Shear Modulus Of Rigidity

What is Shear Modulus?

Shear Modulus of elasticity is one of the measures of mechanical properties of solids. Other elastic moduli are Young’s modulus and Bulk modulus. Shear modulus of a material gives us the ratio of shear stress to shear strain in a body. It can be used to explain how a material resists transverse deformations but this is practical for small deformations only, following which they are able to return to original state. This is because large shearing forces lead to permanent deformations (no longer elastic body).

shear modulus

Shear modulus of elasticity

Calculation of shear modulus:

Shear modulus of rigidity is the elastic coefficient when a shear force is applied resulting in lateral deformation. It gives us a measure of how rigid a body is. The relation is given as follows

G = \(\frac{shear stress}{shear strain}\)

where,

G is the Shear Modulus of a body expressed in \(N/m^{2}\)

The value of G for plywood is \( 6.2 \times 10^{8}\) and for steel it is \( 7.9 \times 10^{10}\) implying that steel is a lot more (really a lot more) rigid than plywood, around 127 times more!

Consider this. A block of unknown material, kept on a table, is under a shearing force. The following information is available to you. The square face is placed on the table.

Dimensions of the block = 60mm x 60mm x 20mm

Shearing Force = 0.245N

Displacement = 5mm

Solution: \(Shear~ Stress ~= ~ \frac{F}{A}~=~\frac{0.245}{60~\times~20~\times~10^{-6}}\)
=\(\frac{2450}{12}~=~204.167~N/m^2\)

\(Shear ~strain~= ~\frac{\Delta x}{L}~=~\frac{5}{60}~=~\frac{1}{12}\)

\(Shear~ modulus~,G~= ~\frac{Shear ~stress}{Shear~ strain}~= ~\frac{2450~\times~12}{12}~=~2.450~KN/m^2\)

The elastic moduli of a material, like Young’s Modulus, Bulk Modulus etc. are specific forms of Hooke’s law, which states that, for an elastic material, the strain experienced by the corresponding stress applied is proportional to that stress. They are related by the following equation:

\(2G(1+\upsilon)=E=3K(1-2\upsilon)\)

where,

\(G\) is the Shear Modulus

\(E\) is the Young’s Modulus

\(K\) is the Bulk Modulus

\(\upsilon\) is Poisson’s Ratio

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Practise This Question

The load versus elongation graph for four wires of the same material is shown in the figure. The thickest wire is represented by the line