Why would density increase when Poisson's ratio > 0.5?

[theBEAST]

Homework Statement

Does anyone understand why density would increase when the poisson ratio is greater than 0.5 as indicated in this slide from my professor:

Does this density increase apply to a system undergoing tension or compression or both?

 

[Simon Bridge]

You have to think about what Poisson's ration means ... how does a compression affect volume for different Poisson's Ratios?

 

[theBEAST]

You have to think about what Poisson's ration means ... how does a compression affect volume for different Poisson's Ratios?

Alright so here calculated what the volume change would be for when Poisson's ratio < 0.5. It told me that the volume would decrease which implies that the density increases. But according to the lecture notes this is not true. Instead when the ratio is < 0.5 the density decreases...

Note: I also calculated for the volume when Poisson's ratio = 0.5, which according to the notes there should be no volume change. But in my calculations I found there is a volume change...

What am I doing wrong?

 

[Chestermiller]

Alright so here calculated what the volume change would be for when Poisson's ratio < 0.5. It told me that the volume would decrease which implies that the density increases. But according to the lecture notes this is not true. Instead when the ratio is < 0.5 the density decreases...

Note: I also calculated for the volume when Poisson's ratio = 0.5, which according to the notes there should be no volume change. But in my calculations I found there is a volume change...

What am I doing wrong?

Let a bar be stretched by ε_x. For a Poisson ratio of 0.5, the strains in the y and z directions are:

ε_y=-0.5 ε_x

ε_z=-0.5 ε_x

The volumetric strain is the sum of the three linear strains, and is equal to zero.

Same problem with Poisson ratio = 0.3

ε_x + ε_y + ε_z = 0.4 ε_x

So the volume increases, and the density decreases when a bar is stretched under tension and the material has a Poisson ratio of <0.5