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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!

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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.

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  • $\begingroup$ Thanks, this is a really helpful answer! I had no idea there were two types of strain, true strain and engineering strain. I also really like the notes you linked, they're very clear and insightful! Once again, thanks a bunch! $\endgroup$
    – 13509
    Commented Aug 25, 2020 at 20:09

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