Homogenized elasticity and domain of linear elasticity of 2D architectured materials

Abstract

This article presents a generic method to determine analytically the equivalent homogeneous linear elastic behavior of any 2D architectured material and to determine its macroscopic linear elasticity domain. The method is applied throughout the paper in the equilateral triangular case, as an example. Additional examples of square, isosceles right-angled triangular and hexagonal architectured materials are presented as a conclusion. The proposed method combines a periodic homogenization for the equivalent homogeneous linear elastic behavior and three criteria to establish the macroscopic linear elasticity domain. The first criterion corresponds to the limit strength of the bars. The second one, which corresponds to the apparition of a periodic buckling, involving a finite number of cells, is studied on a single primitive unit cell using Bloch wave analysis. The buckling modes that appear under equibiaxial compression for the equilateral triangle architectured material are similar to experimental results reported in the literature. The third criterion is the aperiodic buckling, or macroscopic instability, which corresponds to the apparition of shear bands or a macroscopic buckling. The resultant macroscopic linear elasticity domain resulting from these three criteria is presented, for the first time, in the case of the equilateral triangle, isosceles right-angled triangle architectured material and also for two other geometries, square and hexagonal architectured material. It is shown that the theoretical properties of these architectured materials are, in an Ashby chart, in the domain of light but relatively strong materials, a domain in which man-made materials are currently poorly represented.