Graph-Theoretical Predictions of Two-Dimensional Stress Propagation Through Complex Strut Lattices

Predicting stress propagation through mechanical materials is a central question to applications including impact mitigation and architectural load support. One class of materials which has shown great promise in terms of tunability of mechanical properties is disordered network mechanical metamaterials (DNMMs). Previous literature has shown how DNMMs of the same material with different network structures may exhibit very different mechanical properties. Conventional computational methods for property predictions typically require numerical solutions to constitutive relations. However, the use of these methods for DNMMs becomes prohibitively expensive because such structures are large and/or highly granular. In addition to limiting the application of methods, this large computational cost is a barrier to data driven studies.In many fields, the analysis of networks is carried out with graph theory (GT). And GT’s specific application to materials science is becoming increasingly common. In the context of transport phenomena, particular focus is given to networks formed from granular packings of disks. In particular, it has been shown that a group of parameters called “betweenness” may be used to predict areas of concentrated heat transport in packed beds, and failure locations in mechanical networks. However, the suitability of this method for different network architectures has not been investigated.
In this work, we show how a graph theoretic treatment may be generalized to a broad class of networks, by augmenting their topological description with pertinent geometric features. Specifically, we show how incorporating boundary conditions and edge lengths into the analysis achieves higher accuracy for the classification of stressed edges than conventional parameters. We demonstrate this by imaging the stress-induced birefringence of laser-cut acrylic networks while undergoing uniaxial compression. We show classifications for networks from granular packings, Archimedean lattices, auxetic structures, and networks resembling structures obtained from aramid nanofiber microscopy. Finally, we validate the method against finite element analysis. By integrating conventional simulations, graph theory, and experiment, we expect our approach will accelerate the design of DNMMs.