Shear Modulus

The ratio of shearing stress to the corresponding shearing strain is called the shear modulus of the material and is represented by G. It is also called the modulus of rigidity.

G = shearing stress (σs )/shearing strain 

G = (F/A)/(∆x/L) 

G = (F × L)/(A × ∆x)                           (1) 

Similarly, from Eq. (tan θ ≈ θ) 

G = (F/A)/θ 

G = F/(A × θ)                                    (2) 

The shearing stress σ$_s$ can also be expressed as 

σ$_s$ = G × θ                                 (3) 

SI unit of shear modulus is N m$^{–2}$ or Pa. 

The shear moduli of a few common materials are given in Table 1. It can be seen that shear modulus (or modulus of rigidity) is generally less than Young’s modulus (from Table 9.1). For most materials G ≈ Y/3.

Example 1

A square lead slab of side 50 cm and thickness 10 cm is subject to a shearing force (on its narrow face) of 9.0 × 10$^4$  N. The lower edge is riveted to the floor. How much will the upper edge be displaced?

Answer 

The lead slab is fixed and the force is applied parallel to the narrow face as shown in Fig. 1. The area of the face parallel to which this force is applied is

A = 50 cm × 10 cm 

 = 0.5 m × 0.1 m 

 = 0.05 m$^2$

Therefore, the stress applied is 

 = (9.4 × 104 N/0.05 m$^2$) 

 = 1.80 × 10$^6$  N.m$^{–2}$

We know that shearing strain = (∆x/L)= Stress /G. 

Therefore the displacement 

∆x = (Stress × L)/G 

 = (1.8 × 10$^6$  N.m$^{–2}$ × 0.5m)/(5.6 × 10$^9$  N.m$^{–2}$) 

 = 1.6 × 10$^{–2}$ m = 0.16 mm