Stress, strain and the Young modulus flashcards

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Tensile stress
Stretching force per unit cross-sectional area, σ = F/A. Unit Pa (N m⁻²).
Compressive stress
Squashing force per unit cross-sectional area, σ = F/A. Unit Pa.
Strain
Change in length ÷ original length, ε = Δx/x. No unit (may be given as a %).
Young modulus
Stress ÷ strain, for a material within its limit of proportionality. Unit Pa. Measures the stiffness of the material.
Breaking stress (ultimate tensile stress)
The greatest stress a material can withstand before breaking — the highest point of its stress–strain graph.
Stiff
Needs a large stress for a small strain: large Young modulus, steep straight section.
Strong
Needs a large stress to break it: high breaking stress.
Brittle
Breaks with little or no plastic deformation (e.g. glass, cast iron, ceramics).
Ductile
Undergoes large plastic deformation under tension; can be drawn into wires (e.g. copper).
Malleable
Can be hammered, pressed or rolled permanently into a new shape without breaking (plastic deformation under compression).
Hard
Resists scratching and denting of its surface.
Tough
Absorbs a lot of energy before breaking: large area under the stress–strain graph.
Equation for stress
σ = F/A (Yes)
Equation for strain
ε = Δx/x (Yes)
Equation for the Young modulus
E = σ/ε (Yes)
Area of a wire from its diameter d
A = πd²/4 (= πr²) (No — learn it)
Young modulus from raw measurements
E = Fx/(AΔx), from E = (F/A) ÷ (Δx/x). (No — derive it)
Stiffness k of a wire in terms of E
k = F/Δx = EA/x (links to Hooke's law, topic 1.8). (No — derive it)
Gradient of the straight section of a stress–strain graph
The Young modulus E.
Area under a stress–strain graph up to breaking
Energy absorbed per unit volume before breaking — a measure of toughness.
Shape of the stress–strain graph for a brittle material
Straight line from the origin that ends suddenly at fracture, with no curved plastic region.
Shape of the stress–strain graph for a ductile metal
Straight line, then a long curved plastic region rising to a maximum (UTS); the stress falls as the wire necks, then it breaks.
Unloading after plastic deformation
Follows a line parallel to the original straight section; where it meets the strain axis is the permanent strain.
Why does a thicker wire of the same metal have the same Young modulus?
E is a property of the material. A larger area lowers the stress and the strain in the same proportion; only the object's stiffness k changes.
Core Practical 3: why use a long, thin wire?
Long: larger extension for the same strain. Thin: large stress for a modest load. Both make the extension easier to measure, with a smaller percentage uncertainty.
Core Practical 3: how is the diameter measured, and why that way?
Micrometer at several points along the wire, in two directions at right angles; take the mean. It averages out variations and a non-circular cross-section.
Core Practical 3: what is plotted, and how is E found?
Stress (mg/A) against strain (Δx/x) for each load; E is the gradient of the straight section.
Can a material be strong but not tough?
Yes — a strong brittle material (e.g. glass) needs a high stress to break but absorbs little energy, because its strain at breaking is small.
A wire of diameter 1.0 mm carries a tension of 50 N. Stress?
A = π(1.0 × 10⁻³)²/4 = 7.85 × 10⁻⁷ m²; σ = 50 ÷ 7.85 × 10⁻⁷ = 6.4 × 10⁷ Pa.
A 3.0 m wire extends by 1.5 mm. Strain?
1.5 × 10⁻³ ÷ 3.0 = 5.0 × 10⁻⁴ (no unit).
Stress 8.0 × 10⁷ Pa produces a strain of 4.0 × 10⁻⁴. Young modulus?
E = 8.0 × 10⁷ ÷ 4.0 × 10⁻⁴ = 2.0 × 10¹¹ Pa.
A 2.5 m copper wire (E = 1.2 × 10¹¹ Pa) is under a stress of 1.2 × 10⁸ Pa. Extension?
ε = σ/E = 1.0 × 10⁻³; Δx = 1.0 × 10⁻³ × 2.5 = 2.5 mm.
A rod of area 2.0 mm² has a breaking stress of 4.0 × 10⁸ Pa. Greatest load?
2.0 mm² = 2.0 × 10⁻⁶ m²; F = 4.0 × 10⁸ × 2.0 × 10⁻⁶ = 800 N.
Same metal and load; wire B has twice the length and twice the diameter of A. Extension of B ÷ extension of A?
Δx ∝ x/d²: 2 ÷ 4 = 0.5.
Area from a diameter in mm
Convert to metres first, then use πd²/4 — not πd². 1 mm² = 10⁻⁶ m².
Breaking stress on a ductile metal's graph
Take the maximum stress, not the stress where the line ends (the wire necks after the maximum).
Describing a material from a graph
Back each word with a feature: stiff — gradient; strong — maximum stress; brittle — no plastic region; ductile — long plastic region; tough — area.
Which measurement dominates the uncertainty in Core Practical 3?
Usually the diameter: it is small, and it is squared in the area. Take several micrometer readings and use half the range.

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