Elasticity — Question 8

Question 8

Axial stretching can leave volume nearly unchanged

A slender isotropic specimen has $E=100\,\mathrm{GPa}$ and Poisson ratio $\nu=.30$. Under uniaxial tensile stress $100\,\mathrm{MPa}$, transverse surfaces are free. Use $\epsilon=\sigma/E$ and transverse strains $\epsilon_t=-\nu\epsilon$.

1. Find axial strain, fractional area change and fractional volume change to first order.

2. What volume change would the same axial strain produce if $\nu=.50$, and does that imply zero elongation?

Question 8 - Solution

Axial stretching can leave volume nearly unchanged

(a) Derivation and calculation

The axial and transverse strains are $$\boxed{\epsilon=\sigma/E=(100\times10^6)/(100\times10^9)=.00100},$$ $$\epsilon_t=-\nu\epsilon=-.30(.00100)=-.000300.$$ To first order, $$\boxed{\Delta A/A=2\epsilon_t=-.000600},$$ $$\boxed{\Delta V/V=\epsilon+2\epsilon_t=(1-2\nu)\epsilon=.000400}.$$

(b) Result and interpretation

At the same axial strain with $\nu=.50$, $$\boxed{\Delta V/V=(1-2(.50))\epsilon=0}.$$ Nevertheless $\Delta L/L=\epsilon=.00100\ne0$. The transverse area contraction $\Delta A/A=-.00100$ offsets elongation to first order, preserving volume while changing shape.

Physical check

The volume-change statement uses the specified small-strain approximation; zero volume strain does not imply zero axial strain.