Bulk modulus

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"Incompressibility" redirects here. For the topic in fluid dynamics, see Compressibility.

Illustration of uniform compression

The bulk modulus (K) of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to cause a given relative decrease in volume. Its base unit is the pascal.

As an example, suppose an iron cannon ball with bulk modulus 160 GPa is to be reduced in volume by 0.5%. This requires a pressure increase of 0.005н—160 GPa = 0.8 GPa (116,000 psi).

1 Definition
2 Thermodynamic relation
3 Measurement
4 Selected values
5 References

Definition

The bulk modulus K can be formally defined by the equation:

$K=-V\frac{\partial P}{\partial V}$

where P is pressure, V is volume, and в€‚P/в€‚V denotes the partial derivative of pressure with respect to volume. The inverse of the bulk modulus gives a substance's compressibility.

Other moduli describe the material's response (strain) to other kinds of stress: the shear modulus describes the response to shear, and Young's modulus describes the response to linear strain. For a fluid, only the bulk modulus is meaningful. For an anisotropic solid such as wood or paper, these three moduli do not contain enough information to describe its behaviour, and one must use the full generalized Hooke's law.

Thermodynamic relation

Strictly speaking, the bulk modulus is a thermodynamic quantity, and it is necessary to specify how the temperature varies in order to specify a bulk modulus: constant-temperature ( $K T$ ), constant-entropy (adiabatic $K S$ ), and other variations are possible. In practice, such distinctions are usually only relevant for gases.

For a gas, the adiabatic bulk modulus $K S$ is approximately given by

$K_S=\gamma\, P$

and the isothermal bulk modulus $K T$ is approximately given by

$K_T=P\,$

where

Оі is the adiabatic index, sometimes called Оє.

P is the pressure.

In a fluid, the bulk modulus K and the density ПЃ determine the speed of sound c (pressure waves), according to the formula

$c=\sqrt{\frac{K}{\rho}}.$

Solids can also sustain transverse waves: for these materials one additional elastic modulus, for example the shear modulus, is needed to determine wave speeds.

Measurement

It is possible to measure the bulk modulus using powder diffraction under applied pressure.

Selected values

Approximate bulk modulus (K) for common materials
Material	Bulk modulus in G Pa	Bulk modulus in psi
Glass (see also diagram below table)	35 to 55	5.8н—10⁶
Steel	160	23н—10⁶
Diamond^[1]	442	64н—10⁶

Influences of selected glass component additions on the bulk modulus of a specific base glass.^[2]

Approximate bulk modulus (K) for other substances
Water	2.2н—10⁹ Pa (value increases at higher pressures)
Air	1.42н—10⁵ Pa (adiabatic bulk modulus)
Air	1.01н—10⁵ Pa (constant temperature bulk modulus)
Solid helium	5н—10⁷ Pa (approximate)

References

Bulk Elastic Properties on hyperphysics at Georgia State University

^ Phys. Rev. B 32, 7988 - 7991 (1985), Calculation of bulk moduli of diamond and zinc-blende solids
^ Bulk modulus calculation of glasses

v - d - e Elastic moduli for homogeneous isotropic materials

Bulk modulus ( $K$ ) - Young's modulus ( $E$ ) - Lamй's first parameter ( $О»$ ) - Shear modulus ( $G$ ) - Poisson's ratio ( $ОЅ$ ) - P-wave modulus ( $M$ )

Conversion formulas
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these, thus given any two, any other of the elastic moduli can be calculated according to these formulas.
	$(\lambda,\,G)$	$(E,\,G)$	$(K,\,\lambda)$	$(K,\,G)$	$(\lambda,\,\nu)$	$(G,\,\nu)$	$(E,\,\nu)$	$(K,\, \nu)$	$(K,\,E)$	$(M,\,G)$
$K=\,$	$\lambda+ \tfrac{2G}{3}$	$\tfrac{EG}{3(3G-E)}$			$\tfrac{\lambda(1+\nu)}{3\nu}$	$\tfrac{2G(1+\nu)}{3(1-2\nu)}$	$\tfrac{E}{3(1-2\nu)}$			$M - \tfrac{4G}{3}$
$E=\,$	$\tfrac{G(3\lambda + 2G)}{\lambda + G}$		$\tfrac{9K(K-\lambda)}{3K-\lambda}$	$\tfrac{9KG}{3K+G}$	$\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}$	$2G(1+\nu)\,$		$3K(1-2\nu)\,$		$\tfrac{G(3M-4G)}{M-G}$
$\lambda=\,$		$\tfrac{G(E-2G)}{3G-E}$		$K-\tfrac{2G}{3}$		$\tfrac{2 G \nu}{1-2\nu}$	$\tfrac{E\nu}{(1+\nu)(1-2\nu)}$	$\tfrac{3K\nu}{1+\nu}$	$\tfrac{3K(3K-E)}{9K-E}$	$M - 2G\,$
$G=\,$			$\tfrac{3(K-\lambda)}{2}$		$\tfrac{\lambda(1-2\nu)}{2\nu}$		$\tfrac{E}{2(1+\nu)}$	$\tfrac{3K(1-2\nu)}{2(1+\nu)}$	$\tfrac{3KE}{9K-E}$
$\nu=\,$	$\tfrac{\lambda}{2(\lambda + G)}$	$\tfrac{E}{2G}-1$	$\tfrac{\lambda}{3K-\lambda}$	$\tfrac{3K-2G}{2(3K+G)}$					$\tfrac{3K-E}{6K}$	$\tfrac{M - 2G}{2M - 2G}$
$M=\,$	$\lambda+2G\,$	$\tfrac{G(4G-E)}{3G-E}$	$3K-2\lambda\,$	$K+\tfrac{4G}{3}$	$\tfrac{\lambda(1-\nu)}{\nu}$	$\tfrac{2G(1-\nu)}{1-2\nu}$	$\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}$	$\tfrac{3K(1-\nu)}{1+\nu}$	$\tfrac{3K(3K+E)}{9K-E}$

Bulk modulus

Bulk modulus

Bulk modulus

Contents

Definition

Thermodynamic relation

Measurement

Selected values

References

Bulk modulus