The first-order flexibility of a crystallographic framework

Abstract

Four sets of necessary and sufficient conditions are obtained for the first-order rigidity of a periodic bond-node framework C in Rd which is of crystallographic type. In particular, an extremal rank characterisation is obtained which incorporates a multi-variable matrix-valued transfer function ΨC(z) defined on the product space C⁎d=(C﹨{0})d. In general the first-order flex space is the closed linear span of polynomially weighted geometric velocity fields whose geometric multi-factors in C⁎d lie in a finite set. It is also shown that, paradoxically, a first-order rigid crystal framework may possess a nontrivial continuous motion. Examples of this phenomenon are given which are associated with aperiodic displacive phase transitions between periodic states.