Multiscale optimization of specific elastic properties and microscopic frequency band-gaps of architectured microtruss lattice materials

Abstract

This paper proposes the development of lattice materials with the concurrent consideration of their specific elastic mechanical properties and their corresponding phononic wave filtering capabilities. A multi-objective and multiscale design optimization problem is presented where the microscopic geometric parameters of truss-like lattice unit cells are introduced as design variables by means of beams cross-section shape transformers. The optimization problem is posed to maximize the stiffness and strength properties of lattice materials along with their frequency band-gap aspect ratios while minimizing their relative density. Floquet-Bloch theorem is employed for the frequency band-gap analysis while the homogenized mechanical properties are estimated using the Cauchy-Born hypothesis and the Hill-Mandel principle of macro-homogeneity. It is found that the band-gap aspect ratios are directly proportional to the macroscopic mechanical properties of the lattice, therefore posing a trade-off design problem. Two case studies are presented for the multi-objective optimization problem including the triangular and the Kagome patched honeycomb topologies. It is demonstrated that the design objectives are satisfied with “I” shaped, horizontally hollow circular or diamond cross-sections. Design charts are introduced which enable the definition of lattice microscopic geometrical attributes required for lattice material development with optimal static macroscopic and dynamic microscopic characteristics.