Shear modulus

From Wikipedia, the free encyclopedia

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]

$G \ \stackrel{\mathrm{def}}{=}\ \frac {\sigma_{xy}} {\epsilon_{xy}} = \frac{F/A}{\Delta x/h} = \frac{F h}{\Delta x A}$

where

$\sigma_{xy} = F/A \,$ = shear stress;

F

is the force which acts

A

is the area on which the force acts

$\epsilon_{xy} = \Delta x/h = \tan \theta \,$ = shear strain;

Δ x

is the transverse displacement

h

is the initial length (labelled

I

in the diagram opposite)

Shear modulus is usually measured in GPa (gigapascals) or ksi (thousands of pounds per square inch).

Material	Typical values for shear modulus (G Pa) (at room temperature)
Diamond^[2]	478.
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

1 Explanation
2 Waves
3 See also
4 References

[edit] Explanation

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

Young's modulus describes the material's response to linear strain (like pulling on the ends of a wire),
the bulk modulus describes the material's response to uniform pressure, and
the shear modulus describes the material's response to shearing strains.

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 and paper exhibit differing material response to stress or strain when tested in different directions. In this case, when the deformation is small enough so that the deformation is linear, the elastic moduli, including the shear modulus, will then be a tensor, rather than a single scalar value.

Influences of selected glass component additions on the shear modulus of a specific base glass.^[6]

[edit] Waves

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

ρ

is the solid's density.

[edit] See also

[edit] References

^ International Union of Pure and Applied Chemistry. "shear modulus, G". Compendium of Chemical Terminology Internet edition.
^ McSkimin, H.J.; Andreatch, P. (1972). "Elastic Moduli of Diamond as a Function of Pressure and Temperature". J. Appl. Phys. 43: 2944-2948. doi:10.1063/1.1661636.
^ ^a ^b ^c ^d ^e Crandall, Dahl, Lardner (1959). An Introduction to the Mechanics of Solids. McGraw-Hill.
^ Material properties
^ Spanos, Pete (November 2003). "Cure system effect on low temperature dynamic shear modulus of natural rubber". Rubber World.
^ Shear modulus calculation of glasses

v • d • e Elastic moduli for homogeneous isotropic materials

Bulk modulus ( $K$ ) • Young's modulus ( $E$ ) • Lamé's first parameter ( $λ$ ) • Shear modulus ( $μ$ ) • Poisson's ratio ( $ν$ ) • 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.
	$(\lambda,\,\mu)$	$(E,\,\mu)$	$(K,\,\lambda)$	$(K,\,\mu)$	$(\lambda,\,\nu)$	$(\mu,\,\nu)$	$(E,\,\nu)$	$(K,\, \nu)$	$(K,\,E)$	$(M,\,\mu)$
$K=\,$	$\lambda+ \frac{2\mu}{3}$	$\frac{E\mu}{3(3\mu-E)}$			$\lambda\frac{1+\nu}{3\nu}$	$\frac{2\mu(1+\nu)}{3(1-2\nu)}$	$\frac{E}{3(1-2\nu)}$			$M - \frac{8\mu}{3}$
$E=\,$	$\mu\frac{3\lambda + 2\mu}{\lambda + \mu}$		$9K\frac{K-\lambda}{3K-\lambda}$	$\frac{9K\mu}{3K+\mu}$	$\frac{\lambda(1+\nu)(1-2\nu)}{\nu}$	$2\mu(1+\nu)\,$		$3K(1-2\nu)\,$		$\mu\frac{3M-4\mu}{M-\mu}$
$\lambda=\,$		$\mu\frac{E-2\mu}{3\mu-E}$		$K-\frac{2\mu}{3}$		$\frac{2 \mu \nu}{1-2\nu}$	$\frac{E\nu}{(1+\nu)(1-2\nu)}$	$\frac{3K\nu}{1+\nu}$	$\frac{3K(3K-E)}{9K-E}$	$M - 2\mu\,$
$\mu=\,$			$3\frac{K-\lambda}{2}$		$\lambda\frac{1-2\nu}{2\nu}$		$\frac{E}{2+2\nu}$	$3K\frac{1-2\nu}{2+2\nu}$	$\frac{3KE}{9K-E}$
$\nu=\,$	$\frac{\lambda}{2(\lambda + \mu)}$	$\frac{E}{2\mu}-1$	$\frac{\lambda}{3K-\lambda}$	$\frac{3K-2\mu}{2(3K+\mu)}$					$\frac{3K-E}{6K}$	$\frac{M - 2\mu}{2M - 3\mu}$
$M=\,$	$\lambda+2\mu\,$	$\mu\frac{4\mu-E}{3\mu-E}$	$3K-2\lambda\,$	$K+\frac{4\mu}{3}$	$\lambda \frac{1-\nu}{\nu}$	$\mu\frac{2-2\nu}{1-2\nu}$	$E\frac{1-\nu}{(1+\nu)(1-2\nu)}$	$3K\frac{1-\nu}{1+\nu}$	$3K\frac{3K+E}{9K-E}$