Generic rigidity in three-dimensional bond-bending networks

Much progress can be made in studying the mechanical stability of frameworks when they are treated as generic, which lack any special symmetries. This is because testing for rigidity becomes topological in nature rather than geometrical. Generic rigidity, synonymous with graph rigidity, depends only on the connectivity of the network, making it a simpler problem to deal with in principle. A complete combinatorial constraint counting characterization of graph rigidity is given by Laman's theorem for two dimensions. Unfortunately there is no known corresponding theorem for three dimensions. Herein it is proposed that the theorem of Laman generalizes to three dimensions for bar-joint networks that have no implied-hinge joints. Particular attention is given to bond bending networks, having a truss structure with constraints between nearest and next-nearest neighbours, that are suitable for modelling many covalent network glasses and macromolecules. It is shown that implied-hinge joints do not exist in bond-bending networks. Based on the proposition that an all subgraph constraint counting characterization of generic rigidity is recovered in three-dimensional bar-joint networks having no implied hinge joints, an efficient combinatorial algorithm is constructed for bond-bending networks. Complete agreement is found with exact calculations involving diagonalization of dynamical matrices, for systems up to 10(3) degrees of freedom.