Effective stiffness, wave propagation, and yield surface attributes of Menger sponge-like pre-fractal topologies

Abstract

In the current work, the effective stiffness, long-wavelength attributes, and yield limits of Menger sponge-like pre-fractal topologies are investigated for the first time. Three types of Menger sponge-like structures with circular (MC), rhombic (MR), and square (MS) cavities are considered up to their third-level fractal topology. For each level, the elastic, shear, and bulk moduli, as well as the Poisson's ratio values are computed. Moreover, third-level Menger sponge-like fractal structures are additively manufactured and experimentally tested. A clear difference between the elastic stiffness and yield strength of the different Menger sponge-like structures is numerically and experimentally identified. It is observed that the cavity type considerably affects the scaling of elastic stiffness attributes upon increasing fractal level. The MS and MC topologies relate to lower relative density structures compared to the MR topology for all fractal levels investigated. The percentage relative density reductions induced by the fractal topology are a multiple of the corresponding decrease in the magnitude of the propagating shear and longitudinal phase velocities. Furthermore, all Menger sponge levels and inner cavity types yield analogous wave propagation directional dependencies, though with different phase velocity magnitudes. It is also found that increasing fractal topologies diminish the yield strength limits of the Menger fractal structures, with the factor associating the normal and shear loading yield strength of two successive Menger topology levels to be lower, but comparable in magnitude with the fractal dimension number. The fractal level evolution of the yield surface is well-captured by the extended Hill's criterion both for bi-axial and axisymmetric loads. The corresponding coefficients are identified for first, second, and third-level MS structures.