Within the limit of elasticity, the stress is proportional to the strain.

Stress & infi ; Strain

or Stress = E * Strain

where, E is the modulus of elasticity of the material of the body.

Types of Modulus of Elasticity

1. Young’s Modulus of Elasticity

It is defined as the ratio of normal stress to the longitudinal strain Within the elastic limit.

y = Normal stress / Longitudinal strain

y = FΔl / Al = Mg Δl / πr² l

Its unit is N/m2 or Pascal and its dimensional formula is [ML-1T-2].

2. Bulk Modulus of Elasticity

It is defined as the ratio of normal stress to the volumetric strain within the elastic limit.

K = Normal stress / Volumetric strain

K = FV / A ΔV = &Delta ; p V / Δ V

where, Δp = F / A = Change in pressure.

Its unit is N/m2

or Pascal and its dimensional formula is [ML-1T-2].

3. Modulus of Rigidity (η)

It is defined as the ratio of tangential stress to the shearing strain, within the elastic limit.

η = Tangential stress / Shearing strain

Its unit is N/m2 or Pascal and its dimensional formula is [ML^-1 T^-2].

Compressibility

Compressibility of a material is the reciprocal of its bulk modulus of elasticity.

Compressibility (C) = 1 / k

Its SI unit is N^-1 m² and CGS unit is dyne^-1 cm².

Steel is more elastic than rubber. Solids are more elastic and gases are least elastic.

For liquids. Modulus of rigidity is zero.

Young’s modulus (Y) and modulus of rigidity (η) are possessed by solid materials only.

Limit of Elasticity

The maximum value of deforming force for which elasticity is present in the body is called its limit of elasticity.

Breaking Stress

The minimum value of stress required to break a wire, is called breaking stress. Breaking stress is fixed for a material but breaking force varies with area of cross-section of the wire.

Safety factor = Breaking stress / Working stress

Elastic Relaxation Time

The time delay in restoring the original configuration after removal of deforming force is called elastic relaxation time.

For quartz and phosphor bronze this time is negligible.

Elastic After Effect

The temporary delay in regaining the original configuration by the elastic body after the removal of deforming force, is called elastic after effect.

Elastic Fatigue

The property of an elastic body by virtue of which its behaviour becomes less elastic under the action of repeated alternating deforming force is called elastic fatigue.

Ductile Materials

The materials which show large plastic range beyond elastic limit are called ductile materials, e.g., copper, silver, iron, aluminum, etc.

Ductile materials are used for making springs and sheets.

Brittle Materials

The materials which show very small plastic range beyond elastic limit are called brittle materials, e.g., glass, cast iron, etc.

Elastomers

The materials for which strain produced is much larger than the stress applied, with in the limit of elasticity are called elastomers, e.g., rubber, the elastic tissue of aorta, the large vessel carrying blood from heart. etc.

Elastomers have no plastic range.

Elastic Potential Energy in a Stretched Wire

The work done in stretching a wire is stored in form of potential energy of the wire.

Potential energy U = Average force * Increase in length

= 1 / 2 FΔl = 1 / 2 Stress * Strain * Volume of the wire

Elastic potential energy per unit volume

U = 1 / 2 * Stress * Strain

= 1 / 2 (Young’s modulus) * (Strain) ²

Elastic potential energy of a stretched spring = 1 / 2 kx²

Where, k = Force constant of spring and x = Change in length.

Thermal Stress

When temperature of a rod fixed at its both ends is changed, then the produced stress is called thermal stress.

Thermal stress = F / A = yαΔθ

Where, α = coefficient of linear expansion of the material of the rod.

When temperature of a gas enclosed in a vessel is changed, then the thermal stress produced is equal to change in pressure (Δp) of the gas.

Thermal stress = Δ p = Ky Δ θ

Where, K = bulk modulus of elasticity and

γ = coefficient of cubical expansion of the gas.

Interatomic force constant

K = Yr₀

Where, r₀ = interatomic distance.

Poisson’s Ratio

When a deforming force is applied at the free end of a suspended wire of length 1 and radius R, then its length increases by dl but its radius decreases by dR . Now two types of strains are produced by a single force.

• Longitudinal strain = & Delta ;U l
• Lateral strain = – Δ R/ R

• Poisson’s Ratio (σ) = Lateral strain / Longitudinal strain = – Δ R/ R / ΔU l

The theoretical value of Poisson’s ratio lies between – 1 and 0.5. Its practical value lies between 0 and 0.5.

Relation between Y, K, η and σ

(1) Y = 3K (1 – 2σ)_s

(2) Y = 2 η (1 + σ)

(3) σ = 3K – 2η / 2η + 6K

(4) 9 / Y = 1 / K + 3 / η or Y = 9K η / η + 3K

Important Points

• Coefficient of elasticity depends upon the material, its temperature and purity but not on stress or strain.
• For the same material, the three coefficients of elasticity γ, η and K have different magnitudes.
• Isothermal elasticity of a gas ET = ρ where, ρ = pressure of the gas.
• Adiabatic elasticity of a gas E_s = γρ

Where, γ = C_p / C_c ratio of specific heats at constant pressure and at constant volume.

• Ratio between isothermal elasticity and adiabatic elasticity E_S/ E_T= γ = C_P / C_V

Cantilever

A beam clamped at one end and loaded at free end is called a cantilever.

Depression at the free end of a cantilever is given by

δ = wl³ / 3YIG

where, w = load, 1 = length of the cantilever,

y = Young’s modulus of elasticity, and IG = geometrical moment of inertia.

For a beam of rectangular cross-section having breadth b and thickness d.

IG = bd³ / 12

For a beam of circular cross-section area having radius r,