Démonstration

On calcule terme à terme les coordonnées du tensuer des contraintes en coordonnées cylindriques.On rappele que les vecteurs de base se dérivent aussi :

\(\frac{\partial{\vec{i}}_r}{\partial\theta}={\vec{i}}_\theta\) et \(\frac{\partial{\vec{i}}_\theta}{\partial\theta}=-{\vec{i}}_r\)

On calcule chaque terme en commençant par :

\(\vec{\nabla}\vec{u}=\left({\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right)\left({\vec{i}}_ru_r+{\vec{i}}_\theta u_\theta+{\vec{i}}_zu_z\right)\)

que l’on développe :

\(\ldots=\left({\vec{i}}_r\frac{\partial{\vec{i}}_ru_r}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial{\vec{i}}_ru_r}{\partial\theta}+{\vec{i}}_z\frac{\partial{\vec{i}}_ru_r}{\partial z}\right)+\left({\vec{i}}_r\frac{\partial{\vec{i}}_\theta u_\theta}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial{\vec{i}}_\theta u_\theta}{\partial\theta}+{\vec{i}}_z\frac{\partial{\vec{i}}_\theta u_\theta}{\partial z}\right)\) \(+\left({\vec{i}}_r\frac{\partial{\vec{i}}_zu_z}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial{\vec{i}}_zu_z}{\partial\theta}+{\vec{i}}_z\frac{\partial{\vec{i}}_zu_z}{\partial z}\right)\)

Le 1er terme s’écrit donc :

\(\left({\vec{i}}_r\frac{\partial{\vec{i}}_ru_r}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial{\vec{i}}_ru_r}{\partial\theta}+{\vec{i}}_z\frac{\partial{\vec{i}}_ru_r}{\partial z}\right) =\) \({\vec{i}}_r{\vec{i}}_r\frac{\partial u_r}{\partial r}+{\vec{i}}_\theta\left(\frac{{\vec{i}}_r}{r}\frac{\partial u_r}{\partial\theta}+\frac{u_r}{r}\frac{\partial{\vec{i}}_r}{\partial\theta}\right)+{\vec{i}}_z{\vec{i}}_r\frac{\partial u_r}{\partial z}\) \(={\vec{i}}_r{\vec{i}}_r\frac{\partial u_r}{\partial r}+{\vec{i}}_\theta\left(\frac{{\vec{i}}_r1}{r}\frac{\partial u_r}{\partial\theta}+\frac{u_r}{r}{\vec{i}}_\theta\right)+{\vec{i}}_z{\vec{i}}_r\frac{\partial u_r}{\partial z}\)

soit pour le 1er terme:

\(\begin{align} ={\vec{i}}_r{\vec{i}}_r\frac{\partial u_r}{\partial r}+{\vec{i}}_\theta{\vec{i}}_r\frac{1}{r}\frac{\partial u_r}{\partial\theta}+\frac{u_r}{r}{\vec{i}}_\theta{\vec{i}}_\theta+{\vec{i}}_z{\vec{i}}_r\frac{\partial u_r}{\partial z} \end{align}\)

De la même façon, on développe pour obtenir le deuxième terme:

\(\left({\vec{i}}_r\frac{\partial{\vec{i}}_\theta u_\theta}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial{\vec{i}}_\theta u_\theta}{\partial\theta}+{\vec{i}}_z\frac{\partial{\vec{i}}_\theta u_\theta}{\partial z}\right)\) \(={\vec{i}}_r{\vec{i}}_\theta\frac{\partial u_\theta}{\partial r}+{\vec{i}}_\theta\left(\frac{{\vec{i}}_\theta}{r}\frac{\partial u_\theta}{\partial\theta}+\frac{u_\theta}{r}\frac{\partial{\vec{i}}_\theta}{\partial\theta}\right)+{\vec{i}}_z{\vec{i}}_\theta\frac{\partial u_\theta}{\partial z}\) \(={\vec{i}}_r{\vec{i}}_\theta\frac{\partial u_\theta}{\partial r}+{\vec{i}}_\theta\left({\vec{i}}_\theta\frac{1}{r}\frac{\partial u_\theta}{\partial\theta}-\frac{u_\theta}{r}{\vec{i}}_r\right)+{\vec{i}}_z{\vec{i}}_\theta\frac{\partial u_\theta}{\partial z}\)

soit pour le 2ème terme :

\({\vec{i}}_r{\vec{i}}_\theta\frac{\partial u_\theta}{\partial r}+{\vec{i}}_\theta{\vec{i}}_\theta\frac{1}{r}\frac{\partial u_\theta}{\partial\theta}-\frac{u_\theta}{r}{{\vec{i}}_\theta\vec{i}}_r+{\vec{i}}_z{\vec{i}}_\theta\frac{\partial u_\theta}{\partial z}\)

Le troisème terme s’obtient de la même façon :

\(\left({\vec{i}}_r\frac{\partial{\vec{i}}_zu_z}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial{\vec{i}}_zu_z}{\partial\theta}+{\vec{i}}_z\frac{\partial{\vec{i}}_zu_z}{\partial z}\right)=\)

Soit directement pour le 3ème terme :

\({\vec{i}}_r{\vec{i}}_z\frac{\partial u_z}{\partial r}+{\vec{i}}_\theta{\vec{i}}_z\frac{1}{r}\frac{\partial u_z}{\partial\theta}+{\vec{i}}_z{\vec{i}}_z\frac{\partial u_z}{\partial z}=\)

Ce tenseur d’ordre 2 contient des termes sur chacune des dyades suivantes :

\(\vec{\nabla}\vec{u} \equiv {\vec{i}}_r{\vec{i}}_r+{\vec{i}}_\theta{\vec{i}}_r+{\vec{i}}_\theta{\vec{i}}_\theta+{\vec{i}}_z{\vec{i}}_r+{\vec{i}}_r{\vec{i}}_\theta+{\vec{i}}_\theta{\vec{i}}_\theta-{{\vec{i}}_\theta\vec{i}}_r+{\vec{i}}_z{\vec{i}}_\theta+{\vec{i}}_r{\vec{i}}_z+{\vec{i}}_\theta{\vec{i}}_z+{\vec{i}}_z{\vec{i}}_z\)

que l’on met sout la forme matricielle suivante :

\(\vec{\nabla}\vec{u}=\left[\begin{matrix} \frac{\partial u_r}{\partial r}&\frac{\partial u_\theta}{\partial r}&\frac{\partial u_z}{\partial r}\\ (\frac{1}{r}\frac{\partial u_r}{\partial\theta}-\frac{u_\theta}{r})& (\frac{u_r}{r}+\frac{1}{r}\frac{\partial u_\theta}{\partial\theta})&\frac{1}{r}\frac{\partial u_z}{\partial\theta}\\\frac{\partial u_r}{\partial z}&\frac{\partial u_\theta}{\partial z}&\frac{\partial u_z}{\partial z}\\ \end{matrix}\right]\)

la transposée de ce tenseur s’écrit sous forme vectorielle :

\({\vec{\nabla}\vec{u}}^T=\left[\begin{matrix}\frac{\partial u_r}{\partial r}&(\frac{1}{r}\frac{\partial u_r}{\partial\theta}-\frac{u_\theta}{r})&\frac{\partial u_r}{\partial z}\\\frac{\partial u_\theta}{\partial r}&(\frac{u_r}{r}+\frac{1}{r}\frac{\partial u_\theta}{\partial\theta})&\frac{\partial u_\theta}{\partial z}\\\frac{\partial u_z}{\partial r}&\frac{1}{r}\frac{\partial u_z}{\partial\theta}&\frac{\partial u_z}{\partial z}\\\end{matrix}\right]\)

On en déduit ainsi la 1ère partie du tenseur des contraintes :

\(\mu\left(\vec{\nabla}\vec{u}+\vec{\nabla}{\vec{u}}^T\right)=\mu\left[\begin{matrix}2\frac{\partial u_r}{\partial r}&\left(\frac{1}{r}\frac{\partial u_r}{\partial\theta}+\frac{\partial u_\theta}{\partial r}-\frac{u_\theta}{r}\right)&\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\\\left(\frac{1}{r}\frac{\partial u_r}{\partial\theta}-\frac{u_\theta}{r}+\frac{\partial u_\theta}{\partial r}\right)&2\left(\frac{u_r}{r}+\frac{1}{r}\frac{\partial u_\theta}{\partial\theta}\right)&\frac{1}{r}\frac{\partial u_z}{\partial\theta}+\frac{\partial u_\theta}{\partial z}\\\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}&\frac{\partial u_\theta}{\partial z}+\frac{1}{r}\frac{\partial u_z}{\partial\theta}&2\frac{\partial u_z}{\partial z}\\\end{matrix}\right]\)

On remarque aussi que \(\frac{\partial u_\theta}{\partial r}-\frac{u_\theta}{r}=r\frac{\partial\frac {u_\theta}{r}}{\partial r}\).

En effet:

\(r\frac{\partial\frac{u_\theta}{r}}{\partial r}=\ \frac{r}{r}\frac{\partial u_\theta}{\partial r}+\ ru_\theta\frac{\partial\frac{1}{r}}{\partial r}=\ \frac{\partial u_\theta}{\partial r}-\ r\frac{u_\theta}{r^2}\) \(=\frac{\partial u_\theta}{\partial r}-\ \frac{u_\theta}{r}\)

On ajoute le terme lié à la compressibilité :

\(-\frac{2}{3}\mu\left(\vec{\nabla}.\vec{u}\right)\bar{\bar{I}}\) avec \(\vec{\nabla.}\vec{u}=\frac{1}{r}\frac{\partial r\ u_r}{\partial r}+\frac{1}{r}\frac{\partial u_\theta}{\partial\theta}+\frac{\partial u_z}{\partial z}\)

Ce terme s’ajoute sur chaque élément de la diagonale principale du tenseur. On obtient bien les 6 composantes du tenseur des contraintes newtonien.