Geometry, topology, and statistics of mechanical networks

Abstract

Many physical, biological, and engineering systems can be modeled as networks embedded in real Euclidean space, with edges representing the geometrical constraints between individual units (nodes). Understanding of the spatial structure and organization of the degrees of freedom (DoF) in these networks is an issue of critical importance, as learning how to identify, count, actuate, combine, or remove certain DoF will provide the means to understand and control these systems. Here we combine geometric rigidity, algebraic sparsity, and graph theory to develop a framework for calculating the DoF in network-based models and identifying the zero-energy floppy modes via a representation that illuminates the underlying hierarchy and modularity of the network, and thence the control of its nestedness and locality. We demonstrate the power of this framework through three applications. First, we apply this framework to origami materials, a type of novel metamaterials inspired by ancient arts of paper folding. We model origami as 3D networks and investigate how excess folds in a simple origami pattern control the floppiness and rigidity of the structure in a scale-invariant manner. Second, we study the kirigami metamaterials deriving from the art of paper cutting. Instead of repetitive cuts, we investigate how we can control the connectivity and rigidity of kirigami materials with random cuts. In both origami and kirigami materials, percolation transition is observed in the rigidity or connectivity of the material, which can be exploited for geometrical information storage. Finally, we turn to general mechanical networks. We frame the question as one of identifying, representing, and understanding the flexibility (floppy modes) in under-constrained networks. By adapting and generalizing a sparse null-space basis algorithm, we show how to create a sparse representation of the floppy modes that uncovers the hidden modularity in the network and describes the combination of hierarchy and spatial localization within the degrees of freedom of the system. This representation leads to a range of applications including robotic reaching tasks with motion primitives, and predicting the linear and nonlinear response of elastic networks based solely on infinitesimal rigidity and sparsity, which we test using physical experiments. Overall, our framework is likely to be of use broadly in characterizing and optimizing the function and performance of a wide range of systems that can be modeled as networks.