thumb|right|Shear strain
In solid mechanics, the shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to shear strain:
<math display=block>\begin{align}
G &:= \frac {\tau_{xy {\gamma_{xy = \frac{\frac{F}{A{\frac{\Delta x}{l = \frac{F l}{A \Delta x}\\
\tau_{xy} &= \frac{F}{A} = \mathrm{shear\ stress}\\
F &= \mathrm{force}\\
A &= \mathrm{area}\\
\gamma_{xy} &= \frac{\Delta x}{l} = \mathrm{shear\ strain}\\
\Delta x &= \mathrm{transverse\ displacement}\\
l &= \mathrm{initial\ length\ or\ height}
\end{align}</math>
The derived SI unit of shear modulus is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousand pounds per square inch (ksi). Its dimensional form is M<sup>1</sup>L<sup>−1</sup>T<sup>−2</sup>, replacing force by mass times acceleration.
Explanation
{| class="wikitable" align=right
!Material
!Typical values for <br>shear modulus (GPa)<br> <small>(at room temperature)</small>
|-
|Diamond, (111)
|478.0
|-
|Diamond, SC (100)
|443
|79.3
|-
|Iron
|52.5
|-
|Copper
|44.7
|-
|Titanium
|0.0006
|-
|Granite
|24
|-
|Shale
<math display=block> E = 2G(1+\nu) = 3K(1-2\nu)</math>
The shear modulus is concerned with the deformation of a solid when it experiences a force perpendicular to one of its surfaces while its opposite face experiences an opposing force (such as friction). In the case of an object shaped like a rectangular prism, it will deform into a parallelepiped. Anisotropic materials such as wood, paper and also essentially all single crystals exhibit differing material response to stress or strain when tested in different directions. In this case, one may need to use the full tensor-expression of the elastic constants, rather than a single scalar value.
One possible definition of a fluid would be a material with zero shear modulus.
Shear waves
thumb|upright=1.5|Influences of selected glass component additions on the shear modulus of a specific base glass.
In homogeneous and isotropic solids, there are two kinds of waves, pressure waves and shear waves. The velocity of a shear wave, <math>(v_s)</math> is controlled by the shear modulus,
:<math>v_s = \sqrt{\frac {G} {\rho} }</math>
where
:G is the shear modulus
:<math>\rho</math> is the solid's density.
Shear modulus of metals
[[File:CuShearMTS.svg|thumb|upright=1.2|Shear modulus of copper as a function of temperature. The experimental data
Several models exist that attempt to predict the shear modulus of metals (and possibly that of alloys). Shear modulus models that have been used in plastic flow computations include:
- the Varshni-Chen-Gray model developed by and used in conjunction with the Mechanical Threshold Stress (MTS) plastic flow stress model.
- the Steinberg-Cochran-Guinan (SCG) shear modulus model developed by and used in conjunction with the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model.
- the Nadal and LePoac (NP) shear modulus model that uses Lindemann theory to determine the temperature dependence and the SCG model for pressure dependence of the shear modulus.
Varshni-Chen-Gray model
The Varshni-Chen-Gray model (sometimes referred to as the Varshni equation) has the form:
:<math>
\mu(T) = \mu_0 - \frac{D}{\exp(T_0/T) - 1}
</math>
where <math> \mu_0 </math> is the shear modulus at <math> T=0K </math>, and <math>D</math> and <math> T_0 </math> are material constants.
SCG model
The Steinberg-Cochran-Guinan (SCG) shear modulus model is pressure dependent and has the form
:<math>
\mu(p,T) = \mu_0 + \frac{\partial \mu}{\partial p} \frac{p}{\eta^\frac{1}{3 +
\frac{\partial \mu}{\partial T}(T - 300) ; \quad
\eta := \frac{\rho}{\rho_0}
</math>
where, μ<sub>0</sub> is the shear modulus at the reference state (T = 300 K, p = 0, η = 1), p is the pressure, and T is the temperature.
NP model
The Nadal-Le Poac (NP) shear modulus model is a modified version of the SCG model. The empirical temperature dependence of the shear modulus in the SCG model is replaced with an equation based on Lindemann melting theory. The NP shear modulus model has the form:
:<math>
\mu(p,T) = \frac{1}{\mathcal{J}\left(\hat{T}\right)}
\left[
\left(\mu_0 + \frac{\partial \mu}{\partial p} \frac{p}{\eta^\frac{1}{3 \right)
\left(1 - \hat{T}\right) + \frac{\rho}{Cm}~T
\right]; \quad
C := \frac{\left(6\pi^2\right)^\frac{2}{3{3} f^2
</math>
where
:<math>
\mathcal{J}(\hat{T}) := 1 + \exp\left[-\frac{1 + 1/\zeta}
{1 + \zeta/\left(1 - \hat{T}\right)}\right] \quad
\text{for} \quad \hat{T} := \frac{T}{T_m}\in[0, 6+ \zeta],
</math>
and μ<sub>0</sub> is the shear modulus at absolute zero and ambient pressure, ζ is an area, m is the atomic mass, and f is the Lindemann constant.
Shear relaxation modulus
The shear relaxation modulus <math>G(t)</math> is the time-dependent generalization of the shear modulus <math>G</math>:
:<math>G=\lim_{t\to \infty} G(t)</math>.
See also
- Elasticity tensor
- Dynamic modulus
- Impulse excitation technique
- Shear strength
- Seismic moment