Shear modulus Formula

When a force is applied on a body which results in its lateral deformation, the elastic coefficient is called the shear modulus. It is the ratio of shear stress to shear strain in a body. Is written as as:

Shear modulus = (shear stress)/(strain) = (Force * no-stress length) / (Area of a section * change in the lateral length)

The equation is

G= = σ /ϵ = (F L) / (A Δx)

We have:

G: Shear modulus

σ : shear stress

ϵ : strain

F: Force applied

L: lateral length of the material without force applied

A: area of a section of the material

Δx: Change in the lateral length of the material after a force is applied

Shear modulus equation Questions:

1) Calculate the shear modulus of a body that experienced a stress of 5*10⁴ N/m² and a strain 4*10^(-2).

Answer: The shear modulus is calculated using the formula,

G= σ / ϵ

G = (5*10⁴ N/m²)/(4*10^(-2)) = 1.25 *10⁶ N/m²

G = 1.25 *10⁶ N/m²

2) A cylindrical bar of width 10 mm is stretched from its original length to 10 mm using a force of 100 N. What is the Shear modulus of the system?

Answer: The shear modulus is found from the equation:

G= (F L) / (A Δx)

The area is calculated using A=π (d/2)², where d is 10 mm, then A is

A= π (5 mm)² = π (0.005 m)² = 2.5*10^(-5) m²

substituting the value of area, the force and the final lateral length,

G= (100 N 0.1 m) / (2.5*10^(-5) m² * (0.01 m))

G= 10 N*m/0.025*10^(-5) m³ = 40000000 N/m² = 40000 KN/m²

G= 40000 KPa