理想弹性丝的等温杨氏模量和线膨胀系数证明

首先求出等温杨氏模量： $$ \begin{aligned} Y &= \frac{L}{A}\left(\frac{\partial F}{\partial L}\right)_T \ &= \frac{L}{A}bT\left(\frac{\partial}{\partial L}\left(\frac{L}{L_0}-\frac{L_0^2}{L^2}\right)\right)_T \ &= \frac{L}{A}bT\left(\frac{1}{L_0}+\frac{2L_0^2}{L^3}\right) \ &= \frac{bT}{A}\left(\frac{L}{L_0}+\frac{2L_0^2}{L^2}\right) \end{aligned} $$ 所以 (1) 成立。

接着求出线膨胀系数： $$ \begin{aligned} \alpha &= \frac{1}{L}\left(\frac{\partial L}{\partial T}\right)F \ &= \frac{1}{L}\left(\frac{\partial L}{\partial T}\right){L_0}\left(\frac{\partial L_0}{\partial T}\right)_F+\frac{1}{L}\left(\frac{\partial L}{\partial L_0}\right)_T\left(\frac{\partial L_0}{\partial T}\right)_F \ &= \frac{\alpha_0}{L}\left(\frac{\partial L}{\partial L_0}\right)_T+\frac{1}{L}\left(\frac{\partial L}{\partial L_0}\right)_T\frac{\mathrm{d} L_0}{\mathrm{d} T} \ &= \frac{\alpha_0}{L}\left(\frac{\partial L}{\partial L_0}\right)_T\left(1-\frac{L}{L_0}\right)+\frac{1}{L}\left(\frac{\partial L}{\partial L_0}\right)_T\frac{\mathrm{d} L_0}{\mathrm{d} T} \ &= \frac{\alpha_0}{L}\left(\frac{L_0^2}{L^2}-1\right)+\frac{1}{L}\frac{\mathrm{d} L_0}{\mathrm{d} T} \ &= \alpha_0-\frac{1}{T}\frac{L^3}{L_0^3-1}\frac{\mathrm{d} L_0}{\mathrm{d} T} \ &= \alpha_0-\frac{1}{T}\frac{L^3}{L_0^3+2}\frac{\mathrm{d} L_0}{\mathrm{d} T} \end{aligned} $$ 其中在第三行到第四行用到了全微分的性质，即 $\mathrm{d} L = \left(\frac{\partial L}{\partial L_0}\right)T\mathrm{d} L_0+\left(\frac{\partial L}{\partial T}\right){L_0}\mathrm{d} T$，因为 $F$ 是 $L$ 和 $T$ 的函数，所以 $F$ 的全微分可以表示为 $\mathrm{d} F = \left(\frac{\partial F}{\partial L}\right)_T\mathrm{d} L+\left(\frac{\partial F}{\partial T}\right)_L\mathrm{d} T$。另外，在第五行到第六行用到了 $\frac{\mathrm{d} L_0}{\mathrm{d} T}=\frac{L_0}{T}\left(\frac{\partial L_0}{\partial T}\right)_F$。

所以 (2) 成立。