# Bulk modulus

One of the elastic moduli, sometimes called incompressibility, volumetric modulus, or stiffness.

## Definition

${\displaystyle \mathrm {bulk} \ \mathrm {modulus} ={\frac {\mathrm {volume} \ \mathrm {stress} }{\mathrm {volume} \ \mathrm {strain} }}}$
${\displaystyle K={\frac {P}{\Delta V/V}}}$

## In terms of VP and VS

${\displaystyle K=\rho \left(V_{\mathrm {P} }^{2}-{\frac {4}{3}}V_{\mathrm {S} }^{2}\right)}$

## Other expressions

K can also be expressed in terms of Young's modulus, E, and Poisson's ratio, ν. This could be thought of as 'the engineer's perspective':

${\displaystyle K={\frac {E}{3(1-2\nu )}}}$

The 'rock physics perspective' casts K in terms of the 1st Lamé parameter λ and shear modulus μ:

${\displaystyle K=\lambda +{\frac {2}{3}}\mu }$

## Bulk modulus of effective media

Upper and lower bounds on the bulk modulus of mixtures of n materials can be obtained using Voigt–Reuss and Hashin–Shtrikman bounds.

Conversion formulas — edit
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.

${\displaystyle \dagger \ X={\sqrt {9\lambda ^{2}+2E\lambda +E^{2}}}}$

${\displaystyle (V_{\mathrm {P} },\,V_{\mathrm {S} })}$ ${\displaystyle (\mu ,\,\lambda )}$ ${\displaystyle (E,\,\lambda )\,\dagger }$ ${\displaystyle (E,\,\mu )}$ ${\displaystyle (K,\,\lambda )}$ ${\displaystyle (K,\,\mu )}$ ${\displaystyle (K,\,E)}$ ${\displaystyle (\nu ,\,\lambda )}$ ${\displaystyle (\nu ,\,\mu )}$ ${\displaystyle (\nu ,\,E)}$ ${\displaystyle (\nu ,\,K)}$
P-wave velocity
${\displaystyle V_{\mathrm {P} }=\,}$
${\displaystyle V_{\mathrm {P} }}$ ${\displaystyle {\sqrt {\tfrac {\lambda +2\mu }{\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {E-\lambda +X}{2\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {\mu \,(E-4\mu )}{\rho \,(E-3\mu )}}}}$ ${\displaystyle {\sqrt {\tfrac {3K-2\lambda }{\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {K+{\frac {4}{3}}\mu }{\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {3K\left(3K+E\right)}{\rho \left(9K-E\right)}}}}$ ${\displaystyle {\sqrt {\tfrac {\lambda (1-\nu )}{\rho \nu }}}}$ ${\displaystyle {\sqrt {\tfrac {2\mu (1-\nu )}{\rho (1-2\nu )}}}}$ ${\displaystyle {\sqrt {\tfrac {E(1-\nu )}{\rho (1+\nu )(1-2\nu )}}}}$ ${\displaystyle {\sqrt {\tfrac {3K(1-\nu )}{\rho (1+\nu )}}}}$
S-wave velocity
${\displaystyle V_{\mathrm {S} }=\,}$
${\displaystyle V_{\mathrm {S} }}$ ${\displaystyle {\sqrt {\tfrac {\mu }{\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {E-3\lambda +X}{4\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {\mu }{\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {3(K-\lambda )}{2\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {\mu }{\rho }}}}$ ${\displaystyle {\sqrt {-{\tfrac {3EK}{\rho \left(E-9K\right)}}}}}$ ${\displaystyle {\sqrt {{\tfrac {\lambda }{2\nu \rho }}-{\tfrac {\lambda }{\rho }}}}}$ ${\displaystyle {\sqrt {\tfrac {\mu }{\rho }}}}$ ${\displaystyle {\sqrt {\tfrac {E}{2\rho (1+\nu )}}}}$ ${\displaystyle {\sqrt {-{\tfrac {3K(2\nu -1)}{2\rho (\nu +1)}}}}}$
Velocity ratio
${\displaystyle \Gamma =\,}$
${\displaystyle {\frac {V_{\mathrm {P} }}{V_{\mathrm {S} }}}}$ ${\displaystyle {\sqrt {\tfrac {\lambda +2\mu }{\mu }}}}$ ${\displaystyle {\sqrt {\tfrac {3E+3\lambda +X}{2E}}}}$ ${\displaystyle {\sqrt {\tfrac {E-4\mu }{E-3\mu }}}}$ ${\displaystyle {\sqrt {\tfrac {{\tfrac {4}{3}}\lambda -2K}{\lambda -K}}}}$ ${\displaystyle {\sqrt {\tfrac {K+{\frac {4}{3}}\mu }{\mu }}}}$ ${\displaystyle {\sqrt {\tfrac {E+3K}{E}}}}$ ${\displaystyle {\sqrt {\tfrac {2\nu -2}{2\nu -1}}}}$ ${\displaystyle {\sqrt {\tfrac {2\nu -2}{2\nu -1}}}}$ ${\displaystyle {\sqrt {\tfrac {2\nu -2}{2\nu -1}}}}$ ${\displaystyle {\sqrt {\tfrac {2\nu -2}{2\nu -1}}}}$
1st Lamé parameter
${\displaystyle \lambda =\,}$
${\displaystyle \rho (V_{\mathrm {P} }^{2}-2V_{\mathrm {S} }^{2})}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle {\tfrac {\mu (E-2\mu )}{3\mu -E}}}$ ${\displaystyle \lambda }$ ${\displaystyle K-{\tfrac {2\mu }{3}}}$ ${\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}$ ${\displaystyle \lambda }$ ${\displaystyle {\tfrac {2\mu \nu }{1-2\nu }}}$ ${\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}$ ${\displaystyle {\tfrac {3K\nu }{1+\nu }}}$
Shear modulus
${\displaystyle \mu =\,}$
${\displaystyle \rho V_{\mathrm {S} }^{2}}$ ${\displaystyle \mu }$ ${\displaystyle {\tfrac {E-3\lambda +X}{4}}}$ ${\displaystyle \mu }$ ${\displaystyle {\tfrac {3(K-\lambda )}{2}}}$ ${\displaystyle \mu }$ ${\displaystyle {\tfrac {3KE}{9K-E}}}$ ${\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}$ ${\displaystyle \mu }$ ${\displaystyle {\tfrac {E}{2(1+\nu )}}}$ ${\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}$
Young's modulus
${\displaystyle E=\,}$
${\displaystyle {\tfrac {\rho V_{\mathrm {S} }^{2}(3V_{\mathrm {P} }^{2}-4V_{\mathrm {S} }^{2})}{V_{\mathrm {P} }^{2}-V_{\mathrm {S} }^{2}}}}$ ${\displaystyle {\tfrac {\mu (3\lambda +2\mu )}{\lambda +\mu }}}$ ${\displaystyle E\ }$ ${\displaystyle E}$ ${\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}$ ${\displaystyle {\tfrac {9K\mu }{3K+\mu }}}$ ${\displaystyle E\ }$ ${\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}$ ${\displaystyle 2\mu (1+\nu )\,}$ ${\displaystyle E\ }$ ${\displaystyle 3K(1-2\nu )\,}$
Bulk modulus
${\displaystyle K=\,}$
${\displaystyle \rho (V_{\mathrm {P} }^{2}-{\tfrac {4}{3}}V_{\mathrm {S} }^{2})}$ ${\displaystyle \lambda +{\tfrac {2\mu }{3}}}$ ${\displaystyle {\tfrac {E+3\lambda +X}{6}}}$ ${\displaystyle {\tfrac {E\mu }{3(3\mu -E)}}}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}$ ${\displaystyle {\tfrac {2\mu (1+\nu )}{3(1-2\nu )}}}$ ${\displaystyle {\tfrac {E}{3(1-2\nu )}}}$ ${\displaystyle K}$
Poisson's ratio
${\displaystyle \nu =\,}$
${\displaystyle {\tfrac {V_{\mathrm {P} }^{2}-2V_{\mathrm {S} }^{2}}{2(V_{\mathrm {P} }^{2}-V_{\mathrm {S} }^{2})}}}$ ${\displaystyle {\tfrac {\lambda }{2(\lambda +\mu )}}}$ ${\displaystyle {\tfrac {-E-\lambda +X}{4\lambda }}}$ ${\displaystyle {\tfrac {E}{2\mu }}-1}$ ${\displaystyle {\tfrac {\lambda }{3K-\lambda }}}$ ${\displaystyle {\tfrac {3K-2\mu }{2(3K+\mu )}}}$ ${\displaystyle {\tfrac {3K-E}{6K}}}$ ${\displaystyle \nu }$ ${\displaystyle \nu }$ ${\displaystyle \nu }$ ${\displaystyle \nu }$
P-wave modulus
${\displaystyle M=\,}$
${\displaystyle \rho V_{\mathrm {P} }^{2}}$ ${\displaystyle \lambda +2\mu \,}$ ${\displaystyle {\tfrac {E-\lambda +X}{2}}}$ ${\displaystyle {\tfrac {\mu (4\mu -E)}{3\mu -E}}}$ ${\displaystyle 3K-2\lambda \,}$ ${\displaystyle K+{\tfrac {4\mu }{3}}}$ ${\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}$ ${\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}$ ${\displaystyle {\tfrac {2\mu (1-\nu )}{1-2\nu }}}$ ${\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}$ ${\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}$