Stress & Strain – Quick, Friendly Notes
1 ‒ Stress: the “push-back” inside a solid
When you squeeze, pull, or twist a body that stays in static equilibrium, it builds a restoring force that tries to undo the deformation. The restoring force per unit area is called stress and is given by \( \sigma \;=\;\dfrac{F}{A} \) (Eq. 8.1).:contentReference[oaicite:0]{index=0} Its SI unit is pascal (N m-2) and its dimensional formula is [ ML-1T-2 ].:contentReference[oaicite:1]{index=1}
Flavours of stress
- Tensile stress – you pull the ends of a rod outward.:contentReference[oaicite:2]{index=2}
- Compressive stress – you push those ends inward. (Both kinds are also called longitudinal stress.):contentReference[oaicite:3]{index=3}
- Shearing (tangential) stress – you slide one face of a block parallel to the opposite face.:contentReference[oaicite:4]{index=4}
- Hydraulic stress – a fluid presses equally on every point of the surface.:contentReference[oaicite:5]{index=5}
2 ‒ Strain: how much the shape changes
2.1 Longitudinal strain
Pull a rod and its length changes by \( \Delta L \). Longitudinal strain is the fractional change in length: \( \displaystyle \frac{\Delta L}{L} \).:contentReference[oaicite:6]{index=6}
2.2 Shearing strain
Slide the top face of a block by \( \Delta x \) while the height stays \( L \). Shearing strain is \( \displaystyle \frac{\Delta x}{L} = \tan\theta \approx \theta \) (for small angles).:contentReference[oaicite:7]{index=7}
2.3 Volume strain
Under uniform fluid pressure, a solid’s volume drops by \( \Delta V \).
Volume strain is
\( \displaystyle \frac{\Delta V}{V} \).:contentReference[oaicite:8]{index=8}
All strains are pure numbers (no units).
3 ‒ Hooke’s Law: the elastic straight-line zone
For small deformations, stress is directly proportional to strain: \( \sigma \propto \text{strain} \) or \( \sigma = k\,(\text{strain}) \) (Eq. 8.6), where k is the modulus of elasticity.:contentReference[oaicite:9]{index=9} Most materials follow this linear rule only up to a point; beyond that they misbehave!
4 ‒ Making sense of the Stress–Strain curve
The standard tensile test produces a curve like the one sketched below (letters mark key stages).:contentReference[oaicite:10]{index=10}
- O → A: Elastic region – straight line obeying Hooke’s law; remove the load and the material snaps back perfectly.
- B: Yield point / Elastic limit – the last spot where full recovery is still possible. The corresponding stress is the yield strength \( \sigma_y \).:contentReference[oaicite:11]{index=11}
- A → D: Plastic region – strain shoots up with little extra stress; permanent set appears if you unload anywhere between B and D.:contentReference[oaicite:12]{index=12}
- D: Ultimate tensile strength \( \sigma_u \)
- E: Fracture – the sample breaks. If D and E are close, the material is brittle; if far apart, it is ductile.:contentReference[oaicite:13]{index=13}
High-yield NEET nuggets (learn these!)
- Remember the stress formula \( \sigma = \dfrac{F}{A} \) with its unit pascal.:contentReference[oaicite:14]{index=14}
- Hooke’s law governs the O → A linear segment; stress ∝ strain with modulus k.:contentReference[oaicite:15]{index=15}
- Yield point B and yield strength \( \sigma_y \) mark the elastic limit – a favourite exam keyword.:contentReference[oaicite:16]{index=16}
- Ultimate tensile strength \( \sigma_u \) (point D) and the difference between brittle vs ductile failure often appear in questions.:contentReference[oaicite:17]{index=17}
- Know the three strain formulas: \( \dfrac{\Delta L}{L} \) (longitudinal), \( \dfrac{\Delta x}{L} \) (shearing), and \( \dfrac{\Delta V}{V} \) (volume) – all unit-less.
