弹性力学基本方程

笑看风云

平衡（运动）微分方程：

\frac{\partial\sigma_{x}}{\partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{zx}}{\partial z}+F_{x}=0
\frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\sigma_{y}}{\partial y}+\frac{\partial\tau_{zy}}{\partial z}+F_{y}=0
\frac{\partial\tau_{xz}}{\partial x}+\frac{\partial\tau_{yz}}{\partial y}+\frac{\partial\sigma_{z}}{\partial z}+F_{z}=0
几何方程（柯西方程）- 应变和位移的关系：

\varepsilon_{x}=\frac{\partial u}{\partial x} \gamma_{yz}=\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}
\varepsilon_{y}=\frac{\partial v}{\partial y} \gamma_{zx}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}
\varepsilon_{z}=\frac{\partial w}{\partial z} \gamma_{xy}=\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}
物理方程 - 应力和应变的关系：

\varepsilon_{x}=\frac{\sigma_{x}-\mu\left( \sigma_{y}+\sigma_{z} \right)}{E} \gamma_{yz}=\frac{2\left( 1+\mu \right)}{E}\tau_{yz}
\varepsilon_{y}=\frac{\sigma_{y}-\mu\left( \sigma_{x}+\sigma_{z} \right)}{E} \gamma_{xz}=\frac{2\left( 1+\mu \right)}{E}\tau_{xz}
\varepsilon_{z}=\frac{\sigma_{z}-\mu\left( \sigma_{x}+\sigma_{y} \right)}{E} \gamma_{xy}=\frac{2\left( 1+\mu \right)}{E}\tau_{xy}

原文地址：https://zhuanlan.zhihu.com/p/57657300