Unit 4: Elasticity

1. Hooke's Law and Stress-Strain [cite: 1356-1357]

Elasticity is the property of a material to regain its original shape and size after the removal of deforming forces.

Stress

Stress is the internal restoring force (F) per unit area (A) developed inside a deformed body. It is equal in magnitude and opposite in direction to the applied deforming force.

Formula: Stress = Force / Area (Units: N/m² or Pascal)

Strain

Strain is the measure of deformation. It is the fractional change in dimension (length, volume, or shape).

Formula: Strain = (Change in dimension) / (Original dimension)

Strain is a dimensionless quantity.

Hooke's Law

Hooke's Law states that, within the elastic limit, the stress applied to a body is directly proportional to the strain produced.

Hooke's Law: Stress ∝ Strain

Formula: Stress = E * Strain

The constant of proportionality 'E' is called the Modulus of Elasticity. Its value depends on the material and the type of deformation.

Stress-Strain Diagram

A graph of stress vs. strain for a material (like a metal wire) shows its behavior:

1. Proportional Limit (A): The point up to which Stress ∝ Strain (Hooke's Law is obeyed).
2. Elastic Limit (B): The maximum stress a material can withstand and still return to its original shape when the load is removed.
3. Yield Point (C): The point at which the material begins to deform permanently (plastic deformation) with little or no increase in stress.
4. Ultimate Tensile Strength (D): The maximum stress the material can withstand before starting to neck (thin) and break.
5. Fracture Point (E): The point where the material breaks.

2. Elastic Moduli

There are three main types of elastic moduli, corresponding to three types of strain:

1. Young's Modulus (Y)

This relates to longitudinal (length) deformation.

Formula: Y = (Longitudinal Stress) / (Longitudinal Strain) = (F/A) / (ΔL/L)

Use: Describes the "stiffness" of a material against being stretched or compressed.

2. Bulk Modulus (K)

This relates to volume deformation (e.g., an object under pressure in a fluid).

Formula: K = (Volume Stress) / (Volume Strain) = (-ΔP) / (ΔV/V)

The negative sign is because an increase in pressure (ΔP) leads to a decrease in volume (ΔV).

3. Shear Modulus or Modulus of Rigidity (η or G)

This relates to shear (shape) deformation. A tangential force is applied, causing layers to slide.

Formula: η = (Shear Stress) / (Shear Strain) = (F/A) / (θ)

Where θ is the angle of shear (in radians).

3. Poisson's Ratio (σ)

When you stretch a wire (longitudinal strain), it gets thinner (lateral strain).

Poisson's Ratio (σ) is the ratio of the lateral strain to the longitudinal strain, within the elastic limit.

Let original length = L, change in length = ΔL. Longitudinal Strain = ΔL/L
Let original diameter = D, change in diameter = ΔD. Lateral Strain = ΔD/D

Formula: σ = - (Lateral Strain) / (Longitudinal Strain) = - (ΔD/D) / (ΔL/L)

The negative sign is included because lateral and longitudinal strains always have opposite signs (if L increases, D decreases). This makes σ a positive number.

4. Relation between Elastic Constants

The four elastic constants (Y, K, η, σ) are not independent. For a homogeneous, isotropic material, knowing any two allows you to find the other two. The key relations are:

1. Y = 2η(1 + σ)
2. Y = 3K(1 - 2σ)
3. Y = (9Kη) / (3K + η)
4. σ = (3K - 2η) / (6K + 2η)

5. Work Done in Stretching/Twisting

Work Done in Stretching a Wire

When stretching a wire of length L by an amount ΔL, the force is not constant; it increases from 0 to F = kΔL (where k = YA/L).

Work Done (dW) = Force(l) * dl = (k * l) dl

Total Work W = ∫₀^ΔL (k l) dl = (1/2)k(ΔL)²

Since F = kΔL, we can write W = (1/2) * (kΔL) * ΔL

Work Done: W = (1/2) * (Final Force) * (Total Extension)

The Energy Stored per Unit Volume (u) is:

u = W / Volume = [ (1/2) * F * ΔL ] / (A * L) = (1/2) * (F/A) * (ΔL/L)

Energy Density: u = (1/2) * Stress * Strain

Work Done in Twisting a Wire

Similarly, the torque (τ) required to twist a wire by an angle (θ) is proportional to the angle: τ = Cθ, where C is the torsional constant.

Work Done W = ∫₀^θ (Cθ) dθ = (1/2)Cθ²

Work Done: W = (1/2) * (Final Torque) * (Total Angle of Twist)

6. Torsion and Torsional Pendulum

Twisting Couple (Torque) on a Cylinder

For a solid cylinder (or wire) of length L, radius r, and modulus of rigidity η, the torque (τ) required to twist its free end by an angle θ (in radians) is:

Formula (Twisting Couple): τ = (πηr⁴θ) / (2L)

The term C = (πηr⁴) / (2L) is the torsional constant (torque per unit twist) of the wire.

Torsional Pendulum

A torsional pendulum consists of a rigid body (like a disc) suspended by a wire. When twisted and released, it oscillates with angular simple harmonic motion.

The restoring torque is τ = -Cθ (Hooke's Law for twisting).
From Newton's second law for rotation, τ = Iα = I (d²θ/dt²).
I (d²θ/dt²) = -Cθ => d²θ/dt² + (C/I)θ = 0

This is the equation for SHM with angular frequency ω² = C/I.

The time period (T) of the oscillation is:

Formula: T = 2π / ω = 2π √(I / C)

Where I is the moment of inertia of the suspended body and C is the torsional constant of the wire.

7. Bending of Beams and Cantilever

Bending of Beams

When a beam is bent, the "outer" surface is stretched (in tension) and the "inner" surface is compressed. Somewhere in between is the neutral axis, which is neither stretched nor compressed.

The bending moment (M) of the applied forces (the total torque causing the bend) is related to the beam's properties and its radius of curvature (R).

Bending Moment Formula: M = (YI_g) / R

Where:

• Y = Young's Modulus of the beam material.
• R = Radius of curvature of the neutral axis.
• I_g = Geometric Moment of Inertia (or "second moment of area") of the beam's cross-section. This measures how the area is distributed.
  • For a rectangle (breadth b, depth d): I_g = (bd³) / 12
  • For a circle (radius r): I_g = (πr⁴) / 4

The term (YI_g) is called the flexural rigidity of the beam.

Cantilever

A cantilever is a beam that is fixed (clamped) at one end and free at the other.

If a load (W) is applied to the free end, the beam will bend, causing a depression (δ) at that end.

By integrating the bending moment equation, we can find the depression:

Formula (Cantilever Depression): δ = (WL³) / (3YI_g)

Where L is the length of the cantilever. This formula is widely used in engineering and for experimentally determining Young's Modulus (Y).