Elasticity

Longitudinal strain, $ε_L=0.2\%=0.002$Poisson's ratio, $\mu=0.3$We are asked to find the volume strain $(ε_V)$.Step 1: Recall the formula for volume strainFor small deformations,Volume strain $= \varepsilon_V = \varepsilon_L (1-2\mu)$However, to be more precise, the change in volume $\frac{\Delta V}{V}=ε_L+2ε_t$,where $\varepsilon_t = -\mu \varepsilon_L$ (lateral strain is negative if the material stretches longitudinally).So,$\;\frac{\Delta V}{V}=\varepsilon_L+2(-\mu \varepsilon_L)$$\;\Rightarrow ε_V=ε_L(1-2\mu)$Step 2: Substitute the values$\;ε_V=0.002(1-2\times 0.3)$$\;ε_V=0.002(1-0.6)$$\;ε_V=0.002\times 0.4=0.0008$Step 3: Convert to percentage$ε_V=0.0008\times 100=0.08\%$ Final Answer:$0.08\%$