Six fold networks in 2D

Figure 37.1 shows a two dimensional network which has six fold rotational symmetry through the vertices. We analyze this problem by changing the coordinate system from Cartesian coordinates x and y to complex coordinates  and . Then the free energy  contains terms. Now a rotation about the origin of the x, y coordinate by an angle  changes the coordinates from  to  and , in other word from  to   . Since six fold symmetry implies that the modulii remain unchanged because of rotation of the axis through  i.e.  and . Only non-zero components of  that remains unchanged by this transformation are those which contains equal number of times  and  because . Only two components of  satisfy this symmetry  and . The change in free energy density can be then written as

(37.8)

                                 
Which contains results from four combinations involving  and two combinations involving . We can replace the strain components by those in the Cartesian components in which we use components of tensor transform as products of the corresponding coordinates. Since,

,  and

(37.9)

we can write,

,

(37.10)

From equation 37.8, the expression for energy density can be written as,

(37.11)

We can relate the  to modulii to more common forms of modulii, e.g. area compression modulus  and shear modulus :

(37.12)

Hence, equation 37.11 changes to

(37.13)

Equation 37.13 is very similar to that for isotropic deformation implying that in two dimension, both isotropic materials and six fold symmetry are represented by two different elastic modulii.