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4. 本构关系

Course NotesEngineering Mechanics for Civil EngineeringLess than 1 minuteAbout 233 words

4.1 广义胡克定律

4.1.1 各向同性弹性体

逆弹性关系

εx=1E\bqtyσx−ν(σy+σz)=1+νEσx−νEΘεy=1E\bqtyσy−ν(σz+σx)=1+νEσy−νEΘεz=1E\bqtyσz−ν(σx+σy)=1+νEσz−νEΘγxy=1Gτxyγyz=1Gτyzγzx=1Gτzx \begin{align*} & \varepsilon_x = \frac{1}{E} \bqty{\sigma_x - \nu \pqty{\sigma_y + \sigma_z}} = \frac{1 + \nu}{E} \sigma_x - \frac{\nu}{E} \Theta \\ & \varepsilon_y = \frac{1}{E} \bqty{\sigma_y - \nu \pqty{\sigma_z + \sigma_x}} = \frac{1 + \nu}{E} \sigma_y - \frac{\nu}{E} \Theta \\ & \varepsilon_z = \frac{1}{E} \bqty{\sigma_z - \nu \pqty{\sigma_x + \sigma_y}} = \frac{1 + \nu}{E} \sigma_z - \frac{\nu}{E} \Theta \\ & \gamma_{xy} = \frac{1}{G} \tau_{xy} \\ & \gamma_{yz} = \frac{1}{G} \tau_{yz} \\ & \gamma_{zx} = \frac{1}{G} \tau_{zx} \end{align*}

EE — 杨氏模量; ν\nu — 泊松比; GG — 剪切模量

Tips

考试给

G=E2(1+ν) G = \frac{E}{2 \pqty{1 + \nu}}

λ=νE(1+ν)(1−2ν) \lambda = \frac{\nu E}{\pqty{1 + \nu} \pqty{1 - 2 \nu}}

应力-应变关系 / 弹性关系

σij=2Gεij+λεkkδij \sigma_{ij} = 2 G \varepsilon_{ij} + \lambda \varepsilon_{kk} \delta_{ij}

σx=2Gεx+λθ;τxy=Gγxyσy=2Gεy+λθ;τyz=Gγyzσz=2Gεz+λθ;τzx=Gγzx \begin{align*} \sigma_x & = 2 G \varepsilon_x + \lambda \theta; & \tau_{xy} & = G \gamma_{xy} \\ \sigma_y & = 2 G \varepsilon_y + \lambda \theta; & \tau_{yz} & = G \gamma_{y z} \\ \sigma_z & = 2 G \varepsilon_z + \lambda \theta; & \tau_{zx} & = G \gamma_{zx} \end{align*}

弹性常数 GG 和 λ\lambda — 拉梅系数

4.1.2 各向异性的弹性体