Consider a rod of 1 m diameter and 2 m length. The Poisson's ratio of the material is 0.5. Assume the rod is stretched to 2.5 m length.
Now longitudinal strain is 0.5/2 = 0.25
lateral strain = longitudinal strain * Poisson's ratio $= 0.25\cdot 0.5 = 0.125$
Change in diameter is 0.125
Original volume $= \pi (1\cdot 1)/4\cdot 2 = 1.57$
Diameter is reduced to $1-0.125 =0.875$
New volume $= \pi (0.875\cdot 0.875)/4 \cdot 2.5 = 1.502$
The volume change is not zero when in comes to direct calculation. I know by $E=3K(1-2\nu)$, the volume change is zero, though.
Why is it so ?
Also, if you can give a valid explanation to it, Can this apply on a rectangular bar of square cross section?