# 几何方程

<span class="gray-text">
几何方程揭示了应变与位移之间的内在联系。应变的本质源于不同点之间相对位置的变化。因此，通过分析不同方向上两条线段端点的相对位置变化，可以推导出应变与位移的关系方程
</span>

考虑弹性体内任一点 $P$，沿 $x$ 轴和 $y$ 轴的正方向取两个微小线段$\overline{PA} = \mathrm{d}x $ 和 $\overline{PA} = \mathrm{d}y$，假定弹性体受力后 $P,A,B$ 三点分别移动到了 $P^`,A^`,B^`$

```{figure} ../../images/Elasticity/displ_strain.png
---
width: 800px
name: sec4-fig:displ-strain
---
位移和应变关系示意图
```

$$
\begin{equation}
\begin{aligned}
u(P)&=u(x_{P},y_{P}),\\
u(A)&=u(x_{A},y_{A})=u(x_{P}+\mathrm{d}x,y_{P}) \approx u(x_{P},y_{P}) + \frac{\partial u}{\partial x}(x_P,y_P)\cdot\mathrm{d}x,\\
u(B)&=u(x_{B},y_{B})=u(x_{P},y_{P}+\mathrm{d}y) \approx u(x_{P},y_{P}) + \frac{\partial u}{\partial y}(x_P,y_P)\cdot\mathrm{d}y,\\
v(P)&=v(x_{P},y_{P}),\\
v(A)&=v(x_{A},y_{A})=v(x_{P}+\mathrm{d}x,y_{P}) \approx v(x_{P},y_{P}) + \frac{\partial v}{\partial x}(x_P,y_P)\cdot\mathrm{d}x,\\
v(B)&=v(x_{B},y_{B})=v(x_{P},y_{P}+\mathrm{d}y) \approx v(x_{P},y_{P}) + \frac{\partial v}{\partial y}(x_P,y_P)\cdot\mathrm{d}y.\\
\end{aligned}
\end{equation}
$$

```{margin}
建立方程时，采用了**小变形**假设，因此以弹性体**变形前**的尺寸作为微分体的几何尺寸，简化了力学分析
```

于是，$\overline{PA}$ 的线应变为

$$
\varepsilon_{xx} = \frac{\left( u + \frac{\partial u}{\partial x} \, \mathrm{d}x \right) - u}{\mathrm{d}x} = \frac{\partial u}{\partial x},
$$

类似地，$\overline{PB}$ 的线应变为

$$
\varepsilon_{yy} = \frac{\partial v}{\partial y}.
$$

对于 $\overline{PA}$ 的转角 $\alpha$ 有

$$
\alpha = \frac{\left( v + \frac{\partial v}{\partial x} \, \mathrm{d}x \right) - v}{\mathrm{d}x} = \frac{\partial v}{\partial x},
$$

类似地，$\overline{PB}$ 的转角

$$
\beta = \frac{\partial u}{\partial y},
$$

```{margin}
$\gamma_{xy}$ 被称为工程剪切应变
```

于是切应变 $\gamma_{xy}$ 为

$$
\gamma_{xy} = \alpha + \beta = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}.
$$

将几何方程整理如下

$$
\begin{equation}
\begin{aligned}
&\varepsilon_{xx} = \frac{\partial u}{\partial x}, \\
&\varepsilon_{yy} = \frac{\partial v}{\partial y}, \\
&\gamma_{xy} = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}.
\end{aligned}
\end{equation}
$$ (sec4-eq:geometry)

可以写为

$$
\boldsymbol{\varepsilon} = \frac{1}{2}(\nabla \mathbf{u} + \nabla\mathbf{u}^{T}),
$$

其中，$\boldsymbol{\varepsilon}$ 是应变张量，$\mathbf{u} = [u\,v]^{T}$ 是位移向量，即

```{margin}
$\varepsilon_{xy} = \frac{\gamma_{xy}}{2}$ 被称为张量剪切应变
```

$$
\begin{equation}
\boldsymbol{\varepsilon}=
\begin{bmatrix}
\varepsilon_{xx}  &  \varepsilon_{xy} \\
\varepsilon_{yx}  &  \varepsilon_{yy}
\end{bmatrix} = \frac{1}{2}(\begin{bmatrix}
\frac{\partial u}{\partial x}  &  \frac{\partial u}{\partial y} \\
\frac{\partial v}{\partial x}  &  \frac{\partial v}{\partial y}
\end{bmatrix} + 
\begin{bmatrix}
\frac{\partial u}{\partial x}  &  \frac{\partial v}{\partial x} \\
\frac{\partial u}{\partial y}  &  \frac{\partial v}{\partial y}
\end{bmatrix}),
\end{equation}
$$

也可以写为 Voigt 形式

$$
\begin{equation}
\begin{bmatrix}
\varepsilon_{xx} \\ \varepsilon_{yy} \\ 2\varepsilon_{xy} 
\end{bmatrix}
=\begin{bmatrix}
\frac{\partial }{\partial x} & 0 \\
0 & \frac{\partial }{\partial y} \\
\frac{\partial }{\partial y} & \frac{\partial }{\partial x}
\end{bmatrix}
\begin{bmatrix}
u \\ v 
\end{bmatrix}.
\end{equation}
$$

对于三维情形，有

$$
\begin{equation}
\begin{bmatrix}
\varepsilon_{xx}\\\varepsilon_{yy}\\\varepsilon_{zz}\\2\varepsilon_{xy}\\2\varepsilon_{yz}\\2\varepsilon_{zx}
\end{bmatrix}
=\begin{bmatrix}
\frac{\partial}{\partial x} & 0 & 0 \\
0 & \frac{\partial}{\partial y} & 0 \\
0 & 0 & \frac{\partial}{\partial z} \\
\frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \\
0 & \frac{\partial}{\partial z} & \frac{\partial}{\partial y} \\
\frac{\partial}{\partial z} & 0 & \frac{\partial}{\partial x}
\end{bmatrix}
\begin{bmatrix}
u_{x}\\u_{y}\\u_{z}
\end{bmatrix}.
\end{equation}
$$


```{admonition} 刚体运动
:class: tip, dropdown

当位移分量 $u,v$ 完全确定时，应变分量 $\varepsilon_{xx}, \varepsilon_{yy}, \gamma_{xy}$ 也完全确定。反之，应变分量完全确定时，位移分量不能完全确定。
假设

$$
\varepsilon_{xx} = \varepsilon_{yy} = \gamma_{xy} = 0,
$$

则

$$
\frac{\partial u}{\partial x} = 0, \quad
\frac{\partial v}{\partial y} = 0, \quad
\gamma_{xy} = 0,
$$

可得

$$
u = u_{0} - wy,\quad
v = v_{0} + wx.
$$

$u_{0}$ 和 $v_{0}$ 分别表示物体沿 $x$ 轴和 $y$ 轴方向的**刚体平移**，而 $w$ 则为物体绕 $z$ 轴的**刚体转动**。当只有 $w$ 不为 0 时，对于坐标内任一点 $P$，其位移分量为 $u=-wy, v = wx$，满足

$$
\begin{bmatrix}
u & v
\end{bmatrix}
\begin{bmatrix}
x \\ y
\end{bmatrix} = 0,
$$

因此，点 $P$ 的位移方向为以原点 $O$ 为圆心，$\overline{OP}$ 为半径的圆在 $P$ 点的切向。由于位移是微小的（$w$非常小），因此运动表现为刚体转动，位移为

$$
\sqrt{u^2+v^2}=w\sqrt{x^2+y^2}.
$$

```