understand that deformation is caused by tensile or compressive forces (forces and deformations will be assumed to be in one dimension only)

Stress and Strain – Cambridge IGCSE / A‑Level (Syllabus 6.1 & 6.2)

Learning Objective

To understand how one‑dimensional tensile or compressive forces cause deformation of a material, and how this deformation is quantified using stress, strain and Young’s modulus.

Key Definitions

• Load (F) – External force applied to the specimen. It may be tensile (pulling) or compressive (pushing).
• Cross‑sectional area (A) – Area of the specimen perpendicular to the applied load.
• Original length (L₀) – Length of the specimen before any load is applied.
• Extension (ΔL) – Increase in length under a tensile load (ΔL > 0).
• Compression (ΔL) – Decrease in length under a compressive load (ΔL < 0; often written as a negative extension).
• Stress (σ) – Internal resistance to the applied load.

  \$\sigma = \frac{F}{A}\qquad\text{(Pa)}\$

• Strain (ε) – Relative deformation.

  \$\varepsilon = \frac{\Delta L}{L_{0}}\qquad\text{(dimensionless)}\$

• Hooke’s law (elastic region) – Within the elastic limit stress and strain are directly proportional:

  \$\sigma = E\,\varepsilon\$

  where E is Young’s modulus.

• Young’s Modulus (E) – Slope of the straight‑line portion of the stress–strain curve; a measure of material stiffness.

  \$E = \frac{\sigma}{\varepsilon}= \frac{F\,L_{0}}{A\,\Delta L}\qquad\text{(Pa)}\$

• Limit of proportionality (elastic limit) – Maximum stress at which Hooke’s law (σ = Eε) is still obeyed.
• Yield point – Stress at which permanent (plastic) deformation begins. For many metals the yield point occurs shortly after the limit of proportionality and is identified by a noticeable deviation from the linear portion of the curve.
• Plastic region – Portion of the curve where strain is no longer fully recoverable on unloading.
• Ultimate tensile strength (UTS) – Highest stress the material can sustain before necking starts.
• Fracture point – Stress at which the specimen breaks.
• Buckling (compressive loading) – Sudden lateral deflection that can occur in slender columns when the compressive stress reaches a critical value; a behaviour distinct from the necking seen in tension.

Units and Symbols

Behaviour of Materials Under One‑Dimensional Load

1. Elastic region (up to the limit of proportionality)

   • σ ∝ ε; the relationship is linear and described by Hooke’s law (σ = Eε).
   • When the load is removed the specimen returns to its original length.

2. Yield point

   • Stress at which permanent deformation begins.
   • Often identified by a small “yield drop” or by a 0.2 % offset method in exams.

3. Plastic region

   • Strain is no longer fully recoverable; the material will retain a permanent elongation (or shortening) after unloading.

4. Ultimate tensile strength (UTS)

   • Maximum stress the material can sustain.
   • Beyond this point necking begins, leading eventually to fracture.

5. Fracture point

   • Stress at which the specimen breaks.

Typical Stress–Strain Diagram (Tensile Test)

Experimental Determination of Young’s Modulus (Tensile)

1. Apparatus

   • Universal testing machine (UTM) with calibrated load cell.
   • Specimen of known length L₀ and uniform cross‑sectional area A (measure A with a micrometer or vernier caliper).
   • Extensometer or dial gauge to read ΔL accurately.

2. Procedure (step‑by‑step)

   1. Measure and record L₀ and A, noting the instrument uncertainties.
   2. Mount the specimen in the UTM ensuring the load is applied along its longitudinal axis.
   3. Zero the extensometer with the specimen under no load.
   4. Increase the load gradually (e.g., 0.5 kN increments) and record the corresponding extension ΔL after each increment.
   5. Continue loading until the specimen reaches the elastic limit (or a predetermined maximum load).
   6. Plot the recorded stress (σ = F/A) against strain (ε = ΔL/L₀). The initial straight‑line portion gives the slope = Young’s modulus.
   7. Unloading the specimen and re‑measuring the final length checks for permanent deformation.

3. Uncertainty and Error Analysis

   Source of Uncertainty	Typical Magnitude	Effect on E
   Length measurement (L₀)	±0.1 mm (caliper)	Inverse proportional – larger L₀ → smaller E.
   Cross‑sectional area (A)	±1 % (micrometer)	Inverse proportional – over‑estimated A → underestimated E.
   Load reading (F)	±0.5 % (load cell)	Direct proportional – over‑estimated F → over‑estimated E.
   Extension reading (ΔL)	±0.01 mm (extensometer)	Inverse proportional – over‑estimated ΔL → underestimated E.
   Alignment of load	Qualitative	Mis‑alignment introduces bending, giving apparent lower stiffness.

Compressive Loading (One‑Dimensional)

• Stress and strain are defined exactly as for tension: σ = F/A, ε = ΔL/L₀ (ΔL < 0).
• In the elastic region the stress–strain curve is a mirror image of the tensile curve; the slope remains E.
• Buckling – For slender columns, when the compressive stress reaches the critical Euler stress, the specimen may deflect laterally rather than simply shorten. This behaviour is not captured by the simple σ–ε line and must be considered in design.
• Beyond the limit of proportionality many materials crush or shear rather than neck, leading to a different shape of the post‑elastic portion of the curve.

Worked Example (Tensile Loading)

Given

• Load, F = 5.0 kN
• Original length, L₀ = 200 mm
• Cross‑sectional area, A = 10 mm²
• Measured extension, ΔL = 0.40 mm

Calculate stress, strain and Young’s modulus.

1. Stress

   \$\sigma = \frac{5.0\times10^{3}\,\text{N}}{10\times10^{-6}\,\text{m}^{2}} = 5.0\times10^{8}\,\text{Pa}=500\;\text{MPa}\$

2. Strain**

   \$\varepsilon = \frac{0.40\times10^{-3}\,\text{m}}{0.200\,\text{m}} = 2.0\times10^{-3}\$

3. Young’s Modulus**

   \$E = \frac{\sigma}{\varepsilon}= \frac{5.0\times10^{8}\,\text{Pa}}{2.0\times10^{-3}} = 2.5\times10^{11}\,\text{Pa}=250\;\text{GPa}\$

Key Points to Remember

• All forces and deformations are treated as acting along a single axis (the length of the specimen).
• Load → stress (σ = F/A); deformation → strain (ε = ΔL/L₀).
• In the elastic region Hooke’s law holds: σ = Eε. The slope of the linear part of a stress–strain diagram is Young’s modulus.
• The limit of proportionality marks the end of the linear (elastic) region; the yield point marks the start of permanent deformation.
• Compression follows the same linear law up to the elastic limit, but may lead to buckling rather than necking.
• Experimental determination of E requires careful measurement of F, A, L₀ and ΔL, and a systematic assessment of uncertainties.

Typical Exam Questions (Syllabus 6.1 & 6.2)

1. Given a material’s Young’s modulus, original dimensions and a tensile load, calculate the resulting extension and state whether the material remains in the elastic region.
2. Interpret a provided stress–strain diagram: identify the limit of proportionality, yield point, ultimate tensile strength and fracture point.
3. Two rods of different materials are subjected to the same load. Using Young’s modulus, compare their extensions and comment on which is stiffer.
4. For a compressive test, calculate stress and strain, then discuss whether buckling is likely based on the slenderness ratio (L₀ / radius).
5. Describe a step‑by‑step experimental method to determine Young’s modulus and list three possible sources of error, explaining how each would affect the final value of E.