Bulk modulus - Citizendia

"Incompressibility" redirects here. For topic in fluid dynamics, see Compressibility. In Thermodynamics and Fluid mechanics, compressibility is a measure of the relative volume change of a Fluid or Solid as a response

Illustration of uniform compression

Influences of selected glass component additions on the bulk modulus of a specific base glass. Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many ^[1]

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. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically

As an example, suppose an iron cannon ball with bulk modulus 160 GPa (giga pascal) is to be reduced in volume by 0. For other meanings see Giga (disambiguation Giga- (symbol G is a prefix in the SI system of units denoting 109 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:

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

where p is pressure, V is volume, and ∂p/∂V denotes the partial derivative of pressure with respect to volume. In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant The inverse of the bulk modulus gives a substance's compressibility. In Thermodynamics and Fluid mechanics, compressibility is a measure of the relative volume change of a Fluid or Solid as a response

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. Stress is a measure of the average amount of Force exerted per unit Area. In Materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of Shear In Solid mechanics, Young's modulus (E is a measure of the Stiffness of an isotropic elastic material For a fluid, only the bulk modulus is meaningful. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code 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. Anisotropy (pronounced with stress on the third syllable ˌænaɪˈsɒtrəpi is the property of being directionally dependent as opposed to Isotropy, which means homogeneity Wood is hard fibrous lignified structural tissue produced as secondary Xylem in the stems of Woody plants notably trees but also shrubs Paper is thin material mainly used for writing upon printing upon or packaging In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the

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 Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature In Thermodynamics and molecular chemistry, the enthalpy (denoted as H, h, or rarely as χ) is a quotient or description of This article covers adiabatic processes in Thermodynamics. For adiabatic processes in Quantum mechanics, see Adiabatic process (quantum mechanics In practice, such distinctions are usually only relevant for gases. This page is about the physical properties of gas as a state of matter

Bulk modulus values for some example substances
Water	2. Water is a common Chemical substance that is essential for the survival of all known forms of Life. 2×10⁹ Pa (value increases at higher pressures)
Air	1. 42×10⁵ Pa (adiabatic bulk modulus)
Air	1. 01×10⁵ Pa (constant temperature bulk modulus)
Steel	1. Steel is an Alloy consisting mostly of Iron, with a Carbon content between 0 6×10¹¹ Pa
Glass	3. Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many 5×10¹⁰ to 5. 5×10¹⁰ Pa
Solid helium	5×10⁷ Pa (approximate)

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. Helium ( He) is a colorless odorless tasteless non-toxic Inert Monatomic Chemical Ideal gas relations For an ideal gas the heat capacity is constant with temperature

p is the pressure. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface

In a fluid, the bulk modulus K and the density ρ 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 one additional elastic modulus, for example the shear modulus, is needed to determine wave speeds. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different Sound is a vibration that travels through an elastic medium as a Wave. P-wave can also refer to a type of electronic wavefunction in atomic physics see Atomic orbital. A transverse wave is a moving Wave that consists of oscillations occurring perpendicular to the direction of energy transfer An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i In Materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of Shear

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. Powder diffraction is a scientific technique using X-ray, Neutron, or Electron Diffraction on powder or Microcrystalline samples for

References

Bulk Elastic Properties on hyperphysics at Georgia State University

^ Bulk modulus calculation of glasses

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. Georgia State University ( GSU) is an urban Research University in Downtown Atlanta, Georgia, USA.
	$(\lambda,\,\mu)$	$(E,\,\mu)$	$(K,\,\lambda)$	$(K,\,\mu)$	$(\lambda,\,\nu)$	$(\mu,\,\nu)$	$(E,\,\nu)$	$(K,\, \nu)$	$(K,\,E)$	$(M,\,\mu)$
$K=\,$	$\lambda+ \frac{2\mu}{3}$	$\frac{E\mu}{3(3\mu-E)}$			$\lambda\frac{1+\nu}{3\nu}$	$\frac{2\mu(1+\nu)}{3(1-2\nu)}$	$\frac{E}{3(1-2\nu)}$			$M - \frac{8\mu}{3}$
$E=\,$	$\mu\frac{3\lambda + 2\mu}{\lambda + \mu}$		$9K\frac{K-\lambda}{3K-\lambda}$	$\frac{9K\mu}{3K+\mu}$	$\frac{\lambda(1+\nu)(1-2\nu)}{\nu}$	$2\mu(1+\nu)\,$		$3K(1-2\nu)\,$		$\mu\frac{3M-4\mu}{M-\mu}$
$\lambda=\,$		$\mu\frac{E-2\mu}{3\mu-E}$		$K-\frac{2\mu}{3}$		$\frac{2 \mu \nu}{1-2\nu}$	$\frac{E\nu}{(1+\nu)(1-2\nu)}$	$\frac{3K\nu}{1+\nu}$	$\frac{3K(3K-E)}{9K-E}$	$M - 2\mu\,$
$\mu=\,$			$3\frac{K-\lambda}{2}$		$\lambda\frac{1-2\nu}{2\nu}$		$\frac{E}{2+2\nu}$	$3K\frac{1-2\nu}{2+2\nu}$	$\frac{3KE}{9K-E}$
$\nu=\,$	$\frac{\lambda}{2(\lambda + \mu)}$	$\frac{E}{2\mu}-1$	$\frac{\lambda}{3K-\lambda}$	$\frac{3K-2\mu}{2(3K+\mu)}$					$\frac{3K-E}{6K}$	$\frac{M - 2\mu}{2M - 3\mu}$
$M=\,$	$\lambda+2\mu\,$	$\mu\frac{4\mu-E}{3\mu-E}$	$3K-2\lambda\,$	$K+\frac{4\mu}{3}$	$\lambda \frac{1-\nu}{\nu}$	$\mu\frac{2-2\nu}{1-2\nu}$	$E\frac{1-\nu}{(1+\nu)(1-2\nu)}$	$3K\frac{1-\nu}{1+\nu}$	$3K\frac{3K+E}{9K-E}$

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