Bending‐Free Lattices as Underlying Optimal Isotropic Microstructures of the Least‐Compliant 2D Elastic Bodies

Abstract

This study puts forward a method of reconstruction of the suboptimal isotropic truss microstructures for the known distribution of Young's modulus and Poisson's ratio predicted by the Isotropic Material Design (IMD) method of minimizing the compliance (maximizing the stiffness) of a 2D elastic body. The varying underlying microstructures corresponding to the optimal designs are recovered by matching the values of the optimal elastic moduli with the values of the effective moduli of the representative volume elements (RVE) computed by the asymptotic homogenization method. The used hexagonal shape of the RVE with its specific internal symmetry provides the assumed isotropy of the periodic body. Hexagonal periodic cells are made up of 3 or 12 pin‐jointed families of bars with different stiffness characteristics. Combining these families into two or three groups with identical bar stiffnesses in a group allows for the exact reproduction of Poisson's ratio for each point of the body. The ability to model an auxetic behavior within the subdomains where the optimal Poisson's ratio assumes negative values is also shown. The effective value of the Young's modulus is scaled independently of the value of the effective Poisson's ratio by selecting the cross section of the bars in the group.

The Isotropic Material Design of In-Plane Loaded Elasto-Plastic Plates

“…For instance, the hexagonal (or honeycomb in the plane) gridwork is characterized by a very small shear modulus, if ligaments are slender (see [ 5 , 6 ]). On the other hand, spiral microstructures are characterized by very small bulk modulus, which implies the effective Poisson ratio almost attaining its lower 2D limit equal −1 (see [ 7 , 8 , 9 , 10 , 11 , 12 ]). To encompass such a broad class of composites, it is necessary to admit the largest possible range of the bulk and shear moduli.…”

“…The stress-based FEM was already developed in the 1970s (see [ 18 ]) and then extended to elasto-plasticity [ 19 ]. In the present paper, the present authors’ stress-based FE algorithm is proposed, originated in [ 20 ] in the context of Anisotropic Material Design, extended to the elastic IMD setting in [ 3 , 10 , 11 ] and adopted here to the elasto-plastic version of the IMD method. Since the numerical method of solving IMD problems plays an essential role here, it is put forward in detail in Section 5 , while the numerical optimization procedure is explained in Section 6 .…”