Pure Shear Questionthanks for your help

[physicsbuff123]

Homework Statement

Hey everyone. I was practicing some problems, and I came across one that seems pretty simple, yet I am having difficulty proving the answer. The question asks us to show that a state of pure shear causes no volume change in linear elastic deformation. I have a very good understanding on shear stresses and strains, however, this problem seems harder then it actually is. If someone could point me in the right direction I would appreciate it greatly!

Homework Equations

Strain Eq, Stress Eq, Young's Modulus (standard elementary equations)

The Attempt at a Solution

I have literally spent hours reading my book and working out problems, such as these. I seem to understand all of the formulas, however, I am having trouble proving the statement.

 

[radou]

What is the expression for volume deformation, i.e. how does it relate with the principal stresses? What are the values of these stresses for the pure shear state?