Elastic Behaviour of Materials
1. How Materials Compress and Stretch
- Solids hardly compress because neighbouring atoms are tightly linked.
- Liquids compress a little more; the atomic links are looser than in solids.
- Gases compress the most—about a million times more than solids—because molecules barely interact.
2. Worked Example – Water at Ocean Depth
At a depth of 3 000 m in the Indian Ocean, pressure squeezes water slightly.
Use the bulk modulus \(B = 2.2 \times 10^{9}\,\text{N m}^{-2}\)
The pressure from the water column is \(p = h\rho g = 3 \times 10^{7}\,\text{N m}^{-2}\).
The fractional volume change is
\[ \frac{\Delta V}{V} = \frac{\text{stress}}{B} = \frac{3 \times 10^{7}}{2.2 \times 10^{9}} = 1.36 \times 10^{-2}\;(\text{about }1.36\%). \]
3. Poisson’s Ratio \((\nu)\)
When you stretch a wire, it gets thinner sideways. The ratio of sideways (lateral) strain to length-wise (longitudinal) strain is
\[ \nu = \frac{(\Delta d / d)}{(\Delta L / L)} = \frac{\Delta d}{\Delta L}\frac{L}{d}. \]
- Typical steel: \(\nu \approx 0.28 – 0.30\).
- Aluminium alloys: \(\nu \approx 0.33\).
4. Energy Stored in a Stretched Wire
Pulling a wire stores elastic potential energy:
\[ u = \frac{1}{2}\sigma \varepsilon, \]
where \(u\) is energy per unit volume, \(\sigma\) is stress, and \(\varepsilon\) is strain.
5. Everyday Engineering Applications
5.1 Crane Ropes
To lift a 10-tonne load safely you choose a rope cross-section area
\[ A \ge \frac{W}{\sigma_y} = \frac{Mg}{\sigma_y}, \]
with yield strength \(\sigma_y \approx 300 \times 10^{6}\,\text{N m}^{-2}\) for mild steel. This gives \(A \approx 3.3 \times 10^{-4}\,\text{m}^2\) — roughly a 1 cm radius. Engineers add a safety factor (~10×), so real ropes use bundles of thinner wires adding up to about a 3 cm radius.
5.2 Beams in Bridges and Buildings
A beam of length \(l\), breadth \(b\), and depth \(d\) that carries a central load \(W\) sags by
\[ \delta = \frac{W l^{3}}{4 b d^{3} Y}. \]
- Choose a material with a high Young’s modulus \(Y\) (it bends less).
- Increasing depth d is far more effective than increasing breadth b.
- Deep, thin beams can buckle, so engineers use an I-section: lots of depth for stiffness, little weight in the middle.
5.3 Pillars and Columns
Pillars carry heavier loads when their ends spread the force over a wider area instead of meeting the ground in a sharp curve. Distributed ends reduce stress concentrations and the risk of failure.
5.4 Why Mountains Aren’t Taller than ~10 km
Rocks flow when shearing stress exceeds about \(30 \times 10^{7}\,\text{N m}^{-2}\). Setting \(h\rho g = 30 \times 10^{7}\) with rock density \(\rho = 3 \times 10^{3}\,\text{kg m}^{-3}\) gives
\[ h \approx 10\,\text{km}, \]
close to the height of Mount Everest.
6. Quick-Reference Equations
- Bulk compression: \(\displaystyle \frac{\Delta V}{V} = \frac{\text{stress}}{B}\)
- Poisson’s ratio: \(\displaystyle \nu = \frac{(\Delta d / d)}{(\Delta L / L)}\)
- Energy density in a wire: \(\displaystyle u = \frac{1}{2}\sigma \varepsilon\)
- Minimum rope area: \(\displaystyle A \ge \dfrac{Mg}{\sigma_y}\)
- Beam sag: \(\displaystyle \delta = \dfrac{W l^{3}}{4 b d^{3} Y}\)
7. High-Yield Ideas for NEET
- The energy density formula \(u = \tfrac{1}{2}\sigma \varepsilon\) connects stress, strain, and stored energy.
- Poisson’s ratio (\(\nu\)) values and its definition often appear in conceptual questions.
- Minimum cross-section area \(A \ge Mg/\sigma_y\) for safe crane ropes tests practical use of yield strength.
- Beam sag equation \(\delta = W l^{3} / (4 b d^{3} Y)\) highlights how depth controls bending.
- Bulk compression relation \(\Delta V / V = \text{stress} / B\) is a shortcut for quick percentage change problems.
Keep practising these relationships — they pop up everywhere from solving exam questions to understanding real-world structures!
