Shear deformation occurs when a force acts parallel to a surface, causing one layer of material to slide relative
to another. The material’s resistance to this sliding deformation is measured by the shear modulus, also called the
modulus of rigidity.
1. Shear stress
Shear stress is tangential force per unit sheared area:
\[
\tau=\frac{F}{A}.
\]
Here, \(F\) is the tangential force and \(A\) is the area over which the force acts. The SI unit is the pascal:
\[
1\ \mathrm{Pa}=1\ \mathrm{N/m^2}.
\]
2. Shear strain
For a rectangular block, shear strain is the ratio of lateral displacement to separation distance:
\[
\gamma=\frac{\Delta x}{L}.
\]
Shear strain is dimensionless. For small deformations, it is approximately equal to the shear angle in radians.
3. Shear modulus
In the linear elastic region, shear stress is proportional to shear strain:
\[
\tau=G\gamma.
\]
Therefore,
\[
G=\frac{\tau}{\gamma}.
\]
For a block under shear,
\[
G=\frac{F/A}{\Delta x/L}.
\]
\[
\boxed{G=\frac{F L}{A\Delta x}}.
\]
4. Rectangular block shear
If the sheared face is rectangular with width \(w\) and depth \(d\), then
\[
A=w d.
\]
The shear stress is
\[
\tau=\frac{F}{w d}.
\]
The shear strain is
\[
\gamma=\frac{\Delta x}{L}.
\]
5. Torsion of circular shafts
A circular shaft under torque experiences shear stress and shear strain that increase with radius. The maximum
values occur at the outer surface.
For a shaft under torque \(T\), the maximum shear stress is
\[
\tau_{\max}=\frac{T c}{J},
\]
where \(c\) is the outer radius and \(J\) is the polar second moment of area.
For a solid circular shaft,
\[
J=\frac{\pi R^4}{2}.
\]
For a hollow circular shaft,
\[
J=\frac{\pi(R^4-R_i^4)}{2}.
\]
6. Angle of twist
In elastic torsion, the angle of twist is
\[
\phi=\frac{T L}{J G}.
\]
Rearranging gives the shear modulus:
\[
\boxed{G=\frac{T L}{J\phi}}.
\]
The maximum shear strain at the outer radius is
\[
\gamma_{\max}=\frac{c\phi}{L}.
\]
7. Solving design problems
The same formulas can be rearranged depending on the unknown.
Shear force for a block:
\[
F=G A\frac{\Delta x}{L}.
\]
Lateral displacement for a block:
\[
\Delta x=\frac{F L}{A G}.
\]
Required area for a block:
\[
A=\frac{F L}{G\Delta x}.
\]
Torque for a shaft:
\[
T=\frac{GJ\phi}{L}.
\]
Angle of twist for a shaft:
\[
\phi=\frac{T L}{JG}.
\]
Required polar moment:
\[
J=\frac{T L}{G\phi}.
\]
8. Shear stress–strain graph
In the linear elastic region, the shear stress–strain graph is a straight line:
\[
\tau=G\gamma.
\]
The slope of this graph is the shear modulus:
\[
G=\frac{\Delta\tau}{\Delta\gamma}.
\]
A steeper slope means the material is more rigid in shear.
9. Shear elastic limit and safety factor
The linear formula applies only while the material remains elastic. A simple utilization estimate is
\[
\text{utilization}=\frac{|\tau|}{\tau_{\mathrm{limit}}}.
\]
The corresponding safety factor is
\[
\text{safety factor}=\frac{\tau_{\mathrm{limit}}}{|\tau|}.
\]
If utilization exceeds \(1\), the shear stress is above the entered shear elastic-limit estimate.
10. Typical shear modulus values
| Material |
Approximate shear modulus |
Meaning |
| Steel |
\(79\ \mathrm{GPa}\) |
High rigidity in shear |
| Aluminium |
\(26\ \mathrm{GPa}\) |
Lower shear rigidity than steel |
| Copper |
\(44\ \mathrm{GPa}\) |
Moderate shear rigidity |
| Brass |
\(39\ \mathrm{GPa}\) |
Common engineering alloy |
| Rubber-like materials |
\(\sim 0.0003\ \mathrm{GPa}\) |
Very low shear rigidity and often nonlinear |
11. Worked example
A rectangular block has face dimensions
\[
w=0.20\ \mathrm{m},
\qquad
d=0.15\ \mathrm{m}.
\]
The sheared area is
\[
A=w d=(0.20)(0.15)=0.030\ \mathrm{m^2}.
\]
If a tangential force of \(8000\ \mathrm{N}\) is applied, then
\[
\tau=\frac{F}{A}
=
\frac{8000}{0.030}
=
2.67\times10^5\ \mathrm{Pa}.
\]
\[
\tau=0.267\ \mathrm{MPa}.
\]
If the separation distance is \(L=0.15\ \mathrm{m}\) and the lateral displacement is
\(\Delta x=0.8\ \mu\mathrm{m}\), then
\[
\gamma=\frac{\Delta x}{L}
=
\frac{0.8\times10^{-6}}{0.15}
=
5.33\times10^{-6}.
\]
The shear modulus is
\[
G=\frac{\tau}{\gamma}
=
\frac{2.67\times10^5}{5.33\times10^{-6}}
=
5.0\times10^{10}\ \mathrm{Pa}.
\]
\[
G=50\ \mathrm{GPa}.
\]
Key idea: shear stress measures tangential load intensity, shear strain measures angular/sliding deformation, and
shear modulus measures rigidity against shear deformation.
