Elasticity: Stress, Strain, Hooke’s Law, Young’s Modulus, Applications

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## What is Elasticity?
Elasticity is the property of a material to **return to its original shape and size** after the deforming force is removed. Think of a rubber band: when you stretch it and release, it snaps back. But if you stretch it too much, it may break or stay deformed—that’s called **plastic deformation**.

### Key Terms:
- **Elastic Body**: Returns to original shape after force is removed (e.g., rubber, steel springs).
- **Plastic Body**: Does not return to original shape (e.g., clay, putty).
- **Elastic Limit**: Maximum force beyond which a material does not return to its original shape.



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## Stress and Strain
When a force acts on an object, it causes **deformation**. We measure this deformation using two quantities:

### 1. **Stress (σ)**
Stress is the **internal force per unit area** acting inside a material. It is caused by external forces like tension, compression, or shear.
**Formula**:
\[
\sigma = \frac{F}{A}
\]
- \(F\) = Applied force (Newton, N)
- \(A\) = Cross-sectional area (m²)
- **Unit**: Pascal (Pa) or N/m²

**Types of Stress**:
```mermaid
graph TD
    A[Stress] --> B[Tensile Stress]
    A --> C[Compressive Stress]
    A --> D[Shear Stress]
    B -->|Pulling apart| E[Example: Stretching a wire]
    C -->|Pushing together| F[Example: Crushing a can]
    D -->|Sliding forces| G[Example: Cutting with scissors]

2. Strain (ε)

Strain measures how much an object deforms compared to its original size. It is a dimensionless quantity (no units). Formula: [ \epsilon = \frac{\Delta L}{L_0} ]

• (\Delta L) = Change in length (m)
• (L_0) = Original length (m)

Types of Strain:

graph TD
    A["Strain"] --> B["Longitudinal Strain"]
    A --> C["Shear Strain"]
    B -->|"Length change"| D["Example: Stretching a spring"]
    C -->|"Angle change"| E["Example: Twisting a rod"]

Hooke’s Law

Robert Hooke discovered that within the elastic limit, stress is directly proportional to strain: [ \sigma \propto \epsilon \quad \text{or} \quad \sigma = Y \cdot \epsilon ] where (Y) is the Young’s modulus (a property of the material).

Worked Example 1: Calculating Stress and Strain

A steel wire of length 2 m and cross-sectional area (10^{-6}) m² is stretched by 0.5 mm under a force of 500 N. Calculate:

1. Stress on the wire.
2. Strain produced.
3. Young’s modulus of steel.

Solution:

1. Stress: [ \sigma = \frac{F}{A} = \frac{500}{10^{-6}} = 5 \times 10^8 , \text{Pa} ]
2. Strain: [ \epsilon = \frac{\Delta L}{L_0} = \frac{0.5 \times 10^{-3}}{2} = 2.5 \times 10^{-4} ]
3. Young’s Modulus: [ Y = \frac{\sigma}{\epsilon} = \frac{5 \times 10^8}{2.5 \times 10^{-4}} = 2 \times 10^{11} , \text{Pa} ]

Young’s Modulus (Y)

Young’s modulus is a measure of stiffness of a material. It tells us how much stress is needed to produce a given strain.

• High Y: Stiff material (e.g., steel, diamond).
• Low Y: Flexible material (e.g., rubber, rubber bands).

Stress-Strain Graph

The relationship between stress and strain is shown in a stress-strain graph. Here’s what it tells us:

Key Points from the Graph:

1. Elastic Region (O to A): Stress ∝ Strain (Hooke’s law applies).
2. Elastic Limit (A): Beyond this, permanent deformation occurs.
3. Yield Point (B): Material starts to deform plastically.
4. Ultimate Tensile Strength (C): Maximum stress before breaking.
5. Fracture Point (D): Material breaks.

Applications of Elasticity

Elasticity is used in many real-life applications:

Solved Problems (NEB Style)

Problem 1:

A copper wire of length 2.5 m and radius 1 mm is stretched by 1 mm when a force of 100 N is applied. Calculate:

1. Stress in the wire.
2. Strain produced.
3. Young’s modulus of copper.

Solution:

1. Area of wire: $$A=\pi r^2=\pi (1\times 10^{-3}{)}^2=\pi \times 10^{-6}{\text{m}}^2$$ Stress: $$\sigma =\frac{F}{A}=\frac{100}{\pi \times 10^{-6}}=\frac{100\times 10^6}{\pi }\approx 3.18\times 10^7\text{Pa}$$
2. Strain: $$ϵ=\frac{\Delta L}{L_0}=\frac{1\times 10^{-3}}{2.5}=4\times 10^{-4}$$
3. Young’s Modulus: $$Y=\frac{\sigma }{ϵ}=\frac{3.18\times 10^7}{4\times 10^{-4}}=7.95\times 10^{10}\text{Pa}$$

Problem 2:

Define:

1. Elasticity.
2. Plasticity.
3. Elastic Limit.

Solution:

1. Elasticity: The property of a material to return to its original shape and size after the deforming force is removed.
2. Plasticity: The property of a material to retain its deformed shape after the force is removed (permanent deformation).
3. Elastic Limit: The maximum stress that a material can withstand without permanent deformation.

Exam Tip

1. Memorize Formulas:

   • Stress ($\sigma =\frac{F}{A}$)
   • Strain ($ϵ=\frac{\Delta L}{L_0}$)
   • Young’s Modulus ($Y=\frac{\sigma }{ϵ}$)

2. Understand Graphs:

   • Know the regions (elastic, plastic, fracture) in a stress-strain graph.
   • Identify elastic limit, yield point, and ultimate tensile strength.

3. Unit Conversions:

   • Always convert units to SI units (N, m, Pa) before calculations.

4. Real-World Applications:

   • Relate concepts to everyday objects (springs, rubber bands, bridges).

5. Numerical Problems:

   • Practice calculating stress, strain, and Young’s modulus.
   • Watch out for significant figures and unit consistency.

A typical stress-strain graph showing elastic and plastic regions. (Image: Nicoguaro, CC BY 4.0, via Wikimedia Commons)