3D printable spatial fractal structures undergoing auxetic elasticity

Currently simple and conventional structures are unable to meet increasingly critical and diverse applications, as well as establishment of constitutive relationship of complex structure is becoming a major challenge in materials and mechanics disciplines. To overcome these challenges, fractal structures have attracted much attention due to their abilities to establish constitutive relationships of complex structures. In this study, we design a spatial fractal framework by using high elasticity material and 3D printing method. The constitutive relationships of fractal framework are mathematically modeled. Effects of fractal levels, fractal dimension, and spatial pattern of structures on the mechanical behavior of the fractal framework and assembled spatial fractal structures are studied using both finite element analysis and experimental measurements. The constitutive relationships among stress, strain, fractal dimension, fractal levels, and spatial patterns of structures are proposed and discussed. This paper develops a novel method to establish constitutive relationships of complex structures, and provides theoretical and engineering basis for 3D printable spatial fractal structures undergoing auxetic elasticity.