POINT
連続物体の運動方程式には「応力テンソル」が現れます（関連記事 [A]）．- 粘性流体の応力テンソル，弾性体の応力テンソルについて．
粘性流体と弾性体の場合に，応力テンソルの表式を整理します．
【関連記事】
- [A] 流体力学の方程式（運動方程式・連続の方程式・状態方程式） - Notes_JP
- [B] フックの法則/ひずみテンソルの座標変換（極座標・円筒座標） - Notes_JP
- [C] 応力テンソルとは - Notes_JP
- [D] ラメ定数 - Notes_JP
- [E] 熱力学的観点から音速を探る - Notes_JP
粘性流体
変形速度テンソル
$\bm{v}$を流速，$\partial_{i} = \partial / \partial x_{i}$とする．変形速度テンソル (rate-of-strain tensor)を
\begin{aligned}
& e_{ij} = \partial_{j} v_{i} + \partial_{i} v_{j}
\qquad (\text{文献[1] (59.10)})
\end{aligned}
で表す．& e_{ij} = \partial_{j} v_{i} + \partial_{i} v_{j}
\qquad (\text{文献[1] (59.10)})
\end{aligned}
変形速度テンソルについて，次の関係式が成り立つ
\begin{aligned}
e_{jj} &= e_{ij} \delta_{ij} = 2 \partial_{j} v_{j} =2\mathrm{div\,} \bm{v} \\
\partial_{j} e_{ji}
&= \partial_{i} \partial_{j} v_{j} + \partial_{j} \partial_{j} v_{i} \\
&= (\mathrm{grad\,}\mathrm{div\,}\bm{v} + \Delta \bm{v})_{i} \\
&= [\mathrm{grad\,}\mathrm{div\,}\bm{v} + (\mathrm{grad\,}\mathrm{div\,}\bm{v} - \mathrm{rot\,}\mathrm{rot\,} \bm{v})]_{i} \\
&=(2\mathrm{grad\,}\mathrm{div\,}\bm{v} - \mathrm{rot\,}\mathrm{rot\,} \bm{v})_{i}
\end{aligned}
（【参考】ベクトル解析の公式 - Notes_JP）．e_{jj} &= e_{ij} \delta_{ij} = 2 \partial_{j} v_{j} =2\mathrm{div\,} \bm{v} \\
\partial_{j} e_{ji}
&= \partial_{i} \partial_{j} v_{j} + \partial_{j} \partial_{j} v_{i} \\
&= (\mathrm{grad\,}\mathrm{div\,}\bm{v} + \Delta \bm{v})_{i} \\
&= [\mathrm{grad\,}\mathrm{div\,}\bm{v} + (\mathrm{grad\,}\mathrm{div\,}\bm{v} - \mathrm{rot\,}\mathrm{rot\,} \bm{v})]_{i} \\
&=(2\mathrm{grad\,}\mathrm{div\,}\bm{v} - \mathrm{rot\,}\mathrm{rot\,} \bm{v})_{i}
\end{aligned}
応力テンソル
「粘性流体の」応力テンソル (rate-of-strain tensor) は\begin{aligned}
p_{ij}
& = -p\delta_{ij} + \lambda\Theta\delta_{ij} + \mu e_{ij} \\
& = -p\delta_{ij} + \biggl(\mu^{\prime} - \frac{2}{3}\mu \biggr) (\mathrm{div\,}\bm{v}) \delta_{ij} + \mu e_{ij} \\
& (\text{文献[1] (60.2)})
\end{aligned}
p_{ij}
& = -p\delta_{ij} + \lambda\Theta\delta_{ij} + \mu e_{ij} \\
& = -p\delta_{ij} + \biggl(\mu^{\prime} - \frac{2}{3}\mu \biggr) (\mathrm{div\,}\bm{v}) \delta_{ij} + \mu e_{ij} \\
& (\text{文献[1] (60.2)})
\end{aligned}
- $\mu$：粘性率 (coefficient of viscosity)
- $\lambda$：第2粘性率 (second coefficient of viscosity)
- $\mu^{\prime} = \lambda + \frac{2}{3}\mu$：体積粘性率 (bulk viscosity)
運動方程式（関連記事 [A]）では$\partial_{j} p_{ji}$を使う．これは，
\begin{aligned}
\partial_{j} p_{ji}
&= -\partial_{i} p
+ \biggl(\mu^{\prime} + \frac{1}{3}\mu \biggr)
\partial_{i}(\mathrm{div\,}\bm{v})
+ \mu \Delta v_{i} \\
&= -\partial_{i} p
+ \biggl(\mu^{\prime} + \frac{4}{3}\mu \biggr)
\partial_{i}(\mathrm{div\,}\bm{v})
-\mu (\mathrm{rot\,}\mathrm{rot\,} \bm{v})_{i}
\end{aligned}
などと表せる．\partial_{j} p_{ji}
&= -\partial_{i} p
+ \biggl(\mu^{\prime} + \frac{1}{3}\mu \biggr)
\partial_{i}(\mathrm{div\,}\bm{v})
+ \mu \Delta v_{i} \\
&= -\partial_{i} p
+ \biggl(\mu^{\prime} + \frac{4}{3}\mu \biggr)
\partial_{i}(\mathrm{div\,}\bm{v})
-\mu (\mathrm{rot\,}\mathrm{rot\,} \bm{v})_{i}
\end{aligned}
弾性体
歪（ひずみ）テンソル
$\boldsymbol{u}$を弾性体の変位ベクトルとするとき，歪テンソル (strain tensor) は\begin{aligned}
\varepsilon_{ij}
=\frac{1}{2} (\partial_{j} u_{i} + \partial_{i} u_{j})
\qquad (\text{文献[2] (7.5)})
\end{aligned}
\varepsilon_{ij}
=\frac{1}{2} (\partial_{j} u_{i} + \partial_{i} u_{j})
\qquad (\text{文献[2] (7.5)})
\end{aligned}
歪テンソルも変形速度テンソルと同様に，次の関係式が成り立つ
\begin{aligned}
\varepsilon_{jj} &= \mathrm{div\,} \bm{u} \\
\partial_{j} \varepsilon_{ji}
&= \frac{1}{2} (\mathrm{grad\,}\mathrm{div\,}\bm{u} + \Delta \bm{u})_{i} \\
&= \frac{1}{2} (2\mathrm{grad\,}\mathrm{div\,}\bm{u} - \mathrm{rot\,}\mathrm{rot\,} \bm{u})_{i}
\end{aligned}
\varepsilon_{jj} &= \mathrm{div\,} \bm{u} \\
\partial_{j} \varepsilon_{ji}
&= \frac{1}{2} (\mathrm{grad\,}\mathrm{div\,}\bm{u} + \Delta \bm{u})_{i} \\
&= \frac{1}{2} (2\mathrm{grad\,}\mathrm{div\,}\bm{u} - \mathrm{rot\,}\mathrm{rot\,} \bm{u})_{i}
\end{aligned}
応力テンソル
「弾性体の」応力テンソルは\begin{aligned}
\sigma_{ij}
&=2\mu \biggl[\varepsilon_{ij} - \frac{1}{3} (\mathrm{div\,} \bm{u}) \delta_{ij} \biggr]
+\overbrace{K (\mathrm{div\,} \bm{u})}^{=-p} \delta_{ij} \\
&= 2\mu \varepsilon_{ij}
+\biggl(K - \frac{2}{3}\mu \biggr) (\mathrm{div\,} \bm{u}) \delta_{ij} \\
& (\text{文献[2] (7.19)})
\end{aligned}
\sigma_{ij}
&=2\mu \biggl[\varepsilon_{ij} - \frac{1}{3} (\mathrm{div\,} \bm{u}) \delta_{ij} \biggr]
+\overbrace{K (\mathrm{div\,} \bm{u})}^{=-p} \delta_{ij} \\
&= 2\mu \varepsilon_{ij}
+\biggl(K - \frac{2}{3}\mu \biggr) (\mathrm{div\,} \bm{u}) \delta_{ij} \\
& (\text{文献[2] (7.19)})
\end{aligned}
- $\mu$：ずれ弾性率 (shear modulus)
- $K$：体積弾性率 (bulk modulus)
運動方程式（関連記事 [A]）では$\partial_{j} p_{ji}$を使う．これは，
\begin{aligned}
\partial_{j} \sigma_{ji}
&= -\partial_{i} p + \frac{1}{3}\mu \partial_{i} (\mathrm{div\,} \bm{u})
+ \mu \Delta u_{i} \\
&= -\partial_{i} p + \frac{4}{3}\mu \partial_{i} (\mathrm{div\,} \bm{u})
- \mu (\mathrm{rot\,}\mathrm{rot\,} \bm{u})_{i} \\
&= \biggl(K + \frac{1}{3}\mu \biggr) \partial_{i} (\mathrm{div\,} \bm{u})
+ \mu \Delta u_{i} \\
&= \biggl(K + \frac{4}{3}\mu \biggr) \partial_{i} (\mathrm{div\,} \bm{u})
- \mu (\mathrm{rot\,}\mathrm{rot\,} \bm{u})_{i} \\
&= (\lambda + \mu) \partial_{i} (\mathrm{div\,} \bm{u})
+ \mu \Delta u_{i} \\
&= (\lambda + 2\mu) \partial_{i} (\mathrm{div\,} \bm{u})
- \mu (\mathrm{rot\,}\mathrm{rot\,} \bm{u})_{i}
\end{aligned}
などと表せる（$K = \frac{3\lambda + 2\mu}{3} $ (ラメ定数 - Wikipedia)）．\partial_{j} \sigma_{ji}
&= -\partial_{i} p + \frac{1}{3}\mu \partial_{i} (\mathrm{div\,} \bm{u})
+ \mu \Delta u_{i} \\
&= -\partial_{i} p + \frac{4}{3}\mu \partial_{i} (\mathrm{div\,} \bm{u})
- \mu (\mathrm{rot\,}\mathrm{rot\,} \bm{u})_{i} \\
&= \biggl(K + \frac{1}{3}\mu \biggr) \partial_{i} (\mathrm{div\,} \bm{u})
+ \mu \Delta u_{i} \\
&= \biggl(K + \frac{4}{3}\mu \biggr) \partial_{i} (\mathrm{div\,} \bm{u})
- \mu (\mathrm{rot\,}\mathrm{rot\,} \bm{u})_{i} \\
&= (\lambda + \mu) \partial_{i} (\mathrm{div\,} \bm{u})
+ \mu \Delta u_{i} \\
&= (\lambda + 2\mu) \partial_{i} (\mathrm{div\,} \bm{u})
- \mu (\mathrm{rot\,}\mathrm{rot\,} \bm{u})_{i}
\end{aligned}
最後の2式について疎密波の音速（関連記事 [D]）
\begin{aligned}
c_{1}
&=\sqrt{\frac{K + 4\mu / 3}{\rho}}
\end{aligned}
を使うと，c_{1}
&=\sqrt{\frac{K + 4\mu / 3}{\rho}}
\end{aligned}
\begin{aligned}
\partial_{j} \sigma_{ji}
&= (\rho c_{1}^{2} - \mu ) \partial_{i} (\mathrm{div\,} \bm{u})
+ \mu \Delta u_{i} \\
&= \rho c_{1}^{2} \partial_{i} (\mathrm{div\,} \bm{u})
- \mu (\mathrm{rot\,}\mathrm{rot\,} \bm{u})_{i}
\end{aligned}
となる．\partial_{j} \sigma_{ji}
&= (\rho c_{1}^{2} - \mu ) \partial_{i} (\mathrm{div\,} \bm{u})
+ \mu \Delta u_{i} \\
&= \rho c_{1}^{2} \partial_{i} (\mathrm{div\,} \bm{u})
- \mu (\mathrm{rot\,}\mathrm{rot\,} \bm{u})_{i}
\end{aligned}
圧力勾配の変形
$\partial_{j} p_{ji}$の計算では，粘性流体でも弾性体でも$\partial_{i} p$の項が出てきた．熱力学の関係式を用いて，密度$\rho$と温度$T$で表す．$v = 1 / \rho$を比容（単位質量あたりの体積）とする．
圧力について
\begin{aligned}
\mathrm{d}p
& = \biggl(\frac{\partial p}{\partial v}\biggr)_{T} \mathrm{d}v
+ \biggl(\frac{\partial p}{\partial T}\biggr)_{v} \mathrm{d}T
\end{aligned}
より\mathrm{d}p
& = \biggl(\frac{\partial p}{\partial v}\biggr)_{T} \mathrm{d}v
+ \biggl(\frac{\partial p}{\partial T}\biggr)_{v} \mathrm{d}T
\end{aligned}
\begin{aligned}
0
&= \biggl(\frac{\partial p}{\partial v}\biggr)_{T}
\biggl(\frac{\partial v}{\partial T}\biggr)_{p}
+ \biggl(\frac{\partial p}{\partial T}\biggr)_{v}
\end{aligned}
が得られる．よって，熱膨張率（Thermal expansion - Wikipedia）0
&= \biggl(\frac{\partial p}{\partial v}\biggr)_{T}
\biggl(\frac{\partial v}{\partial T}\biggr)_{p}
+ \biggl(\frac{\partial p}{\partial T}\biggr)_{v}
\end{aligned}
\begin{aligned}
\alpha
&=\frac{1}{v} \biggl(\frac{\partial v}{\partial T}\biggr)_{p}
\end{aligned}
を使うと\alpha
&=\frac{1}{v} \biggl(\frac{\partial v}{\partial T}\biggr)_{p}
\end{aligned}
\begin{aligned}
\biggl(\frac{\partial p}{\partial T}\biggr)_{v}
&= - \frac{\alpha}{\rho} \biggl(\frac{\partial p}{\partial v}\biggr)_{T}
\end{aligned}
となる．\biggl(\frac{\partial p}{\partial T}\biggr)_{v}
&= - \frac{\alpha}{\rho} \biggl(\frac{\partial p}{\partial v}\biggr)_{T}
\end{aligned}
関連記事[E]の関係式
\begin{aligned}
\biggl(\frac{\partial p}{\partial v} \biggr)_{T}
&= -\frac{\rho^{2} c^{2}}{\gamma}
\end{aligned}
を使うと，\biggl(\frac{\partial p}{\partial v} \biggr)_{T}
&= -\frac{\rho^{2} c^{2}}{\gamma}
\end{aligned}
\begin{aligned}
\partial_{i} p
&=\biggl(\frac{\partial p}{\partial v}\biggr)_{T}
\biggl(\frac{\partial v}{\partial \rho}\biggr)_{T}
\partial_{i} \rho
+ \biggl(\frac{\partial p}{\partial T}\biggr)_{v}
\partial_{i} T \\
&=\biggl(\frac{\partial p}{\partial v}\biggr)_{T}
\biggl[ \biggl(\frac{\partial v}{\partial \rho}\biggr)_{T} \partial_{i} \rho
- \frac{\alpha}{\rho} \partial_{i} T
\biggr] \\
&=-\frac{\rho^{2} c^{2}}{\gamma}
\biggl(-\frac{1}{\rho^{2}} \partial_{i} \rho
- \frac{\alpha}{\rho} \partial_{i} T \biggr) \\
&=\frac{c^{2}}{\gamma}
\biggl( \partial_{i} \rho
+ \rho \alpha \partial_{i} T \biggr)
\end{aligned}
となる．\partial_{i} p
&=\biggl(\frac{\partial p}{\partial v}\biggr)_{T}
\biggl(\frac{\partial v}{\partial \rho}\biggr)_{T}
\partial_{i} \rho
+ \biggl(\frac{\partial p}{\partial T}\biggr)_{v}
\partial_{i} T \\
&=\biggl(\frac{\partial p}{\partial v}\biggr)_{T}
\biggl[ \biggl(\frac{\partial v}{\partial \rho}\biggr)_{T} \partial_{i} \rho
- \frac{\alpha}{\rho} \partial_{i} T
\biggr] \\
&=-\frac{\rho^{2} c^{2}}{\gamma}
\biggl(-\frac{1}{\rho^{2}} \partial_{i} \rho
- \frac{\alpha}{\rho} \partial_{i} T \biggr) \\
&=\frac{c^{2}}{\gamma}
\biggl( \partial_{i} \rho
+ \rho \alpha \partial_{i} T \biggr)
\end{aligned}
ただし，
\begin{aligned}
c^{2}
&=\biggl(\frac{\partial p}{\partial \rho} \biggr)_{\mathrm{ad}}
\end{aligned}
であり，$c$は等方的な変形をする場合（$\sigma_{ij} = -p \delta_{ij}$，「球面」圧縮波（疎密波））の音速である．$p = -K \mathrm{div\,} \bm{u}$であり，c^{2}
&=\biggl(\frac{\partial p}{\partial \rho} \biggr)_{\mathrm{ad}}
\end{aligned}
\begin{aligned}
0&=\frac{\partial \rho}{\partial t} + \rho_{0} \, \mathrm{div\,} \dot{\bm{u}} \\
&=\frac{\partial }{\partial t} (\rho + \rho_{0} \mathrm{div\,} \bm{u})
\end{aligned}
だから$\rho + \rho_{0} \mathrm{div\,} \bm{u} = f(x, y, z)$．したがって，0&=\frac{\partial \rho}{\partial t} + \rho_{0} \, \mathrm{div\,} \dot{\bm{u}} \\
&=\frac{\partial }{\partial t} (\rho + \rho_{0} \mathrm{div\,} \bm{u})
\end{aligned}
\begin{aligned}
c^{2}
&=\biggl(\frac{\partial p}{\partial \rho} \biggr)_{\mathrm{ad}} \\
&=\biggl(\frac{\partial p}{\partial (\mathrm{div\,} \bm{u})} \biggr)_{\mathrm{ad}}
\biggl(\frac{\partial (\mathrm{div\,} \bm{u})}{\partial \rho} \biggr)_{\mathrm{ad}} \\
&=\frac{K}{\rho_{0}}
\end{aligned}
となる．c^{2}
&=\biggl(\frac{\partial p}{\partial \rho} \biggr)_{\mathrm{ad}} \\
&=\biggl(\frac{\partial p}{\partial (\mathrm{div\,} \bm{u})} \biggr)_{\mathrm{ad}}
\biggl(\frac{\partial (\mathrm{div\,} \bm{u})}{\partial \rho} \biggr)_{\mathrm{ad}} \\
&=\frac{K}{\rho_{0}}
\end{aligned}
参考文献
- [1]流体力学 (前編) (今井 功)：$\text{\sect} 60.$「変形速度と応力の関係」
- [2] 弾性体と流体 (恒藤 敏彦)：7「弾性体」
- [3] 振動・波動 (基礎物理学選書 8)：$\text{\sect} 6.7$弾性波，$\text{\sect} 6.8$音波
