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En rhéologie, le module de relaxation permet de rendre compte de la relaxation de contrainte, la déformation étant maintenue constante.
La contrainte σ {\displaystyle \sigma } à un temps t {\displaystyle t} ne dépend pour un fluide newtonien que du taux de déformation à ce même temps :
Par contre, pour un fluide viscoélastique, cette même contrainte va dépendre de l'histoire des taux de déformation via le module de relaxation G ( t ) {\displaystyle G(t)} (ou E ( t ) {\displaystyle E(t)} ) :
Physiquement, on s'attend à ce que cette fonction tende vers 0 lorsque t tend vers l'infini ; c'est la perte de mémoire des états les plus anciens.
Dans le cadre du modèle de Maxwell, on montre que le module de relaxation G ( t ) {\displaystyle G(t)} vaut :
où τ = η E {\displaystyle \tau ={\frac {\eta }{E}}} est le temps de relaxation du modèle de Maxwell.
Expérimentalement, on applique en DMA des déformations sinusoïdales. On définit une déformation complexe :
ce qui amène à une contrainte complexe :
avec :
où :
Le facteur de perte indique la capacité d'une matière viscoélastique à dissiper de l'énergie mécanique en chaleur. Il est donné par l'équation :
où δ {\displaystyle \delta } est l'angle de phase ou de perte.
Une valeur faible du facteur de perte traduit un comportement élastique marqué : le matériau étant soumis à une sollicitation, la dissipation d'énergie par frottement interne est faible.
Il est par ailleurs possible de définir une viscosité complexe de la manière suivante :
formules en 3D
( λ , G ) {\displaystyle (\lambda ,G)}
( E , G ) {\displaystyle (E,G)}
( K , λ ) {\displaystyle (K,\lambda )}
( K , G ) {\displaystyle (K,G)}
( λ , ν ) {\displaystyle (\lambda ,\nu )}
( G , ν ) {\displaystyle (G,\nu )}
( E , ν ) {\displaystyle (E,\nu )}
( K , ν ) {\displaystyle (K,\nu )}
( K , E ) {\displaystyle (K,E)}
( M , G ) {\displaystyle (M,G)}
K [ P a ] = {\displaystyle K\,[\mathrm {Pa} ]=}
λ + 2 G 3 {\displaystyle \lambda +{\tfrac {2G}{3}}}
E G 3 ( 3 G − E ) {\displaystyle {\tfrac {EG}{3(3G-E)}}}
λ ( 1 + ν ) 3 ν {\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}
2 G ( 1 + ν ) 3 ( 1 − 2 ν ) {\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}
E 3 ( 1 − 2 ν ) {\displaystyle {\tfrac {E}{3(1-2\nu )}}}
M − 4 G 3 {\displaystyle M-{\tfrac {4G}{3}}}
E [ P a ] = {\displaystyle E\,[\mathrm {Pa} ]=}
G ( 3 λ + 2 G ) λ + G {\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}
9 K ( K − λ ) 3 K − λ {\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}
9 K G 3 K + G {\displaystyle {\tfrac {9KG}{3K+G}}}
λ ( 1 + ν ) ( 1 − 2 ν ) ν {\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}
2 G ( 1 + ν ) {\displaystyle 2G(1+\nu )\,}
3 K ( 1 − 2 ν ) {\displaystyle 3K(1-2\nu )\,}
G ( 3 M − 4 G ) M − G {\displaystyle {\tfrac {G(3M-4G)}{M-G}}}
λ [ P a ] = {\displaystyle \lambda \,[\mathrm {Pa} ]=}
G ( E − 2 G ) 3 G − E {\displaystyle {\tfrac {G(E-2G)}{3G-E}}}
K − 2 G 3 {\displaystyle K-{\tfrac {2G}{3}}}
2 G ν 1 − 2 ν {\displaystyle {\tfrac {2G\nu }{1-2\nu }}}
E ν ( 1 + ν ) ( 1 − 2 ν ) {\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}
3 K ν 1 + ν {\displaystyle {\tfrac {3K\nu }{1+\nu }}}
3 K ( 3 K − E ) 9 K − E {\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}
M − 2 G {\displaystyle M-2G}
G [ P a ] = {\displaystyle G\,[\mathrm {Pa} ]=}
3 ( K − λ ) 2 {\displaystyle {\tfrac {3(K-\lambda )}{2}}}
λ ( 1 − 2 ν ) 2 ν {\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}
E 2 ( 1 + ν ) {\displaystyle {\tfrac {E}{2(1+\nu )}}}
3 K ( 1 − 2 ν ) 2 ( 1 + ν ) {\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}
3 K E 9 K − E {\displaystyle {\tfrac {3KE}{9K-E}}}
ν [ 1 ] = {\displaystyle \nu \,[1]=}
λ 2 ( λ + G ) {\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}
E 2 G − 1 {\displaystyle {\tfrac {E}{2G}}-1}
λ 3 K − λ {\displaystyle {\tfrac {\lambda }{3K-\lambda }}}
3 K − 2 G 2 ( 3 K + G ) {\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}
3 K − E 6 K {\displaystyle {\tfrac {3K-E}{6K}}}
M − 2 G 2 M − 2 G {\displaystyle {\tfrac {M-2G}{2M-2G}}}
M [ P a ] = {\displaystyle M\,[\mathrm {Pa} ]=}
λ + 2 G {\displaystyle \lambda +2G}
G ( 4 G − E ) 3 G − E {\displaystyle {\tfrac {G(4G-E)}{3G-E}}}
3 K − 2 λ {\displaystyle 3K-2\lambda \,}
K + 4 G 3 {\displaystyle K+{\tfrac {4G}{3}}}
λ ( 1 − ν ) ν {\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}
2 G ( 1 − ν ) 1 − 2 ν {\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}
E ( 1 − ν ) ( 1 + ν ) ( 1 − 2 ν ) {\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}
3 K ( 1 − ν ) 1 + ν {\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}
3 K ( 3 K + E ) 9 K − E {\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}
formules en 2D
( λ 2 D , G 2 D ) {\displaystyle (\lambda _{\mathrm {2D} },G_{\mathrm {2D} })}
( E 2 D , G 2 D ) {\displaystyle (E_{\mathrm {2D} },G_{\mathrm {2D} })}
( K 2 D , λ 2 D ) {\displaystyle (K_{\mathrm {2D} },\lambda _{\mathrm {2D} })}
( K 2 D , G 2 D ) {\displaystyle (K_{\mathrm {2D} },G_{\mathrm {2D} })}
( λ 2 D , ν 2 D ) {\displaystyle (\lambda _{\mathrm {2D} },\nu _{\mathrm {2D} })}
( G 2 D , ν 2 D ) {\displaystyle (G_{\mathrm {2D} },\nu _{\mathrm {2D} })}
( E 2 D , ν 2 D ) {\displaystyle (E_{\mathrm {2D} },\nu _{\mathrm {2D} })}
( K 2 D , ν 2 D ) {\displaystyle (K_{\mathrm {2D} },\nu _{\mathrm {2D} })}
( K 2 D , E 2 D ) {\displaystyle (K_{\mathrm {2D} },E_{\mathrm {2D} })}
( M 2 D , G 2 D ) {\displaystyle (M_{\mathrm {2D} },G_{\mathrm {2D} })}
K 2 D [ N / m ] = {\displaystyle K_{\mathrm {2D} }\,[\mathrm {N/m} ]=}
λ 2 D + G 2 D {\displaystyle \lambda _{\mathrm {2D} }+G_{\mathrm {2D} }}
G 2 D E 2 D 4 G 2 D − E 2 D {\displaystyle {\tfrac {G_{\mathrm {2D} }E_{\mathrm {2D} }}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
λ 2 D ( 1 + ν 2 D ) 2 ν 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}
G 2 D ( 1 + ν 2 D ) 1 − ν 2 D {\displaystyle {\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} }}}}
E 2 D 2 ( 1 − ν 2 D ) {\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1-\nu _{\mathrm {2D} })}}}
M 2 D − G 2 D {\displaystyle M_{\mathrm {2D} }-G_{\mathrm {2D} }}
E 2 D [ N / m ] = {\displaystyle E_{\mathrm {2D} }\,[\mathrm {N/m} ]=}
4 G 2 D ( λ 2 D + G 2 D ) λ 2 D + 2 G 2 D {\displaystyle {\tfrac {4G_{\mathrm {2D} }(\lambda _{\mathrm {2D} }+G_{\mathrm {2D} })}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}
4 K 2 D ( K 2 D − λ 2 D ) 2 K 2 D − λ 2 D {\displaystyle {\tfrac {4K_{\mathrm {2D} }(K_{\mathrm {2D} }-\lambda _{\mathrm {2D} })}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}
4 K 2 D G 2 D K 2 D + G 2 D {\displaystyle {\tfrac {4K_{\mathrm {2D} }G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}
λ 2 D ( 1 + ν 2 D ) ( 1 − ν 2 D ) ν 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} }}}}
2 G 2 D ( 1 + ν 2 D ) {\displaystyle 2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,}
2 K 2 D ( 1 − ν 2 D ) {\displaystyle 2K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}
4 G 2 D ( M 2 D − G 2 D ) M 2 D {\displaystyle {\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} }}}}
λ 2 D [ N / m ] = {\displaystyle \lambda _{\mathrm {2D} }\,[\mathrm {N/m} ]=}
2 G 2 D ( E 2 D − 2 G 2 D ) 4 G 2 D − E 2 D {\displaystyle {\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
K 2 D − G 2 D {\displaystyle K_{\mathrm {2D} }-G_{\mathrm {2D} }}
2 G 2 D ν 2 D 1 − ν 2 D {\displaystyle {\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}
E 2 D ν 2 D ( 1 + ν 2 D ) ( 1 − ν 2 D ) {\displaystyle {\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}
2 K 2 D ν 2 D 1 + ν 2 D {\displaystyle {\tfrac {2K_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}
2 K 2 D ( 2 K 2 D − E 2 D ) 4 K 2 D − E 2 D {\displaystyle {\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
M 2 D − 2 G 2 D {\displaystyle M_{\mathrm {2D} }-2G_{\mathrm {2D} }}
G 2 D [ N / m ] = {\displaystyle G_{\mathrm {2D} }\,[\mathrm {N/m} ]=}
K 2 D − λ 2 D {\displaystyle K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}
λ 2 D ( 1 − ν 2 D ) 2 ν 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}
E 2 D 2 ( 1 + ν 2 D ) {\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1+\nu _{\mathrm {2D} })}}}
K 2 D ( 1 − ν 2 D ) 1 + ν 2 D {\displaystyle {\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} }}}}
K 2 D E 2 D 4 K 2 D − E 2 D {\displaystyle {\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} }}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
ν 2 D [ 1 ] = {\displaystyle \nu _{\mathrm {2D} }\,[1]=}
λ 2 D λ 2 D + 2 G 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}
E 2 D 2 G 2 D − 1 {\displaystyle {\tfrac {E_{\mathrm {2D} }}{2G_{\mathrm {2D} }}}-1}
λ 2 D 2 K 2 D − λ 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}
K 2 D − G 2 D K 2 D + G 2 D {\displaystyle {\tfrac {K_{\mathrm {2D} }-G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}
2 K 2 D − E 2 D 2 K 2 D {\displaystyle {\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} }}{2K_{\mathrm {2D} }}}}
M 2 D − 2 G 2 D M 2 D {\displaystyle {\tfrac {M_{\mathrm {2D} }-2G_{\mathrm {2D} }}{M_{\mathrm {2D} }}}}
M 2 D [ N / m ] = {\displaystyle M_{\mathrm {2D} }\,[\mathrm {N/m} ]=}
λ 2 D + 2 G 2 D {\displaystyle \lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}
4 G 2 D 2 4 G 2 D − E 2 D {\displaystyle {\tfrac {4G_{\mathrm {2D} }^{2}}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
2 K 2 D − λ 2 D {\displaystyle 2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}
K 2 D + G 2 D {\displaystyle K_{\mathrm {2D} }+G_{\mathrm {2D} }}
λ 2 D ν 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\nu _{\mathrm {2D} }}}}
2 G 2 D 1 − ν 2 D {\displaystyle {\tfrac {2G_{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}
E 2 D ( 1 − ν 2 D ) ( 1 + ν 2 D ) {\displaystyle {\tfrac {E_{\mathrm {2D} }}{(1-\nu _{\mathrm {2D} })(1+\nu _{\mathrm {2D} })}}}
2 K 2 D 1 + ν 2 D {\displaystyle {\tfrac {2K_{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}
4 K 2 D 2 4 K 2 D − E 2 D {\displaystyle {\tfrac {4K_{\mathrm {2D} }^{2}}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
![{\displaystyle K\,[\mathrm {Pa} ]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3ab31b3a3865235dccbc488812ff4f37eebbab)
![{\displaystyle E\,[\mathrm {Pa} ]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/750cbb7f7d67f888ecb19e183ac4d7410006ef9b)
![{\displaystyle \lambda \,[\mathrm {Pa} ]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/feb97a3834f075f32114f08046be6251bd037800)
![{\displaystyle G\,[\mathrm {Pa} ]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2326ec29a9bfa73a5703f0571334e02c57bdd833)
![{\displaystyle \nu \,[1]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b31963444a16a4993d1f3289b9a5e86d2a224934)
![{\displaystyle M\,[\mathrm {Pa} ]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49c2ea87a079fbcc6987591a1669001a144e61ed)
![{\displaystyle K_{\mathrm {2D} }\,[\mathrm {N/m} ]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e5562fda79e2283d607f220ec8aab1a0fc9ee74)
![{\displaystyle E_{\mathrm {2D} }\,[\mathrm {N/m} ]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4026a07ee80c1281e875b1bbed75cd5687a2b13)
![{\displaystyle \lambda _{\mathrm {2D} }\,[\mathrm {N/m} ]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19e94c73fca89904175ad30491fc448f274b8f2c)
![{\displaystyle G_{\mathrm {2D} }\,[\mathrm {N/m} ]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7aefa2eb44d967723c47e26dd09918f1850d553f)
![{\displaystyle \nu _{\mathrm {2D} }\,[1]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5fcb0958432c6617c834d1e36885be1790ac09)
![{\displaystyle M_{\mathrm {2D} }\,[\mathrm {N/m} ]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a0381f833f359a1b0c12a557d1c4455a05d0b7)
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