§12.5 — Elastic Moduli

Why is steel stiffer than rubber? Because under the same stress, rubber deforms more. This difference in material property is captured by the elastic modulus.

General Definition

The elastic modulus is the ratio of stress to strain:

\[ E = \frac{\text{Stress}}{\text{Strain}} = \frac{\sigma}{\epsilon} \]

Unit: Pa or N/m² (same as stress).

Key point: the elastic modulus is a material property, not specific to any object. All steel has the same Young's modulus.

Three Types of Elastic Modulus

1. Young's Modulus

For tensile or compressive stress:

\[ Y = \frac{\sigma}{\epsilon} = \frac{F/A}{\Delta L / L_0} = \frac{F L_0}{A \Delta L} \]

Unit: Pa (typically: GPa = 10⁹ Pa).

Physical meaning: how stiff the material is under tension or compression. Large Young's modulus = stiff material.

Typical values:

• Steel: 200–210 GPa (very stiff)
• Copper: 120–130 GPa
• Wood: 10–15 GPa
• Rubber: 0.1–1 GPa (very soft)

2. Shear Modulus

For shear stress:

\[ G = \frac{\tau_s}{\gamma} = \frac{F/A}{\Delta x / L} \]

where Δx is displacement and L is height.

Meaning: stiffness of material under shear. Typically smaller than Young's modulus.

Examples:

• Steel: 80–85 GPa
• Copper: 40–50 GPa

3. Bulk Modulus

For hydrostatic pressure (compression from all sides):

\[ K = -\frac{\Delta P}{\Delta V / V_0} \]

where:

• ΔP = pressure change
• ΔV = volume change
• V₀ = original volume

Meaning: resistance to compression. Bulk modulus applies to liquids and gases too.

Examples:

• Steel: 160–170 GPa
• Water: 2.2 GPa (more compressible than steel!)

Hooke's Law

For small stresses, stress and strain are linear:

\[ \sigma = E \epsilon \]

or:

\[ F = k \Delta L \]

where k = EA/L₀ is the spring constant.

Sensitive point: Hooke's law holds only for small stresses. Beyond that, the stress-strain curve becomes nonlinear.

Numerical Example: Steel Rod

Given:

• Original length: L₀ = 1 m
• Diameter: d = 10 mm = 0.01 m
• Cross-sectional area: A = π(d/2)² = π × (0.005)² ≈ 7.85 × 10⁻⁵ m²
• Applied tension: F = 10,000 N
• Young's modulus for steel: Y = 200 GPa = 200 × 10⁹ Pa

Calculation:

\[ \sigma = \frac{F}{A} = \frac{10,000}{7.85 \times 10^{-5}} ≈ 127 \times 10^6 \text{ Pa} = 127 \text{ MPa} \]

\[ \epsilon = \frac{\sigma}{Y} = \frac{127 \times 10^6}{200 \times 10^9} ≈ 6.35 \times 10^{-4} = 0.0635\% \]

\[ \Delta L = \epsilon \times L_0 = 6.35 \times 10^{-4} \times 1 = 0.635 \text{ mm} \]

Result: the steel rod stretches about 0.6 mm. Remarkably small!

Elastic Energy

When an object deforms under stress, energy is stored:

\[ U = \frac{1}{2} k (\Delta L)^2 = \frac{1}{2} \frac{EA}{L_0} (\Delta L)^2 \]

Or in terms of stress and strain:

\[ u = \frac{1}{2} \sigma \epsilon = \frac{1}{2} E \epsilon^2 \]

where u is the energy density (energy per unit volume).

Limitations of Hooke's Law

Hooke's law (σ = Eε) holds only in the elastic region:

1. Elastic region: σ and ε are linear; if load is removed, object returns to original shape
2. Plastic region: stress increases further, but strain grows faster; if load is removed, permanent deformation remains
3. Fracture point: if stress is too high, material breaks

Comparison of Materials

Material	Young's Modulus	Stiffness	Ductility
Diamond	1,200 GPa	very stiff	very brittle
Steel	200 GPa	stiff	somewhat ductile
Copper	120 GPa	moderate	ductile
Wood	12 GPa	soft	ductile
Rubber	1 GPa	very soft	very elastic

Special note: large Young's modulus = stiffer, but not necessarily more elastic. Rubber is soft but extremely elastic.

What You Should Know

• Elastic modulus: ratio of stress to strain E = σ/ε
• Three types: Young's (tension/compression), shear, bulk (hydrostatic)
• Hooke's law: σ = Eε (valid for small stresses only)
• Elastic energy: u = (1/2)Eε²
• Regimes: elastic → plastic → fracture

Preview of §12.6

Now we know how material behaves under stress. But in reality, when does equilibrium break down? When material deforms plastically or fractures. Let's solve real problems of equilibrium and elasticity.

📚 See also: Halliday Vol 1, Ch 12, §12.5 — Elastic moduli. 🔗 Reference: §12.4 (Stress and strain), §7 (Elastic energy).

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