Elastic Moduli — Fast & Friendly Notes

When you stretch, squeeze, or twist something, its shape or size changes. Inside the elastic limit, the change is proportional to the applied force. That neat proportionality lets us define three elastic moduli that characterise every material.

1. Young’s Modulus (Y) — stretching or squeezing along the length

  • Definition: \( Y = \dfrac{\sigma}{\varepsilon} \) where \( \sigma \) is tensile / compressive stress and \( \varepsilon \) is longitudinal strain (8.7) :contentReference[oaicite:0]{index=0}
  • Alternate form: \( Y = \dfrac{F/A}{\Delta L/L} = \dfrac{F\,L}{A\,\Delta L} \) (8.8) :contentReference[oaicite:1]{index=1}
  • Unit: pascal (Pa) — same as stress.
  • Typical values: steel ≈ \(2.0\times10^{11}\,\text{Pa}\); copper ≈ \(1.1\times10^{11}\,\text{Pa}\); bone ≈ \(9.4\times10^{9}\,\text{Pa}\)
  • Why it matters: a 0.1 cm² steel wire needs ≈ 2000 N to stretch just 0.1 %. Aluminium, brass, and copper need far less force, so steel feels “stiffer.” :contentReference[oaicite:3]{index=3}
  • Worked idea (Example 8.1): A 1 m steel rod (radius 10 mm) under 100 kN shows a stress of \(3.18\times10^{8}\,\text{Pa}\), elongates 1.59 mm, and gains a strain of \(1.59\times10^{-3}\) (≈ 0.16 %). :contentReference[oaicite:4]{index=4}

2. Shear Modulus (G) — sliding layers sideways

  • Definition: \( G = \dfrac{\text{shearing stress}}{\text{shearing strain}} = \dfrac{F/A}{\Delta x/L} = \dfrac{F\,L}{A\,\Delta x} \) (8.10) and \( G = \dfrac{F}{A\,\theta} \) (8.11) :contentReference[oaicite:5]{index=5}
  • Shearing stress also obeys \( \sigma_s = G\,\theta \). :contentReference[oaicite:6]{index=6}
  • Unit: pascal.
  • Rule of thumb: for many solids, \( G \approx \tfrac{1}{3}Y \). :contentReference[oaicite:7]{index=7}
  • Snapshot of values:
    • Steel 84 GPa
    • Aluminium 25 GPa
    • Lead 5.6 GPa
    • Tungsten 150 GPa
    :contentReference[oaicite:8]{index=8}
  • Worked idea (Example 8.4): A 50 cm × 10 cm lead slab under a 9 × 104 N sideways push slides only 0.16 mm — tiny but measurable. :contentReference[oaicite:9]{index=9}

3. Bulk Modulus (B) — uniform squeezing from all sides

  • Definition: \( B = -\dfrac{p}{\Delta V/V} \) (8.12); negative sign reminds us that pressure increase shrinks volume. :contentReference[oaicite:10]{index=10}
  • Compressibility \( k = \dfrac{1}{B} \) tells how easy it is to squeeze something. :contentReference[oaicite:11]{index=11}
  • Unit: pascal.
  • Hierarchy of stiffness: solids ≫ liquids ≫ gases. Air is ~a million times more compressible than solids. :contentReference[oaicite:12]{index=12}
  • Typical values:
    • Steel 160 GPa
    • Copper 140 GPa
    • Water 2.2 GPa
    • Air 1 × 10−4 GPa
    :contentReference[oaicite:13]{index=13}
  • Worked idea (Example 8.5): At the 3000 m-deep ocean floor, water is compressed by ~1.36 %. :contentReference[oaicite:14]{index=14}

4. Poisson’s Ratio (ν)

  • Lateral strain is the side-kick contraction/expansion that happens when you stretch/compress a rod.
  • Poisson’s ratio: \( \nu = \dfrac{\text{lateral strain}}{\text{longitudinal strain}} \). Pure number, no unit. :contentReference[oaicite:15]{index=15}
  • Typical range: steel ≈ 0.28 – 0.30, aluminium alloys ≈ 0.33. :contentReference[oaicite:16]{index=16}

5. Elastic Potential Energy in a Stretched Wire

Pulling a wire stores energy:

\[ u \;=\; \tfrac12\,\sigma\,\varepsilon \]

That’s half the product of stress and strain per unit volume (8.14) :contentReference[oaicite:17]{index=17}

6. Big-Picture Comparison (At a Glance)

  • Stretching deals with \( Y \) (Young’s modulus).
  • Sliding deals with \( G \) (shear or rigidity modulus).
  • Squeezing everywhere deals with \( B \) (bulk modulus).
  • An engineer often checks all three to ensure safety and minimal deformation. :contentReference[oaicite:18]{index=18}

7. Everyday Applications

  • Steel beams, cables, and machine parts rely on high \( Y \) for minimal stretch.
  • I-shaped girders reduce weight while keeping bending stiffness high.
  • Bone compression in acrobatic “human pyramids” is tiny thanks to bone’s respectable \( Y \). :contentReference[oaicite:19]{index=19}
  • Sand dunes and hills settle into shapes that balance shear and gravitational stresses. :contentReference[oaicite:20]{index=20}

High-Yield Ideas for NEET

  1. The three defining equations: \( Y=\dfrac{\sigma}{\varepsilon},\; G=\dfrac{\text{shearing stress}}{\text{shearing strain}},\; B=-\dfrac{p}{\Delta V/V} \). Remember units and when to use each.
  2. Relation \( G\approx Y/3 \) for many solids — a quick shortcut for MCQs. :contentReference[oaicite:22]{index=22}
  3. Poisson’s ratio values and the idea that it links side-ways squeeze to lengthwise stretch. :contentReference[oaicite:23]{index=23}
  4. Energy stored in a stretched wire: \( u=\tfrac12\sigma\varepsilon \) — watch for questions on work done in stretching. :contentReference[oaicite:24]{index=24}
  5. Order of compressibility: solids < liquids < gases (and the huge difference!). :contentReference[oaicite:25]{index=25}