Stability of Equilibrium

Let us consider a cube of an incompressible, neo-Hookean material with unit edges in the unstrained state. It is strained to dimensions  under the action of three forces ,  and so that the strain energy stored in it can be written as,

(23.5)

and the work done by these forces is

(23.6)

The net energy can be expressed as the summation of the above two:

(23.7)

The equilibrium of forces for all possible values of ,  and  can be obtained by finding the minima of the above total energy, such that the following conditions are satisfied:

and

(23.8)

Writing  we can obtain the allowable variations ,  and  of ,  and  respectively, which are to satisfy the following condition:

(23.9)

Then  can be rewritten as a function of  and :

(23.10)

Then the quantities  and  can be written as,

(23.11)