Course description

2.2 Review of elastic properties of solids:

Elasticity: It is a property of a matter for which a material body able to regain its initial state of condition on removal of the external forces that applied on it.

Perfectly elastic body: The body that can regain its original state completely on the removal of the force (Figs 7 a, b).

Perfectly plastic body: The body that shows no tendency to regain its original condition on the removal of the deforming force (Fig. 7 c).

(a)                                                                    (b)                                                         (c)

                                                    Fig. 7

Load: Load = External force + Own weight along the force (Fig. 8).

                          (a)

(b)

                                                   Fig. 8

Stress:

In the elastic limit,

i) Stress = force applied/ area = F/A (if F is applied uniformly)

When the deforming force F be inclined to the surface then: (Fig. 9 a)

ii) Tangential  (shearing stress) = Fcosφ/A  (Fig. 9 b).

iii) Normal stress  (tensile stress) = Fsinφ/A  (Fig. 9 b).

Strain: Under a stress a body undergoes a deformation in respect of length or volume or shape. The change in the dimension is described by the quantity strain.

 i) Longitudinal or tensile strain = change in length/ original length = δl/L (Fig. 10 a).

ii) Volume strain = change in volume/ original volume = δv/V (Fig. 10 b).

iii) Shearing strain, θ = tanθ (Fig. 10 c).

(a)                                       (b)                                             (c)

                                                               Fig. 10

Hooke’s law (fundamental law of elasticity):

Within the elastic limit the stress is proportional to strain (Fig. 11). Therefore, stress/strain = constant (E). This constant is called modulus of elasticity.

Fig. 11

Stress-strain diagram of a material:

OA: The wire is perfectly elastic.

AB: Stress and strain are not proportional. But OB region still exhibits elastic behavior.

Beyond B: From D the wire does not come back to its original length at O but come to the position at C.

Fig. 12

Three types of elasticity:

      (a)                                         (b)                                                    (c)

                                                                                Fig. 13

If tan θ = θ = l/L then µ =(F/A)/(l/L)=FL/AI  same as the Young’s modulus. But the difference that, in the case of modulus of rigidity, F is the tangential stress not a linear one, and the displacement l take place at right angles to L, not along it.

Poisson’s ratio: The idea is that on being stretched, a wire becomes longer but thinner.

From Fig. 14:

σ =(∆D/D)/(∆l/l)

Lateral and linear (or tangential) strains per unit stress are denoted by β and α respectively then σ = β/α.    

Fig. 14

Problem 1: A wire of 2 meters long and 0.5 mm in diameter and a mass of 10 kg is suspended. It is stretched by 2.33 mm. Find out the value of the Young’s modulus of the wire.

Soln.: Let  be the length of the wire that can hang vertically without breaking.

If A is the area of cross-section and ρ is the density of the material of the wire,

then the mass of the wire = volume × density =  A ρ

Breaking stress = F/A = mg/A =  1A ρ g /A =  1  ρ g =  7.9 × 〖10〗^8

or,  

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