Auxetic regions in large deformations of periodic frameworks

In materials science, auxetic behavior refers to lateral widening upon stretching. We investigate the problem of finding domains of auxeticity in global deformation spaces of periodic frameworks. Case studies include planar periodic mechanisms constructed from quadrilaterals with diagonals as periods and other frameworks with two vertex orbits. We relate several geometric and kinematic descriptions.Comment: Presented at the International Conference on "Interdisciplinary Applications of Kinematics" (IAK18), Lima, Peru, March 201

The rigidity of periodic body-bar frameworks on the three-dimensional fixed torus

We present necessary and sufficient conditions for the generic rigidity of body-bar frameworks on the three-dimensional fixed torus. These frameworks correspond to infinite periodic body-bar frameworks in $\mathbb{R}^3$ with a fixed periodic lattice.Comment: 31 pages, 12 figure

Polynomials for Crystal Frameworks and the Rigid Unit Mode Spectrum

To each discrete translationally periodic bar-joint framework \C in \bR^d we associate a matrix-valued function \Phi_\C(z) defined on the d-torus. The rigid unit mode spectrum \Omega(\C) of \C is defined in terms of the multi-phases of phase-periodic infinitesimal flexes and is shown to correspond to the singular points of the function z \to \rank \Phi_\C(z) and also to the set of wave vectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium, which corresponds to \Phi_\C(z) being square, the determinant of \Phi_\C(z) gives rise to a unique multi-variable polynomial p_\C(z_1,\dots,z_d). For ideal zeolites the algebraic variety of zeros of p_\C(z) on the d-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealised framework rigidity and flexibility and in particular leads to an explicit formula for the number of supercell-periodic floppy modes. In the case of certain zeolite frameworks in dimensions 2 and 3 direct proofs are given to show the maximal floppy mode property (order $N$). In particular this is the case for the cubic symmetry sodalite framework and some other idealised zeolites.Comment: Final version with new examples and figures, and with clearer streamlined proof