# Does the volume change under the tensile or compressive stress?

This confusion arise because my teacher says that volume does not change under tensile/compressive stress within the elastic limit of the material ( consider metal here). But as far as I can see via applying Poisson's ratio it does ( otherwise lateral strain would be proportional to square root of longitudinal strain). So does the volume chaneg under tensile stress ? What might be the molecular basis of such?

I defined a quantity (cause it was simple to calculate) $$\frac {\Delta A}{A}$$ and then found that

via **Poisson's ratio

$$\frac {\Delta A}{A} = \alpha \epsilon _{lon}( \alpha \epsilon _{lon} +2)$$

whereas the one derived via assuming the volume to be constant (i.e., $$AL = A_0 L_0$$) was $$\frac {\Delta A}{A} = \frac {\epsilon _{lon}}{\epsilon _{lon} +1}$$

Here $$\alpha = \frac {\epsilon_{lat}}{\epsilon_{lon}}$$

• If the length changes under compression for a bar then will the diameter increase? Nov 23 '19 at 16:41
• @Solar yes it will (Poisson's ratio)
– user23622
Nov 23 '19 at 16:43
• May I see your equations ? Nov 23 '19 at 16:44
• In fact, Poisson thought this was correct, and therefore his ratio was 0.5 for all materials. And one of his colleagues Cagniard-Latour measured this incorrect value for brass, and so "proved" Poisson was right. Well, everybody makes mistakes sometimes - including your teacher :) Nov 23 '19 at 18:36
• @SolarMike "the net volume change is????" - either positive, negative, or zero. I'm not sure where you are trying to get to here. Nov 23 '19 at 18:40

Yes, the volume changes.

The relative change of volume $$ΔV/V$$of a cube due to the stretch of the material : Using $$V = L^3$$ and

$$V + \Delta V = (L + \Delta L)\left(L + \Delta L'\right)^2$$:

:$$\frac{\Delta V}{V} = \left(1 + \frac{\Delta L}{L} \right)\left(1 + \frac{\Delta L'}{L} \right)^2 - 1$$

Using the above derived relationship between $$\Delta L$$ and $$\Delta L'$$:

:$$\frac {\Delta V} {V} = \left(1+\frac{\Delta L}{L} \right)^{1-2\nu} - 1$$

and for very small values of $$\Delta L \ and \ \Delta L'$$, the first-order approximation yields:

$$\frac {\Delta V} {V} \approx (1-2\nu)\frac{\Delta L}{L}$$

For isotropic materials, we can use Lamé parameters

$$\frac{1}{2} - \frac{E}{6K}$$

where K is bulk modulus and E is elastic modulus or Young's modulus. wikipedia link