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Lamé parametersFrom Wikipedia, the free encyclopedia

Lamé Constants

 

values that relate the components of elastic stress at some point of a solid isotropic deformable body to th

e components ofstrain at this point:

σx = 2μεxx + !εxx + εyy + ε""#

$ = μεxy

%here σ  and $ are the normal and tan&ential components of stress' ε are the components of strain' and th

e coefficients and μ are Lamé constants' named after the (rench mathematician )* Lamé* he Lamé co

nstants depend on the materialand its temperature* hey are related to the modulus of elasticity !,oun&-s

modulus# and .oisson-s ratio v  by

%here E  is the modulus of elasticity and G is the shear modulus*

/n continuum mechanics' the Lamé parameters !also called the Lamé coefficients or Lamé

constants# are t%o material0dependent 1uantities denoted by and μ that arise in strain0

stress relationships*34 /n &eneral' and μ are individually referred to as Lamé's first

 parameter  and Lamé's second parameter ' respectively* 5ther names are sometimes employed for

one or both parameters' dependin& on context* (or example' the parameter μ is referred to in fluid

dynamics as the dynamic viscosity of a fluid6 %hereas in the context of elasticity' μ is called

the shear modulus'24:p*777 and is sometimes denoted by G instead of μ* ypically the notation ) is seen

paired %ith the use of ,oun&8s modulus' and the notation μ is paired %ith the use of *

/n homo&eneous and isotropic materials' these define 9ooe8s la% in 7;'

%here σ is the stress' ε the strain tensor ' the identity matrix and the trace function*

he t%o parameters to&ether constitute a parameteri"ation of the elastic moduli for

homo&eneous isotropic media' popular in mathematical literature' and are thus related to the

other elastic moduli6 for instance' the bul modulus can be expressed as *

https://en.wikipedia.org/wiki/Continuum_mechanicshttps://en.wikipedia.org/wiki/Continuum_mechanicshttps://en.wikipedia.org/wiki/Deformation_(mechanics)https://en.wikipedia.org/wiki/Deformation_(mechanics)https://en.wikipedia.org/wiki/Stress_(mechanics)https://en.wikipedia.org/wiki/Lam%C3%A9_parameters#cite_note-Weisstein-1https://en.wikipedia.org/wiki/Lam%C3%A9_parameters#cite_note-Weisstein-1https://en.wikipedia.org/wiki/Fluid_dynamicshttps://en.wikipedia.org/wiki/Fluid_dynamicshttps://en.wikipedia.org/wiki/Fluid_dynamicshttps://en.wikipedia.org/wiki/Viscosity#Dynamic_.28shear.29_viscosityhttps://en.wikipedia.org/wiki/Elasticity_(physics)https://en.wikipedia.org/wiki/Shear_modulushttps://en.wikipedia.org/wiki/Shear_modulushttps://en.wikipedia.org/wiki/Shear_modulushttps://en.wikipedia.org/wiki/Lam%C3%A9_parameters#cite_note-Salencon-2https://en.wikipedia.org/wiki/Lam%C3%A9_parameters#cite_note-Salencon-2https://en.wikipedia.org/wiki/Isotropichttps://en.wikipedia.org/wiki/Isotropichttps://en.wikipedia.org/wiki/Hooke's_lawhttps://en.wikipedia.org/wiki/Stress_(physics)https://en.wikipedia.org/wiki/Stress_(physics)https://en.wikipedia.org/wiki/Strain_tensorhttps://en.wikipedia.org/wiki/Identity_matrixhttps://en.wikipedia.org/wiki/Identity_matrixhttps://en.wikipedia.org/wiki/Identity_matrixhttps://en.wikipedia.org/wiki/Trace_(linear_algebra)https://en.wikipedia.org/wiki/Trace_(linear_algebra)https://en.wikipedia.org/wiki/Elastic_modulushttps://en.wikipedia.org/wiki/Bulk_modulushttps://en.wikipedia.org/wiki/Deformation_(mechanics)https://en.wikipedia.org/wiki/Stress_(mechanics)https://en.wikipedia.org/wiki/Lam%C3%A9_parameters#cite_note-Weisstein-1https://en.wikipedia.org/wiki/Fluid_dynamicshttps://en.wikipedia.org/wiki/Fluid_dynamicshttps://en.wikipedia.org/wiki/Viscosity#Dynamic_.28shear.29_viscosityhttps://en.wikipedia.org/wiki/Elasticity_(physics)https://en.wikipedia.org/wiki/Shear_modulushttps://en.wikipedia.org/wiki/Lam%C3%A9_parameters#cite_note-Salencon-2https://en.wikipedia.org/wiki/Isotropichttps://en.wikipedia.org/wiki/Hooke's_lawhttps://en.wikipedia.org/wiki/Stress_(physics)https://en.wikipedia.org/wiki/Strain_tensorhttps://en.wikipedia.org/wiki/Identity_matrixhttps://en.wikipedia.org/wiki/Trace_(linear_algebra)https://en.wikipedia.org/wiki/Elastic_modulushttps://en.wikipedia.org/wiki/Bulk_modulushttps://en.wikipedia.org/wiki/Continuum_mechanics

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 *0C* ?hi' Mathematical Theory of Elastic Structures' ?prin&er @e% ,or' /?A@ B0

7D0E372F0G' !3H3#

• )* Iavo' * IuerJi' K* ;vorin' The Rock Physics and!ook ' Cambrid&e niversity .ress

!paperbac#' /?A@ B0E230EG7GG0G' !2BB7#

• M*?* ?lau&hter' The Lineari"ed Theory of Elasticity ' AirhNuser' /?A@ B03DF0G33D07' !2BB2#

Referencesedit4

3* Jump up ̂  OLamé ConstantsO* Meisstein' Pric* Pric Meisstein8s Morld of ?cience' < MolframMeb Qesource* Qetrieved 2B3E0B2022*

2* Jump up ̂  Kean ?alencon !2BB3#' O9andboo of Continuum Iechanics: )eneral Concepts'hermoelasticityO* ?prin&er ?cience R Ausiness Iedia /?A@ 70EGB0G3GG70F

• v

• t

• e

Elastic modu

Bulk modulus ( )

Young's modulus ( )

Lamé's first parameter ( )

Shear modulus ( )

Poisson's ratio ( )

P!ave modulus ( )

hide4Conversion formulas

9omo&eneous isotropic linear elastic materials have their elastic properties uni1uely determined by anyt%o moduli amon& these6 thus' &iven any t%o' any other of the elastic moduli can be calculatedaccordin& to these formulas*

@otes

https://en.wikipedia.org/wiki/Gabriel_Lam%C3%A9https://en.wikipedia.org/wiki/Gabriel_Lam%C3%A9https://en.wikipedia.org/wiki/Gabriel_Lam%C3%A9https://en.wikipedia.org/w/index.php?title=Lam%C3%A9_parameters&action=edit&section=1https://en.wikipedia.org/wiki/Special:BookSources/0387513264https://en.wikipedia.org/wiki/Special:BookSources/0387513264https://en.wikipedia.org/wiki/Special:BookSources/0521543444https://en.wikipedia.org/wiki/Special:BookSources/0817641173https://en.wikipedia.org/wiki/Special:BookSources/0817641173https://en.wikipedia.org/w/index.php?title=Lam%C3%A9_parameters&action=edit&section=2https://en.wikipedia.org/wiki/Lam%C3%A9_parameters#cite_ref-Weisstein_1-0http://scienceworld.wolfram.com/physics/LameConstants.htmlhttps://en.wikipedia.org/wiki/Lam%C3%A9_parameters#cite_ref-Salencon_2-0http://books.google.com/books?id=H3xIED8ctfUC&pg=PA333http://books.google.com/books?id=H3xIED8ctfUC&pg=PA333https://en.wikipedia.org/wiki/Special:BookSources/3540414436https://en.wikipedia.org/wiki/Template:Elastic_modulihttps://en.wikipedia.org/wiki/Template_talk:Elastic_modulihttps://en.wikipedia.org/w/index.php?title=Template:Elastic_moduli&action=edithttps://en.wikipedia.org/wiki/Elastic_modulushttps://en.wikipedia.org/wiki/Bulk_modulushttps://en.wikipedia.org/wiki/Young's_modulushttps://en.wikipedia.org/wiki/Shear_modulushttps://en.wikipedia.org/wiki/Poisson's_ratiohttps://en.wikipedia.org/wiki/P-wave_modulushttps://en.wikipedia.org/wiki/P-wave_modulushttps://en.wikipedia.org/wiki/Lam%C3%A9_parametershttps://en.wikipedia.org/wiki/Lam%C3%A9_parametershttps://en.wikipedia.org/wiki/Lam%C3%A9_parametershttps://en.wikipedia.org/wiki/Gabriel_Lam%C3%A9https://en.wikipedia.org/w/index.php?title=Lam%C3%A9_parameters&action=edit&section=1https://en.wikipedia.org/wiki/Special:BookSources/0387513264https://en.wikipedia.org/wiki/Special:BookSources/0387513264https://en.wikipedia.org/wiki/Special:BookSources/0521543444https://en.wikipedia.org/wiki/Special:BookSources/0817641173https://en.wikipedia.org/w/index.php?title=Lam%C3%A9_parameters&action=edit&section=2https://en.wikipedia.org/wiki/Lam%C3%A9_parameters#cite_ref-Weisstein_1-0http://scienceworld.wolfram.com/physics/LameConstants.htmlhttps://en.wikipedia.org/wiki/Lam%C3%A9_parameters#cite_ref-Salencon_2-0http://books.google.com/books?id=H3xIED8ctfUC&pg=PA333http://books.google.com/books?id=H3xIED8ctfUC&pg=PA333https://en.wikipedia.org/wiki/Special:BookSources/3540414436https://en.wikipedia.org/wiki/Template:Elastic_modulihttps://en.wikipedia.org/wiki/Template_talk:Elastic_modulihttps://en.wikipedia.org/w/index.php?title=Template:Elastic_moduli&action=edithttps://en.wikipedia.org/wiki/Elastic_modulushttps://en.wikipedia.org/wiki/Bulk_modulushttps://en.wikipedia.org/wiki/Young's_modulushttps://en.wikipedia.org/wiki/Shear_modulushttps://en.wikipedia.org/wiki/Poisson's_ratiohttps://en.wikipedia.org/wiki/P-wave_modulushttps://en.wikipedia.org/wiki/Lam%C3%A9_parameters

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here are t%o validsolutions*

he plus si&n leads

to *he minus si&n leads

to *

Cannot be used

%hen