§12.6 — Elastic Limit & Breaking

All materials can be stretched, but how far? A limit exists beyond which material doesn't snap back. This is the elastic limit.

The Stress-Strain Curve

When we observe material behavior under stress, the graph of stress versus strain typically looks like this:

    Stress
       ↑
       |     Fracture
       |        ●
       |       /
       |      / Plastic region
       |     /
       |    / Yield point (elastic limit)
       |   ●────────
       |  /|        ╲
       | / |elastic   ╲ strain hardening
       |/  |region     ╲
       |___|____________╲___→ Strain
       0

1. Elastic Region

• Straight line (linear)
• Stress and strain proportional: σ = Eε
• If load is removed, material fully recovers
• Elastic limit at the end of this region

Example: steel under ordinary tension

2. Yield Point

The transition from linear to nonlinear behavior.

At this point: σ_yield (yield stress).

Beyond this:

• The curve bends
• Strain increases faster
• Behavior becomes nonlinear

3. Plastic Region

• The curve curves (nonlinear)
• Strain hardening: the more you stretch, the more the material resists
• If load is removed, material doesn't return to original shape
• Permanent deformation remains

Example: steel under extreme tension

4. Fracture Point

Stress reaches a point where atomic bonds break and material suddenly ruptures.

Fracture stress: σ_fracture.

Two Types of Failure: Ductile and Brittle

1. Ductile Materials

• Large plastic region (can be stretched far)
• Visible bending before fracture
• Permanent deformation, but some warning signs
• Gradual failure (can be squeezed or bent)

Examples: steel, copper, aluminum

2. Brittle Materials

• Small or no plastic region
• Little warning before fracture
• Fractures suddenly without much deformation
• Fracture is clean and abrupt

Examples: glass, stone, granite, diamond

Physical Difference

Ductile

Atomic bonds gradually slip and shift:

• High stress → atoms slide
• More sliding → deformation
• More stress → more sliding
• Bonds don't yet break

Brittle

Atomic bonds suddenly break:

• Low stress → intact
• High stress → suddenly fails
• Internal flaws → weak points
• Fracture initiates at weak point

Bending

When you bend a beam:

• Top surface is under tension (stretched)
• Bottom surface is under compression (squeezed)
• Center is neutral (no stress)

Bending Formula

Deflection of a beam:

\[ y = \frac{F L^3}{3 E I} \]

where:

• y = vertical deflection (meters)
• F = applied force
• L = beam length
• E = Young's modulus
• I = second moment of area (geometric)

Key point: deflection is inversely proportional to Young's modulus and the second moment.

Numerical Example

Steel beam:

• Length: L = 2 m
• Circular diameter: d = 0.1 m → I ≈ 4.91 × 10⁻⁴ m⁴
• Young's modulus: E = 200 GPa
• Force: F = 1000 N

\[ y = \frac{1000 \times 2^3}{3 \times 200 \times 10^9 \times 4.91 \times 10^{-4}} \]

\[ y ≈ 0.0068 \text{ m} = 6.8 \text{ mm} \]

The beam deflects about 7 mm.

Stress Concentration Factor

If a beam has a small flaw (like a tiny notch), the stress locally at that point is much higher.

Concentration factor:

\[ K_t = \frac{\sigma_{\text{max}}}{\sigma_{\text{nominal}}} \]

For a notch:

• K_t ≈ 2–5 (stress increases 2–5 times!)

Practical consequence: beam failure often starts at weak points, not at the average stress!

What You Should Know

• Stress-strain curve: elastic → yield → plastic → fracture
• Elastic limit: transition to nonlinear behavior
• Ductile: large plastic region; visible deformation before fracture
• Brittle: little plastic region; sudden fracture
• Bending: y ∝ F/E
• Stress concentration: flaws are weak points

Preview of §12.7

Now we have all the tools for equilibrium and elasticity. Let's solve real problems.

📚 See also: Halliday Vol 1, Ch 12, §12.6 — Elastic limit and breaking. 🔗 Reference: §12.4–12.5 (Stress, strain, moduli).

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