Properties of Materials

This lesson covers AQA Spec 3.4.2.1–3.4.2.2 in detail: elastic and plastic behaviour, brittle/ductile/polymeric materials, loading–unloading curves, and hysteresis.

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1 Hooke's Law

Hooke's Law: The extension of a spring (or wire) is directly proportional to the applied force, provided the limit of proportionality is not exceeded.

F = kx

where k is the spring constant (or stiffness) in N m⁻¹, and x is the extension (or compression).

Worked Example 1 — Spring Constant

A spring extends by 40 mm when a force of 5.0 N is applied. Find (a) the spring constant and (b) the extension when a 12 N force is applied (assuming Hooke's law still applies).

Solution:

(a) k = F/x = 5.0 / (40 × 10⁻³) = 125 N m⁻¹

(b) x = F/k = 12 / 125 = 0.096 m = 96 mm

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2 Springs in Series and Parallel

Springs in Series

For two springs with constants k₁ and k₂ in series:

1/k_total = 1/k₁ + 1/k₂

The total spring is softer (lower k).

Springs in Parallel

For two springs in parallel:

k_total = k₁ + k₂

The total spring is stiffer (higher k).

Worked Example 2

Two springs, k₁ = 100 N m⁻¹ and k₂ = 200 N m⁻¹, support a 6.0 N load. Find the extension when arranged (a) in parallel and (b) in series.

Solution:

(a) Parallel: k_total = 100 + 200 = 300 N m⁻¹

x = F/k = 6.0/300 = 0.020 m = 20 mm

(b) Series: 1/k_total = 1/100 + 1/200 = 2/200 + 1/200 = 3/200

k_total = 200/3 = 66.7 N m⁻¹

x = F/k = 6.0/66.7 = 0.090 m = 90 mm

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3 Elastic and Plastic Deformation

Elastic deformation: The material returns to its original shape when the load is removed. No permanent deformation. The atoms are displaced from their equilibrium positions but return when the force is removed.

Plastic deformation: The material is permanently deformed when the load is removed. The atoms have moved to new equilibrium positions — planes of atoms have slipped past each other.

The elastic limit is the maximum stress (or load) that can be applied without causing permanent deformation. Beyond this, some deformation is plastic and the material does not fully recover.

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4 Elastic Potential Energy

The energy stored in a stretched spring (or wire) that obeys Hooke's law:

E = ½Fx = ½kx²

This is the area under the force–extension graph (a triangle for a linear spring).

Worked Example 3

A spring with k = 80 N m⁻¹ is compressed by 0.15 m. Find the elastic potential energy stored.

Solution:

E = ½kx² = ½ × 80 × 0.15² = ½ × 80 × 0.0225 = 0.90 J

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5 Force–Extension Graphs for Different Materials

Ductile Material (e.g., Copper)

Described diagram — Force–extension graph for copper wire:

• Linear region (Hooke's law) up to the limit of proportionality.
• The curve bends over as plastic deformation begins.
• Large plastic extension before fracture — the material "necks" (cross-section reduces locally).
• On unloading from the plastic region, the wire follows a line parallel to the original loading line, but offset — showing a permanent extension.

Brittle Material (e.g., Glass)

Described diagram — Force–extension graph for glass:

• Straight line (linear, elastic) right up to the fracture point.
• No plastic deformation — the material snaps suddenly.
• Glass obeys Hooke's law until it breaks.

Polymeric Material (e.g., Rubber)

Described diagram — Force–extension graph for a rubber band (loading and unloading):

• Non-linear curve — rubber can stretch to several times its original length.
• The unloading curve lies below the loading curve — this is called hysteresis.
• The area between the loading and unloading curves represents the energy dissipated as heat within the rubber.
• Rubber returns to its original length (elastic), but energy is lost.

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6 Hysteresis

Key Definition: Hysteresis is the phenomenon where the loading and unloading curves on a force–extension (or stress–strain) graph do not coincide. The area enclosed between the curves represents the energy dissipated per cycle.

Applications of Hysteresis