Conversion between the elastic constants

In the case of isotropic materials, the relationships between the moduli of elasticity allow the other moduli of stiffness to be calculated from any two material parameters. Accordingly, in the theory of elasticity, the elastic properties of linear-elastic, homogeneous, isotropic materials are clearly determined by two material parameters.

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. ↑ G. Mavko, T. Mukerji, J. Dvorkin: The Rock Physics Handbook . Cambridge University Press, 2003, ISBN 0-521-54344-4 (paperback).