Bulk modulus

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"Incompressibility" redirects here. For topic in fluid dynamics, see Compressibility.
Illustration of uniform compression
Illustration of uniform compression
Influences of selected glass component additions on the bulk modulus of a specific base glass.
Influences of selected glass component additions on the bulk modulus of a specific base glass.[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.

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:

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 (KT), constant-enthalpy (adiabatic KS), and other variations are possible. In practice, such distinctions are usually only relevant for gases.

Bulk modulus values for some example substances
Water 2.2×109 Pa (value increases at higher pressures)
Air 1.42×105 Pa (adiabatic bulk modulus)
Air 1.01×105 Pa (constant temperature bulk modulus)
Steel 1.6×1011 Pa
Glass 3.5×1010 to 5.5×1010 Pa
Solid helium 5×107 Pa (approximate)


For a gas, the adiabatic bulk modulus KS is approximately given by

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

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.

[edit] 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.


[edit] References

  1. ^ Bulk modulus calculation of glasses
v • d • e
Elastic moduli for homogeneous isotropic materials
Bulk modulus (K) • Young's modulus (E) • Lamé's first parameter (λ) • Shear modulus (μ) • Poisson's ratio (ν) • 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,\,\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}