Question

A material has Poisson's ratio \(\nu=0.5\) and Young's modulus \(E=2500\ \text{MPa}\). Under a hydrostatic stress of magnitude \(10\ \text{MPa}\), the percentage change in its volume is _________,(answer in integer).

Hint:

Remember \(\varepsilon_v=3(1-2\nu)\sigma/E\). If \(\nu=0.5\), the material is (ideally) incompressible and volume change under hydrostatic loading is zero.

Answer 1

Step 1: Volumetric strain under hydrostatic stress.
For an isotropic linear elastic solid under equal principal stresses \(\sigma\) (tension \(+\)), the volumetric strain is \[ \varepsilon_v=\varepsilon_x+\varepsilon_y+\varepsilon_z =3\,\frac{1-2\nu}{E}\,\sigma. \] Step 2: Substitute the given \(\nu\).
Here \(\nu=0.5 \Rightarrow (1-2\nu)=0\). Hence \[ \varepsilon_v = 3\cdot \frac{0}{E}\cdot \sigma = 0 \quad \Rightarrow \quad \frac{\Delta V}{V}=0. \] Therefore percentage change \(= 0\times 100\%=\boxed{0\%}\). (Note: \(\nu=0.5\) implies incompressible behavior; hydrostatic stress changes pressure but not volume.)