Bulk modulus
The bulk modulus (K) of a substance 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 (gigapascal) is to be reduced in volume by 0.5%. This requires a pressure increase of 0.005×160 GPa = 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 100 MPa/160 GPa = 0.000625, or 0.0625%.
The bulk modulus K can be formally defined by the equation:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): K_T ), constant-enthalpy (adiabatic Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): K_S ), and other variations are possible. In practice, such distinctions are usually only relevant for gases.
Bulk modulus values for some example substances | |
---|---|
Water | 2.2×10^{9} Pa (value increases at higher pressures) |
Air | 1.42×10^{5} Pa (adiabatic bulk modulus) |
Air | 1.01×10^{5} Pa (constant temperature bulk modulus) |
Steel | 1.6×10^{11} Pa |
Glass | 3.5×10^{10} to 5.5×10^{10} Pa |
Solid helium | 5×10^{7} Pa (approximate) |
For a gas, the adiabatic bulk modulus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): K_S
is approximately given by
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): K_S=\kappa\, p
where
- κ is the adiabatic index, sometimes called γ.
- p is the pressure.
In a fluid, the bulk modulus K and the density ρ determine the speed of sound c (pressure waves), according to the formula
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): c=\sqrt{\frac{K}{\rho}}.
Solids can also sustain transverse waves, for these one additional elastic modulus, for example the shear modulus, is needed to determine wave speeds.
Anisotropy
For crystalline solids with a symmetry lower than cubic the bulk modulus is not the same in all directions and needs to be described with a tensor with more than one independent value. It is possible to study the tensor elements using powder diffraction under applied pressure.
References
da:Kompressibilitetsmodul de:Kompressionsmodul ms:Modulus pukal nl:Compressiemodulus uk:Модуль всебічного стиску