Shear modulus

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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]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): G \ \stackrel{\mathrm{def}}{=}\ \frac {\sigma} {\tau} = \frac{F/A}{\Delta x/h} = \frac{F h}{\Delta x A}

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \sigma = F/A \, = shear stress;
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): F is the force which acts
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): A is the area on which the force acts
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \tau = \Delta x/h \, = shear strain;
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \Delta x is the transverse displacement
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): h is the initial length

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

Material Typical values for
shear modulus (GPa)
(at room temperature)
Diamond[2] 478.
Steel[3] 79.3
Copper[3] 63.4
Titanium[3] 41.4
Glass[3] 26.2
Aluminium[3] 25.5
Polyethylene[3] 0.117
Rubber[4] 0.0006

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.

File:SpiderGraph ShearModulus.GIF
Influences of selected glass component additions on the shear modulus of a specific base glass.[5]

Sound waves

In homogeneous and isotropic solids, there are two kinds of sound waves, pressure waves and shear waves. The speed of sound for shear waves, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): v_s is controlled by the shear modulus,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): v_s = \sqrt{\frac {G} {\rho} }

where

G is the shear modulus
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \rho is the solid's density.

See also

References

  1. International Union of Pure and Applied Chemistry. "shear modulus, G". Compendium of Chemical Terminology Internet edition.
  2. Template:Cite Journal
  3. 3.0 3.1 3.2 3.3 3.4 3.5 Crandall, Dahl, Lardner (1959). An Introduction to the Mechanics of Solids. McGraw-Hill. 
  4. Template:Cite Journal
  5. Shear modulus calculation of glasses

Template:Elastic moduli

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