Predicting stress propagation in two-dimensional mechanical metamaterials using graph theory

In materials with extensive percolation networks, the mechanical properties are dependent on both connectivity patterns and material properties. Interplay of local and global mechanics of the network as well as the correlation between topology, dynamic stress transmission and failure behavior make it difficult to predict mechanical responses. For the case of impact, global descriptors such as elastic modulus and density define the acoustic wave speed but are not sufficient to describe how stresses propagate locally within a discretized material. In addition, current predictive numerical approaches can be inefficient. Therefore, new tools that can accurately and efficiently predict the location and the magnitude of stresses in ordered and disordered systems are needed. Here, we use graph theory (GT) as a general methodology to describe mechanical metamaterials as sets of nodes (n) and edges (e) to predict the propagation and magnitude of stresses. Our experimental platform is based on macroscale 2D samples tested in quasistatic and dynamic conditions. High-speed polarized videography is used to characterize stress propagation and the data is compared against GT predictions and Finite Element models (FEM). We show that predictions of deformations closely matching experiments are achieved when classical GT parameters are modified to include geometric information that is usually ignored. We use these insights to propose stress wave management strategies in architected strut-based materials.