広告
応力のまとめ
まとめると次の様になります。
x軸に垂直な面に作用する剪断応力の成分
\[
\begin{aligned}
\tau_{xx}&=-2\mu\frac{\partial v_x}{\partial x}+\frac{2}{3}\mu(\nabla\cdot\vec v)\\[6pt]
\tau_{xy}&=-\mu\left(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right)\\[6pt]
\tau_{xz}&=-\mu\left(\frac{\partial v_x}{\partial z}+\frac{\partial v_z}{\partial x}\right)
\end{aligned}
\]
y軸に垂直な面に作用する剪断応力の成分
\[
\begin{aligned}
\tau_{yx}&=-\mu\left(\frac{\partial v_y}{\partial x}+\frac{\partial v_x}{\partial y}\right)\\[6pt]
\tau_{yy}&=-2\mu\frac{\partial v_y}{\partial y}+\frac{2}{3}\mu(\nabla\cdot\vec v)\\[6pt]
\tau_{yz}&=-\mu\left(\frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\right)
\end{aligned}
\]
z軸に垂直な面に作用する剪断応力の成分
\[
\begin{aligned}
\tau_{zx}&=-\mu\left(\frac{\partial v_z}{\partial x}+\frac{\partial v_x}{\partial z}\right)\\[6pt]
\tau_{zy}&=-\mu\left(\frac{\partial v_z}{\partial y}+\frac{\partial v_y}{\partial z}\right)\\[6pt]
\tau_{zz}&=-2\mu\frac{\partial v_z}{\partial z}+\frac{2}{3}\mu(\nabla\cdot\vec v)
\end{aligned}
\]
一般的に次の様に表されます。
\[
\tau_{ij}
=
-\mu\left(
\frac{\partial v_i}{\partial x_j}
+\frac{\partial v_j}{\partial x_i}
\right)
+\delta_{ij}\frac{2}{3}\mu(\nabla\cdot\vec v)
\]
また、
\[
\begin{aligned}
\sigma_{ij}
&=-\delta_{ij}p_j-\tau_{ij}\\[6pt]
&=-\delta_{ij}p_j
+\mu\left(
\frac{\partial v_i}{\partial x_j}
+\frac{\partial v_j}{\partial x_i}
\right)
-\delta_{ij}\frac{2}{3}\mu(\nabla\cdot\vec v)
\end{aligned}
\]
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