材料力学 第3章 应变状态分析

3-1 应变的概念

线应变：某一方向上单位长度的相对伸长或缩短 （伸长为正，缩短为负）

\[\varepsilon_x =\lim_{\Delta x\to 0}\frac{\Delta u}{\Delta x} =\frac{\partial u}{\partial x}\]

切应变：互相垂直的两条线段之间直角的改变量 （使直角减小的切应变为正）

\[\gamma_{xy} = \lim_{\Delta x\to 0,\ \Delta y\to 0} (\alpha+\beta) = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\]

体积应变： 单位体积的体积改变

\[\theta = \frac{V'-V}{V} = \varepsilon_1+\varepsilon_2+\varepsilon_3 = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\]

$u$和$v$是单元体的位置，是和位置有关的函数

3-2 平面应变状态分析

3.2.1 斜方向应变

\[\varepsilon_{x'} = \frac{\varepsilon_x+\varepsilon_y}{2} + \frac{\varepsilon_x-\varepsilon_y}{2}\cos 2\alpha + \frac{\gamma_{xy}}{2}\sin 2\alpha\] \[\frac{\gamma_{x'y'}}{2} = -\frac{\varepsilon_x-\varepsilon_y}{2}\sin 2\alpha + \frac{\gamma_{xy}}{2}\cos 2\alpha\]

3.2.2 主应变和主切应变

\[\left. \begin{aligned} \varepsilon_{\max}\\ \varepsilon_{\min} \end{aligned} \right\} = \frac{\varepsilon_x+\varepsilon_y}{2} \pm \sqrt{ \left( \frac{\varepsilon_x-\varepsilon_y}{2} \right)^2 + \left( \frac{\gamma_{xy}}{2} \right)^2 } \quad \qquad \tan 2\alpha_{\varepsilon} = \frac{\gamma_{xy}} {\varepsilon_x-\varepsilon_y}\] \[\left. \begin{aligned} \frac{\gamma_{\max}}{2}\\ \frac{\gamma_{\min}}{2} \end{aligned} \right\} = \pm \sqrt{ \left( \frac{\varepsilon_x-\varepsilon_y}{2} \right)^2 + \left( \frac{\gamma_{xy}}{2} \right)^2 } \quad \qquad \tan 2\alpha_{\gamma} = -\frac{\varepsilon_x-\varepsilon_y} {\gamma_{xy}}\]

为什么切应变要除2而切应力不用？

材料力学中使用的$\gamma_{xy}$是“工程切应变”，而应变张量中的实际切应变分量为$\frac{\gamma_{xy}}{2}$

3.2.3 直角应变花公式

\[\varepsilon_{x'}=\varepsilon_0\] \[\varepsilon_{y'}=\varepsilon_{90}\] \[\gamma_{x'y'}=2\varepsilon_{45}-\left(\varepsilon_0+\varepsilon_{90}\right)\]