Lamé constants

The two Lamé constants and (according to Gabriel Lamé ) are material constants that define all components of the elasticity tensor of an isotropic material in the context of continuum mechanics . Their dimensions correspond to a pressure ( force per area , in SI units ). ${\displaystyle \lambda }$ ${\displaystyle \mu }$ ${\displaystyle \mathrm {N} /\mathrm {m} ^{2}}$

Elasticity theory

In the linear elasticity theory , the linear dependence of the stress tensor on the strain tensor is described by the elasticity tensor ( generalized Hooke's law ). In component notation and with the help of Einstein's sum convention, this relationship reads : ${\displaystyle \sigma }$ ${\displaystyle \varepsilon }$ ${\displaystyle C}$

${\displaystyle \sigma _{ij}=C_{ijkl}\,\varepsilon _{kl}.}$

The stress and strain tensors are 2nd level and the elasticity tensor is a 4th level tensor.

In the case of an isotropic material this can be simplified to:

${\displaystyle \sigma _{ij}=2\,\mu \,\varepsilon _{ij}+\lambda \;\mathrm {Spur} (\varepsilon )\,\delta _{ij}}$

With

• the first Lamé constant ${\displaystyle \lambda ={\frac {\nu }{1-2\nu }}\cdot {\frac {1}{1+\nu }}\cdot E}$
• the second Lamé constant or the shear modulus ${\displaystyle \mu =G={\frac {1}{2}}\cdot {\frac {1}{1+\nu }}\cdot E}$
  • the Poisson's ratio (Poisson's ratio) ${\displaystyle \nu }$
  • the modulus of elasticity ${\displaystyle E}$
• the Kronecker Delta ${\displaystyle \delta _{ij}}$
• the track .

For further formulas depending on the Lamé constants see section.

Derivation

In the case of an isotropic, linearly elastic material, i. H. the stress tensor depends linearly on the components of the strain tensor, one can define a scalar potential that indicates the energy density of the material as a function of the strain and through the relationship ${\displaystyle U_{0}(\varepsilon _{ij})}$

${\displaystyle \sigma _{ij}={\frac {\partial U_{0}}{\partial \varepsilon _{ij}}}}$

defines a stress-distortion relation. This function may only depend on invariants of the strain tensor, since the choice of the coordinate system must not change the energy density of the described strain state. The strain tensor is symmetric, therefore it has the following invariants (in the notation with Einstein's sum convention)

${\displaystyle I_{1}=\varepsilon _{ii},}$

${\displaystyle I_{2}={\frac {1}{2}}\varepsilon _{ij}\varepsilon _{ji},}$

${\displaystyle I_{3}={\frac {1}{3}}\varepsilon _{ij}\varepsilon _{jk}\varepsilon _{ki}.}$

In order to obtain a linear strain-voltage relation, the potential may only depend on the square of the components of the strain tensor. Therefore, and because of the coordinate invariance of the potential, it must be the form

${\displaystyle U_{0}=C_{1}I_{1}^{2}+C_{2}I_{2}}$

have, with arbitrary constants and . If one uses this potential approach in the stress-distortion relation and carries out some transformations, the relation results ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$

${\displaystyle \sigma _{ij}=2C_{1}\varepsilon _{kk}\delta _{ij}+C_{2}\varepsilon _{ij}.}$

With the definitions

${\displaystyle 2C_{1}=\lambda }$ and

${\displaystyle C_{2}=2\mu }$

are now called and first and second Lamé constants. ${\displaystyle \lambda }$ ${\displaystyle \mu }$

Fluid mechanics

In the Navier-Stokes equations of fluid mechanics , the symbol of the second Lamé constant is often used for the dynamic shear viscosity (unit ) and the symbol of the first Lamé constant for the volume viscosity . However, these viscosities should not be confused with the above Lame constants, which represent the elasticity measures of a solid. ${\displaystyle \mathrm {N} \cdot \mathrm {s} /\mathrm {m} ^{2}}$ ${\displaystyle \mu }$ ${\displaystyle \lambda }$

Conversion between the elastic constants

The module ...	... results from:
	${\displaystyle (K,\,E)}$	${\displaystyle (K,\,\lambda )}$	${\displaystyle (K,\,G)}$	${\displaystyle (K,\,\nu )}$	${\displaystyle (E,\,\lambda )}$	${\displaystyle (E,\,G)}$	${\displaystyle (E,\,\nu )}$	${\displaystyle (\lambda ,\,G)}$	${\displaystyle (\lambda ,\,\nu )}$	${\displaystyle (G,\,\nu )}$	${\displaystyle (G,\,M)}$
Compression module ${\displaystyle K\,}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle (E+3\lambda )+}$ ${\displaystyle {\tfrac {\sqrt {(E+3\lambda )^{2}-4\lambda E}}{6}}}$	${\displaystyle {\tfrac {EG}{3(3G-E)}}}$	${\displaystyle {\tfrac {E}{3(1-2\nu )}}}$	${\displaystyle \lambda +}$ ${\displaystyle {\tfrac {2G}{3}}}$	${\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}$	${\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}$	${\displaystyle M-}$ ${\displaystyle {\tfrac {4G}{3}}}$
modulus of elasticity ${\displaystyle E\,}$	${\displaystyle E}$	${\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}$	${\displaystyle {\tfrac {9KG}{3K+G}}}$	${\displaystyle 3K(1-2\nu )\,}$	${\displaystyle E}$	${\displaystyle E}$	${\displaystyle E}$	${\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}$	${\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}$	${\displaystyle 2G(1+\nu )\,}$	${\displaystyle {\tfrac {G(3M-4G)}{M-G}}}$
1. Lamé constant ${\displaystyle \lambda \,}$	${\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}$	${\displaystyle \lambda }$	${\displaystyle K-}$ ${\displaystyle {\tfrac {2G}{3}}}$	${\displaystyle {\tfrac {3K\nu }{1+\nu }}}$	${\displaystyle \lambda }$	${\displaystyle {\tfrac {G(E-2G)}{3G-E}}}$	${\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}$	${\displaystyle \lambda }$	${\displaystyle \lambda }$	${\displaystyle {\tfrac {2G\nu }{1-2\nu }}}$	${\displaystyle M-2G\,}$
Shear modulus or (2nd Lamé constant) ${\displaystyle G}$ ${\displaystyle \mu }$	${\displaystyle {\tfrac {3KE}{9K-E}}}$	${\displaystyle {\tfrac {3(K-\lambda )}{2}}}$	${\displaystyle G}$	${\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}$	${\displaystyle (E-3\lambda )+}$ ${\displaystyle {\tfrac {\sqrt {(E-3\lambda )^{2}+8\lambda E}}{4}}}$	${\displaystyle G}$	${\displaystyle {\tfrac {E}{2(1+\nu )}}}$	${\displaystyle G}$	${\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}$	${\displaystyle G}$	${\displaystyle G}$
Poisson's number ${\displaystyle \nu \,}$	${\displaystyle {\tfrac {3K-E}{6K}}}$	${\displaystyle {\tfrac {\lambda }{3K-\lambda }}}$	${\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}$	${\displaystyle \nu }$	${\displaystyle -(E+\lambda )+}$ ${\displaystyle {\tfrac {\sqrt {(E+\lambda )^{2}+8\lambda ^{2}}}{4\lambda }}}$	${\displaystyle {\tfrac {E}{2G}}}$ ${\displaystyle -1}$	${\displaystyle \nu }$	${\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}$	${\displaystyle \nu }$	${\displaystyle \nu }$	${\displaystyle {\tfrac {M-2G}{2M-2G}}}$
Longitudinal module ${\displaystyle M\,}$	${\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}$	${\displaystyle 3K-2\lambda \,}$	${\displaystyle K+}$ ${\displaystyle {\tfrac {4G}{3}}}$	${\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}$	${\displaystyle {\tfrac {E-\lambda +{\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}{2}}}$	${\displaystyle {\tfrac {G(4G-E)}{3G-E}}}$	${\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}$	${\displaystyle \lambda +2G\,}$	${\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}$	${\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}$	${\displaystyle M}$

Individual evidence

1. ↑ Tribikram Kundu: Ultrasonic and Electromagnetic NDE for Structure and Material Characterization . CRC Press, 2012, ISBN 1-4398-3663-9 , pp. 27 ff . ( limited preview in Google Book search).
2. ↑ Emmanuil G. Sinaiski: Hydromechanics . John Wiley & Sons, 2011, ISBN 978-3-527-63378-4 , pp. 30 ( limited preview in Google Book search).
3. ↑ G. Mavko, T. Mukerji, J. Dvorkin: The Rock Physics Handbook . Cambridge University Press, 2003, ISBN 0-521-54344-4 (paperback).