Exercices
Vérifier la relation suivante: \[ rot \left( \overrightarrow{A} \wedge \overrightarrow{B} \right)\, =\, \overrightarrow{A} \; div\overrightarrow{B}\, - \,\overrightarrow{B} \; div\overrightarrow{A}\, + \nabla\overrightarrow{A} \cdot \overrightarrow{B} \,-\, \nabla \overrightarrow{B} \cdot \overrightarrow{A}\]
\[\overrightarrow{A} \wedge \overrightarrow{B} =\varepsilon_{ijk} \, A_i\, B_j \,\overrightarrow{e}_k\] \[rot \overrightarrow{U} = \varepsilon_{pqr} \, U_{r,q}\,\overrightarrow{e}_p\] \[\begin{aligned} rot \left( \overrightarrow{A} \wedge \overrightarrow{B} \right) &= \varepsilon_{pqr} \left( \varepsilon_{ijr} \, A_i\, B_j \right)_{,q} \,\overrightarrow{e}_p \\ &= \varepsilon_{rpq} \varepsilon_{rij}\left( \varepsilon_{ijr} \, A_i\, B_j \right)_{,q} \,\overrightarrow{e}_p \\ &= \left( \delta_{pi} \delta_{qj} - \delta_{pj} \delta_{qi} \right) \left( A_i B_j \right)_{,q} \,\overrightarrow{e}_p \\ &= \delta_{pi} \delta_{qj} \left( A_i B_j \right)_{,q} \,\overrightarrow{e}_p - \delta_{pj} \delta_{qi} \left( A_i B_j \right)_{,q} \,\overrightarrow{e}_p \\ &= \left( A_i B_j \right)_{,j} \,\overrightarrow{e}_i - \left( A_i B_j \right)_{,i} \,\overrightarrow{e}_j \\ &= A_{i,j} B_j \,\overrightarrow{e}_i \; + \; A_{i} B_{j,j}\,\overrightarrow{e}_i\; - \; A_{i,i} B_{j}\,\overrightarrow{e}_j\;-\; A_{i} B_{j,i}\,\overrightarrow{e}_j \\ &= \, \nabla\overrightarrow{A} \cdot \overrightarrow{B} \; + \; \overrightarrow{A} \; div\overrightarrow{B}\, - \,\overrightarrow{B} \; div\overrightarrow{A} \,-\, \nabla \overrightarrow{B} \cdot \overrightarrow{A} \end{aligned}\] car \[\nabla\overrightarrow{A} = A_{i,j} \,\overrightarrow{e}_i \otimes \overrightarrow{e}_j\] \[\nabla\overrightarrow{A} \cdot \overrightarrow{B} = A_{i,j} B_k \,\overrightarrow{e}_i \otimes \overrightarrow{e}_j \cdot \overrightarrow{e}_k = A_{i,j} B_j \,\overrightarrow{e}_i\]
Vérifier la relation suivante: \[ div \left( \overline{\overline{A}} \overrightarrow{u} \right)\, =\, div\overline{\overline{A}}^T \cdot \overrightarrow{u}\, + \,\nabla\overrightarrow{u} \;: \; \overline{\overline{A}}\]
\[\overline{\overline{A}} \overrightarrow{u} \,=\, \left( A_{ij}\;\overrightarrow{e}_i\otimes\overrightarrow{e}_j \right) \cdot \left( u_{k}\;\overrightarrow{e}_k \right)\,=\, A_{ij} u_k \,\overrightarrow{e}_i \otimes \overrightarrow{e}_j \cdot \overrightarrow{e}_k = A_{ij} u_j \,\overrightarrow{e}_i\] \[div \left( \overline{\overline{A}} \overrightarrow{u} \right) \,=\, \left( A_{ij} u_j \right)_i \,=\, A_{ij,i} u_j\,+\, A_{ij} u_j,i\] \[\overline{\overline{A}}^T \,=\, A_{pq} \,\overrightarrow{e}_q \otimes \overrightarrow{e}_p\] \[div\overline{\overline{A}}^T \,=\, A_{pq,p} \,\overrightarrow{e}_q\] \[div\overline{\overline{A}}^T \cdot \overrightarrow{u}\,=\, A_{pq,p}u_q \,=\, A_{ij,i} u_j\] \[\nabla\overrightarrow{u} = u_{i,j} \,\overrightarrow{e}_i \otimes \overrightarrow{e}_j\] \[\nabla\overrightarrow{u} \;: \; \overline{\overline{A}} = u_{i,j} A_{pq} \,\overrightarrow{e}_i \otimes \overrightarrow{e}_j\,:\, \overrightarrow{e}_p \otimes \overrightarrow{e}_q \,=\, u_{i,j} A_{jq} \,\overrightarrow{e}_i \,\cdot\, \overrightarrow{e}_q\,=\, u_{i,j} A_{ji}\]
Soient \(\overline{\overline{A}}\) et \(\overline{\overline{B}}\) deux tenseurs d’ordre 2 et \(\overline{\overline{\overline{\overline{C}}}}\) un tenseur d’ordre 4, tels que : \(\overline{\overline{B}}\,=\,\overline{\overline{\overline{\overline{C}}}}\,:\,\overline{\overline{A}}\) et \(\overline{\overline{\overline{\overline{C}}}}\,=\,\left( \lambda \,\delta_{ij}\,\delta_{kl} \,+\, \mu \,\delta_{ik}\,\delta_{jl} \,+\,\mu \,\delta_{il}\,\delta_{jk} \, \right) \overrightarrow{e}_i \otimes\overrightarrow{e}_j \otimes\overrightarrow{e}_k \otimes\overrightarrow{e}_l\). où \(\lambda\) et \(\mu\) sont des constantes.
Déterminer \(Tr\overline{\overline{B}}\) en fonction de \(Tr\overline{\overline{A}}\).
Déterminer \(Tr\left(\overline{\overline{B}}^2\right)\) en fonction de \(\left(Tr\overline{\overline{A}}\right)^2\), \(\overline{\overline{A}}:\overline{\overline{A}}\) et \(||\overline{\overline{A}}||^2\).
\[\overline{\overline{B}}\,=\,\overline{\overline{\overline{\overline{C}}}}\,:\,\overline{\overline{A}} \,=\, C_{ijkl} \,\overrightarrow{e}_i \otimes\overrightarrow{e}_j \otimes\overrightarrow{e}_k \otimes\overrightarrow{e}_l \;:\; A_{pq} \, \overrightarrow{e}_p \otimes\overrightarrow{e}_q \] \[\overline{\overline{B}}\,=\,\overline{\overline{\overline{\overline{C}}}}\,:\,\overline{\overline{A}} \,=\, C_{ijkp} \,A_{pq} \, \overrightarrow{e}_i \otimes\overrightarrow{e}_j \otimes\overrightarrow{e}_k \cdot \overrightarrow{e}_q \,=\, C_{ijqp} \,A_{pq} \, \overrightarrow{e}_i \otimes \overrightarrow{e}_j \] \[Tr\overline{\overline{B}} = \overline{\overline{B}} : \overline{\overline{I}} = B_{ii}\] \[B_{ii} = C_{iiqp} \,A_{pq} = \left( \lambda \,\delta_{ii}\,\delta_{qp} \,+\, \mu \,\delta_{iq}\,\delta_{ip} \,+\,\mu \,\delta_{ip}\,\delta_{iq} \right)\,A_{pq}\] \[Tr\overline{\overline{B}} = \lambda \,\delta_{ii}\,\delta_{qp}\,A_{pq} \,+\, \mu \,\delta_{iq}\,\delta_{ip}\,A_{pq} \,+\,\mu \,\delta_{ip}\,\delta_{iq}\,A_{pq}\] \[Tr\overline{\overline{B}} = \lambda \,\delta_{ii}\,A_{pp} \,+\, \mu \,A_{ii} \,+\,\mu \,A_{ii}\] \[Tr\overline{\overline{B}} = \lambda \,3\,Tr\overline{\overline{A}} \,+\, \mu \,Tr\overline{\overline{A}} \,+\,\mu \,Tr\overline{\overline{A}}\;=\; \left( 3\lambda + 2 \mu \right) Tr\overline{\overline{A}}\]
\[\overline{\overline{B}}\cdot \overline{\overline{B}} = C_{ijkl} \,A_{lk} \,\overrightarrow{e}_i \otimes\overrightarrow{e}_j \cdot C_{pqmn} A_{nm} \,\overrightarrow{e}_p \otimes\overrightarrow{e}_q \] \[\overline{\overline{B}}\cdot \overline{\overline{B}}= C_{ijkl} \, C_{jqmn} \,A_{lk} \,A_{nm } \, \overrightarrow{e}_i \otimes \overrightarrow{e}_q\] \[Tr\left(\overline{\overline{B}}^2\right) = C_{ijkl} \, C_{jimn} \,A_{lk} \,A_{nm }\] \[\begin{aligned} Tr\left(\overline{\overline{B}}^2\right) &= \lambda \delta_{ij} \delta_{kl} A_{lk} \lambda \delta_{ji} \delta_{mn} A_{mn} + \lambda \delta_{ij} \delta_{kl} A_{lk} \mu \delta_{jm} \delta_{in} A_{nm} + \lambda \delta_{ij} \delta_{kl} A_{lk} \mu \delta_{im} \delta_{jn} A_{nm} \\&+ \mu \delta_{ik} \delta_{jl} A_{lk} \lambda \delta_{ji} \delta_{mn} A_{mn} + \mu \delta_{ik} \delta_{jl} A_{lk} \mu \delta_{jm} \delta_{in} A_{nm} + \mu \delta_{ik} \delta_{jl} A_{lk} \mu \delta_{im} \delta_{jn} A_{nm} \\&+ \mu \delta_{il} \delta_{jk} A_{lk} \lambda \delta_{ji} \delta_{mn} A_{mn} + \mu \delta_{il} \delta_{jk} A_{lk} \mu \delta_{jm} \delta_{in} A_{nm} + \mu \delta_{il} \delta_{jk} A_{lk} \mu \delta_{im} \delta_{jn} A_{nm} \\&= \lambda \delta_{ii} A_{kk} \lambda A_{mm} + \lambda A_{kk} \mu A_{ii} + \lambda A_{kk} \mu A_{ii} \\&+ \mu A_{ii} \lambda A_{mm} + \mu A_{ji} \mu A_{ij} + \mu A_{ji} \mu A_{ji} \\&+ \mu A_{ii} \lambda A_{nn} + \mu A_{ij} \mu A_{ij} + \mu A_{ij} \mu A_{ji} \\&= 3 \lambda^2 \left( A_{kk} \right)^2 + \lambda \mu \left( A_{kk} \right)^2 + \lambda \mu \left( A_{kk} \right)^2 \\&+ \lambda \mu \left( A_{kk} \right)^2 + \mu^2 A_{ji} A_{ij} + \mu^2 A_{ji} A_{ij} \\&+ \lambda \mu \left( A_{kk} \right)^2 + \mu^2 A_{ij} A_{ij} + \mu^2 A_{ji} A_{ij} \\&= \left(3 \lambda^2 + 4 \lambda \mu \right) \left( Tr\overline{\overline{A}} \right)^2 + 2 \mu^2 \overline{\overline{A}}:\overline{\overline{A}} + 2 \mu^2 ||\overline{\overline{A}}||^2 \end{aligned}\]