极坐标下求解

平衡微分方程

径向的平衡方程：

$$\begin{aligned} 0=&-\sigma_{\rho}\cdot\rho d \varphi + \left(\sigma_{\rho}+\frac{\partial \sigma_{\rho}}{\partial \rho}d\rho\right)\left(\rho+d\rho\right)d\varphi -\\ &\left(2\sigma_{\varphi}+\frac{\partial\sigma_{\varphi}}{\partial \varphi}d\varphi\right) d\rho\sin\frac{d\varphi}{2} +\\ &\frac{\partial\tau_{\varphi\rho}}{\partial\varphi}d\varphi d\rho \cos\frac{d\varphi}{2} + f_{\rho}\rho d\varphi d\rho \\ =& \sigma_{\rho}d\rho d\varphi + \frac{\partial \sigma_{\rho}}{\partial \rho}\rho d\rho d\varphi - \sigma_{\varphi}d\rho d\varphi + \frac{\partial \tau_{\varphi\rho}}{\partial \varphi}d\varphi d\rho + f_{\rho}\rho d\varphi d\rho \end{aligned}$$

即

$$\frac{\partial\sigma_{\rho}}{\partial\rho} + \frac{1}{\rho}\frac{\partial\tau_{\varphi\rho}}{\partial\varphi} + \frac{\sigma_{\rho}-\sigma_{\varphi}}{\rho} + f_{\rho} = 0$$