4. 本构关系

4.1 广义胡克定律

4.1.1 各向同性弹性体

逆弹性关系

$$\begin{array}{rl}& {\epsilon }_x=\frac{1}{E}\text{bqty}{\sigma }_x-\nu \text{pqty}{\sigma }_y+{\sigma }_z=\frac{1+\nu }{E}{\sigma }_x-\frac{\nu }{E}\mathrm{\Theta }\\ & {\epsilon }_y=\frac{1}{E}\text{bqty}{\sigma }_y-\nu \text{pqty}{\sigma }_z+{\sigma }_x=\frac{1+\nu }{E}{\sigma }_y-\frac{\nu }{E}\mathrm{\Theta }\\ & {\epsilon }_z=\frac{1}{E}\text{bqty}{\sigma }_z-\nu \text{pqty}{\sigma }_x+{\sigma }_y=\frac{1+\nu }{E}{\sigma }_z-\frac{\nu }{E}\mathrm{\Theta }\\ & {\gamma }_{xy}=\frac{1}{G}{\tau }_{xy}\\ & {\gamma }_{yz}=\frac{1}{G}{\tau }_{yz}\\ & {\gamma }_{zx}=\frac{1}{G}{\tau }_{zx}\end{array}$$

$E$ — 杨氏模量; $\nu$ — 泊松比; $G$ — 剪切模量

Tips

考试给

$$G=\frac{E}{2\text{pqty}1+\nu }$$$$\lambda =\frac{\nu E}{\text{pqty}1+\nu \text{pqty}1-2\nu }$$

应力 - 应变关系 / 弹性关系

$${\sigma }_{ij}=2G{\epsilon }_{ij}+\lambda {\epsilon }_{kk}{\delta }_{ij}$$$$\begin{array}{rlrl}{\sigma }_x& =2G{\epsilon }_x+\lambda \theta ;& {\tau }_{xy}& =G{\gamma }_{xy}\\ {\sigma }_y& =2G{\epsilon }_y+\lambda \theta ;& {\tau }_{yz}& =G{\gamma }_{yz}\\ {\sigma }_z& =2G{\epsilon }_z+\lambda \theta ;& {\tau }_{zx}& =G{\gamma }_{zx}\end{array}$$

弹性常数 $G$ 和 $\lambda$ — 拉梅系数

4.1.2 各向异性的弹性体