Elastic Modulus—

The ratio of stress and strain is known as Elastic Modulus.

Elastic Modulus (E) = Stress (σ) / Strain (ϵ)

 

The constant of proportionality depends on the material being deformed and the nature of the deformation. This constant is called the elastic modulus.

The elastic modulus determines the amount of force required per unit deformation.

Ø  A material with large elastic modulus is difficult to deform.

Ø  With small elastic modulus is easier to deform.

 

HOOKE’S LAW—

Within the elastic limit, Stress- Strain ratio is constant. [Stress/Strain = Constant]


“In an elastic member stress (σ) is directly proportional to the strain (ϵ) within elastic Limit.”

σ ∝ ϵ

σ = E. ϵ

or, E = σ / ϵ

Where,

σ = Stress

ϵ = Strain

E = Constant (modulus of Elasticity / Young’s Modulus)

ϵ à
 
                          


Initially, a stress-strain curve is a straight line. As the stress increases, the curve is no longer straight. When the stress exceeds the elastic limit, the object is permanently distorted and does not return to its original shape after the stress is removed. Hence, the shape of the object is permanently changed. As the stress is increased even further, the material ultimately breaks.

 

Types of Elastic Modulus

1.      Young’s Modulus (y)

2.      Bulk Modulus (B)

3.      Rigidity Modulus / Shear modulus (η)

 

Young’s Modulus (y)

Within the elastic limit, the ratio of longitudinal stress (tensile/ compressive) and longitudinal strain (tensile/ Compressive) is called young’s modulus of elasticity (y).

 

Young’s modulus (y) = Longitudinal stress / Longitudinal strain

y = σ / ϵ

y = (F/ A)/(ΔL/L) = [FL/AΔL]

 

Bulk Modulus (B)

Within elastic limit the ratio of the volume stress (F/A) and the volume strain (ΔV/V) is called bulk modulus of elasticity.

 

B = (σvol / ϵvol) = (F/A) / (-ΔV/V) = ΔP / (-ΔV/V)

B = = ΔP / (-ΔV/V)

Where,

ΔP = Pressure applied

 

Rigidity Modulus / Shear modulus (η)

Within elastic limit, the ratio of shearing stress (or) tangential stress and shearing strain (or) tangential strain is called modulus of rigidity.

 

η = Shearing stress / shearing strain

η = τ / ϕ

Where,

η = shear modulus

τ = shear force (F/A)

ϕ = angle of shear

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