Shear modulus - Citizendia

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). Materials Science or Materials Engineering is an interdisciplinary field involving the properties of matter and its applications to various areas of Science and A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material Shear strain is a strain that acts parallel to the face of a material that it is acting on

Material	Typical values for shear modulus (G Pa) (at room temperature)
Diamond^[2]	478. For other meanings see Giga (disambiguation Giga- (symbol G is a prefix in the SI system of units denoting 109 In Mineralogy, diamond is the allotrope of carbon where the carbon atoms are arranged in
Steel^[3]	79. Steel is an Alloy consisting mostly of Iron, with a Carbon content between 0 3
Copper^[4]	44. Copper (ˈkɒpɚ is a Chemical element with the symbol Cu (cuprum and Atomic number 29 7
Titanium^[3]	41. Titanium (taɪˈteɪniəm is a Chemical element with the symbol Ti and Atomic number 22 4
Glass^[3]	26. Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many 2
Aluminium^[3]	25. WikipediaNaming 5
Polyethylene^[3]	0. Polyethylene or polythene ( IUPAC name poly(ethene) is a Thermoplastic commodity heavily used in consumer products (notably the 117
Rubber^[5]	0. 0006

1 Explanation
2 Waves
3 See also
4 References

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. In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the In Solid mechanics, Young's modulus (E is a measure of the Stiffness of an isotropic elastic material Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface

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. Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism Anisotropic materials such as wood and paper exhibit differing material response to stress or strain when tested in different directions. Anisotropy (pronounced with stress on the third syllable ˌænaɪˈsɒtrəpi is the property of being directionally dependent as opposed to Isotropy, which means homogeneity Wood is hard fibrous lignified structural tissue produced as secondary Xylem in the stems of Woody plants notably trees but also shrubs Paper is thin material mainly used for writing upon printing upon or packaging 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. Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many ^[6]

Waves

In homogeneous and isotropic solids, there are two kinds of waves, pressure waves and shear waves. Isotropy is uniformity in all directions Precise definitions depend on the subject area P-wave can also refer to a type of electronic wavefunction in atomic physics see Atomic orbital. S-wave can also refer to the lowest energy electronic wavefunction in atomic physics see Atomic orbital. 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. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different

References

^ International Union of Pure and Applied Chemistry. A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material Shear strain is a strain that acts parallel to the face of a material that it is acting on Shear strength in Engineering is a term used to describe the strength of a material or component against the type of yield or Structural failure where the Dynamic modulus is the ratio of stress to strain under vibratory conditions (calculated from data obtained from either free or forced vibration tests in shear compression In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the Practical The impulse excitation technique is a Nondestructive test method that uses Natural frequency, dimensions and mass of a test-piece to determine Young's The International Union of Pure and Applied Chemistry ( IUPAC) (aɪjuːpæk or ay-yoo-pec) is an international Non-governmental organization "shear modulus, G". Compendium of Chemical Terminology Internet edition. Compendium of Chemical Terminology (ISBN 0-86542-684-8 is a book published by IUPAC containing internationally accepted definitions for terms in Chemistry.
^ 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 digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ ^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

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.
	$(\lambda,\,\mu)$	$(E,\,\mu)$	$(K,\,\lambda)$	$(K,\,\mu)$	$(\lambda,\,\nu)$	$(\mu,\,\nu)$	$(E,\,\nu)$	$(K,\, \nu)$	$(K,\,E)$	$(M,\,\mu)$
$K=\,$	$\lambda+ \frac{2\mu}{3}$	$\frac{E\mu}{3(3\mu-E)}$			$\lambda\frac{1+\nu}{3\nu}$	$\frac{2\mu(1+\nu)}{3(1-2\nu)}$	$\frac{E}{3(1-2\nu)}$			$M - \frac{8\mu}{3}$
$E=\,$	$\mu\frac{3\lambda + 2\mu}{\lambda + \mu}$		$9K\frac{K-\lambda}{3K-\lambda}$	$\frac{9K\mu}{3K+\mu}$	$\frac{\lambda(1+\nu)(1-2\nu)}{\nu}$	$2\mu(1+\nu)\,$		$3K(1-2\nu)\,$		$\mu\frac{3M-4\mu}{M-\mu}$
$\lambda=\,$		$\mu\frac{E-2\mu}{3\mu-E}$		$K-\frac{2\mu}{3}$		$\frac{2 \mu \nu}{1-2\nu}$	$\frac{E\nu}{(1+\nu)(1-2\nu)}$	$\frac{3K\nu}{1+\nu}$	$\frac{3K(3K-E)}{9K-E}$	$M - 2\mu\,$
$\mu=\,$			$3\frac{K-\lambda}{2}$		$\lambda\frac{1-2\nu}{2\nu}$		$\frac{E}{2+2\nu}$	$3K\frac{1-2\nu}{2+2\nu}$	$\frac{3KE}{9K-E}$
$\nu=\,$	$\frac{\lambda}{2(\lambda + \mu)}$	$\frac{E}{2\mu}-1$	$\frac{\lambda}{3K-\lambda}$	$\frac{3K-2\mu}{2(3K+\mu)}$					$\frac{3K-E}{6K}$	$\frac{M - 2\mu}{2M - 3\mu}$
$M=\,$	$\lambda+2\mu\,$	$\mu\frac{4\mu-E}{3\mu-E}$	$3K-2\lambda\,$	$K+\frac{4\mu}{3}$	$\lambda \frac{1-\nu}{\nu}$	$\mu\frac{2-2\nu}{1-2\nu}$	$E\frac{1-\nu}{(1+\nu)(1-2\nu)}$	$3K\frac{1-\nu}{1+\nu}$	$3K\frac{3K+E}{9K-E}$

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Contents

Explanation

Waves

See also

References