Stress-Strain Graphs and Material Behaviour

Different materials respond to tensile stress in very different ways: some bend, some snap, some keep on stretching for ages before finally giving up. A well-constructed stress-strain graph tells you, at a glance, which class a material belongs to and what engineering role it can fulfil. Steel holds up the Tyne Bridge because of its shape; glass holds up a wine glass because of its brittleness; nylon holds up a climbing rope because of its stretchiness. This lesson — the final one of OCR Module 3 — studies brittle, ductile and polymeric material behaviour, the distinctions between elastic and plastic deformation, and the interpretation of key points on the stress-strain graph.

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The Generic Stress-Strain Graph

A typical ductile metal like mild steel has the following stress-strain curve:

flowchart LR
  A[Origin] --> B[P: Limit of proportionality]
  B --> C[E: Elastic limit]
  C --> D[Y1: Upper yield point]
  D --> E[Y2: Lower yield point]
  E --> F[UTS: Ultimate tensile stress]
  F --> G[Necking begins]
  G --> H[Breaking / fracture stress]

Key features:

Point	Name	Meaning
O–P	Linear region	σ ∝ ε (Hooke's law), gradient = E (Young modulus)
P	Limit of proportionality	Beyond here, σ is no longer proportional to ε
E	Elastic limit	Beyond here, deformation becomes permanent (plastic)
Y	Yield point	Large plastic flow begins at near-constant stress
UTS	Ultimate tensile stress	Maximum stress the material can withstand
Fracture	Breaking stress	The sample finally breaks

For many materials P, E and Y are very close together, and "yield stress" is often used to mean all three.

Exam Tip: You should be able to label at least the limit of proportionality, elastic limit and UTS on a given stress-strain curve. OCR mark schemes are strict about which point has which name.

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Elastic vs Plastic Deformation

Elastic deformation:

• Reversible — the sample returns to its original length when the load is removed.
• Occurs below the elastic limit.
• Stored energy is released when unloaded (like a spring).
• Atomic bonds are stretched but not broken.

Plastic deformation:

• Permanent — the sample retains some extension after unloading.
• Occurs above the elastic limit.
• Energy is dissipated as heat (and lattice defects).
• Atoms have moved to new equilibrium positions — bonds broken and re-formed.

A key practical test: load the material, then unload it. If it returns exactly to its original length, the deformation was elastic. If it retains some extension (called "permanent set"), plastic deformation has occurred.

On the stress-strain graph, unloading from a point beyond the elastic limit produces a new straight line parallel to the original linear region (slope = E), intersecting the ε-axis at the permanent strain. This is why cold-working a metal (bending, hammering) permanently deforms it.

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Ductile Materials

Ductile materials — including copper, aluminium, mild steel and gold — undergo large plastic deformation before breaking. Typical features of their stress-strain curves:

• Long plastic region between yield and UTS.
• Significant strain (often >20%) before fracture.
• Area under the curve (energy per unit volume absorbed) is large — they are tough.
• After UTS, a "neck" forms where the cross-section locally thins, reducing the load needed and causing the graph to tip downward before fracture.

Ductile materials are ideal for:

• Drawing into wires (copper wire for electrical cables, steel piano wire).
• Rolling into sheets (aluminium foil, car body panels).
• Pressing and forging (ship hulls, machine parts).
• Absorbing impacts — which is why cars are designed to "crumple" in a crash, absorbing kinetic energy through plastic deformation rather than passing it to the occupants.

Why Ductility?

In metals, ductility arises from the movement of dislocations through the crystal lattice. Under stress, planes of atoms slip past each other, but only one line of bonds at a time needs to break — drastically reducing the required stress compared with breaking the whole plane at once. This is why real metals yield at around 0.1% of the theoretical strength calculated from bond energies.

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Brittle Materials

Brittle materials — including glass, cast iron, ceramics, concrete and most rocks — show little or no plastic deformation before fracture. Features:

• Nearly straight line all the way to fracture.
• Very small strain at breaking (often < 0.5%).
• Small area under the curve — they are not tough.
• Can still have very high Young modulus (they are stiff) and high UTS (they are strong).

Don't confuse "brittle" with "weak". Diamond and tungsten carbide are both extremely brittle, but also among the strongest materials known. Brittleness refers to the inability to deform plastically, not to a low failure stress.

Brittle materials fail suddenly, usually from a tiny crack or surface flaw that concentrates stress at its tip. A glass rod can be snapped easily after scoring — the scratch acts as a stress concentrator. Griffith's analysis of crack propagation showed that the fracture stress scales as 1/√(crack length), explaining why even fine scratches can dramatically weaken brittle materials.

Engineering strategies to make brittle materials usable:

• Toughened glass — deliberately stressed during manufacture to put the surface in compression, so small cracks cannot grow.
• Reinforced concrete — steel rods embedded inside take tension (which concrete cannot handle) while the concrete handles compression.
• Glass fibre in polymers (GRP) — the matrix prevents crack propagation in the fibres.

Worked Example — Area Under a Brittle Curve

A brittle material has a straight-line stress-strain curve from origin to fracture at σ_f = 150 MPa, ε_f = 0.002. Find the energy stored per unit volume at the moment of fracture.

Energy per unit volume = ½ σ ε = ½ × 150 × 10⁶ × 0.002 = 1.5 × 10⁵ J m⁻³

Compare this with a ductile metal with σ_f = 400 MPa and ε_f = 0.30: energy ≈ 60 × 10⁶ J m⁻³, which is 400 times larger. This is why a steel kitchen knife bends rather than shatters, while a ceramic knife splinters.

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Polymeric Materials

Polymers are long-chain molecules (plastics, rubbers, proteins). They show behaviour intermediate between the crystalline regularity of metals and the amorphous disorder of glass. Common examples: polyethylene, polystyrene, nylon, PVC, rubber, Kevlar.

Polymers split into two broad families:

Thermoplastics (e.g. polyethylene, nylon)