Shear modulus

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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{F l}{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
in engineering, \gamma_{xy} = \Delta x/l = \tan \theta \, = shear strain. Elsewhere, \gamma_{xy} = \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.

The shear modulus is always positive.

Explanation

Material Typical values for
shear modulus (GPa)
(at room temperature)
Diamond2 478.0
Steel3 79.3
Copper4 44.7
Titanium3 41.4
Glass3 26.2
Aluminium3 25.5
Polyethylene3 0.117
Rubber5 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 describes the material's response to linear stress (like pulling on the ends of a wire or 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 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.6

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 data78 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.9

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:

  1. the MTS shear modulus model developed by10 and used in conjunction with the Mechanical Threshold Stress (MTS) plastic flow stress model.1112
  2. the Steinberg-Cochran-Guinan (SCG) shear modulus model developed by13 and used in conjunction with the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model.
  3. the Nadal and LePoac (NP) shear modulus model8 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 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 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, kb is the Boltzmann constant, m is the atomic mass, and f is the Lindemann constant.

See also

References

  1. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "shear modulus, G".
  2. ^ 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. 
  3. ^ a b c d e Crandall, Dahl, Lardner (1959). An Introduction to the Mechanics of Solids. Boston: McGraw-Hill. ISBN 0-07-013441-3. 
  4. ^ Material properties
  5. ^ Spanos, Pete (2003). "Cure system effect on low temperature dynamic shear modulus of natural rubber". Rubber World. 
  6. ^ Shear modulus calculation of glasses
  7. ^ 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. 
  8. ^ a b 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. 
  9. ^ March, N. H., (1996), Electron Correlation in Molecules and Condensed Phases, Springer, ISBN 0-306-44844-0 p. 363
  10. ^ 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. 
  11. ^ 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. 
  12. ^ 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. 
  13. ^ 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. 
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{2 G \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