# Template:Elastic modulus for seismic

Conversion formulas
The elastic properties of homogeneous isotropic linear elastic materials are uniquely determined by any two moduli. Given any two, the others can thus be calculated. Key reference: Mavko, G, T Mukerji and J Dvorkin (2003), The Rock Physics Handbook, Cambridge University Press.
${\displaystyle E}$ ${\displaystyle \nu }$ ${\displaystyle K}$ ${\displaystyle \mu }$ ${\displaystyle \lambda }$ ${\displaystyle V_{P}}$ ${\displaystyle V_{S}}$ ${\displaystyle V_{P}/V_{S}}$
Young's modulus, Poisson's ratio
${\displaystyle E,\nu }$
${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle {\tfrac {E}{3(1-2\nu )}}}$ ${\displaystyle {\tfrac {E}{2(1+\nu )}}}$ ${\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}$ ${\displaystyle {\sqrt {\tfrac {E(1-\nu )}{\rho (1-\nu )(1-2\nu )}}}}$ ${\displaystyle {\sqrt {\tfrac {E}{2\rho (1-\nu )}}}}$ ${\displaystyle {\sqrt {\tfrac {1-\nu }{{\tfrac {1}{2}}-\nu }}}}$
Bulk modulus, Shear modulus
${\displaystyle K,\mu }$
${\displaystyle {\tfrac {9K\mu }{3\kappa +\mu }}}$ ${\displaystyle {\tfrac {3K+2\mu }{2(3\kappa +\mu )}}}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle \kappa -{\tfrac {2}{3}}\mu }$ ${\displaystyle {\sqrt {\tfrac {\kappa +{\tfrac {4}{3}}\mu }{\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {\mu }{\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {\kappa +{\tfrac {4}{3}}\mu }{\mu }}}}$
Shear modulus, 1st Lamé parameter
${\displaystyle \mu ,\lambda }$
${\displaystyle {\tfrac {\mu (3\lambda +2\mu )}{\lambda +\mu }}}$ ${\displaystyle {\tfrac {\lambda }{2(\lambda +\mu )}}}$ ${\displaystyle \lambda +{\tfrac {2}{3\mu )}}}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle {\sqrt {\tfrac {\lambda +2\mu }{\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {\mu }{\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {\lambda +2\mu }{\mu }}}}$
P-wave velocity, S-wave velocity
${\displaystyle V_{P},V_{S}}$
${\displaystyle {\tfrac {\rho V_{S}^{2}(3V_{P}^{2}-4V_{S}^{2})}{V_{P}^{2}-V_{S}^{2}}}}$ ${\displaystyle {\tfrac {V_{P}^{2}-2V_{S}^{2}}{2(V_{P}^{2}-V_{S}^{2})}}}$ ${\displaystyle \rho (V_{P}^{2}-{\tfrac {4}{3}}V_{S}^{2})}$ ${\displaystyle \rho V_{S}^{2}}$ ${\displaystyle \rho V_{P}^{2}-2V_{S}^{2}}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$