understand and use the terms load, extension, compression and limit of proportionality

Stress, Strain and the Limit of Proportionality

Learning Objectives

• Define load (force), extension, compression and the limit of proportionality (LoP).
• Distinguish between tensile and compressive loading.
• Apply Hooke’s law to calculate stress, strain, extension and the spring constant for a rod.
• Read and interpret a stress–strain graph: locate the linear (elastic) region, LoP, elastic limit, yield point, ultimate tensile strength and fracture.
• Explain how Young’s modulus relates to the slope of the linear region and to material stiffness.
• Identify key experimental considerations (area measurement, safety limits, avoiding permanent deformation).

Key Definitions

Fundamental Formulas

• Stress: $\displaystyle \sigma = \frac{F}{A}$ (where $A$ is the original cross‑sectional area).
• Strain: $\displaystyle \varepsilon = \frac{\Delta L}{L_{0}}$.
• Hooke’s law (elastic region): $\displaystyle \sigma = E\,\varepsilon$.
• Spring‑constant form for a straight rod (derived from Hooke’s law):
  $\displaystyle F = k\,\Delta L\qquad\text{with}\qquad k = \frac{EA}{L_{0}}$.

Derivation of $k = EA/L_{0}$

1. Start with $\sigma = E\varepsilon$.
2. Replace $\sigma$ by $F/A$ and $\varepsilon$ by $\Delta L/L_{0}$:
   $\displaystyle \frac{F}{A}=E\frac{\Delta L}{L_{0}}$.
3. Re‑arrange to obtain $F = \left(\frac{EA}{L_{0}}\right)\Delta L$; the term in brackets is the spring constant $k$ for the rod.

Typical Values of Young’s Modulus

Stress–Strain Graph (Cambridge A‑Level)

Graph‑Reading Exercise

1. Estimate the stress at the LoP ($\sigma_{\text{LoP}}$) from the vertical axis.
2. Using the data in the example below, calculate the stress produced by the given load. Compare it with $\sigma_{\text{LoP}}$ and state whether the load lies within the linear region.
3. Read the strain at the yield point. Comment on the difference between the LoP and the yield point for a ductile metal such as steel.

Worked Example – Elastic Extension of a Steel Rod

Given:

• Original length $L_{0}=1.20\ \text{m}$
• Diameter $d=1.80\ \text{cm}$ → $A=\dfrac{\pi d^{2}}{4}=2.54\times10^{-4}\ \text{m}^{2}$
• Young’s modulus $E=2.0\times10^{11}\ \text{Pa}$
• Tensile load $F=5.0\times10^{4}\ \text{N}$ (assumed to be within the elastic region)

1. Stress: $\displaystyle \sigma=\frac{F}{A}= \frac{5.0\times10^{4}}{2.54\times10^{-4}}=1.97\times10^{8}\ \text{Pa}$.
2. Strain (Hooke’s law): $\displaystyle \varepsilon=\frac{\sigma}{E}= \frac{1.97\times10^{8}}{2.0\times10^{11}}=9.85\times10^{-4}$ (≈ 0.098 %).
3. Extension: $\displaystyle \Delta L = \varepsilon L_{0}= (9.85\times10^{-4})(1.20)=1.18\times10^{-3}\ \text{m}=1.18\ \text{mm}$.
4. From the typical steel curve, $\sigma_{\text{LoP}}\approx2.0\times10^{8}\ \text{Pa}$. Since $\sigma=1.97\times10^{8}\ \text{Pa}<\sigma_{\text{LoP}}$, the load is safely within the linear region.

Distinguishing Tensile and Compressive Loading

• Tensile loading: pulls the material, giving a positive extension ($\Delta L>0$) and a positive stress ($\sigma>0$).
• Compressive loading: pushes the material, giving a negative extension ($\Delta L<0$) and a negative stress ($\sigma<0$). In practice the magnitude of compressive stress is quoted as a positive number, but the loading type must be noted.
• Both obey $\sigma = E\varepsilon$ while the material remains in the elastic region.

Common Misconceptions – Clarified

• Load vs. Stress: Load is the total force $F$ (N). Stress is the force per unit area ($\sigma = F/A$) and has units of pressure (Pa).
• Extension vs. Strain: Extension $\Delta L$ is an absolute change in length (m). Strain $\varepsilon$ is the relative change ($\Delta L/L_{0}$) and is dimensionless.
• LoP vs. Yield Point: LoP marks the end of the proportional (linear) relationship. The yield point marks the onset of permanent deformation; they may be close for many metals but are distinct concepts.
• Units: Stress → Pa (or N mm⁻²). Strain → dimensionless (often expressed as a %). They are not interchangeable.

Typical Linear‑Elastic Strain Ranges

• Steel: up to ≈ 0.20 % strain ($2\times10^{-3}$) before noticeable deviation.
• Aluminium: up to ≈ 0.15 % strain.
• Copper: up to ≈ 0.25 % strain.
• Polymers (e.g., PMMA): up to 1 %–2 % strain.

Experimental & Safety Considerations

• Measure the cross‑sectional area accurately (calipers for diameter, then $A=\pi d^{2}/4$ for circular rods; $A=wh$ for rectangular bars).
• Apply loads gradually; record extension after each increment to avoid sudden fracture.
• Never exceed the ultimate tensile strength shown on the stress–strain curve – failure is catastrophic.
• When using a universal testing machine, set the maximum load just below the estimated $\sigma_{\text{LoP}}$ for a first trial.
• Always wear safety goggles and keep a safe distance from the testing apparatus.

Symbol Summary