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剪断成分の値
下図より剪断速度を導出します。
Da[rad]は次式となります。
\[
\begin{aligned}
\Delta\alpha &\cong \tan\Delta\alpha\\[6pt]
&=\frac{\frac{\partial v_y}{\partial x}\Delta x\Delta t}
{\Delta x+\frac{\partial v_x}{\partial x}\Delta x\Delta t}\\[6pt]
&=\frac{\frac{\partial v_y}{\partial x}}
{\frac{1}{\Delta t}+\frac{\partial v_x}{\partial x}}
\end{aligned}
\]
また、Db[rad]は次式となります。
\[
\begin{aligned}
\Delta\beta &\cong \tan\Delta\beta\\[6pt]
&=\frac{\frac{\partial v_x}{\partial y}\Delta y\Delta t}
{\Delta y+\frac{\partial v_y}{\partial y}\Delta y\Delta t}\\[6pt]
&=\frac{\frac{\partial v_x}{\partial y}}
{\frac{1}{\Delta t}+\frac{\partial v_y}{\partial y}}
\end{aligned}
\]
従って、xy面の剪断速度g z [m/s] は次式となります。
\[
\begin{aligned}
\gamma_z
&=\lim_{\Delta t\to0}\frac{\Delta\alpha+\Delta\beta}{\Delta t}\\[6pt]
&=\lim_{\Delta t\to0}\frac{1}{\Delta t}
\left(
\frac{\frac{\partial v_y}{\partial x}}
{\frac{1}{\Delta t}+\frac{\partial v_x}{\partial x}}
+
\frac{\frac{\partial v_x}{\partial y}}
{\frac{1}{\Delta t}+\frac{\partial v_y}{\partial y}}
\right)\\[6pt]
&=\lim_{\Delta t\to0}
\left(
\frac{\frac{\partial v_y}{\partial x}}
{1+\frac{\partial v_x}{\partial x}\Delta t}
+
\frac{\frac{\partial v_x}{\partial y}}
{1+\frac{\partial v_y}{\partial y}\Delta t}
\right)\\[6pt]
&=\frac{\partial v_y}{\partial x}+\frac{\partial v_x}{\partial y}
\end{aligned}
\]
同様に、 yz面 , zx面の剪断速度 gy , g z [m/s] は次式となります。
\[
\begin{aligned}
\gamma_y&=\frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\\[6pt]
\gamma_z&=\frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}
\end{aligned}
\]
従って、
\[
\begin{aligned}
\tau_{xy}&=\tau_{yx}=-\mu\left(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right)\\[6pt]
\tau_{yz}&=\tau_{zx}=-\mu\left(\frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\right)\\[6pt]
\tau_{zx}&=\tau_{yz}=-\mu\left(\frac{\partial v_z}{\partial x}+\frac{\partial v_x}{\partial z}\right)
\end{aligned}
\]
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