# 胡克定律

$\sigma = E \varepsilon$

$\Delta L = \frac{1}{E} \times L \times \frac{F}{A} = \frac{1}{E} \times L \times \sigma$

## 弹簧方程

$F=-kx$

$U={1\over2}kx^2$

$\omega_n = \sqrt{k \over m}$

1. 最大强度
2. 屈服强度
3. 破坏点
4. 应变硬化
5. 颈缩区

$\sigma_{ij} = \sum_{kl} c_{ijkl} \cdot \varepsilon_{kl}$

## 胡克定律的张量形式

（在牛顿流体中的类比参见粘性词条。）

$\varepsilon_{ij}=\left(\frac{1}{3}\varepsilon_{kk}\delta_{ij}\right) +\left(\varepsilon_{ij}-\frac{1}{3}\varepsilon_{kk}\delta_{ij}\right)$

$\sigma_{ij}=3K\left(\frac{1}{3}\varepsilon_{kk}\delta_{ij}\right) +2G\left(\varepsilon_{ij}-\frac{1}{3}\varepsilon_{kk}\delta_{ij}\right)$

$\begin{cases} \varepsilon_{11} = \cfrac{1}{Y}\left( \sigma_{11} - \nu(\sigma_{22}+\sigma_{33}) \right)\\ \varepsilon_{22} = \cfrac{1}{Y}\left( \sigma_{22} - \nu(\sigma_{11}+\sigma_{33}) \right)\\ \varepsilon_{33} = \cfrac{1}{Y}\left( \sigma_{33} - \nu(\sigma_{11}+\sigma_{22}) \right)\\ \varepsilon_{12} = \cfrac{\sigma_{12}}{2G}\\ \varepsilon_{13} = \cfrac{\sigma_{13}}{2G}\\ \varepsilon_{23} = \cfrac{\sigma_{23}}{2G} \end{cases}$

### 正交各向异性材料

$\begin{pmatrix} \sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{23}\\ \sigma_{31}\\ \end{pmatrix}= \begin{pmatrix} C_{11}&C_{12}&C_{13}&0&0&0\\ C_{12}&C_{22}&C_{23}&0&0&0\\ C_{13}&C_{23}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{55}&0\\ 0&0&0&0&0&C_{66} \end{pmatrix} \begin{pmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{33}\\ \varepsilon_{12}\\ \varepsilon_{23}\\ \varepsilon_{31}\\\end{pmatrix}$

## 参考文献

• [1] Y. C. Fung （冯元桢）, Foundations of Solid Mechanics, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1965
• [2] A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed