Module 4 : Nonlinear elasticity
  Lecture:23 : Homogeneuos Strain
 

 

Now consider the following situation in which a cube is extended along the three axes so that we have the following extension ratios:

and
  (23.4)
 

What can we say about the uniqueness of the pressure, or, can it be uniquely defined? Let us say, for simplicity that . The we can think about the following two equations, at their intersections lie the solution of .

Needless to say that  will intersect  at and . Furthermore, when, , and when , . Hence, the  curve look like the one plotted in figure.

 Since is positive, the line will lie beneath the line, so that curve intersect  either at one location or at three locations  and . The figure shows that one of the solutions, for example  should be less than , whereas, other two solutions  and  should be between  and . Then from equation (23.4)

and

 is real and  and are complex numbers. Therefore, it is obvious that there is only solution possible for .