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3.1 Rigidity of Polyhedral Structural Units
ОглавлениеAn isolated single regular polyhedral unit can be specified by two parameters: the dimension (δ) of the unit and its number of vertices (V). The dimension of the unit is the minimum dimension necessary to embed it. For example, δ = 1 for a rod, 2 for a triangular unit, and 3 for a tetrahedron. It is clear that δ ≤ d (the dimension of the network) and that V ≥ (δ + 1).
When a regular polyhedral structural unit is rigid, the total number, Nu, of independent constraints in the unit satisfies the following relation:
(3)
Table 1 Degrees of freedom (f) of d‐dimensional TD networks of rigid units (δ, V) with C units sharing a vertex (with the assumption h = θ = 0) based on Eqs. (4) and (5).
| Structural unit | δ | V | n u | d | C | f |
|---|---|---|---|---|---|---|
| Rod | 1 | 2 | 0.5 | 2 | 3 | 0.5 |
| 3 | 4 | 1 | ||||
| 3 | 6 | 0 | ||||
| Triangle | 2 | 3 | 1 | 2 | 2 | 0 |
| 3 | 2 | 1 | ||||
| Square | 2 | 4 | 1.25 | 2 | 2 | −0.5 |
| Tetrahedron | 3 | 4 | 1.5 | 3 | 2 | 0 |
| 3 | 3 | −1.5 | ||||
| Octahedron | 3 | 6 | 2 | 3 | 2 | −1 |
| Cube | 3 | 8 | 2.25 | 3 | 2 | −1.5 |
Therefore, the number of independent constraints per vertex (nu = Nu/V) in a rigid unit is
(4)
A major advantage of PCT is that Eq. (3) counts correctly the number of independent constraints in a rigid unit. From Eq. (4) and the values of nu listed in Table 1 for several simple polyhedral units, one sees that nu increases with both V (for fixed δ) and δ (for fixed V > δ). When a structural unit is non‐regular and rigid, other parameters, in addition to δ and V, are needed to specify the structural unit.
