Shear Modulus - Wikipedia, The Free Encyclopedia

Shear modulus

Common symbols G

SI unit pascal

Derivations fromother quantities

G = τ / γ

Shear strain

Shear modulusFrom Wikipedia, the free encyclopedia

In materials science, shear modulus or modulus of rigidity,denoted by G, or sometimes S or μ, is defined as the ratio of shear

stress to the shear strain:[1]

where

= shear stress;

is the force which acts

is the area on which the force acts

in engineering, = shear strain. Elsewhere,

is the transverse displacement

is the initial length

Shear modulus' derived SI unit is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in

thousands of pounds per square inch (ksi). Its dimensional form is M1L−1T−2.

The shear modulus is always positive.

Contents

1 Explanation

2 Waves

3 Shear modulus of metals

3.1 MTS shear modulus model

3.2 SCG shear modulus model

3.3 NP shear modulus model

4 See also

5 References

Explanation

The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in thegeneralized Hooke's law:

Young's modulus describes the material's response to uniaxial stress (like pulling on the ends of a wire or

Shear modulus - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Shear_modulus

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Material

Typical values for

shear modulus (GPa)

(at room temperature)

Diamond[2] 478.0

Steel[3] 79.3

Copper[4] 44.7

Titanium[3] 41.4

Glass[3] 26.2

Aluminium[3] 25.5

Polyethylene[3] 0.117

Rubber[5] 0.0006

Influences of selected glass component

additions on the shear modulus of a specific

base glass.[6]

putting a weight on top of a column),

the bulk modulus describes the material's response to uniform

pressure (like the pressure at the bottom of the ocean or a deep

swimming pool)

the shear modulus describes the material's response to shear stress

(like cutting it with dull scissors).

The shear modulus is concerned with the deformation of a solid when itexperiences a force parallel to one of its surfaces while its opposite faceexperiences an opposing force (such as friction). In the case of an objectthat's shaped like a rectangular prism, it will deform into a parallelepiped.Anisotropic materials such as wood, paper and also essentially all singlecrystals exhibit differing material response to stress or strain when testedin different directions. In this case one may need to use the full tensor-expression of the elastic constants, rather than a single scalar value.

One possible definition of a fluid would be a material with zero shearmodulus.

Waves

In homogeneous and isotropic solids, there are two kinds ofwaves, pressure waves and shear waves. The velocity of a shearwave, is controlled by the shear modulus,

where

G is the shear modulus

is the solid's density.

Shear modulus of metals

The shear modulus of metals is usually observed to decrease with increasing temperature. At high pressures, theshear 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.[9]

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

the MTS shear modulus model developed by[10] and used in conjunction with the Mechanical Threshold

Stress (MTS) plastic flow stress model.[11][12]

1.

the Steinberg-Cochran-Guinan (SCG) shear modulus model developed by[13] and used in conjunction with

the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model.

2.

the Nadal and LePoac (NP) shear modulus model[8] that uses Lindemann theory to determine the3.


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Shear modulus of copper as a function of

temperature. The experimental data[7][8] are

shown with colored symbols.

temperature dependence and the SCG model for pressure

dependence of the shear modulus.

MTS shear modulus model

The MTS shear modulus model has the form:

where µ0 is the shear modulus at 0 K, and D and T0 are material

constants.

SCG shear modulus model

The Steinberg-Cochran-Guinan (SCG) shear modulus model ispressure dependent and has the form

where, µ0 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 shear modulus model

The Nadal-Le Poac (NP) shear modulus model is a modified version of the SCG model. The empirical temperaturedependence of the shear modulus in the SCG model is replaced with an equation based on Lindemann meltingtheory. The NP shear modulus model has the form:

where

and µ0 is the shear modulus at 0 K and ambient pressure, ζ is a material parameter, kb is the Boltzmann constant,

m is the atomic mass, and f is the Lindemann constant.

See also

Shear strength

Dynamic modulus

Impulse excitation technique


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References

IUPAC, Compendium of Chemical Terminology, 2nd ed.

(the "Gold Book") (1997). Online corrected version:

(2006–) "shear modulus, G (http://goldbook.iupac.org

/S05635.html)".

1.

McSkimin, H.J.; Andreatch, P. (1972). "Elastic Moduli of

Diamond as a Function of Pressure and Temperature".

J. Appl. Phys. 43 (7): 2944–2948.

Bibcode:1972JAP....43.2944M

(http://adsabs.harvard.edu/abs/1972JAP....43.2944M).

doi:10.1063/1.1661636 (https://dx.doi.org

/10.1063%2F1.1661636).

2.

Crandall, Dahl, Lardner (1959). An Introduction to the

Mechanics of Solids. Boston: McGraw-Hill.

ISBN 0-07-013441-3.

3.

Material properties (http://homepages.which.net

/~paul.hills/Materials/MaterialsBody.html)

4.

Spanos, Pete (2003). "Cure system effect on low

temperature dynamic shear modulus of natural rubber"

(http://www.thefreelibrary.com

/Cure+system+effect+on+low+temperature+dynamic

+shear+modulus+of...-a0111451108). Rubber World.

5.

Shear modulus calculation of glasses

(http://www.glassproperties.com/shear_modulus/)

6.

Overton, W.; Gaffney, John (1955). "Temperature

Variation of the Elastic Constants of Cubic Elements. I.

Copper". Physical Review 98 (4): 969.

Bibcode:1955PhRv...98..969O

(http://adsabs.harvard.edu/abs/1955PhRv...98..969O).

doi:10.1103/PhysRev.98.969 (https://dx.doi.org

/10.1103%2FPhysRev.98.969).

7.

Nadal, Marie-Hélène; Le Poac, Philippe (2003).

"Continuous model for the shear modulus as a function

of pressure and temperature up to the melting point:

Analysis and ultrasonic validation". Journal of Applied

Physics 93 (5): 2472. Bibcode:2003JAP....93.2472N

(http://adsabs.harvard.edu/abs/2003JAP....93.2472N).

doi:10.1063/1.1539913 (https://dx.doi.org

/10.1063%2F1.1539913).

8.

March, N. H., (1996), Electron Correlation in Molecules

and Condensed Phases (http://books.google.com

/books?id=PaphaJhfAloC&pg=PA363), Springer, ISBN

0-306-44844-0 p. 363

9.

Varshni, Y. (1970). "Temperature Dependence of the

Elastic Constants". Physical Review B 2 (10): 3952.

Bibcode:1970PhRvB...2.3952V

(http://adsabs.harvard.edu/abs/1970PhRvB...2.3952V).

doi:10.1103/PhysRevB.2.3952 (https://dx.doi.org

/10.1103%2FPhysRevB.2.3952).

10.

Chen, Shuh Rong; Gray, George T. (1996). "Constitutive

behavior of tantalum and tantalum-tungsten alloys".

Metallurgical and Materials Transactions A 27 (10):

2994. Bibcode:1996MMTA...27.2994C

(http://adsabs.harvard.edu

/abs/1996MMTA...27.2994C). doi:10.1007/BF02663849

(https://dx.doi.org/10.1007%2FBF02663849).

11.

Goto, D. M.; Garrett, R. K.; Bingert, J. F.; Chen, S. R.;

Gray, G. T. (2000). "The mechanical threshold stress

constitutive-strength model description of HY-100

steel". Metallurgical and Materials Transactions A 31

(8): 1985–1996. doi:10.1007/s11661-000-0226-8

(https://dx.doi.org/10.1007%2Fs11661-000-0226-8).

12.

Guinan, M; Steinberg, D (1974). "Pressure and

temperature derivatives of the isotropic polycrystalline

shear modulus for 65 elements". Journal of Physics and

Chemistry of Solids 35 (11): 1501.

Bibcode:1974JPCS...35.1501G

(http://adsabs.harvard.edu/abs/1974JPCS...35.1501G).

doi:10.1016/S0022-3697(74)80278-7

(https://dx.doi.org

/10.1016%2FS0022-3697%2874%2980278-7).

13.


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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.

Notes

There are two valid solutions.The plus sign leads to .

The minus sign leads to .

Cannot be used when

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