A new class of auxetic lattices

In this work we have studied the mechanical and thermal behavior of a new class of auxetic lattices. We have started introducing a novel two-dimensional lattice family with three different realizations that leads to a Poisson's ratio arbitrarily close to the thermodynamic limit corresponding to ν = -1. The effect is achieved by the superposition of clockwise and anti-clockwise internal rotations leading to a macroscopic non-chiral effect. We have detailed the kinematic of the mechanical system for three types of periodic lattices and we have determined analytically the macroscopic constitutive properties of these structures. The dependence of the effective properties on the constitutive and geometrical parameters of the micro-structure is given and a comparative analysis with hexagonal, triangular and square honeycombs is also performed. We have present also experimental evidence that the new lattices can achieve a negative Poisson's ratio ν=-0.9931± 0.0025. Later we have extended our study to a three-dimensional model that can reproduce the same particular auxetic behavior. The system, we have introduced, has an elementary cell with cubic shape. We have proposed two different versions, these two different systems are generally anisotropic, or more precisely cubic according to symmetry consideration, but one of them can be modified in its structural properties so that the macroscopic behavior becomes isotropic. The anisotropic elastic character of the material is obviously reflected on the effective Young's modulus E and on the effective Poisson's ratio ν, which are therefore a function of the direction in the solid. We have given a complete numerical study of these two elastic constants in particular focusing on the Poisson's ratio ν. The maximum and minimum global Poisson's ratio ν obtained, for the anisotropic tridimensional lattice, are -1 and 1. In the case of the isotropic tridimensional lattice, Poisson's ratio ν ranges from -0.58 to 0.14. After having considered the mechanical behavior of the lattices we have studied their thermal behavior. The particular geometry of the micro-structures makes possible to have dissimilar expansions of the elements within the lattice, leading to an effective coefficient of thermal expansion different from that of the constituent elements. It is possible to reach strongly negative, null, or positive coefficient of thermal expansion using conventional constituent materials with positive thermal expansion. A theoretical expression of the macroscopic coefficient of thermal expansion of the lattices has been provided in full analytical form as a function of the coefficients of thermal expansion of the constituent materials and of the geometrical configuration of the lattices. We have tested experimentally a typology of lattice to prove that achieves a negative coefficient of thermal expansion. The novel systems are constructible at any length-scale, including the micro-scale, from conventional components with different mechanical and thermal properties, which may be combined to ensure the required macroscopic behavior. For this reason they can be used in several engineering fields: bio-medic, structural, aeronautics, civil, etc. An interesting aspect, that concerns the thermal behavior, is that the macroscopic thermal expansion is not anymore an intrinsic property of the material but it is correlated to the micro-structure and to the ratio between the coefficients of thermal expansion of the constituent materials. This allow for example to realize structure insensitive to temperature changes or with a specific coefficient of thermal expansion depending on the application, without no need to use special materials generally very expensive.