Stress, Strain, and Young's Modulus

$\boxed{\sigma = \frac{F}{A}}$

• $\sigma$ = tensile stress (Pa or N/m²)
• $F$ = force applied (N)
• $A$ = cross-sectional area (m²)

Stress tells us the force per unit area — it's independent of the size of the sample.

$\boxed{\varepsilon = \frac{\Delta L}{L}}$

• $\varepsilon$ = strain (no units — it's a ratio)
• $\Delta L$ = extension (m)
• $L$ = original length (m)

Strain tells us the fractional change in length.

$\boxed{E = \frac{\sigma}{\varepsilon} = \frac{F/A}{\Delta L/L} = \frac{FL}{A\Delta L}}$

• $E$ = Young's modulus (Pa)
• Measures the stiffness of a material
• High $E$ → stiff (hard to stretch) e.g., steel ($E \approx 200$ GPa)
• Low $E$ → flexible e.g., rubber ($E \approx 0.01$ GPa)

Typical Values

Key Features for a Ductile Material (e.g., copper)

1. Linear region (O to P): Obeys Hooke's Law ($\sigma \propto \varepsilon$). Gradient = Young's modulus.
2. Limit of proportionality (P): Beyond this, stress and strain are no longer proportional.
3. Elastic limit (E): Beyond this, the material won't return to original length when unloaded — permanent deformation begins.
4. Yield point (Y): Material suddenly extends significantly with little extra stress.
5. Plastic region: Material deforms permanently.
6. Ultimate tensile stress (UTS): Maximum stress the material can withstand.
7. Fracture point: Material breaks.

Elastic vs Plastic Deformation

Elastic deformation: Material returns to original shape when force is removed. Energy is fully recovered. Bonds stretch but don't break.

Plastic deformation: Material is permanently deformed. Energy is NOT fully recovered (some converted to heat). Atomic planes slide past each other.

$F = k\Delta L$

Where $k$ = spring constant (N/m).

Valid up to the limit of proportionality.

Energy Stored in a Stretched Wire/Spring

$E = \frac{1}{2}F\Delta L = \frac{1}{2}k(\Delta L)^2$

On a force-extension graph, energy stored = area under the curve (up to the loading line).

Energy from Stress-Strain Graph

Energy stored per unit volume: $\frac{E}{V} = \frac{1}{2}\sigma\varepsilon$

(Area under the stress-strain graph)

Brittle Materials (e.g., glass, ceramics)

• Break suddenly with little or no plastic deformation
• Stress-strain graph is almost straight until fracture
• No significant yield point

Ductile Materials (e.g., copper, mild steel)

• Show significant plastic deformation before breaking
• Can be drawn into wires
• Clear yield point

Polymers (e.g., rubber, polythene)

• Very large strains at low stress
• Rubber: elastic but non-linear (Hooke's Law doesn't apply)
• Loading and unloading curves differ → hysteresis (energy lost as heat)

Method

1. Clamp a long, thin wire (e.g., copper) horizontally or vertically
2. Measure original length $L$ with a tape measure
3. Measure diameter with a micrometer at several points → calculate area $A = \pi(d/2)^2$
4. Add known masses (forces $F = mg$) and measure extension $\Delta L$ with a ruler or travelling microscope
5. Plot a graph of stress ($F/A$) against strain ($\Delta L/L$)
6. Gradient = Young's modulus

Improvements

• Use a long wire (larger extension, easier to measure)
• Use a thin wire (larger extension for same force)
• Use a comparison wire to eliminate thermal expansion
• Use a vernier scale or travelling microscope for accurate extension measurement
• Take diameter measurements at multiple points and average

• $\sigma = F/A$ (stress), $\varepsilon = \Delta L/L$ (strain), $E = \sigma/\varepsilon$ (Young's modulus)
• Young's modulus = gradient of linear region of stress-strain graph
• Elastic limit: beyond this → permanent deformation
• Brittle: breaks with no plastic deformation; Ductile: large plastic region
• Energy stored per unit volume = area under stress-strain graph