Shear stress
A force acting parallel to a surface of an immobilised object might cause shearing. The stress and strain in this case are called shear stress and shear strain, respectively. The corresponding elastic modulus is the shear modulus.
Shear strain
Shear strain is the ratio of the distance by which a surface of an object is displaced to the original value of the perpendicular dimension of the object. For an object shorn horizontally, this means the vertical size, that is, the height. Using the notations of the figure above,
\[\epsilon = \frac{\Delta x}{y}.\]
Shear strain, as a ratio of quantities having the same unit, is unitless.
Shear stress
Shear stress is the ratio of the force \(F\) causing the object to shear to the area \(A\) of the surface being displaced:
\[\tau := \frac{F}{A}.\]
The SI unit of shear stress is the newton per square metre, called the pascal:
\[\left[\tau\right] = 1\,\frac{\mathrm{N}}{\mathrm{m}^2} = 1\,\mathrm{Pa}.\]
Shear modulus
For relatively small shear strains, the shear stress is proportional to the shear strain. The constant of proportionality is called the shear modulus \(S\):
\[\tau = S \cdot \epsilon.\]
The SI unit of shear modulus is the newton per square metre, also called the pascal:
\[\left[S\right] = 1\,\frac{\mathrm{N}}{\mathrm{m}^2} = 1\,\mathrm{Pa}.\]