# Shear modulus

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]

${\displaystyle G\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\sigma }{\tau }}={\frac {F/A}{\Delta x/h}}={\frac {Fh}{\Delta xA}}}$

where

${\displaystyle \sigma =F/A\,}$ = shear stress;
${\displaystyle F}$ is the force which acts
${\displaystyle A}$ is the area on which the force acts
${\displaystyle \tau =\Delta x/h\,}$ = shear strain;
${\displaystyle \Delta x}$ is the transverse displacement
${\displaystyle h}$ 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, ${\displaystyle v_{s}}$ is controlled by the shear modulus,

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

where

G is the shear modulus
${\displaystyle \rho }$ is the solid's density.

2. Crandall, Dahl, Lardner (1959). An Introduction to the Mechanics of Solids. McGraw-Hill. Unknown parameter |city= ignored (help)