原理的塑性变形部分第二篇，有点赶，不废话了
小应变几何方程

$$ \begin{aligned} \varepsilon_{xx} &= \frac{\partial u}{\partial x}, \\ \varepsilon_{yy} &= \frac{\partial v}{\partial y}, \\ \varepsilon_{zz} &= \frac{\partial w}{\partial z}, \\ \varepsilon_{xy} &= \frac{1}{2} \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right), \\ \varepsilon_{yz} &= \frac{1}{2} \left( \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} \right), \\ \varepsilon_{zx} &= \frac{1}{2} \left( \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} \right). \end{aligned} $$

体积不变条件

$$ \varepsilon_x + \varepsilon_y + \varepsilon_z = 0 $$

应变张量不变量

$$ \begin{aligned} I_1 &= \varepsilon_x + \varepsilon_y + \varepsilon_z = \varepsilon_1 + \varepsilon_2 + \varepsilon_3, \\ I_2 &= -(\varepsilon_x \varepsilon_y + \varepsilon_y \varepsilon_z + \varepsilon_z \varepsilon_x) + \gamma_{xy}^2 + \gamma_{yz}^2 + \gamma_{zx}^2 = \varepsilon_1 \varepsilon_2 + \varepsilon_2 \varepsilon_3 + \varepsilon_3 \varepsilon_1, \\ I_3 &= \varepsilon_x \varepsilon_y \varepsilon_z + 2 \gamma_{xy} \gamma_{yz} \gamma_{zx} - (\varepsilon_x \gamma_{yz}^2 + \varepsilon_y \gamma_{zx}^2 + \varepsilon_z \gamma_{xy}^2) = \varepsilon_1 \varepsilon_2 \varepsilon_3. \end{aligned} $$

等效应变

$$ \bar{\varepsilon} = \frac{\sqrt{2}}{3} \sqrt{ (\varepsilon_x - \varepsilon_y)^2 + (\varepsilon_y - \varepsilon_z)^2 + (\varepsilon_z - \varepsilon_x)^2 + 6 \left( \gamma_{xy}^2 + \gamma_{yz}^2 + \gamma_{zx}^2 \right) } $$

等效应变是一个不变量，在塑性变形时，数值上等于单向均匀拉伸或均匀压缩方向上的线应变。

应变连续方程

$$ \begin{cases} \frac{\partial \gamma_{xy}}{\partial x \partial y} = \frac{1}{2} \left( \frac{\partial^2 \varepsilon_x}{\partial y^2} + \frac{\partial^2 \varepsilon_y}{\partial x^2} \right) \\ \frac{\partial \gamma_{yz}}{\partial y \partial z} = \frac{1}{2} \left( \frac{\partial^2 \varepsilon_y}{\partial z^2} + \frac{\partial^2 \varepsilon_z}{\partial y^2} \right) \\ \frac{\partial \gamma_{zx}}{\partial z \partial x} = \frac{1}{2} \left( \frac{\partial^2 \varepsilon_z}{\partial x^2} + \frac{\partial^2 \varepsilon_x}{\partial z^2} \right) \end{cases} $$

平面应力状态下，无应力的方向为主方向，所有有关的应力均为0。(注意：是平面应力状态，区别于应变）
只有在纯切应力状态，没有应力的方向才没有应变。

平面变形的应力状态是纯切应力状态叠加一球应力状态。各分量与z无关，仅有$\sigma_z$不独立。

$$ \sigma_z=\frac{1}{2}(\sigma_x+\sigma_y)=\sigma_m $$

轴对称状态各应力分量与$\theta$坐标无关，偏导为0。