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!

Chemomechanics · Answer

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.

Answered by Chemomechanics on January 29, 2021

Young Modulus in Finn's Thermal Physics

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