速度勾配tensor

$ \bm l:=\bm\nabla\bm v

性質

変形勾配tensor$ \bm Fとの関係

$ \bm l=\left.(\dot{\bm{F}}\cdot{\bm{F}^{-1}})\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}

導出

$ \bm{l}:=\bm{\nabla}\bm{v}

$ =\bm{\nabla}\dot{\bm{\phi}}(\bm{\phi}^{-1}(\bm{x},t),t)

$ =\left.\bm{\nabla}\dot{\bm{\phi}}\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}\cdot\bm{\nabla}\bm{\phi}^{-1}(\bm{x},t)

$ =\dot{\bm{F}}(\bm{\phi}^{-1}(\bm{x},t),t)\cdot{\bm{F}(\bm{\phi}^{-1}(\bm{x},t),t)}^{-1}

$ =\left.(\dot{\bm{F}}\cdot{\bm{F}^{-1}})\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}

$ \bm l=\bm d+\bm w