A symmetry extension of Maxwell's rule for rigidity of frames

Flexibility in MOFs: do scalar and group-theoretical counting rules work?

“…Several symmetry extended versions of the counting rules have been developed by Fowler and Guest, for bar-and-joint frameworks 38 , for body-and-joint systems 28 , with and without taking periodicity into account 31 . From the previous section on scalar rules, we have seen that in order to treat the joints between carboxylate linkers and SBUs correctly as chiefly hinge-like, a body-and-joint description is necessary.…”

Symmetry Perspectives on Some Auxetic Body-Bar Frameworks

“…The calculation of the net mobility Γ(m) − Γ(s) for the single-link hexagonal framework (shown in Figure 1a) is set out below in the usual tabular form [5]:…”

“…In recent years, extensions of these classic counting rules have been made, employing arguments based on point-group symmetry. It turns out that "counting with symmetry" can often detect mechanisms and/or states of self-stress that cancel out 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].…”

“…It turns out that "counting with symmetry" can often detect mechanisms and/or states of self-stress that cancel out 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]. Symmetry analyses have dealt with classes of system such as rotating rings of tetrahedra [8], [N ]-loops [9] toroidal deltahedra [10] and various mechanical toys and models [11][12][13].…”

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“…However, in the particular planar configuration shown, the structure attains D 3h symmetry, where it has a single state of self-stress and four mechanisms. The symmetry form of the Maxwell rule for pin-jointed frameworks [Fowler and Guest 2000] gives a full account, and yields As, by inspection, (s) = A 1 ≡ 0 , the four mechanisms span…”