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| Bulk modulus | |
|---|---|
Common symbols | K,B,k |
| SI unit | Pa |
Derivations from other quantities | K =E / [3(1 - 2ν)] |
Thebulk modulus ( or or) of a substance is a measure of the resistance of a substance to bulkcompression. It is defined as the ratio of theinfinitesimalpressure increase to the resultingrelative decrease of thevolume.[1]
Other moduli describe the material's response (strain) to other kinds ofstress: theshear modulus describes the response toshear stress, andYoung's modulus describes the response to normal (lengthwise stretching) stress. For afluid, only the bulk modulus is meaningful. For a complexanisotropic solid such aswood orpaper, these three moduli do not contain enough information to describe its behaviour, and one must use the full generalizedHooke's law. The reciprocal of the bulk modulus at fixed temperature is called the isothermalcompressibility.
The bulk modulus (which is usually positive) can be formally defined by the equation
where is pressure, is the initial volume of the substance, and denotes thederivative of pressure with respect to volume. Since the volume is inversely proportional to the density, it follows that
where is the initialdensity and denotes the derivative of pressure with respect to density. The inverse of the bulk modulus gives a substance'scompressibility. Generally the bulk modulus is defined at constanttemperature as the isothermal bulk modulus, but can also be defined at constantentropy as theadiabatic bulk modulus.
Strictly speaking, the bulk modulus is athermodynamic quantity, and in order to specify a bulk modulus it is necessary to specify how the pressure varies during compression: constant-temperature (isothermal), constant-entropy (isentropic), and other variations are possible. Such distinctions are especially relevant forgases.
For anideal gas, an isentropic process has:
where is theheat capacity ratio. Therefore, the isentropic bulk modulus is given by
Similarly, an isothermal process of an ideal gas has:
Therefore, the isothermal bulk modulus is given by
When the gas is not ideal, these equations give only an approximation of the bulk modulus. In a fluid, the bulk modulus and thedensity determine thespeed of sound (pressure waves), according to the Newton-Laplace formula
In solids, and have very similar values. Solids can also sustaintransverse waves: for these materials one additionalelastic modulus, for example the shear modulus, is needed to determine wave speeds.
It is possible to measure the bulk modulus usingpowder diffraction under applied pressure.It is a property of a fluid which shows its ability to change its volume under its pressure.
| Material | Bulk modulus in GPa | Bulkmodulus inMpsi |
|---|---|---|
| Diamond (at 4K)[2] | 443 | 64 |
| Alumina (γ phase)[3] | 162 ± 14 | 23.5 |
| Steel | 160 | 23.2 |
| Limestone | 65 | 9.4 |
| Granite | 50 | 7.3 |
| Glass (see also diagram below table) | 35 to55 | 5.8 |
| Graphite 2H (single crystal)[4] | 34 | 4.9 |
| Sodium chloride | 24.42 | 3.542 |
| Shale | 10 | 1.5 |
| Chalk | 9 | 1.3 |
| Rubber[5] | 1.5 to2 | 0.22 to0.29 |
| Sandstone | 0.7 | 0.1 |
A material with a bulk modulus of 35 GPa loses one percent of its volume when subjected to an external pressure of 0.35 GPa (~3500 bar) (assumed constant or weakly pressure dependent bulk modulus).
| β-Carbon nitride | 427±15 GPa[7] (predicted) |
| Water | 2.2 GPa (0.32 Mpsi) (value increases at higher pressures) |
| Methanol | 823 MPa (at 20 °C and 1 Atm) |
| Solidhelium | 50 MPa (approximate) |
| Air | 142 kPa (adiabatic bulk modulus [orisentropic bulk modulus]) |
| Air | 101 kPa (isothermal bulk modulus) |
| Spacetime | 4.5×1031 Pa (for typical gravitational wave frequencies of 100Hz)[8] |
Since linear elasticity is a direct result of interatomic interaction, it is related to the extension/compression of bonds. It can then be derived from theinteratomic potential for crystalline materials.[9] First, let us examine the potential energy of two interacting atoms. Starting from very far points, they will feel an attraction towards each other. As they approach each other, their potential energy will decrease. On the other hand, when two atoms are very close to each other, their total energy will be very high due to repulsive interaction. Together, these potentials guarantee an interatomic distance that achieves a minimal energy state. This occurs at some distance r0, where the total force is zero:
Where U is interatomic potential and r is the interatomic distance. This means the atoms are in equilibrium.
To extend the two atoms approach into solid, consider a simple model, say, a 1-D array of one element with interatomic distance of r, and the equilibrium distance isr0. Its potential energy-interatomic distance relationship has similar form as the two atoms case, which reaches minimal atr0, The Taylor expansion for this is:
At equilibrium, the first derivative is 0, so the dominant term is the quadratic one. When displacement is small, the higher order terms should be omitted. The expression becomes:
Which is clearly linear elasticity.
Note that the derivation is done considering two neighboring atoms, so the Hook's coefficient is:
This form can be easily extended to 3-D case, with volume per atom(Ω) in place of interatomic distance.
{{cite book}}: CS1 maint: multiple names: authors list (link)| 3D Formulae | |||||||
|---|---|---|---|---|---|---|---|
| Knowns | Bulk modulus(K) | Young's modulus(E) | Lamé's first parameter(λ) | Shear modulus(G) | Poisson's ratio(ν) | P-wave modulus(M) | Notes |
| (K,E) | 3K(1 +6K/E − 9K) | E/3 −E/3K | 1/2 −E/6K | 3K +E/3 −E/3K | |||
| (K, λ) | 9K(K − λ)/3K − λ | 3(K − λ)/2 | λ/3K − λ | 3K − 2λ | |||
| (K,G) | 9KG/3K +G | K −2G/3 | 3K − 2G/6K + 2G | K +4G/3 | |||
| (K,ν) | 3K(1 − 2ν) | 3Kν/1 +ν | 3K(1 − 2ν)/2(1 +ν) | 3K(1 −ν)/1 +ν | |||
| (K,M) | 9K(M −K)/3K +M | 3K −M/2 | 3(M −K)/4 | 3K −M/3K +M | |||
| (E, λ) | E + 3λ + R/6 | E − 3λ +R/4 | −E +R/4λ −1/4 | E − λ +R/2 | R = ±(E2 + 9λ2 + 2Eλ)1/2 | ||
| (E,G) | EG/3(3G −E) | G(E − 2G)/3G −E | E/2G − 1 | G(4G −E)/3G −E | |||
| (E,ν) | E/3 − 6ν | Eν/(1 +ν)(1 − 2ν) | E/2(1 +ν) | E(1 −ν)/(1 +ν)(1 − 2ν) | |||
| (E,M) | 3M −E +S/6 | M −E +S/4 | 3M +E −S/8 | E +S/4M −1/4 | S = ±(E2 + 9M2 − 10EM)1/2 | ||
| (λ,G) | λ +2G/3 | G(3λ + 2G)/λ +G | λ/2(λ +G) | λ + 2G | |||
| (λ,ν) | λ/3(1 +1/ν) | λ(1/ν − 2ν − 1) | λ(1/2ν − 1) | λ(1/ν − 1) | |||
| (λ,M) | M + 2λ/3 | (M − λ)(M+2λ)/M + λ | M − λ/2 | λ/M + λ | |||
| (G,ν) | 2G(1 +ν)/3 − 6ν | 2G(1 +ν) | 2Gν/1 − 2ν | 2G(1 −ν)/1 − 2ν | |||
| (G,M) | M −4G/3 | G(3M − 4G)/M −G | M − 2G | M − 2G/2M − 2G | |||
| (ν,M) | M(1 +ν)/3(1 −ν) | M(1 +ν)(1 − 2ν)/1 −ν | Mν/1 −ν | M(1 − 2ν)/2(1 −ν) | |||
| 2D Formulae | |||||||
| Knowns | (K) | (E) | (λ) | (G) | (ν) | (M) | Notes |
| (K2D,E2D) | 2K2D(2K2D −E2D)/4K2D −E2D | K2DE2D/4K2D −E2D | 2K2D −E2D/2K2D | 4K2D^2/4K2D −E2D | |||
| (K2D, λ2D) | 4K2D(K2D − λ2D)/2K2D − λ2D | K2D − λ2D | λ2D/2K2D − λ2D | 2K2D − λ2D | |||
| (K2D,G2D) | 4K2DG2D/K2D +G2D | K2D −G2D | K2D −G2D/K2D +G2D | K2D +G2D | |||
| (K2D,ν2D) | 2K2D(1 −ν2D) | 2K2Dν2D/1 +ν2D | K2D(1 −ν2D)/1 +ν2D | 2K2D/1 +ν2D | |||
| (E2D,G2D) | E2DG2D/4G2D −E2D | 2G2D(E2D − 2G2D)/4G2D −E2D | E2D/2G2D − 1 | 4G2D^2/4G2D −E2D | |||
| (E2D,ν2D) | E2D/2(1 −ν2D) | E2Dν2D/(1 +ν2D)(1 −ν2D) | E2D/2(1 +ν2D) | E2D/(1 +ν2D)(1 −ν2D) | |||
| (λ2D,G2D) | λ2D +G2D | 4G2D(λ2D +G2D)/λ2D + 2G2D | λ2D/λ2D + 2G2D | λ2D + 2G2D | |||
| (λ2D,ν2D) | λ2D(1 +ν2D)/2ν2D | λ2D(1 +ν2D)(1 −ν2D)/ν2D | λ2D(1 −ν2D)/2ν2D | λ2D/ν2D | |||
| (G2D,ν2D) | G2D(1 +ν2D)/1 −ν2D | 2G2D(1 +ν2D) | 2G2Dν2D/1 −ν2D | 2G2D/1 −ν2D | |||
| (G2D,M2D) | M2D −G2D | 4G2D(M2D −G2D)/M2D | M2D − 2G2D | M2D − 2G2D/M2D | |||
