§12.4 — Stress and Strain

A steel wire under 1000 N tension — does it collapse? No! A thin wire (just its own weight) sags more. Why? Because the tension is distributed over the cross-sectional area.

Stress

Stress is the internal force per unit area:

\[ \sigma = \frac{F}{A} \]

where:

• σ = stress (Pa or N/m²)
• F = applied force (N)
• A = cross-sectional area (m²)

Unit: Pascals (Pa) or Newtons per square meter (N/m²).

• 1 atm ≈ 10⁵ Pa
• Typical steel stress: 10⁶–10⁹ Pa

Three Types of Stress

1. Tensile Stress

Object is pulled:

\[ \sigma_t = \frac{F_{\text{pulling}}}{A} \]

Example: hanging rope, spring wire, bridge cable

2. Compressive Stress

Object is pushed:

\[ \sigma_c = \frac{F_{\text{pushing}}}{A} \]

Example: stone column, arch bridge

3. Shear Stress

Parallel forces on opposite surfaces:

\[ \tau_s = \frac{F_{\text{parallel}}}{A} \]

Example: knife cutting paper, car gears

Strain

Strain is the fractional change in dimension (or shape):

\[ \epsilon = \frac{\Delta L}{L_0} \]

where:

• ε = strain (dimensionless)
• ΔL = change in length (m)
• L₀ = original length (m)

Key point: strain is dimensionless. A strain of 0.01 means 1% change in length.

Three Types of Strain

1. Tensile Strain

Change in length:

\[ \epsilon_t = \frac{\Delta L}{L_0} \]

If ΔL > 0, object is stretched.

2. Compressive Strain

Change in length under compression:

\[ \epsilon_c = \frac{\Delta L}{L_0} < 0 \]

If ΔL < 0, object shortens.

3. Shear Strain

Change in shape (angle):

\[ \gamma = \tan\theta \approx \theta \]

where θ is the shear angle (radians).

Numerical Example

Steel rod:

• Original length: L₀ = 1 m
• Cross-sectional area: A = 0.01 m² (10 cm square)
• Tension: F = 10,000 N
• Length change: ΔL = 0.005 m = 5 mm

Stress: \[ \sigma = \frac{F}{A} = \frac{10,000}{0.01} = 1,000,000 \text{ Pa} = 1 \text{ MPa} \]

Strain: \[ \epsilon = \frac{\Delta L}{L_0} = \frac{0.005}{1} = 0.005 = 0.5\% \]

Stress-Strain Relationship

Key point: stress and strain are related — stress is the external cause (applied force), strain is the internal response (deformation).

For most materials, stress and strain are proportional (even if nonlinear overall):

• Larger stress → larger strain
• Smaller stress → smaller strain

This linear relationship (for small stresses) is Hooke's law, covered in the next section.

Why Stress and Strain Matter

Stress

Stress is the true internal force in material. If stress exceeds the material's limit, the material:

• Deforms permanently (plastically)
• or breaks (fractures)

Strain

Strain shows how much the material has deformed. Small strain is usually reversible (material bounces back), large strain is permanent (material doesn't recover).

Typical Material Values

Material	Breaking Stress	Breaking Strain
Steel	400–550 MPa	0.15–0.25
Copper	200–400 MPa	0.10–0.50
Aluminum	70–110 MPa	0.10–0.30
Wood	20–50 MPa	0.01–0.05
Granite	50–100 MPa	0.001–0.005

Special note: granite resists compression well but is weak in tension.

What You Should Know

• Stress: internal force per unit area σ = F/A
• Strain: fractional change in dimension ε = ΔL/L₀
• Three types: tensile, compressive, shear
• Strain is dimensionless: just a ratio
• Scales: typical stress 10⁶–10⁹ Pa

Preview of §12.5

Now we know how to define stress and strain. But what's the relationship between them? That's the material property: elastic modulus.

📚 See also: Halliday Vol 1, Ch 12, §12.4 — Stress and strain. 🔗 Reference: §12.3 (Equilibrium), §7 (Work and energy).

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