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# Shear modulus

In materials science, shear modulus, G, or sometimes S or μ, sometimes referred to as the modulus of rigidity, is defined as the ratio of shear stress to the shear strain:

Material Typical values for
shear modulus (GPa)
(at room temperature)
Steel 79.3
Copper 63.4
Titanium 41.4
Glass 26.2
Aluminium 25.5
Polyethylene 0.117
Rubber 0.0003 $G \ \stackrel{\mathrm{def}}{=}\ \frac{F/A}{\Delta x/h} = \frac{F h}{\Delta x A}$

where

F / A = shear stress;
force F acts on area A;
Δx / h = shear strain;
with initial length h and transverse displacement Δx.

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

## 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. Anisotropic materials such as wood and paper exhibit differing material response to stress or strain when tested in different directions.

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.

In solids, there are two kinds of sound waves, pressure waves and shear waves. The speed of sound for shear waves is controlled by the shear modulus.

## References

1. ^ International Union of Pure and Applied Chemistry. "shear modulus, G". Compendium of Chemical Terminology Internet edition.
2. ^ Crandall, Dahl, Lardner (1959). An Introduction to the Mechanics of Solids. McGraw-Hill.
3. ^ 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. $(\lambda,\,\mu)$ $(E,\,\mu)$ $(K,\,\lambda)$ $(K,\,\mu)$ $(\lambda,\,\nu)$ $(\mu,\,\nu)$ $(E,\,\nu)$ $(K,\, \nu)$ $(K,\,E)$ $K=\,$ $\lambda+ \frac{2\mu}{3}$ $\frac{E\mu}{3(3\mu-E)}$ $\lambda\frac{1+\nu}{3\nu}$ $\frac{2\mu(1+\nu)}{3(1-2\nu)}$ $\frac{E}{3(1-2\nu)}$ $E=\,$ $\mu\frac{3\lambda + 2\mu}{\lambda + \mu}$ $9K\frac{K-\lambda}{3K-\lambda}$ $\frac{9K\mu}{3K+\mu}$ $\frac{\lambda(1+\nu)(1-2\nu)}{\nu}$ $2\mu(1+\nu)\,$ $3K(1-2\nu)\,$ $\lambda=\,$ $\mu\frac{E-2\mu}{3\mu-E}$ $K-\frac{2\mu}{3}$ $\frac{2 \mu \nu}{1-2\nu}$ $\frac{E\nu}{(1+\nu)(1-2\nu)}$ $\frac{3K\nu}{1+\nu}$ $\frac{3K(3K-E)}{9K-E}$ $\mu=\,$ $3\frac{K-\lambda}{2}$ $\lambda\frac{1-2\nu}{2\nu}$ $\frac{E}{2+2\nu}$ $3K\frac{1-2\nu}{2+2\nu}$ $\frac{3KE}{9K-E}$ $\nu=\,$ $\frac{\lambda}{2(\lambda + \mu)}$ $\frac{E}{2\mu}-1$ $\frac{\lambda}{3K-\lambda}$ $\frac{3K-2\mu}{2(3K+\mu)}$ $\frac{3K-E}{6K}$ $M=\,$ $\lambda+2\mu\,$ $\mu\frac{4\mu-E}{3\mu-E}$ $3K-2\lambda\,$ $K+\frac{4\mu}{3}$ $\lambda \frac{1-\nu}{\nu}$ $\mu\frac{2-2\nu}{1-2\nu}$ $E\frac{1-\nu}{(1+\nu)(1-2\nu)}$ $3K\frac{1-\nu}{1+\nu}$ $3K\frac{3K+E}{9K-E}$