Abstract
A crystallographic bar-joint framework, C in Rd, is shown to be almost periodically infinitesimally rigid if and only if it is strictly periodically infinitesimally rigid and the rigid unit mode (RUM) spectrum, Ω (C), is a singleton. Moreover, the almost periodic infinitesimal flexes of C are characterised in terms of a matrix-valued function, ΦC(z), on the d-torus, Td, determined by a full rank translation symmetry group and an associated motif of joints and bars.
Highlights
The rigidity of a crystallographic bar-joint framework, C, in the Euclidean spaces, Rd, with respect to periodic first order flexes is determined by a finite matrix, the associated periodic rigidity matrix.For essentially generic frameworks of this type in two dimensions, there is a deeper combinatorial characterisation, which is a counterpart of Laman’s characterisation of the infinitesimal rigidity of generic placements of finite graphs in the plane
In Owen and Power [6] and Power [8], the rigid unit mode (RUM) spectrum was formalised in mathematical terms as a subset, Ω(C), of the d-torus, Td, or as an equivalent subset of [0, 1)d, which arises from a choice of translation group T
We show that for crystal frameworks whose RUM spectrum decomposes into a finite union of linear components, there is a corresponding vector space decomposition of the almost periodic flex space
Summary
The rigidity of a crystallographic bar-joint framework, C, in the Euclidean spaces, Rd , with respect to periodic first order flexes is determined by a finite matrix, the associated periodic rigidity matrix. In Owen and Power [6] and Power [8], the RUM spectrum was formalised in mathematical terms as a subset, Ω(C), of the d-torus, Td , or as an equivalent subset of [0, 1)d , which arises from a choice of translation group T This set of multi-phases is determined by a matrix-valued function, ΦC (z), on. We show that for crystal frameworks whose RUM spectrum decomposes into a finite union of linear components, there is a corresponding vector space decomposition of the almost periodic flex space. The flexes in these subspaces are periodic in specific directions associated with certain symmetries of the crystallographic point group.
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