Network with six-fold symmetry under stress

Let us consider a network which is made of identical springs of length  under zero stress. The network is then subjected to isotropic two dimensional tension so that the area of each triangle increases although it remains equilateral. The length of each spring increases to  which can be calculated by minimizing the enthalpy of the network

(38.1)

Here  is the energy of the springs and is the area of the network. Notice, there are three springs per vertex and energy for each vertex, so that energy per vertex is

(38.2)

And the area per vertex is

(38.3)

Hence the enthalpy per vertex is obtained by substituting in 38.1 the expressions for 37.16 and 38.3

(38.4)

The spring length that minimizes  for a given tension can be obtained by equating  to zero, which yields

(38.5)

Substituting  in equation 17.5, we obtain the minimum value for ,

(38.6)

And the area per vertex is derived as,

(38.7)

The area diverges as the tension approaches a critical value .