We have derived earlier the following general relation of strain tensor in terms of the rate of change of displacement vector  with position vector 

(37.1)

The subscripts and k represent the axes in Cartesian coordinates. In two dimension there are four components of and in three dimension there are nine components. It was shown earlier that  is symmetric with respect to the indices i and j. We derived also that for small deformation we can neglect the last term in equation 37.1 yielding

(37.2)

                                 
Using Hooke’s law, the stress components was related to the strains as;

(37.3)

In which are called the material constants, elastic modulii. The corresponding expression for strain energy density is a quadratic function of strain tensor components and hence can be written as,

(37.4)

Symmetry considerations greatly reduce the number of independent constants from for three dimension and for two dimension to much smaller numbers, minimum number of constants being required for isotropic systems for which all directions are equivalent. For example, since  is symmetric for exchange between i and j,  is symmetric for pair exchange between i and j, k and l, so that,

(37.5)

Further since product  are symmetric for interchange of indices ij and kl, so that

(37.6)

These two symmetry conditions decreases the number of constants to 21 for three dimension and 6 for two dimensions. For 2D, these constants are,

Symmetry in the material can further reduce the number of constants. We will now discuss two dimensional elastic networks with four fold and six fold symmetry to demonstrate this fact.