Connections Between Topology and Macroscopic Mechanical Properties of Three-Dimensional Open-Pore Materials

Abstract

This work addresses a number of fundamental questions regarding the topological description of materials characterized by a highly porous three-dimensional structure with bending as the major deformation mechanism. Highly efficient finite-element beam models were used for generating data on the mechanical behavior of structures with different topologies, ranging from highly coordinated bcc to Gibson–Ashby structures. Random cutting enabled a continuous modification of average coordination numbers ranging from the maximum connectivity to the percolation-cluster transition of the 3D network. The computed macroscopic mechanical properties – Young’s modulus, yield strength, and Poisson’s ratio – combined with the cut fraction, average coordination number, and statistical information on the local coordination numbers formed a database consisting of more than 100 different structures. Via data mining, the interdependencies of topological parameters and relationships between topological parameters with mechanical properties were discovered. A scaled genus density could be identified, which assumes a linear dependency on the average coordination number. Feeding statistical information about the local coordination numbers of detectable junctions with coordination number of 3 and higher to an artificial neural network enables the determination the average coordination number without any knowledge of the fully connected structure. This parameter serves as a common key for determining the cut fraction, the scaled genus density, and the macroscopic mechanical properties. The dependencies of macroscopic Young’s modulus, yield strength, and Poisson’s ratio on the cut fraction (or average coordination number) could be represented as master curves, covering a large range of structures from a coordination number of 8 (bcc reference) to 1.5, close to the percolation-cluster transition. The suggested fit functions with a single adjustable parameter agree with the numerical data within a few percent error. Artificial neural networks allow a further reduction of the error by at least a factor of 2. All data for macroscopic Young’s modulus and yield strength are covered by a single master curve. This leads to the important conclusion that the relative loss of macroscopic strength due to pinching-off of ligaments corresponds to that of macroscopic Young’s modulus. Experimental data in literature support this unexpected finding.