Donald Jacobs - Graph rigidity: applications to material science and proteins

The mechanical stability of network glasses and proteins can be effectively studied by modeling the microstructure as a generic bar-joint framework. Graph rigidity for bar-joint frameworks in two dimensions is completely characterized by Laman's theorem [1]. Recently, an efficient algorithm has been constructed to study generic rigidity percolation in two dimensions [2]. The desire to construct a similar algorithm in three dimensions is impeded by the lack of an analogous theorem. It has been proposed [3] that an all subgraph constraint counting characterization of generic rigidity is recovered in three-dimensional bar-joint networks having no implied hinge joints. Based on this proposition, an efficient combinatorial algorithm has been constructed for bond-bending networks, which have no implied-hinge joints. Complete agreement is found with exact calculations involving diagonalization of dynamical matrices, for systems up to 1000degrees of freedom.

References

1.  G. Laman, J. Engrg. Math. 4(1970), 331. 2.  D.J. Jacobs and B. Hendrickson, J. Comput. Phys. 137(1997), 346.

3.  D.J. Jacobs, J. Phys. A: Math. Gen. 31(1998), 6653.