme in equilibrium with zero external loads and X=1. The deformed configurations of the isostatic equivalent structure violate the internal constraint in the two springs suppressed in the structure made isostatic. M