Editing Bulk modulus

{{Short description|Resistance of a material to uniform pressure}}
{{Confusing|date=December 2023}}
[[Image:Isostatic pressure deformation.svg|thumb|300px|right|Illustration of uniform compression]]

The '''bulk modulus''' (<math>K</math> or <math>B</math> or <math>k</math>) of a substance is a measure of the resistance of a substance to bulk [[compression (physics)|compression]]. It is defined as the ratio of the [[infinitesimal]] [[pressure]] increase to the resulting ''relative'' decrease of the [[volume]].<ref>{{cite web| url= http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html|title= Bulk Elastic Properties|work=hyperphysics|publisher=Georgia State University}}</ref>

Other moduli describe the material's response ([[Strain (materials science)|strain]]) to other kinds of [[Stress (physics)|stress]]: the [[shear modulus]] describes the response to [[shear stress]], and [[Young's modulus]] describes the response to normal (lengthwise stretching) stress.  For a [[fluid]], only the bulk modulus is meaningful.  For a complex [[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]]. The reciprocal of the bulk modulus at fixed temperature is called the isothermal [[compressibility]].

==Definition==
The bulk modulus <math>K</math> (which is usually positive) can be formally defined by the equation
:<math>K=-V\frac{dP}{dV} ,</math>
where <math>P</math> is pressure, <math>V</math> is the initial volume of the substance, and <math>dP/dV</math> denotes the [[derivative]] of pressure with respect to volume.  Since the volume is inversely proportional to the density, it follows that 
:<math>K=\rho \frac{dP}{d\rho} ,</math>
where <math>\rho</math> is the initial [[density]] and <math>dP/d\rho</math> denotes the derivative of pressure with respect to density. The inverse of the bulk modulus gives a substance's [[compressibility]]. Generally the bulk modulus is defined at constant [[temperature]] as the isothermal bulk modulus, but can also be defined at constant [[entropy]] as the [[adiabatic]] bulk modulus.

==Thermodynamic relation==
Strictly speaking, the bulk modulus is a [[thermodynamic]] quantity, and in order to specify a bulk modulus it is necessary to specify how the pressure varies during compression: constant-[[temperature]] (isothermal <math>K_T</math>), constant-[[entropy]] ([[isentropic process|isentropic]] <math>K_S</math>), and other variations are possible. Such distinctions are especially relevant for [[gas]]es.

For an [[Ideal gas#Speed of sound|ideal gas]], an isentropic process has:

:<math>PV^\gamma=\text{constant} \Rightarrow P\propto \left(\frac{1}{V}\right)^\gamma\propto \rho ^\gamma,
</math>

where <math>\gamma </math> is the [[heat capacity ratio]]. Therefore, the isentropic bulk modulus <math>K_S</math> is given by

:<math>K_S=\gamma P.</math>

Similarly, an isothermal process of an ideal gas has:

:<math>PV=\text{constant} \Rightarrow P\propto \frac{1}{V} \propto \rho,
</math>

Therefore, the isothermal bulk modulus <math>K_T</math> is given by

:<math>K_T = P </math> .

When the gas is not ideal, these equations give only an approximation of the bulk modulus. In a fluid, the bulk modulus <math>K</math> and the [[density]] <math>\rho</math> determine the [[speed of sound]] <math>c</math> ([[P-wave|pressure waves]]), according to the Newton-Laplace formula

:<math>c=\sqrt{\frac{K_S}{\rho}}.</math>

In solids, <math>K_S</math> and <math>K_T</math> have very similar values. Solids can also sustain [[transverse waves]]: for these materials one additional [[elastic modulus]], for example the shear modulus, is needed to determine wave speeds.

== Measurement ==
It is possible to measure the bulk modulus using  [[Powder diffraction#Expansion tensors.2C bulk modulus|powder diffraction]] under applied pressure.
It is a property of a fluid which shows its ability to change its volume under its pressure.

==Selected values==
{| class="wikitable" style="text-align:center"
|+ Approximate bulk modulus (''K'') for common materials
!Material
!Bulk modulus in GPa
!Bulk '''modulus''' in [[Pounds per square inch#psi|Mpsi]] 
|-
| align="left" |[[Diamond]] (at 4K) <ref>Page 52 of "[[Introduction to Solid State Physics]], 8th edition" by Charles Kittel, 2005, {{ISBN|0-471-41526-X}}</ref>
| {{val|443}}
| {{val|64}}
|-
| align="left" |[[Alumina]] (γ phase)<ref>{{Cite journal|last=Gallas|first=Marcia R.|last2=Piermarini|first2=Gasper J.|date=1994|title=Bulk Modulus and Young's Modulus of Nanocrystalline γ-Alumina|url=https://ceramics.onlinelibrary.wiley.com/doi/abs/10.1111/j.1151-2916.1994.tb04524.x|journal=Journal of the American Ceramic Society|language=en|volume=77|issue=11|pages=2917–2920|doi=10.1111/j.1151-2916.1994.tb04524.x|issn=1551-2916|url-access=subscription}}</ref>
| {{val|162}} ± 14
| {{val|23.5}}
|-
| align="left" |[[Steel]]<!--http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html-->
| {{val|160}}
| {{val|23.2}}
|-
| align="left" |[[Limestone]] 
| {{val|65}}
| {{val|9.4}}
|-
| align="left" |[[Granite]] 
| {{val|50}}
| {{val|7.3}}
|-
| align="left" |[[Glass]] (see also diagram below table)
| {{val|35}} to {{val|55}}
| {{val|5.8}}
|-
| align="left" |[[Graphite]] 2H ([[single crystal]])<ref>{{Cite web|title=Graphite Properties Page by John A. Jaszczak|url=https://pages.mtu.edu/~jaszczak/graphprop.html|access-date=2021-07-16|website=pages.mtu.edu}}</ref>
| {{val|34}}
| {{val|4.9}}
|-
| align="left" |[[Sodium chloride]]
| {{val|24.42}}
| {{val|3.542}}
|-
| align="left" |[[Shale]] 
| {{val|10}}
| {{val|1.5}}
|-
| align="left" |[[Chalk]] 
| {{val|9}}
| {{val|1.3}}
|-
|align=left|[[Rubber]]<ref>{{cite web|url=https://www.azom.com/properties.aspx?ArticleID=920|title= Silicone Rubber| work = AZO materials}}</ref>
| {{val|1.5}} to {{val|2}}
| {{val|0.22}} to {{val|0.29}}
|-
|align=left|[[Sandstone]] 
| {{val|0.7}}
| {{val|0.1}}
|}

[[Image:SpiderGraph BulkModulus.gif|thumb|Influences of selected glass component additions on the bulk modulus of a specific base glass.<ref>{{cite web| url=http://www.glassproperties.com/bulk_modulus/|title= Bulk modulus calculation of glasses| work = glassproperties.com |first= Alexander |last= Fluegel}}</ref>]]

A material with a bulk modulus of 35&nbsp;GPa loses one percent of its volume when subjected to an external pressure of 0.35&nbsp;GPa (~{{val|3500|u=bar}}) (assumed constant or weakly pressure dependent bulk modulus).
{| class="wikitable"
|+ Approximate bulk modulus (''K'') for other substances
|-
| [[β-Carbon nitride]] 
|  {{val|427|15|u=GPa}}<ref>Liu, A. Y.; Cohen, M. L. (1989). "Prediction of New Low Compressibility Solids". Science. 245 (4920): 841–842.</ref> (predicted)
|-
| [[Water]]
| {{val|2.2|u=GPa}}<!--http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html--> ({{val|0.32|u=Mpsi}}) (value increases at higher pressures)<!--REF:  Perry's Handbook, Seventh Ed'n, p. 2-149-->
|-
| Methanol
| {{val|823|u=MPa}} (at 20&nbsp;°C and 1 Atm)
|-
| Solid [[helium]] 
|  {{val|50|u=MPa}} (approximate)<!--http://www3.interscience.wiley.com/cgi-bin/abstract/105558571/ABSTRACT-->
|-
| Air  
| {{val|142|u=kPa}} (adiabatic bulk modulus [or [[isentropic]] bulk modulus])
|-
| Air 
| {{val|101|u=kPa}} (isothermal bulk modulus)
|-
| [[Spacetime]] 
|  {{val|4.5e31|u=Pa}} (for typical gravitational wave frequencies of 100Hz) <ref>Beau, M. R. (2018). "On the nature of space-time, cosmological inflation, and expansion of the universe". Preprint. DOI:10.13140/RG.2.2.16796.95364</ref>
|}

== Microscopic origin ==
=== Interatomic potential and linear elasticity ===
[[File:Interatomic potentual.png|alt=The left one shows the interatomic potential and equilibrium position, while the right one shows the force|thumb|440x440px|Interatomic potential and force]]
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 the [[interatomic potential]] for crystalline materials.<ref>{{Cite book|title=Mechanical Behavior of Materials|last=H.|first=Courtney, Thomas|date=2013|publisher=McGraw Hill Education (India)|isbn=978-1259027512|edition=2nd ed. Reimp|location=New Delhi| oclc=929663641}}</ref> 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 r<sub>0</sub>, where the total force is zero:

:<math>F=-{\partial U \over \partial r}=0</math>

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 is ''r''<sub>0</sub>. Its potential energy-interatomic distance relationship has similar form as the two atoms case, which reaches minimal at ''r''<sub>0</sub>, The Taylor expansion for this is:

:<math>u(r)=u(r_0)+ \left({\partial u \over \partial r} \right )_{r=r_0}(r-r_0)+{1 \over 2} \left ({\partial^2\over\partial r^2}u \right )_{r=r_0}(r-r_0)^2+O \left ((r-r_0)^3 \right )</math>

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:

:<math>u(r)=u(r_0)+{1 \over 2} \left ({\partial^2\over\partial r^2}u \right )_{r=r_0} (r-r_0)^2</math>

:<math>F(a)=-{\partial u \over \partial r}= \left ({\partial^2\over\partial r^2}u \right )_{r=r_0}(r-r_0)</math>

Which is clearly linear elasticity.

Note that the derivation is done considering two neighboring atoms, so the Hook's coefficient is:

:<math>K=r_0{dF \over dr}=r_0 \left ({\partial^2\over\partial r^2}u \right )_{r=r_0}</math>

This form can be easily extended to 3-D case, with volume per atom(Ω) in place of interatomic distance.

:<math>K=\Omega_0 \left ({\partial^2\over\partial \Omega^2}u \right )_{\Omega=\Omega_0}</math>

== See also ==
* [[Elasticity tensor]]
* [[Volumetric strain]]

== References ==
{{Reflist}}

==Further reading==
*{{cite journal|last1 = De Jong| first1 = Maarten| last2 = Chen | first2 = Wei| year = 2015| title = Charting the complete elastic properties of inorganic crystalline compounds| journal = [[Scientific Data (journal)|Scientific Data]]| volume = 2 | doi = 10.1038/sdata.2015.9| pages = 150009|bibcode = 2013NatSD...2E0009D | pmc = 4432655 | pmid=25984348}}

{{Elastic moduli}}
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[[Category:Elasticity (physics)]]
[[Category:Mechanical quantities]]