A continuous crystallographic approach to generate cubic lattices and its effect on relative stiffness of architectured materials

Abstract

A B S T R A C TThis original work proposes to investigate the transposition of crystallography rules to cubic lattice architectured materials to generate new 3D porous structures. The application of symmetry operations provides a complete and convenient way to configure the lattice architecture with only two parameters. New lattice structures were created by slipping from the conventional Bravais lattice toward non-compact complex structures. The resulting stiffness of the porous materials was thoroughly evaluated for all the combinations of architecture parameters. This exhaustive study revealed attractive structures having high specific stiffness, up to twice as large as the usual octet-truss for a given relative density. It results in a relationship between effective Young modulus and relative density for any lattice structure. It also revealed the opportunity to generate auxetic structures at will, with a controlled Poisson ratio. The collection of the elastic properties for all the cubic structures into 3D maps provides a convenient tool for lattice materials design, for research, and for mechanical engineering. The resulting mechanical properties are highly variable according to architecture, and can be easily tailored for specific applications using the simple yet powerful formalism developed in this work.

Stress Concentration and Mechanical Strength of Cubic Lattice Architectures

“…Then the use of symmetry operations of the space group on points A and B results in an infinite network of points corresponding to the possible positions of atoms in a crystal. The whole procedure is not described in this study, but interested readers should turn to the work of Favre et al [ 10 ]. The case x = 0.0 and y = 0.0 results in a primitive cubic lattice ( Figure 1 ), and the progressive increase of both x and y forms an auxetic re-entrant structure, called a hexatruss [ 12 , 13 ].…”

“…The deformation applied on the lattice is a small compressive strain of ε = 1%. The modeling technique is further detailed by Favre et al [ 10 ]. The opposite facets of the representative volume are symmetry planes, corresponding to a frictionless sliding.…”

“…During the elastic compression of the lattice structures, the macroscopic stress applied on the outer surface of the cube is proportional to the macroscopic strain, and the proportionality coefficient is the apparent Young’s modulus E, reported by Favre et al [ 10 ]. The lattice is considered as a homogeneous equivalent material, and the mechanical design relies on the value of , applied on the representative elementary volume.…”

“…The use of Bravais lattices provides simple rules to produce periodic structures [ 9 ]. Some recent works have shown that it is possible to use crystallography formalism to design very versatile complex architectures by just using a point group and two architectural parameters [ 10 ]. It was possible to define a large range of structures with variable stiffness and Poisson ratio.…”

“…The specific stiffness of the lattice structures varies with the square of the specific density [ 10 , 11 ]. However, it was shown that the power exponent could deviate significantly from its nominal value of two, depending on the architecture [ 10 ]. Likewise, the yield strength and fatigue strength are expected to follow a power law of density.…”