Network with four-fold symmetry

The free energy density of deformation of a network with four-fold symmetry has already been derived as

(39.1)

This equation suggests that when and , , which is called pure shear. Similarly, we can have , so that , which is called simple shear. We can have also  and , leading to  which is known as area compression. Thus, it is possible that for same , the network can deform in either of all these three different ways. In other words, without any other constraint, the network can have degenerate states. Notice that the network has 2 springs and 1 vertex per rectangular unit called plaquette, so that when each spring stretches by equal amount from unstressed length  to , the enthalpy per vertex is obtained as

(39.2)

The spring length that minimizes  is obtained as

(39.3)

Equation 39.3 suggests that the spring length and the area per vertex of the plaquette diverges for

(39.4)

because beyond this critical value the tension term dominates. Under positive pressure, the fourfold network collapses because all plaquettes have the same energy