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

• Published in: physica status solidi (b) 2021-12, Vol.258 (12), p.n/a
• Main Authors: Czarnecki, Sławomir, Łukasiak, Tomasz
• Format: Article
• Language: English
• Subjects: compliance minimization; homogenization; isotropic composites; topology optimization
• ISSN: 0370-1972; 1521-3951
• DOI: 10.1002/pssb.202100344

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 recovery of microstructures for given nonhomogeneous optimal layouts of elastic moduli maximizing the stiffness of a 2D isotropic elastic body is analyzed. Two adopted, pin‐jointed, representative volume elements with a nonuniform topology allow to construct properly graded microstructures. Effective properties are calculated by the asymptotic homogenization theory. The ability to model the auxetic, heterogeneous isotropic structure is also shown.