Shear modulus

Shear modulus
Common symbols	G
SI unit	pascal
Derivations from other quantities	G = τ / γ

Shear strain

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]

G\ {\stackrel {{\mathrm {def}}}{=}}\ {\frac {\tau _{{xy}}}{\gamma _{{xy}}}}={\frac {F/A}{\Delta x/l}}={\frac {Fl}{A\Delta x}}

where

\tau _{{xy}}=F/A\,

= shear stress

F

is the force which acts

A

is the area on which the force acts

\gamma _{xy}

= shear strain. In engineering

:=\Delta x/l=\tan \theta

, elsewhere

:=\theta

\Delta x

is the transverse displacement

l

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, replacing force by mass times acceleration.

The shear modulus is always positive.

Explanation

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
Iron^[5]	52.5
Polyethylene^[3]	0.117
Rubber^[6]	0.0006

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

Young's modulus E describes the material's strain response to uniaxial stress in the direction of this stress (like pulling on the ends of a wire or putting a weight on top of a column, with the wire getting longer and the column losing height),
the Poisson's ratio ν describes the response in the directions orthogonal to this uniaxial stress (the wire getting thinner and the column thicker),
the bulk modulus K describes the material's response to (uniform) hydrostatic pressure (like the pressure at the bottom of the ocean or a deep swimming pool),
the shear modulus G describes the material's response to shear stress (like cutting it with dull scissors).
These moduli are not independent, and for isotropic materials they are connected via the equations $2G(1+\nu) = E = 3K(1-2\nu)$ .^[7]

The shear modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force (such as friction). In the case of an object that's shaped like a rectangular prism, it will deform into a parallelepiped. Anisotropic materials such as wood, paper and also essentially all single crystals exhibit differing material response to stress or strain when tested in 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 shear modulus.

Waves

Influences of selected glass component additions on the shear modulus of a specific base glass.^[8]

In homogeneous and isotropic solids, there are two kinds of waves, pressure waves and shear waves. The velocity of a shear wave, $(v_{s})$ is controlled by the shear modulus,

v_{s}={\sqrt {{\frac {G}{\rho }}}}

where

G is the shear modulus

\rho

is the solid's density.

Shear modulus of metals

Shear modulus of copper as a function of temperature. The experimental data^[9]^[10] 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.^[11]

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 MTS shear modulus model developed by^[12] and used in conjunction with the Mechanical Threshold Stress (MTS) plastic flow stress model.^[13]^[14]
the Steinberg-Cochran-Guinan (SCG) shear modulus model developed by^[15] and used in conjunction with the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model.
the Nadal and LePoac (NP) shear modulus model^[10] that uses Lindemann theory to determine the 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:

\mu (T)=\mu _{0}-{\frac {D}{\exp(T_{0}/T)-1}}

where $\mu_0$ is the shear modulus at $T=0K$ , and $D$ and $T_{0}$ are material constants.

SCG shear modulus model

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

\mu (p,T)=\mu _{0}+{\frac {\partial \mu }{\partial p}}{\frac {p}{\eta ^{{1/3}}}}+{\frac {\partial \mu }{\partial T}}(T-300);\quad \eta :=\rho /\rho _{0}

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 temperature dependence of the shear modulus in the SCG model is replaced with an equation based on Lindemann melting theory. The NP shear modulus model has the form:

\mu (p,T)={\frac {1}{{\mathcal {J}}({\hat {T}})}}\left[\left(\mu _{0}+{\frac {\partial \mu }{\partial p}}{\cfrac {p}{\eta ^{{1/3}}}}\right)(1-{\hat {T}})+{\frac {\rho }{Cm}}~k_{b}~T\right];\quad C:={\cfrac {(6\pi ^{2})^{{2/3}}}{3}}f^{2}

where

{\mathcal {J}}({\hat {T}}):=1+\exp \left[-{\cfrac {1+1/\zeta }{1+\zeta /(1-{\hat {T}})}}\right]\quad {\text{for}}\quad {\hat {T}}:={\frac {T}{T_{m}}}\in [0,1+\zeta ],

and µ₀ is the shear modulus at 0 K and ambient pressure, ζ is a material parameter, k_b is the Boltzmann constant, m is the atomic mass, and f is the Lindemann constant.

References

↑ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "shear modulus, G".
↑ 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. doi:10.1063/1.1661636.
1 2 3 4 5 Crandall, Dahl, Lardner (1959). An Introduction to the Mechanics of Solids. Boston: McGraw-Hill. ISBN 0-07-013441-3.
↑ Material properties
↑ Rayne, J.A. (1961). "Elastic constants of Iron from 4.2 to 300 ° K". Physical Review. 122 (6): 1714. Bibcode:1961PhRv..122.1714R. doi:10.1103/PhysRev.122.1714.
↑ Spanos, Pete (2003). "Cure system effect on low temperature dynamic shear modulus of natural rubber". Rubber World.
↑ [Landau LD, Lifshitz EM. Theory of Elasticity, vol. 7. Course of Theoretical Physics. (2nd Ed) Pergamon: Oxford 1970 p13]
↑ Shear modulus calculation of glasses
↑ 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. doi:10.1103/PhysRev.98.969.
1 2 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. doi:10.1063/1.1539913.
↑ March, N. H., (1996), Electron Correlation in Molecules and Condensed Phases, Springer, ISBN 0-306-44844-0 p. 363
↑ Varshni, Y. (1970). "Temperature Dependence of the Elastic Constants". Physical Review B. 2 (10): 3952. Bibcode:1970PhRvB...2.3952V. doi:10.1103/PhysRevB.2.3952.
↑ 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. doi:10.1007/BF02663849.
↑ 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.
↑ 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. doi:10.1016/S0022-3697(74)80278-7.

Elastic moduli for homogeneous isotropic materials

Bulk modulus ( $K$ ) Young's modulus ( $E$ ) Lamé's first parameter ( $\lambda$ ) Shear modulus ( $G,\mu$ ) Poisson's ratio ( $\nu$ ) 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.
	$K=\,$	$E=\,$	$\lambda =\,$	$G=\,$	$\nu =\,$	$M=\,$	Notes
$(K,\,E)$	$K$	$E$	${\tfrac {3K(3K-E)}{9K-E}}$	${\tfrac {3KE}{9K-E}}$	${\tfrac {3K-E}{6K}}$	${\tfrac {3K(3K+E)}{9K-E}}$
$(K,\,\lambda )$	$K$	${\tfrac {9K(K-\lambda )}{3K-\lambda }}$	$\lambda$	${\tfrac {3(K-\lambda )}{2}}$	${\tfrac {\lambda }{3K-\lambda }}$	$3K-2\lambda \,$
$(K,\,G)$	$K$	${\tfrac {9KG}{3K+G}}$	$K-{\tfrac {2G}{3}}$	$G$	${\tfrac {3K-2G}{2(3K+G)}}$	$K+{\tfrac {4G}{3}}$
$(K,\,\nu )$	$K$	$3K(1-2\nu )\,$	${\tfrac {3K\nu }{1+\nu }}$	${\tfrac {3K(1-2\nu )}{2(1+\nu )}}$	$\nu$	${\tfrac {3K(1-\nu )}{1+\nu }}$
$(K,\,M)$	$K$	${\tfrac {9K(M-K)}{3K+M}}$	${\tfrac {3K-M}{2}}$	${\tfrac {3(M-K)}{4}}$	${\tfrac {3K-M}{3K+M}}$	$M$
$(E,\,\lambda )$	${\tfrac {E+3\lambda +R}{6}}$	$E$	$\lambda$	${\tfrac {E-3\lambda +R}{4}}$	${\tfrac {2\lambda }{E+\lambda +R}}$	${\tfrac {E-\lambda +R}{2}}$	$R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}$
$(E,\,G)$	${\tfrac {EG}{3(3G-E)}}$	$E$	${\tfrac {G(E-2G)}{3G-E}}$	$G$	${\tfrac {E}{2G}}-1$	${\tfrac {G(4G-E)}{3G-E}}$
$(E,\,\nu )$	${\tfrac {E}{3(1-2\nu )}}$	$E$	${\tfrac {E\nu }{(1+\nu )(1-2\nu )}}$	${\tfrac {E}{2(1+\nu )}}$	$\nu$	${\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}$
$(E,\,M)$	${\tfrac {3M-E+S}{6}}$	$E$	${\tfrac {M-E+S}{4}}$	${\tfrac {3M+E-S}{8}}$	${\tfrac {E-M+S}{4M}}$	$M$	$S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}$ There are two valid solutions. The plus sign leads to $\nu \geq 0$ . The minus sign leads to $\nu \leq 0$ .
$(\lambda ,\,G)$	$\lambda +{\tfrac {2G}{3}}$	${\tfrac {G(3\lambda +2G)}{\lambda +G}}$	$\lambda$	$G$	${\tfrac {\lambda }{2(\lambda +G)}}$	$\lambda +2G\,$
$(\lambda ,\,\nu )$	${\tfrac {\lambda (1+\nu )}{3\nu }}$	${\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}$	$\lambda$	${\tfrac {\lambda (1-2\nu )}{2\nu }}$	$\nu$	${\tfrac {\lambda (1-\nu )}{\nu }}$	Cannot be used when $\nu =0\Leftrightarrow \lambda =0$
$(\lambda ,\,M)$	${\tfrac {M+2\lambda }{3}}$	${\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}$	$\lambda$	${\tfrac {M-\lambda }{2}}$	${\tfrac {\lambda }{M+\lambda }}$	$M$
$(G,\,\nu )$	${\tfrac {2G(1+\nu )}{3(1-2\nu )}}$	$2G(1+\nu )\,$	${\tfrac {2G\nu }{1-2\nu }}$	$G$	$\nu$	${\tfrac {2G(1-\nu )}{1-2\nu }}$
$(G,\,M)$	$M-{\tfrac {4G}{3}}$	${\tfrac {G(3M-4G)}{M-G}}$	$M-2G\,$	$G$	${\tfrac {M-2G}{2M-2G}}$	$M$
$(\nu ,\,M)$	${\tfrac {M(1+\nu )}{3(1-\nu )}}$	${\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}$	${\tfrac {M\nu }{1-\nu }}$	${\tfrac {M(1-2\nu )}{2(1-\nu )}}$	$\nu$	$M$

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Shear modulus

Explanation

Waves

Shear modulus of metals

MTS shear modulus model

SCG shear modulus model

NP shear modulus model

See also

References