Bulk modulus

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

Definition

$\mathrm {bulk} \ \mathrm {modulus} ={\frac {\mathrm {volume} \ \mathrm {stress} }{\mathrm {volume} \ \mathrm {strain} }}$

$K={\frac {P}{\Delta V/V}}$

In terms of V_P and V_S

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

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':

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

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

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

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