Counting components, and then comparing the number of constraints with number of degrees of freedom available to a structure, is a good first step in evaluating likely structural behaviour. Maxwell first described this in 1864 when he stated that, in general, a structure with j joints would require 3j-6 bars to make it rigid. Later Calladine generalised this idea by pointing out that the difference between the number of bars and 3j-6 counts the difference between the number of mechanisms and the number of states of self-stress. Sometimes, just simple counting can lead to profound insights, such as showing that any stiff repetitive structure must necessarily be overconstrained.
This talk will introduce the idea that any rule that involves counting components can be expanded to a more general symmetry version that involves counting the symmetries of sets of components, and that this counting can practically be done by simply considering the number of components that are unshifted by particular symmetry operations.
This provides useful insight into why certain symmetric structures are able to move despite apparently having enough members to make them rigid, or that tensengrity structures can be rigid without having enough members.
The talk will describe a recent result on 'auxetic' materials: a symmetry criterion that shows when a periodic system made up of bars, bodies and joints has an 'equiauxetic' mechanism, that is, show the limiting behaviour of Poisson ratio equal to -1, with equal expansion/contraction in all directions. Such systems can provide good models for the design of lattice materials with high, stretching-dominated, shear modulus, but low, bending-dominated, bulk modulus.