Question

What are shearing stress and shearing strain? Define modulus of rigidity of a material and write its S.I. unit.

Accepted Answer

Shearing stress:

When a deforming force acts tangentially to the surface of a body, it produces a change in the shape of the body. The tangential force applied per unit area is equal to the tangential stress or shearing stress.

Shearing stress S.I. $= \frac{F}{A} = N / m^{2}$

When a deforming force is applied on a body along its surface in such a way that the volume of the body remains constant but the shape is changed, the body is said to be sheared and the strain produced in the body is called the shearing strain.

Within elastic limit, it is measured by the ratio of the relative displacement of one plane to its distance from the fixed plane.

Shearing strain $= \frac{A A'}{A B} \frac{a r c}{r a d i u s} = ϕ$

$ϕ = \frac{A A'}{A B}$ , $ϕ$ is very small so we can write $ϕ = \tan ϕ$

Shearing strain $= ϕ = \tan ϕ = \frac{A A'}{A B} = \frac{Relative displacement of two surfaces}{Distance \frac{b}{w} two surfaces}$

Thus the shearing strain is also define as the ratio of displacement of a surface under a tangent force to the perpendicular distance of the displaced surface

Modulus of rigidity or shear modulus $(η)$ within the elastic limit, the ratio of tangential stress to shear strain is called modulus of rigidity.

As shown in fig. consider a rectangular block whose lower face is fixed and a tengental force $F$ is applied over its upper face of area A. On equal and opposite force $F$ come into play on its lower fixed face. The two equal and opposite forces form a couple which exerts a torque. As the lower face of the block is fixed, the couple shears the block into a parallelepiped by displacing its upper face through distance $A A' = ∆ l$

Let,

$A B = D C = l$ and $∠ A B A' = θ$

Tangential stress $= \frac{F}{A}$

Shear strain $= ϕ = \tan ϕ = \frac{A A'}{A B} = \frac{∆ l}{l} . . . . . . . . . . . . . . . . . . . (i)$

The modulus of rigidity is given by $η = \frac{T e n g e n t i a l s t r e s s}{s h e a r s t r a i n} = \frac{\frac{F}{A}}{ϕ} = \frac{F}{A ϕ}$

From equation (i)

$η = \frac{F}{A \frac{∆ l}{l}} = \frac{F l}{A ∆ l}$

Units and dimension S.I. unit $= N / m^{2}$

Dimensional formula $= {ML}^{- 1} T^{- 2}$

G L Mittal and TARUN MITTAL Solutions for Chapter: Elasticity, Exercise 1: QUESTIONS

G L Mittal Physics Solutions for Exercise - G L Mittal and TARUN MITTAL Solutions for Chapter: Elasticity, Exercise 1: QUESTIONS

Questions from G L Mittal and TARUN MITTAL Solutions for Chapter: Elasticity, Exercise 1: QUESTIONS with Hints & Solutions

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