🔧Ductile materials Describe plastic deformation before fracture
🪟Brittle materials Describe fracture without plastic deformation
Tensile Stress and Tensile Strain
When we simply measure force and extension (F and x), we are describing the behaviour of a specific sample. To characterise the material itself — independent of dimensions — we use stress and strain.
Tensile stress: σ = F / A
Tensile strain: ε = ΔL / L
Symbol
Quantity
Unit
σ (sigma)
Tensile stress
Pa (N m⁻²)
F
Applied force (tension)
N
A
Cross-sectional area
m²
ε (epsilon)
Tensile strain
No unit (dimensionless)
ΔL
Extension
m
L
Original length
m
Strain is dimensionless — it is a ratio of two lengths. Stress has units of pascals (Pa = N m⁻²), the same as pressure.
Stress characterises how much force per unit area the material experiences. Strain characterises how much it deforms per unit of original length. Both are intensive properties of the material, not the sample.
When calculating cross-sectional area of a wire, you must measure the diameter (using a micrometer) and use A = π(d/2)² = πd²/4. Small errors in d are doubled when squaring.
Young's Modulus
Within the elastic limit, stress and strain are proportional. The ratio of stress to strain is called the Young's modulus (E), a property of the material itself.
E = σ / ε = (F/A) / (ΔL/L) = FL / (AΔL)
Symbol
Quantity
Unit
E
Young's modulus
Pa (N m⁻²) or GPa
σ
Tensile stress
Pa
ε
Tensile strain
Dimensionless
Young's modulus is the gradient of the linear (Hookean) region of a stress-strain graph. Typical values:
Steel: ~200 GPa (very stiff)
Copper: ~120 GPa
Aluminium: ~70 GPa
Rubber: ~0.01–0.1 GPa (flexible)
Glass: ~70 GPa (stiff but brittle)
A material with a high Young's modulus requires a large stress to produce a small strain — it is stiff. A low Young's modulus indicates a flexible material.
Young's modulus E = gradient of the stress-strain graph in the linear region. Units: Pa or GPa. It is a material constant — independent of sample shape or size.
Stress-Strain Graphs
A stress-strain graph reveals the mechanical properties of a material:
Gradient of linear region = Young's modulus E
Limit of proportionality: end of the linear section
Elastic limit: end of recoverable deformation
Yield stress (yield point): stress at which plastic deformation begins easily
Ultimate tensile stress (UTS): maximum stress the material can withstand
Breaking stress: stress at fracture (may be less than UTS due to necking)
The area under a stress-strain graph represents the energy stored per unit volume of the material (J m⁻³). In the elastic region: energy per unit volume = ½σε = ½E ε².
Ultimate tensile stress (UTS) / Breaking stress: The maximum tensile stress a material can withstand before fracturing. Reported in Pa or MPa.
Very little or no plastic deformation before fracture
Fracture occurs suddenly, close to the elastic limit
Cracks propagate rapidly once initiated
Strong under compression, weak under tension
Polymeric materials (e.g. rubber, nylon):
Non-linear stress-strain curve
Can undergo very large strains
Hysteresis loop during loading-unloading (energy dissipation)
Young's modulus much lower than metals
Ductile: large plastic region, draws into wires. Brittle: snaps suddenly at elastic limit. Rubber: non-linear, large strains, hysteresis.
A steel wire of length 2.0 m and diameter 0.50 mm is subjected to a tensile force of 40 N. Calculate (a) the stress, (b) the strain, (c) the extension. (E_steel = 200 GPa)
1Cross-sectional area: A = π × (0.25 × 10⁻³)² = π × 6.25 × 10⁻⁸ = 1.963 × 10⁻⁷ m²
4Extension: ΔL = ε × L = 1.019 × 10⁻³ × 2.0 = 2.04 × 10⁻³ m
σ = 204 MPa; ε = 1.02 × 10⁻³; ΔL = 2.04 mm
A copper wire of cross-sectional area 2.0 × 10⁻⁷ m² extends by 3.0 mm under a load of 25 N. Its original length was 1.5 m. Calculate Young's modulus for copper.
3Young's modulus: E = σ/ε = (1.25 × 10⁸) / (2.0 × 10⁻³) = 6.25 × 10¹⁰ Pa
E = 6.25 × 10¹⁰ Pa = 62.5 GPa
The UTS of a material is 350 MPa. A cylindrical rod of diameter 8.0 mm is made from this material. What is the maximum load it can support before breaking?
2F = σ_UTS × A = 350 × 10⁶ × 5.027 × 10⁻⁵ = 17 590 N
Maximum load = 17 600 N ≈ 17.6 kN
Q1. Tensile strain is measured in which units?
Q2. A wire has length 3.0 m, diameter 1.0 mm, and is stretched by 1.5 mm under a force of 60 N. Calculate the Young's modulus.
Q3. Which material would show a large plastic region on its stress-strain graph before fracture?
Q4. What does the gradient of the linear region of a stress-strain graph represent?
Q5. A steel wire (E = 200 GPa) has a strain of 8.0 × 10⁻⁴. Calculate the stress in the wire.
Challenge Q1. A guitar string of length 0.65 m, diameter 0.40 mm, and E = 200 GPa is tightened until the tension is 80 N. Calculate the extension and the stress in the string.
Challenge Q2. Two wires X and Y have the same length and are made of the same material. Wire X has twice the diameter of wire Y. Compare (a) their spring constants and (b) the extensions when the same load is applied.
Challenge Q3. Explain why a brittle material is dangerous in structural applications subjected to tensile loads, but can be useful under compressive loads. Refer to crack propagation in your answer.
🔬 Required Practical Determine the Young's modulus of a metal by stretching a wire
Aim
To measure the Young's modulus of a metal wire by applying known loads and measuring the resulting extensions.
Equipment
Long thin metal wire (~2 m, e.g. steel or copper), fixed firmly at ceiling or upper clamp
Micrometer screw gauge (to measure wire diameter)
Metre rule or vernier scale (to measure extension)
Reference wire alongside the test wire (to correct for temperature drift and sag of support)
Mass hanger and slotted masses (100 g increments)
Small adhesive marker on wire aligned with the scale
Method
Measure the diameter d of the wire at several points along its length using a micrometer. Calculate A = πd²/4.
Measure the original length L of the wire from the fixed point to the marker.
Record the initial scale reading (zero extension).
Add masses in increments (e.g. 100 g = 0.981 N). Record the new scale reading and calculate the extension ΔL for each load.
Continue until you have 6–8 data points within the elastic region.
Remove masses and verify the wire returns to its original length (confirming elastic behaviour).
Safety
Wear safety glasses — wires under tension can snap and whip. Ensure masses cannot fall on feet; use a tray below. Do not overload the wire beyond its elastic limit.
Analysis
Load F (N)
Extension ΔL (mm)
Stress σ (Pa)
Strain ε
0
0
0
0
0.981
1.962
2.943
3.924
4.905
Plot a graph of stress (y-axis) against strain (x-axis). The gradient of the best-fit straight line through the origin gives the Young's modulus E.
Alternatively, plot F (y-axis) against ΔL (x-axis). The gradient = EA/L, so E = gradient × L / A.
Sources of Uncertainty
Diameter measurement — use average of multiple readings; small error in d → larger error in A (since A ∝ d²)
Length measurement — parallax error; zero error on micrometer
Wire may not be perfectly uniform along its length
Kinking or bending of wire — ensure wire is taut before zero reading
Temperature changes causing thermal expansion — use a reference wire
Analysis Questions
1. Why is it important to take the diameter measurement at several positions along the wire?
2. Why is a long wire preferable to a short one for this experiment?