Four fold symmetry

In this case the number of independent is reduced from 6 by the condition that the relation remains independent of the co-ordinate inversions:  and . Since, components of a tensor transform as products of the corresponding co-ordinates, the component of  with odd number of x and  y should change sign. Hence these components should vanish because stress-strain should not change sign due to single inversion. Secondly, upon clock-wise rotation by an angle of  about an axis with four-fold symmetry yields  and , implying . Thus these symmetry considerations result in three different modulii:  and . Combining these modulii three different constants are defined:

	(37.14)
(pure shear)
(simple shear)

The expression for energy density can be written as:

(37.15)