Abstract
Scalar mobility counting rules and their symmetry extensions are reviewed for finite frameworks and also for infinite periodic frameworks of the bar-and-joint, body-joint and body-bar types. A recently published symmetry criterion for the existence of equiauxetic character of an infinite framework is applied to two long known but apparently little studied hinged-hexagon frameworks, and is shown to detect auxetic behaviour in both. In contrast, for double-link frameworks based on triangular and square tessellations, other affine deformations can mix with the isotropic expansion mode.
Highlights
Use of counting rules in the study of rigidity and mobility of frameworks has a venerable history, going back to Maxwell’s 1864 rule for bar-and-joint frameworks [1], and its extension to account for the balance between the numbers of mechanisms (m) and states of self-stress (s)
An interesting feature of the calculation for this framework is that Γ(m) − Γ(s) evaluates to exactly (ΓT × ΓT ) − ΓT − ΓR, the representation of the affine deformations of the unit cell denoted by Γa in previous work [19]. This results from the exact cancellation of body freedoms and bar constraints, which is an instance of the symmetry extension of the notion of local isostaticity introduced in [18]
Analysis of some examples of hinged frameworks using periodic symmetry reveal the essential equiauxetic mechanism shared by these systems
Summary
Use of counting rules in the study of rigidity and mobility of frameworks has a venerable history, going back to Maxwell’s 1864 rule for bar-and-joint frameworks [1], and its extension (due to Calladine [2]) to account for the balance between the numbers of mechanisms (m) and states of self-stress (s). It turns out that “counting with symmetry” can often detect mechanisms and/or states of self-stress that cancel out Symmetry 2014, 6 in purely numerical terms in the balance m − s, but have different characteristic symmetries and do not cancel in the reducible representation Γ(m) − Γ(s) [5] Examples of this symmetry-based approach include symmetry-adapted versions of the Maxwell Rule [5], the mobility criterion for body and joint assemblies [6], and for bar-body systems [7]. Direct extension of symmetric Maxwell and mobility rules from point groups and finite objects to unit-cell symmetry in infinite repetitive systems [18] gives criteria for detection of repetitive (zero-wavevector) mechanisms and states of self-stress. An “equiauxetic mode” is a zero stiffness eigenvector that corresponds to equal expansion/contraction in perpendicular directions
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
