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Ligne 139 :
[https://books.google.fr/books?id=PdxOsIkhZCcC&pg=PA24&dq=equation+de+continuit%C3%A9+bilan+masse&hl=fr&sa=X&ved=0ahUKEwi_2Z_soqPOAhUJ2RoKHZ2dABQQ6AEIMDAD#v=onepage&q=equation%20de%20continuit%C3%A9%20bilan%20masse&f=false][http://perso.mines-albi.fr/~louisnar/MECADEF/PolyMecaDef.pdf]
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----[[Fichier:Composantes tenseur des contraintes.png|vignette|400x400px|Illustration]]Les forces surfaciques qui s'appliquent sur les faces d'un élément de volume sont modélisées par ▼
== Tenseur des contraintes ==
▲
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[http://res-nlp.univ-lemans.fr/NLP_C_M02_G02/co/Contenu_21.html tenseur de contraintes]
== Recherche ː travail des forces surfaciques ==
[https://books.google.fr/books?id=YEgrAwAAQBAJ&pg=PA126&dq=conservation+de+l%27%C3%A9nergie+interne+mecanique&hl=fr&sa=X&ved=0ahUKEwjc4I6Y8obOAhWDSRoKHfZ7Dr8Q6AEILDAB#v=onepage&q=conservation%20de%20l'%C3%A9nergie%20interne%20mecanique&f=false eq de l'énergie] ; [http://res-nlp.univ-lemans.fr/NLP_C_M02_G02/co/Contenu_21.html Tenseur des contraintes]
[[w:Produit_matriciel|Produit matriciel]] ; [[w:Matrice_transposée|Matrice transposée]]
<math>\overrightarrow \hbox{div}\ \overline\overline\mathrm{\tau} = \begin{pmatrix}
\frac{\partial\sigma_{xx}}{\partial x} + \frac{\partial\tau_{xy}}{\partial y} + \frac{\partial\tau_{xz}}{\partial z}\\
\frac{\partial\tau_{yx}}{\partial x} + \frac{\partial\sigma_{yy}}{\partial y} + \frac{\partial\tau_{yz}}{\partial z}\\
\frac{\partial\tau_{zx}}{\partial x} + \frac{\partial\tau_{zy}}{\partial y} + \frac{\partial\sigma_{zz}}{\partial z}\\
\end{pmatrix}
</math>
Travail des forces de surface
forces dont la direction est selon l'axe y
<math>\begin{alignat}{2} \sum W_{S_y}=
& \left [ (\tau_{yx}\,v_y)_{x+dx} - (\tau_{yx}\,v_y)_{x}\right ] \mathrm d y \,\mathrm d z
\\ & + \left [ (\sigma_{yy}\,v_y)_{y+dy} - (\sigma_{yy}\,v_y)_{y}\right ] \mathrm d x \,\mathrm d z
\\ & + \left [ (\tau_{yz}\,v_y)_{z+dz} - (\tau_{yz}\,v_y)_{z}\right ] \mathrm d x \,\mathrm d y
\end{alignat}</math>
<math>\sum W_{S_y}= \left(\frac{\partial (\tau_{yx}\,v_y)}{\partial x}
+ \frac{\partial (\sigma_{yy}\,v_y)}{\partial y}
+ \frac{\partial (\tau_{yz}\,v_y)}{\partial z}
\right)\ \mathrm d V</math>
<math>\begin{align} \sum W_{S}=
& \biggl(\frac{\partial (\sigma_{xx}\,v_x)}{\partial x}
+ \frac{\partial (\tau_{xy}\,v_x)}{\partial y}
+ \frac{\partial (\tau_{xz}\,v_x)}{\partial z}
\\ & + \frac{\partial (\tau_{yx}\,v_y)}{\partial x}
+ \frac{\partial (\sigma_{yy}\,v_y)}{\partial y}
+ \frac{\partial (\tau_{yz}\,v_y)}{\partial z}
\\ & + \frac{\partial (\tau_{zx}\,v_z)}{\partial x}
+ \frac{\partial (\tau_{zyy}\,v_z)}{\partial y}
+ \frac{\partial (\sigma_{zz}\,v_z)}{\partial z}\biggr) \mathrm d V
\end{align}
</math>
<math>\sum W_{S}= \hbox{div}\left ( {}^t \overline\overline{\tau} \times \overrightarrow v \right )\mathrm d V
= \hbox{div}
\begin{pmatrix} \sigma_{xx}\,v_x + \tau_{yx}\,v_y + \tau_{zx}\,v_z
\\ \tau_{xy}\,v_x + \sigma_{yy}\,v_y + \tau_{zy}\,v_z
\\ \tau_{xz}\,v_x + \tau_{yz}\,v_y + \sigma_{zz}\,v_z
\end{pmatrix} \mathrm d V
</math>
<math>\begin{align} \sum W_{S}=
& \biggl(\frac{\partial \sigma_{xx}}{\partial x}\,v_x+\sigma_{xx}\,\frac{\partial v_x}{\partial x}
+ \frac{\partial \tau_{xy}}{\partial y}\,v_x+ \tau_{xy}\,\frac{\partial v_x}{\partial y}
+ \frac{\partial \tau_{xz}}{\partial z}\,v_x+ \tau_{xz}\,\frac{\partial v_x}{\partial z}
\\ & + \frac{\partial \tau_{yx}}{\partial x} \,v_y+ \tau_{yx}\,\frac{\partial v_y}{\partial x}
+ \frac{\partial \sigma_{yy}}{\partial y}\,v_y+ \sigma_{yy}\,\frac{\partial v_y}{\partial y}
+ \frac{\partial \tau_{yz}}{\partial z}\,v_y+ \tau_{yz}\,\frac{\partial v_y}{\partial z}
\\ & + \frac{\partial \tau_{zx}}{\partial x}\,v_z+ \tau_{zx}\,\frac{\partial v_z}{\partial x}
+ \frac{\partial \tau_{zyy}}{\partial y}\,v_z+ \tau_{zyy}\,\frac{\partial v_z}{\partial y}
+ \frac{\partial \sigma_{zz}}{\partial z}\,v_z+ \sigma_{zz}\,\frac{\partial v_z}{\partial z}\biggr) \mathrm d V
\end{align}
</math>
<math>\begin{align} \sum W_{S}
= \overrightarrow v \cdot \overrightarrow \hbox{div}\ \overline\overline \mathbf\tau +
& \biggl(\sigma_{xx}\,\frac{\partial v_x}{\partial x}
+ \tau_{xy}\,\frac{\partial v_x}{\partial y}
+ \tau_{xz}\,\frac{\partial v_x}{\partial z}
\\ & + \tau_{yx}\,\frac{\partial v_y}{\partial x}
+ \sigma_{yy}\,\frac{\partial v_y}{\partial y}
+ \tau_{yz}\,\frac{\partial v_y}{\partial z}
\\ & + \tau_{zx}\,\frac{\partial v_z}{\partial x}
+ \tau_{zyy}\,\frac{\partial v_z}{\partial y}
+ \sigma_{zz}\,\frac{\partial v_z}{\partial z}\biggr) \mathrm d V
\end{align}
</math>
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