材料力学 第2章 应力状态分析

2-1 应力状态

2.1.1 一点的应力状态

\[\boldsymbol{\sigma} = \begin{bmatrix} \sigma_x & \tau_{xy} & \tau_{xz} \\ \tau_{xy} & \sigma_y & \tau_{yz} \\ \tau_{xz} & \tau_{yz} & \sigma_z \end{bmatrix}\]

$\tau_{ij}$的第一个下标为截面法线方向，第二个下标为应力分量方向

2.1.2 切应力互等定律

\[\tau_{ij}=\tau_{ji}\]

2.1.3 任意截面的应力公式

\[\begin{cases} \sigma_{x'} = \dfrac{\sigma_x+\sigma_y}{2} + \dfrac{\sigma_x-\sigma_y}{2}\cos 2\alpha + \tau_{xy}\sin 2\alpha, \\[8pt] \tau_{x'y'} = -\dfrac{\sigma_x-\sigma_y}{2}\sin 2\alpha + \tau_{xy}\cos 2\alpha. \end{cases}\]

推导过程

\[\boldsymbol{\sigma}' = \boldsymbol{Q}\boldsymbol{\sigma}\boldsymbol{Q}^{\mathrm T}\] \[\boldsymbol{Q} = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}\] \[\begin{bmatrix} \sigma_{x'} & \tau_{x'y'} \\ \tau_{x'y'} & \sigma_{y'} \end{bmatrix} = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix} \begin{bmatrix} \sigma_x & \tau_{xy} \\ \tau_{xy} & \sigma_y \end{bmatrix} \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}\]

2.1.4 主应力和主平面

\[\frac{\mathrm{d}\sigma_{x'}}{\mathrm{d}\alpha} = 2\left( -\frac{\sigma_x-\sigma_y}{2}\sin 2\alpha + \tau_{xy}\cos 2\alpha \right) =0\]

可得正应力取得极值时切应力为0

切应力为零的平面是主平面，主平面的外法线方向为主方向，主方向上的正应力称为主应力

\[\left. \begin{array}{c} \sigma_{\max} \\[6pt] \sigma_{\min} \end{array} \right\} = \frac{\sigma_x+\sigma_y}{2} \pm \sqrt{ \left( \frac{\sigma_x-\sigma_y}{2} \right)^2 + \tau_{xy}^{\,2} }\] \[\tan 2\alpha_{\sigma} = \frac{2\tau_{xy}}{\sigma_x-\sigma_y} \quad\Longrightarrow\quad \left\{ \begin{array}{c} \alpha_{\sigma} \\[6pt] \alpha_{\sigma}+90^\circ \end{array} \right.\]

2.1.5 主切应力和主切平面

\[\left. \begin{array}{c} \tau' \\[6pt] \tau'' \end{array} \right\} = \pm \sqrt{ \left( \frac{\sigma_x-\sigma_y}{2} \right)^2 + \tau_{xy}^{\,2} }\] \[\sigma' = \frac{\sigma_x+\sigma_y}{2}\] \[\tan 2\alpha_{\tau} = -\frac{\sigma_x-\sigma_y}{2\tau_{xy}} \quad\Longrightarrow\quad \left\{ \begin{array}{c} \alpha_{\tau} \\[6pt] \alpha_{\tau}+90^\circ \end{array} \right.\]

2-2 扩展知识

2.2.1 主应力和正应力关系

\[\sigma_{\max}+\sigma_{\min} = \sigma_x+\sigma_y\]

2.2.2 主应力和主切应力关系

\[\left. \begin{array}{c} \tau' \\[6pt] \tau'' \end{array} \right\} = \pm\frac{1}{2} \left( \sigma_{\max}-\sigma_{\min} \right)\]

2.2.3 y方向上的正应力

\[\sigma_{y'} = \frac{\sigma_x+\sigma_y}{2} - \frac{\sigma_x-\sigma_y}{2}\cos 2\alpha - \tau_{xy}\sin 2\alpha\]

其实我觉得直接代入$\alpha+\frac{\pi}{2}$即可

2.2.4 主应力方位角的判断方法

\[\sigma_x>\sigma_y, \qquad \alpha\text{ 即为主方位角 }\alpha_{\sigma}\]