Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов. Читать онлайн. Newlib. NEWLIB.NET

Автор: Группа авторов
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: Техническая литература
Год издания: 0
isbn: 9781118799499
Скачать книгу
is fortunately not the case in noncrystalline TD networks.

Schematic illustration of the deformation of a finite network of four nodes. Lines represent linear constraints. (Top) floppy network with only four constraints originating from the edges. (Middle) addition of a diagonal constraint makes the network rigid. (Bottom) addition of a second, dependent diagonal constraint does not change the rigidity of the network.

      2.4 Degrees of Freedom and the Network Deformation Modes

      Without constraints (as, for example, in an ideal gas), each atom has d coordinate degrees of freedom where d is the dimension of the network. As constraints are added at a vertex, its coordinate degrees of freedom decrease. If n is the average number of independent constraints per vertex, then the average degrees of freedom per vertex, f, in a network is given by

      (2)equation

      If n < d, then f is positive and the network can deform without expenditure of energy. Such a network is termed “floppy” (or hypostatic) and has exactly f floppy (soft or low frequency) modes per vertex. The number of degrees of freedom decreases as n increases. When n > d, the network is over‐constrained and is termed “stressed‐rigid” (or hyperstatic). The excess (nd) constraints in a stressed‐rigid network are dependent if such a network exists. The transition from floppy to stressed‐rigid takes place at n = d (i.e. at f = 0), which marks the disappearance of floppy modes and the onset of network rigidity. Such a network is called isostatic (Figure 1).

      As mentioned before, PCT considers only chemically ordered networks which, according to Zachariasen [2], are TD networks of structural units made up of corner‐sharing rigid polyhedra. In such networks, it is convenient to treat the shared corners of the polyhedra as the vertices and the polyhedral structural units as links. The rigidity of a network then arises from the rigidity of the structural units as well as from the vertex‐connectivity condition (i.e. the fact that all corners of polyhedral units are shared among a certain number of units). These constraints resulting from the rigidity of structural units and connectivity are termed polyhedral constraints.

      3.1 Rigidity of Polyhedral Structural Units

      An 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 δd (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:

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

e-mail: [email protected]