Editing Shear modulus (section)

==Shear modulus of metals==
[[File:CuShearMTS.svg|thumb|upright=1.2|Shear modulus of copper as a function of temperature. The experimental data<ref name=Overton55>{{cite journal|last1=Overton|first1=W.|last2=Gaffney|first2=John|title=Temperature Variation of the Elastic Constants of Cubic Elements. I. Copper|journal=Physical Review|volume=98|pages=969|year=1955|doi=10.1103/PhysRev.98.969|issue=4|bibcode = 1955PhRv...98..969O }}</ref><ref name=Nadal03/> are shown with colored symbols.]]

The shear modulus of metals is usually observed to decrease with increasing temperature. At high pressures, the shear modulus also appears to increase with the applied pressure. Correlations between the melting temperature, vacancy formation energy, and the shear modulus have been observed in many metals.<ref name=March>March, N. H., (1996), [https://books.google.com/books?id=PaphaJhfAloC&pg=PA363 ''Electron Correlation in Molecules and Condensed Phases''], Springer, {{ISBN|0-306-44844-0}} p. 363</ref>

Several models exist that attempt to predict the shear modulus of metals (and possibly that of alloys). Shear modulus models that have been used in plastic flow computations include:

# the Varshni-Chen-Gray model developed by<ref name=Varshni70>{{cite journal|last1=Varshni|first1=Y.|title=Temperature Dependence of the Elastic Constants|journal=Physical Review B|volume=2|pages=3952–3958|year=1970|doi=10.1103/PhysRevB.2.3952|issue=10|bibcode = 1970PhRvB...2.3952V }}</ref> and used in conjunction with the Mechanical Threshold Stress (MTS) plastic flow stress model.<ref name=Chen96>{{cite journal|last1=Chen|first1=Shuh Rong|last2=Gray|first2=George T.|title=Constitutive behavior of tantalum and tantalum-tungsten alloys|journal=Metallurgical and Materials Transactions A|volume=27|pages=2994|year=1996|doi=10.1007/BF02663849|issue=10|bibcode = 1996MMTA...27.2994C |s2cid=136695336|url=https://zenodo.org/record/1232556}}</ref><ref name=Goto00>{{cite journal|doi=10.1007/s11661-000-0226-8|title=The mechanical threshold stress constitutive-strength model description of HY-100 steel|year=2000|last1=Goto|first1=D. M.|last2=Garrett|first2=R. K.|last3=Bingert|first3=J. F.|last4=Chen|first4=S. R.|last5=Gray|first5=G. T.|journal=Metallurgical and Materials Transactions A|volume=31|issue=8|pages=1985–1996 |bibcode=2000MMTA...31.1985G |s2cid=136118687|url=https://apps.dtic.mil/sti/pdfs/ADA372816.pdf|archive-url=https://web.archive.org/web/20170925012725/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA372816|url-status=live|archive-date=September 25, 2017}}</ref>
# the Steinberg-Cochran-Guinan (SCG) shear modulus model developed by<ref name=Guinan74>{{cite journal|last1=Guinan|first1=M|last2=Steinberg|first2=D|title=Pressure and temperature derivatives of the isotropic polycrystalline shear modulus for 65 elements|journal=Journal of Physics and Chemistry of Solids|volume=35|pages=1501|year=1974|doi=10.1016/S0022-3697(74)80278-7|bibcode=1974JPCS...35.1501G|issue=11}}</ref> and used in conjunction with the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model.
# the Nadal and LePoac (NP) shear modulus model<ref name=Nadal03>{{cite journal|last1=Nadal|first1=Marie-Hélène|last2=Le Poac|first2=Philippe|title=Continuous model for the shear modulus as a function of pressure and temperature up to the melting point: Analysis and ultrasonic validation|journal=Journal of Applied Physics|volume=93|pages=2472|year=2003|doi=10.1063/1.1539913|issue=5|bibcode = 2003JAP....93.2472N }}</ref> that uses [[Lindemann criterion|Lindemann theory]] to determine the temperature dependence and the SCG model for pressure dependence of the shear modulus.

===Varshni-Chen-Gray model===
The Varshni-Chen-Gray model (sometimes referred to as the Varshni equation) has the form:

:<math> 
 \mu(T) = \mu_0 - \frac{D}{\exp(T_0/T) - 1}
 </math>

where <math> \mu_0 </math> is the shear modulus at <math> T=0K </math>, and <math>D</math> and <math> T_0 </math> are material constants.

===SCG model===
The Steinberg-Cochran-Guinan (SCG) shear modulus model is pressure dependent and has the form
:<math> 
 \mu(p,T) = \mu_0 + \frac{\partial \mu}{\partial p} \frac{p}{\eta^\frac{1}{3}} +
 \frac{\partial \mu}{\partial T}(T - 300) ; \quad
 \eta := \frac{\rho}{\rho_0}
 </math>
where, μ<sub>0</sub> is the shear modulus at the reference state (''T'' = 300 K, ''p'' = 0, η = 1), ''p'' is the pressure, and ''T'' is the temperature.

===NP model===
The Nadal-Le Poac (NP) shear modulus model is a modified version of the SCG model. The empirical temperature dependence of the shear modulus in the SCG model is replaced with an equation based on [[Lindemann criterion|Lindemann melting theory]]. The NP shear modulus model has the form:

:<math>
 \mu(p,T) = \frac{1}{\mathcal{J}\left(\hat{T}\right)}
 \left[
   \left(\mu_0 + \frac{\partial \mu}{\partial p} \frac{p}{\eta^\frac{1}{3}} \right)
   \left(1 - \hat{T}\right) + \frac{\rho}{Cm}~T
 \right]; \quad
 C := \frac{\left(6\pi^2\right)^\frac{2}{3}}{3} f^2
 </math>

where

:<math>
 \mathcal{J}(\hat{T}) := 1 + \exp\left[-\frac{1 + 1/\zeta}
 {1 + \zeta/\left(1 - \hat{T}\right)}\right] \quad
 \text{for} \quad \hat{T} := \frac{T}{T_m}\in[0, 6+ \zeta],
 </math>

and μ<sub>0</sub> is the shear modulus at absolute zero and ambient pressure, ζ is an area, ''m'' is the [[atomic mass]], and ''f'' is the [[Lindemann criterion|Lindemann constant]].