Young Modulus in Finn's Thermal Physics

Question

In Finn's Thermal Physics (equation 2.4), the Young modulus $Y$ of a stretched wire with tension $F$ is given to be

$$Y = \frac{L}{A} \left( \frac{\partial F}{\partial L}\right)_T$$

However, usually the Young modulus is defined to be the ratio of stress and strain, specifically

$$Y = \frac{\sigma}{\varepsilon} = \frac{F L_0}{A \Delta L} = \frac{FL_0}{A(L-L_0)}\implies dF = \frac{AY}{L_0} dL$$

this would suggest that the relationship given in Finn's textbook should actually be

$$Y = \frac{L_0}{A} \left( \frac{\partial F}{\partial L}\right)_T$$So I wondered why they have used $L$ instead of $L_0$ in the definition. Is it a mistake? Thanks!

Answer 1

When assuming infinitesimal strains, one can use either $Y=\frac{L}{A}\left(\frac{\partial F}{\partial L}\right)_T$ or $Y=\frac{L_0}{A}\left(\frac{\partial F}{\partial L}\right)_T$, depending on what's convenient; the difference is negligible. The former is related to the definition of the true strain $e$ from $de=dL/L$ (so that $e=\ln(1+\Delta L/L)$) and the latter to the definition of the engineering strain $\varepsilon=\Delta L/L$. The difference between the true strain and engineering strain is, of course, also negligible in this context.