Network with six-fold symmetry under stress (contd...)

Similarly analysis shows that the effective shear modulus of the network can be deducted as, , which shows that the effective shear modulus increases with the stress . The following table summarizes the above results.

The stress dependent modulii can be used to estimate the Poisson ratio  defined as,

(38.13)

in which stress is applied along the x axis. For a three dimensional isotropic material we had shown earlier the following relations for the shear and bulk modulii:

,

(38.14)

From which the Poisson ratio can be obtained as,

(38.15)

It can be shown that the Poisson ratio in two dimension can be obtained as,

(38.16)

For a triangular network with zero stress with  and , the Poisson is defined as, . However, for stress dependent network, the Poisson ratio can be defined as,

(38.17)

Thus, the poisson ratio becomes negative for six-fold network under tension for , implying that such a network will expand laterally when stretched in the longitudinal direction.