Équation de Navier

On note ${\displaystyle \mathbf {u} \left(\mathbf {x} ,t\right)}$ le champ des déplacements et ${\displaystyle \mathbf {f} _{v}}$ la force volumique qui s'exerce.

On a :

${\displaystyle \rho _{0}{\frac {\partial ^{2}\mathbf {u} }{\partial t^{2}}}=\left(\lambda +\mu \right)\mathbf {grad} \left(\mathrm {div} \;\mathbf {u} \right)+\mu \Delta \mathbf {u} +\mathbf {f} _{v}}$

où ${\displaystyle \lambda }$ et ${\displaystyle \mu }$ sont les coefficients de Lamé du solide et ${\displaystyle \rho _{0}}$ sa masse volumique. On peut écrire cette équation en fonction du module d'Young E et du coefficient de Poisson ${\displaystyle \nu }$ :

${\displaystyle \rho _{0}{\frac {\partial ^{2}\mathbf {u} }{\partial t^{2}}}={\frac {E}{2(1+\nu )(1-2\nu )}}\mathbf {grad} \left(\mathrm {div} \;\mathbf {u} \right)+{\frac {E}{2(1+\nu )}}\Delta \mathbf {u} +\mathbf {f} _{v}}$

On note ${\displaystyle {\boldsymbol {\sigma }}~}$ le tenseur des contraintes et ${\displaystyle {\boldsymbol {e}}~}$ le tenseur des déformations. La relation fondamentale de la dynamique s'écrit :

${\displaystyle \rho _{0}{\frac {\partial ^{2}\mathbf {u} }{\partial t^{2}}}=\mathbf {div} {\boldsymbol {\sigma }}+\mathbf {f} _{v}}$

D'autre part, on a la loi de Hooke :

${\displaystyle {\boldsymbol {\sigma }}=\lambda \mathrm {Tr} \left({\boldsymbol {e}}\right)\mathbf {I} +2\mu {\boldsymbol {e}}}$

d'où (en appliquant la sommation sur les indices (Convention de sommation d'Einstein)) :

${\displaystyle {\begin{aligned}\left(\mathbf {div} {\boldsymbol {\sigma }}\right)_{i}&=\lambda \left(\mathbf {div} \left(\mathrm {Tr} \left({\boldsymbol {e}}\right)\mathbf {I} \right)\right)_{i}+2\mu \left(\mathbf {div} {\boldsymbol {e}}\right)_{i}\\\ &=\lambda {\frac {\partial }{\partial x_{j}}}\left(\mathrm {Tr} \left({\boldsymbol {e}}\right)\mathbf {I} \right)_{ij}+2\mu {\frac {\partial e_{ij}}{\partial x_{j}}}\\\ &=\lambda {\frac {\partial e_{ll}}{\partial x_{i}}}+2\mu {\frac {\partial e_{ij}}{\partial x_{j}}}\\\ &=\lambda {\frac {\partial ^{2}u_{l}}{\partial x_{i}\partial x_{l}}}+\mu {\frac {\partial }{\partial x_{j}}}\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right)\\\ &=(\lambda +\mu ){\frac {\partial ^{2}u_{j}}{\partial x_{i}\partial x_{j}}}+\mu {\frac {\partial ^{2}u_{i}}{\partial x_{j}^{2}}}\\\ &=\left(\lambda +\mu \right)(\mathbf {grad} \left(\mathrm {div} \;\mathbf {u} \right))_{i}+\mu (\Delta \mathbf {u} )_{i}\end{aligned}}}$

ce qui donne la relation cherchée.