Network with four-fold symmetry (contd...)

Following the procedure described earlier, the area modulus of the network can be related to the stresses network as,

,

(39.5)

Notice that vanishes at zero implying that the spring energy of the network under simple shear deformation vanishes under zero tension. That is the plaquette is not stable under simple shear mode of deformation.

The square plaquette approximation also implies that the area of the plaquette at zero temperature is  at . However, in practice this estimation does not appear to be correct, in fact, in molecular simulation too this value is calculated to be . Further difference is observed with the elastic modulii of the network. This difference is actually linked to the degeneracy of the system at low stress. The square form is the largest area of the plaquette while the system can explore many other configurations. An estimation of the area per vertex can be made at low temperature condition in which the springs can be assumed to have lengths very close to :

(39.6)

Here the plaquette is in the form of a parallelogram with the angle between adjacent sides  which varies from  to . The mean area is estimated to be  which corroborates very well with that calculated from rigorous MD simulation.