Three dimensional networks

We will consider here free energy of deformation of both isotropic materials in three dimension and those with four-fold symmetry. We have shown earlier that in a generalized situation, a three dimensional system under stress is characterized by  number of elastic moduli ; for two dimension, the number of moduli is . The number of these elastic constants however decreases because of symmetry in the system. For example, since the components of strain  are symmetric with respect to the exchange between the subscript pairs  ,  and , we have the relations  and  which diminish the number of independent moduli to 21. The number of moduli gets further reduced for isotropic materials for which the strain components remain invariant for arbitrary rotations. In fact, there are only two quadratic combinations for strain tensor that remain independent with respect to rotation of the axes:  and . Therefore the free energy density function can be written as

(40.1)

Where  is the volume compression modulus and  is the shear modulus of the material. Thus the isotropic system requires two moduli in order to describe its strain energy density function. It can be shown that a system with hexagonal symmetry requires five independent elastic constants. Similarly a cubic system requires three moduli as observed in its expression of free energy,

(40.2)

Combining elastic moduli and strain tensor components, above equation is written as,

(40.3)

where, , ,