Bulk modulus

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“Incompressibility” redirects here. For topic in fluid dynamics, see Compressibility.

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

As an example, suppose an iron cannon ball with bulk modulus 160 GPa (giga pascal) is to be reduced in volume by 0.5%. This requires a pressure increase of 0.005×160GPa = 0.8 GPa. If the cannon ball is subjected to a pressure increase of only 100 MPa, it will decrease in volume by a factor of 100MPa/160GPa = 0.000625, or 0.0625%.

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 ∂p/∂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.

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-enthalpy (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

$K_S=\kappa\, p$

where

κ is the adiabatic index, sometimes called γ.

p is the pressure.

The adiabatic bulk modulus K and the density ρ of a substance determine the speed of sound c (for pressure waves) in a material, according to the formula

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