# SimpleFEM

**Repository Path**: sjzMDL/SimpleFEM

## Basic Information

- **Project Name**: SimpleFEM
- **Description**: sssss
- **Primary Language**: C
- **License**: Not specified
- **Default Branch**: master
- **Homepage**: None
- **GVP Project**: No

## Statistics

- **Stars**: 0
- **Forks**: 0
- **Created**: 2023-07-14
- **Last Updated**: 2023-07-14

## Categories & Tags

**Categories**: Uncategorized

**Tags**: None

## README

### 1 坐标
初始构型：$^0x_i$，即时构型：$^tx_i$，则：
$$d^tx_i=\frac{\partial{^tx_i}}{\partial{^0x_j}}d^0x_j=^t_0x_{i,j}d^0x_j$$
类似的：
$$d^0x_i=\frac{\partial{^0x_i}}{\partial{^tx_j}}d^tx_j=^t_0x_{i,j}d^tx_j$$
### 2 线段长度变化
$$
\begin{aligned}
(^tds)^2-(^0ds)^2&=d^tx_kd^tx_k-d^0x_kd^0x_k\\
&={^t_0x_{k,i}d^0x_i}{^t_0x_{k,j}d^0x_j}-d^0x_kd^0x_k \\
&={^t_0x_{k,i}d^0x_i}{^t_0x_{k,j}d^0x_j}-\delta_{ij} d^0x_id^0x_j\\
&=(^t_0x_{k,i}{^t_0x_{k,j}}-\delta_{ij})d^0x_id^0x_j \\
&=2^t_0\varepsilon_{ij}d^0x_id^0x_j
\end{aligned}
$$
同理
$$
\begin{aligned}
(^tds)^2-(^0ds)^2&=d^tx_kd^tx_k-d^0x_kd^0x_k\\
&=(\delta_{ij}-^0_tx_{k,i}{^0_tx_{k,j}})d^0x_id^0x_j \\
&=2^0_t\varepsilon_{ij}d^tx_id^tx_j
\end{aligned}
$$
### 3 应变
#### 3.1 Green strain
$$^t_0\varepsilon_{ij}=\frac{1}{2}(^t_0x_{k,i}{^t_0x_{k,j}}-\delta_{ij})$$
#### 3.2 Almansi strain
$${^0_t}\varepsilon_{ij}=\frac{1}{2}(\delta_{ij}-{^0_t}x_{k,i}{{^0_t}x_{k,j}})$$

### 4 位移
位移定义为（$^tu_i={^t}x_i-{^0}x_i$）

则梯度 ${^t_0}x_{i,j}$有：
$${^t_0}x_{i,j}=\frac{\partial{^tx_i}}{\partial{^0x_j}}=\frac{\partial{^tu_i+^0x_i}}{\partial{^0x_j}}={^t_0}u_{i,j}+\delta_{ij}$$
同样的 ${^0_t}x_{i,j}$有：
$${^0_t}x_{i,j}=\delta_{ij}-\frac{\partial{^tu_i}}{\partial{^tx_j}}=\delta_{ij}-{^t_t}u_{i,j}$$

用位移表示应变:
$$^t_0\varepsilon_{ij}=\frac{1}{2}(^t_0x_{k,i}{^t_0x_{k,j}}-\delta_{ij})=\frac{1}{2}(^t_0u_{i,j}+^t_0u_{j,i}+^t_0u_{k,i}{^t_0u_{k,j}})$$
$$^0_t\varepsilon_{ij}=\frac{1}{2}(\delta_{ij}-^0_tx_{k,i}{^0_tx_{k,j}})=\frac{1}{2}(^t_tu_{i,j}+^t_tu_{j,i}+^t_tu_{k,i}{^t_tu_{k,j}})$$
### 5 应力
$$^t\sigma=\frac{^tdT}{^tdS}$$
$$^0\sigma=\frac{^0dT}{^0dS}$$

Lagrange 前后相等 :   $^0dT_i={^t}dT_i$
$${^t}dT_i={{^t_0}T_{ij}\ {^t}d{S_{j}^\perp}}={^0}dT_i$$
$${^t_0}S_{ij}=J\sigma F^{-T}$$
第一类piola-Kirchhoff：${^t_0}T_{ij}$ 不对称

Kirchhoff 前后不相等：$^0dT_i={^0_t}x_{i,j}{^t}dT_i$ 
$^0dT_i={^t}dT_i$

$${^t}dT_i={{^t_0}S_{ij}\ {^t}d{S_{j}^\perp}}={^0_tx_{i,j}}{^0}dT_i$$
$${^t_0}S_{ij}=JF^{-1}\sigma F^{-T}$$
第二类piola-Kirchhoff：${^t_0}S_{ij}$ 对称
### 6 虚功原理(UL)

$$\int_{^tV}
{{{^{t+\Delta t}}_t}S_{ij}}\delta({{{^{t+\Delta t}}_t}\varepsilon_{ij}}){{^t}dv}={{{^{t+\Delta t}}_t}R}
$$

其中${{{^{t+\Delta t}}_t}S_{ij}}$是2nd Piola-Kirchhoff应力张量；
${{{^{t+\Delta t}}_t}\varepsilon_{ij}}$是Green应变张量。
$t$表示定义在$t$时刻的初始构型，在$\Delta t+t$为即时构型。

将$\delta({{{^{t+\Delta t}}_t}\varepsilon_{ij}})$进行分解：
$$
\begin{aligned}
\delta({{{^{t+\Delta t}}_t}\varepsilon_{ij}})&=\delta({{^{t}_t}\varepsilon_{ij}}+{{_t}\varepsilon_{ij}})=\delta({{_t}\varepsilon_{ij}}) \\
&=\delta({{_t}e_{ij}}+{{_t}\eta_{ij}}) \\
&=\frac{1}{2}\delta({_t}u_{i,j}+{_t}u_{j,i})+\frac{1}{2}\delta({_t}u_{k,i}\ {_t}u_{k,j})
\end{aligned}
$$

应力张量分解
$${{^{t+\Delta t}}_t}S_{ij}={^t}\tau_{ij}+{_t}S_{ij}$$

初始构型上
$${_t}S_{ij}={_t}C_{ijrs}\ {_t}\varepsilon_{rs}$$

带入虚功方程，得到：
$$
\begin{aligned}
\int_{^tV}
{_t}C_{ijrs}\ {_t}\varepsilon_{rs}\ \delta({_t}\varepsilon_{ij}){{^t}dv}+
\int_{^tV}{^t}\tau_{ij} \ \delta({_t}\eta_{ij}){{^t}dv}
={{^{t+\Delta t}}_t}R-
\int_{^tV}{^t}\tau_{ij} \ \delta({_t}e_{ij}){{^t}dv}
\end{aligned}
$$
### 7 平衡方程(UL)
$$(^t_t\rm{K}_L+^t_t\rm{K}_{NL})\rm{u}={{^{t+\Delta t}}_t}R-^t_t\rm{F}$$
其中：
$$^t_t\rm{K}_L=\int_{^tV}{^t_t}\rm{B}^T_L\rm{C}\rm{B}_L \ {^t}dV$$

$$^t_t\rm{K}_{NL}=\int_{^tV}{^t_t}\rm{B}^T_{NL} \ {^t}\rm{\tau} \ \rm{B}_{NL} \ {^t}dV$$

$$^t_t\rm{F}=\int_{^tV}{^t_t}\rm{B}^T_{L}\ {^t}\rm{\hat\tau} \ {^t}dV$$

$$\left[^t_t\rm{\tau}\right]=
\begin{bmatrix}
^t_0\tau_{11} & ^t_0\tau_{12} & 0 & 0 \\
\\
^t_0\tau_{21} & ^t_0\tau_{22} & 0 & 0 \\ 
\\
0 & 0 & ^t_0\tau_{11} & ^t_0\tau_{12} \\
\\
0 & 0 & ^t_0\tau_{21} & ^t_0\tau_{22} \\
\end{bmatrix}
$$
$$\left[^t_t\rm{\hat\tau}\right]=
\begin{bmatrix}
^t_0\tau_{11}\\
\\
^t_0\tau_{22}\\ 
\\
^t_0\tau_{12} \\
\end{bmatrix}
$$


其中：
$$\left[^t_t\rm{B}_{L}\right]_i=
\begin{bmatrix}
\dfrac{\partial N_i}{\partial x_1} & 0 \\
\\
0 & \dfrac{\partial N_i}{\partial x_2} \\
\\
\dfrac{\partial N_i}{\partial x_2} & \dfrac{\partial N_i}{\partial x_1}  \\
\end{bmatrix}
$$
$$\left[^t_t\rm{B}_{NL}\right]_i=
\begin{bmatrix}
\dfrac{\partial N_i}{\partial x_1} & 0 \\
\\
\dfrac{\partial N_i}{\partial x_2} & 0 \\
\\
0 & \dfrac{\partial N_i}{\partial x_1} \\
\\
0 & \dfrac{\partial N_i}{\partial x_2} \\
\end{bmatrix}
$$


### 8 本构模型