Coefficient de Lamé

En mécanique des milieux continus, et plus précisément en élasticité linéaire, les coefficients de Lamé sont les deux coefficients suivants :

  • λ {\displaystyle \lambda } , ou premier coefficient de Lamé ;
  • μ {\displaystyle \mu } , le module de cisaillement, aussi appelé second coefficient de Lamé. Ce coefficient est aussi parfois noté G {\displaystyle G} .

Ces deux coefficients sont homogènes à une contrainte et ont donc pour unité le pascal (Pa) ou newton par mètre carré (N/m²). Ils portent le nom de Gabriel Lamé.

Dans un matériau homogène, isotrope, satisfaisant la loi de Hooke en 3 {\displaystyle 3} dimensions, soit:

σ = 2 μ ε + λ tr ⁡ ( ε ) I 3 , {\displaystyle {\boldsymbol {\sigma }}=2\mu {\boldsymbol {\varepsilon }}+\lambda \operatorname {tr} ({\boldsymbol {\varepsilon }}){\boldsymbol {I}}_{3},}
où σ {\displaystyle {\boldsymbol {\sigma }}} est le tenseur des contraintes, ε {\displaystyle {\boldsymbol {\varepsilon }}} le tenseur des déformations, I 3 {\displaystyle {\boldsymbol {I}}_{3}} le tenseur identité et tr ⁡ ( ⋅ ) {\displaystyle \operatorname {tr} (\cdot )} la trace (voir aussi notation de Voigt). Le premier paramètre λ {\displaystyle \lambda } n'a pas d'interprétation physique, mais il sert à simplifier la matrice de raideur dans la loi de Hooke ci-dessus. Les deux paramètres constituent un paramétrage des modules élastiques pour les matériaux homogènes isotropes, et sont donc liés aux autres modules. Selon les cas, on pourra choisir un autre paramétrage.

En particulier, les coefficients de Lamé s'expriment en fonction du module de Young E {\displaystyle E} et du coefficient de Poisson ν {\displaystyle \nu }  :

λ = E ν ( 1 + ν ) ( 1 − 2 ν ) , μ = E 2 ( 1 + ν ) . {\displaystyle \lambda ={\frac {E\nu }{(1+\nu )(1-2\nu )}},\quad \mu ={\frac {E}{2(1+\nu )}}.}
Et inversement :
ν = λ 2 ( λ + μ ) , 1 E = λ + μ μ ( 3 λ + 2 μ ) . {\displaystyle \nu ={\frac {\lambda }{2(\lambda +\mu )}},\quad {\frac {1}{E}}={\frac {\lambda +\mu }{\mu (3\lambda +2\mu )}}.}

Sources

  • F. Kang, S. Zhong-Ci, Mathematical Theory of Elastic Structures, Springer New York, (ISBN 0-387-51326-4), (1981)
  • G. Mavko, T. Mukerji, J. Dvorkin. The Rock Physics Handbook. Cambridge University Press 2003 (paperback). (ISBN 0-521-54344-4)
v · m
Modules d'élasticité pour des matériaux homogènes et isotropes
Formules de conversion
Les propriétés élastiques des matériaux homogènes, isotropes et linéaires sont déterminées de manière unique par deux modules quelconques parmi ceux-ci. Ainsi, on peut calculer chacun à partir de deux d'entre eux en utilisant ces formules.

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} }}}}

 

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