Hooke’s Law & the Stress–Strain Story
1 · Stress, Strain, and Volume Changes
When a fluid squeezes an object from all sides, the object pushes back with an internal restoring force per unit area called hydraulic stress. This hydraulic stress has the same numerical value as the fluid’s pressure. :contentReference[oaicite:0]{index=0}
The fluid also changes the object’s volume. We describe that change with volume strain:
\( \text{Volume strain} \;=\; \frac{\Delta V}{V} \) (Eq. 8.5) :contentReference[oaicite:1]{index=1}
Because it is a ratio, volume strain has no units. :contentReference[oaicite:2]{index=2}
2 · Hooke’s Law—Elasticity Made Simple
For small stretches or compressions, stress and strain stay in step:
\( \text{stress} \;\propto\; \text{strain} \quad\Longrightarrow\quad \text{stress} \;=\; k \times \text{strain} \) (Eq. 8.6) :contentReference[oaicite:3]{index=3}
Here k is the modulus of elasticity—a material’s personal stiffness constant. Experiments confirm this rule for most common solids, though a few quirky materials refuse to follow the line. :contentReference[oaicite:4]{index=4}
3 · Tensile Test & the Famous Stress–Strain Curve
In the lab, engineers hang a wire or rod, pull it harder and harder, and watch how its length changes. Plotting stress (vertical axis) against strain (horizontal axis) produces the classic curve shown in textbooks. :contentReference[oaicite:5]{index=5}
Key Stops Along the Curve
- O → A Elastic & Linear. Stress and strain rise together in a straight line. Remove the load and the sample snaps back to its original size. Hooke’s law rules here. :contentReference[oaicite:6]{index=6}
- A → B Elastic but Non-linear. The straight line bends, yet the material still springs back. Point B is the yield point / elastic limit; the stress at B is the yield strength \( \sigma_y \). :contentReference[oaicite:7]{index=7}
- B → D Plastic Region. Stress just beyond \( \sigma_y \) produces huge extra strain. Even if you ease off at some point C between B and D, the sample keeps its new, longer length—a permanent set. :contentReference[oaicite:8]{index=8}
- D Ultimate Tensile Strength. Point D marks the highest stress the material can withstand: \( \sigma_u \). :contentReference[oaicite:9]{index=9}
- D → E Necking & Fracture. After D, even a smaller load stretches the sample until it breaks at E. If D and E lie close together, the material is brittle; if far apart, it is ductile. :contentReference[oaicite:10]{index=10}
4 · Quick-Reference Definitions
- Hydraulic stress: restoring force per unit area inside an object submerged in a fluid; numerically equal to the fluid’s pressure. :contentReference[oaicite:11]{index=11}
- Volume strain: \( \Delta V / V \); dimensionless. :contentReference[oaicite:12]{index=12}
- Modulus of elasticity k: slope of the stress–strain line in the elastic region. :contentReference[oaicite:13]{index=13}
- Yield strength \( \sigma_y \): stress at the yield point where permanent deformation begins. :contentReference[oaicite:14]{index=14}
- Ultimate tensile strength \( \sigma_u \): maximum stress before necking. :contentReference[oaicite:15]{index=15}
5 · High-Yield NEET Takeaways
- For small deformations, remember \( \text{stress} = k \times \text{strain} \); spotting this relationship in graphs is an easy mark.
- Locate the yield point B on a stress–strain curve; questions often ask for the corresponding yield strength \( \sigma_y \).
- The highest point on the curve is the ultimate tensile strength \( \sigma_u \); it signals the start of necking.
- Volume strain \( \Delta V / V \) carries no units—an ideal quick check for dimensional-analysis items.
- If D and E nearly coincide, the sample is brittle; if they’re well separated, it’s ductile. Expect conceptual questions on this distinction.
You’ve got this—keep practicing those curves and equations, and the elastic region of your brain will stretch just fine!
