Compression module

Deformation under uniform pressure

Compression modulus of some fabrics

material	Compression modulus in GPa
Air (under normal conditions )	001.01 · 10 −4 (isothermal) 1.42 · 10 −4 (adiabatic) 00
Helium (solid)	000.05 (estimated)
Methanol	000.823
Ethanol	000.896
acetone	000.92
oil	001… 1.6
Cesium	001.6
water	002.08 ( 000.1 MPa) 002.68 (100 , 0MPa)
Rubidium	002.5
Glycerin	004.35
sodium	006.3
Iodine	007.7
Methane hydrate	009.1 (mean value in the range 10 ... 100 MPa)
barium	009.6
lithium	011
mercury	028.5
Bismuth	031
Glass	035… 55
lead	046
aluminum	076
steel	160
gold	180
Boron carbide	271
Magnesium oxide	277
boron	320
Rhodium	380
diamond	442
osmium	462
Aggregated diamond nanorods (ADNR)	491 (hardest known material in 2008)
Background colors: gases liquids solids  

The compression modulus ( symbol  K ) is an intensive and material-specific physical variable from the theory of elasticity . It describes which pressure change on all sides is necessary to cause a certain change in volume ( no phase transition may occur). The SI unit of the compression module is therefore Pascal or Newton per square meter .

The fact that substances oppose compression (compression, compression) resistance is primarily based on the interactions of the electrons they contain.

General

The compression is a (all-round) compression of a body / mass-filled space, which reduces its volume and increases its density (mass density) . Bodies are only referred to as compressible if the pressure changes that occur are sufficient to cause noticeable changes in density, which is usually (only) the case with gases. If there are no noticeable changes in density, the body is called incompressible (see also incompressible fluid ).

In strength theory , every solid is generally assumed to be deformable (in the form (pure thrust) as well as hydrostatic volume changes (compressible)). After the process, the body is compressed (compressed). As a rule, there is only elastic deformation ; d. H. when the pressure is released, the compression is reversed and the body expands again (expansion). Depending on the material, a permanent change in the structure can occur (e.g. plastic deformation , crumbling concrete, grain rearrangement in the foundation ).

The compression modulus describes only the spontaneously elastic part (the hydrostatic part) of the volume change, neither plastic, fracture-mechanical nor viscoelastic parts are included, any thermal deformations are also deducted beforehand.

The relationship between the volume of a solid and the external hydrostatic pressure acting on it is described by the equations according to Murnaghan and Birch .

definition

The compression modulus is defined by the spontaneous elastic change in volume (and therefore density) as a result of pressure or mechanical tension:

${\displaystyle K:=-V\cdot \underbrace {\frac {\mathrm {d} p}{\mathrm {d} V}} _{<0}=-{\frac {\mathrm {d} p}{\mathrm {d} V/V}}>0}$

The individual symbols represent the following values:

${\displaystyle V}$        - Volume

${\displaystyle \mathrm {d} p}$       - infinitesimal pressure change

${\displaystyle \mathrm {d} V}$      - infinitesimal volume change

${\displaystyle \mathrm {d} V/V}$ - relative change in volume.

The negative sign was chosen because the increase in pressure reduces the volume ( is negative), but should be positive for practical purposes . The compression module may hang. a. on temperature and pressure. ${\displaystyle \mathrm {d} V/\mathrm {d} p}$ ${\displaystyle K}$

The compression modulus represents a stress or that notional pressure at which the volume would become zero if linear elasticity, i.e. H. , and geometric linearity would be given in the spatial coordinates (thus not in the material coordinates), i.e. the compression modulus would not increase at higher pressures. ${\displaystyle \mathrm {d} p/\mathrm {d} V=\mathrm {const} }$

Compressibility

Compressibility is a redirect to this article. For the compressibility factor Z as a measure of the deviation from the equation of state of the ideal gas, see compressibility factor .

Influences of the addition of selected glass components on the compression module of a special base glass.

For gases and liquids , its reciprocal is often used instead of the compression module. This is called compressibility (symbol: κ or χ ) or also the compressibility coefficient:

${\displaystyle \kappa ={\frac {1}{K}}=-{\frac {\mathrm {d} V/V}{\mathrm {d} p}}=-{\frac {1}{V}}{\frac {\mathrm {d} V}{\mathrm {d} p}}}$.

One distinguishes

• isothermal compressibility (at constant temperature and constant number of particles ), wherethe free energy is: ${\displaystyle \kappa _{T}}$ ${\displaystyle T}$ ${\displaystyle N}$ ${\displaystyle F(T,V,N)}$

  ${\displaystyle \kappa _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial p}}\right)_{T,N}={\frac {1}{V}}\left({\frac {\partial ^{2}F}{\partial V^{2}}}\right)_{T,N}^{-1}}$

• adiabatic compressibility (with constant entropy and constant number of particles), wherethe internal energy is: ${\displaystyle \kappa _{S}}$ ${\displaystyle S}$ ${\displaystyle N}$ ${\displaystyle U(S,V,N)}$

  ${\displaystyle \kappa _{S}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial p}}\right)_{S,N}={\frac {1}{V}}\left({\frac {\partial ^{2}U}{\partial V^{2}}}\right)_{S,N}^{-1}\;.}$

In the approximation of an ideal gas is calculated

• the isothermal compressibility according to the Boyle-Mariotte law : ${\displaystyle \kappa _{T}={\frac {1}{p}}}$
• the adiabatic compressibility according to the adiabatic equation for an ideal gas : ${\displaystyle \kappa _{S}={\frac {1}{\gamma \cdot p}}\;,}$

where (often referred to as ) is the isentropic exponent . ${\displaystyle \gamma }$ ${\displaystyle \kappa }$

The compressibility of liquids was long doubted until John Canton in 1761 , Jacob Perkins in 1820 and Hans Christian Oersted in 1822 were able to prove it by measurements.

Compression modulus of solids with isotropic material behavior

Assuming linear-elastic behavior and isotropic material, the compression modulus can be calculated from other elastic constants: ${\displaystyle K}$

${\displaystyle K={\frac {E}{3-6\nu }}={\frac {GE}{9G-3E}}={\frac {2G(1+\nu )}{3(1-2\nu )}}}$

With

${\displaystyle E}$- Young's modulus

${\displaystyle G}$- shear modulus

${\displaystyle \nu }$  - Poisson's number

water

Water pressure with and without compressibility

The compression modulus of water at a temperature of 10 ° C. under normal pressure is 2.08 · 10 ⁹  Pa at 0.1 MPa and 2.68 · 10 ⁹  Pa at 100 MPa.

If the compressibility of the water is included in the calculation of the pressure , the result is the compressibility

${\displaystyle \kappa =-{\frac {\mathrm {d} V}{V\cdot \mathrm {d} p}}=0{,}5{\frac {1}{\mathrm {GPa} }}}$

the right diagram.

With a density of 1000 kg / m³ on the surface, the compressibility of the water increases the density at a depth of 12 km to 1051 kg / m³ there. The additional pressure caused by the higher density of water in the depths amounts to about 2.6 percent compared to the value if the compressibility is neglected. However, the influences of temperature, gas and salt content that continue to prevail in the sea are not taken into account.

Neutron stars

In neutron stars , all atomic shells collapsed under the pressure of gravity and neutrons were created from electrons in the shells and protons in the atomic nuclei . Neutrons are the most incompressible form of matter known. Their modulus of compression is 20 orders of magnitude higher than that of diamond under normal conditions.

Conversion between the elastic constants

The module ...	... results from:
	${\displaystyle (K,\,E)}$	${\displaystyle (K,\,\lambda )}$	${\displaystyle (K,\,G)}$	${\displaystyle (K,\,\nu )}$	${\displaystyle (E,\,\lambda )}$	${\displaystyle (E,\,G)}$	${\displaystyle (E,\,\nu )}$	${\displaystyle (\lambda ,\,G)}$	${\displaystyle (\lambda ,\,\nu )}$	${\displaystyle (G,\,\nu )}$	${\displaystyle (G,\,M)}$
Compression module ${\displaystyle K\,}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle (E+3\lambda )+}$ ${\displaystyle {\tfrac {\sqrt {(E+3\lambda )^{2}-4\lambda E}}{6}}}$	${\displaystyle {\tfrac {EG}{3(3G-E)}}}$	${\displaystyle {\tfrac {E}{3(1-2\nu )}}}$	${\displaystyle \lambda +}$ ${\displaystyle {\tfrac {2G}{3}}}$	${\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}$	${\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}$	${\displaystyle M-}$ ${\displaystyle {\tfrac {4G}{3}}}$
modulus of elasticity ${\displaystyle E\,}$	${\displaystyle E}$	${\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}$	${\displaystyle {\tfrac {9KG}{3K+G}}}$	${\displaystyle 3K(1-2\nu )\,}$	${\displaystyle E}$	${\displaystyle E}$	${\displaystyle E}$	${\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}$	${\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}$	${\displaystyle 2G(1+\nu )\,}$	${\displaystyle {\tfrac {G(3M-4G)}{M-G}}}$
1. Lamé constant ${\displaystyle \lambda \,}$	${\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}$	${\displaystyle \lambda }$	${\displaystyle K-}$ ${\displaystyle {\tfrac {2G}{3}}}$	${\displaystyle {\tfrac {3K\nu }{1+\nu }}}$	${\displaystyle \lambda }$	${\displaystyle {\tfrac {G(E-2G)}{3G-E}}}$	${\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}$	${\displaystyle \lambda }$	${\displaystyle \lambda }$	${\displaystyle {\tfrac {2G\nu }{1-2\nu }}}$	${\displaystyle M-2G\,}$
Shear modulus or (2nd Lamé constant) ${\displaystyle G}$ ${\displaystyle \mu }$	${\displaystyle {\tfrac {3KE}{9K-E}}}$	${\displaystyle {\tfrac {3(K-\lambda )}{2}}}$	${\displaystyle G}$	${\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}$	${\displaystyle (E-3\lambda )+}$ ${\displaystyle {\tfrac {\sqrt {(E-3\lambda )^{2}+8\lambda E}}{4}}}$	${\displaystyle G}$	${\displaystyle {\tfrac {E}{2(1+\nu )}}}$	${\displaystyle G}$	${\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}$	${\displaystyle G}$	${\displaystyle G}$
Poisson's number ${\displaystyle \nu \,}$	${\displaystyle {\tfrac {3K-E}{6K}}}$	${\displaystyle {\tfrac {\lambda }{3K-\lambda }}}$	${\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}$	${\displaystyle \nu }$	${\displaystyle -(E+\lambda )+}$ ${\displaystyle {\tfrac {\sqrt {(E+\lambda )^{2}+8\lambda ^{2}}}{4\lambda }}}$	${\displaystyle {\tfrac {E}{2G}}}$ ${\displaystyle -1}$	${\displaystyle \nu }$	${\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}$	${\displaystyle \nu }$	${\displaystyle \nu }$	${\displaystyle {\tfrac {M-2G}{2M-2G}}}$
Longitudinal module ${\displaystyle M\,}$	${\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}$	${\displaystyle 3K-2\lambda \,}$	${\displaystyle K+}$ ${\displaystyle {\tfrac {4G}{3}}}$	${\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}$	${\displaystyle {\tfrac {E-\lambda +{\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}{2}}}$	${\displaystyle {\tfrac {G(4G-E)}{3G-E}}}$	${\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}$	${\displaystyle \lambda +2G\,}$	${\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}$	${\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}$	${\displaystyle M}$

See also

• thermal equation of state
• Continuum mechanics
• Elasticity theory

Individual evidence

1. ↑ Dieter Will, Norbert Gebhardt: Hydraulics: Basics, components, circuits . Springer DE, 1 June 2011, ISBN 978-3-642-17242-7 , p. 21–.
2. ^ Natalia Dubrovinskaia, Leonid Dubrovinsky, Wilson Crichton, Falko Langenhorst, Asta Richter. Aggregated diamond nanorods, the densest and least compressible form of carbon. Applied Physics Letters, August 22, 2005.
3. ↑ Glassproperties.com Calculation of the bulk modulus for Glasses
4. ↑ G. Mavko, T. Mukerji, J. Dvorkin: The Rock Physics Handbook . Cambridge University Press, 2003, ISBN 0-521-54344-4 (paperback).