Stress-Strain Graphs and Material Behaviour

Spec mapping: OCR H556 Module 3.4 — Materials (full stress-strain curves; limit of proportionality, elastic limit, yield, ultimate tensile stress, breaking stress; elastic vs plastic deformation; behaviour of brittle, ductile and polymeric materials; hysteresis). PAG 2 anchor lesson. Refer to the official OCR H556 specification document for exact wording.

Different materials respond to tensile loading in remarkably different ways. A copper wire stretches by tens of percent before finally breaking; a glass rod snaps cleanly at a tiny extension; a rubber band can be stretched to several times its natural length yet recovers fully on release. A well-drawn stress-strain graph captures this character in a single picture — and it is, as much as the chemical composition, what defines a material's engineering role. Steel holds up the Tyne Bridge because it can carry stress without fracturing; glass holds up a wine glass because of (and despite) its brittleness; nylon holds up a climbing rope because of its huge stretchiness combined with toughness. This final lesson of the Materials module studies brittle, ductile and polymeric behaviour, the distinction between elastic and plastic deformation, and the interpretation of every landmark on a real stress-strain curve.

Key Definition: A stress-strain graph plots tensile stress $\sigma = F/A$ on the y-axis against tensile strain $\varepsilon = \Delta\ell/\ell_0$ on the x-axis. Because both axes are normalised by sample geometry, the graph is shape-independent — it depicts an intrinsic property of the material. Key landmarks (in increasing strain): limit of proportionality (P), elastic limit (E), yield point (Y), ultimate tensile stress (UTS), and fracture / breaking stress.

The Generic Stress-Strain Graph for a Ductile Metal

A representative ductile metal — mild steel, copper, aluminium — shows the following landmarks on its stress-strain curve:

flowchart LR
  A[Origin] --> B[P: limit of proportionality]
  B --> C[E: elastic limit, very close to P]
  C --> D[Y1: upper yield point]
  D --> E[Y2: lower yield point]
  E --> F[Plastic plateau: cold-drawing region]
  F --> G[UTS: ultimate tensile stress]
  G --> H[Necking onset]
  H --> I[Fracture / breaking]

Point	Name	Meaning
O–P	Linear (Hookean) region	$\sigma \propto \varepsilon$ , gradient = Young modulus $E$
P	Limit of proportionality	Beyond P, $\sigma$ is no longer proportional to $\varepsilon$
E	Elastic limit	Beyond E, deformation becomes permanent (plastic)
Y	Yield point	Large plastic strain at nearly constant stress
Plateau	Plastic flow / cold-drawing	The material elongates at nearly constant stress
UTS	Ultimate tensile stress	Maximum stress the material can sustain (engineering stress, $F/A_0$ )
Necking	Cross-section narrows locally	Engineering stress drops; true stress at the neck still rises
Fracture	Breaking stress	Final rupture

For many materials P, E and Y are very close together, and the term "yield stress" is informally used for all three. OCR mark schemes, however, are strict about which point has which name — so always label them in the correct order: $\sigma_P \leq \sigma_E \leq \sigma_Y$ .

Common Exam Mistake: Labelling P, E, Y, UTS in the wrong order, or confusing UTS with breaking stress. Memorise the canonical sequence and the relationship $\sigma_P \leq \sigma_E \leq \sigma_Y \leq \sigma_{\text{UTS}}$ , with breaking stress $\sigma_{\text{break}}$ at or below $\sigma_{\text{UTS}}$ (lower for ductile metals due to necking, essentially equal for brittle materials).

Elastic vs Plastic Deformation

The distinction between elastic and plastic deformation is the central conceptual divide in materials science:

Elastic deformation:

Reversible — the sample returns exactly to its original length when the load is removed.
Occurs below the elastic limit (sometimes called the yield strain for materials where P, E, Y nearly coincide).
Energy stored in the deformation is fully recoverable on unloading (no hysteresis).
Atomic-level picture: bonds are stretched but not broken; atoms remain in their original lattice sites and snap back to equilibrium positions.

Plastic deformation:

Permanent — the sample retains some extension ("permanent set") after the load is removed.
Occurs beyond the elastic limit.
Energy is dissipated as heat (and as energy stored in defects in the crystal lattice — dislocations, vacancies).
Atomic-level picture: planes of atoms have slipped past one another via dislocation motion. Bonds have broken and re-formed in new equilibrium positions, and the atomic arrangement has been permanently rearranged.

A practical test: load a sample, then unload. If it returns to its original length, the deformation was elastic. If it retains any extension, plastic deformation has occurred.

On the stress-strain graph, unloading from any point produces a straight line of slope $E$ (the Young modulus) — parallel to the original linear (Hookean) portion. The unloading line intersects the strain axis at the permanent strain $\varepsilon_{\text{perm}}$ , and the recovered (elastic) strain is $\varepsilon_{\text{recov}} = \sigma / E$ . This is why cold-working a metal (bending, hammering, rolling) permanently deforms it; the metal "remembers" plastic strains forever.

Ductile Materials

Ductile materials — copper, aluminium, mild steel, gold, lead — undergo large plastic deformation before fracture. Characteristic features:

A clear, well-defined yield point where plastic flow begins at near-constant stress.
A long plastic plateau between yield and UTS — sometimes called the cold-drawing region — during which the material elongates while the stress hardly rises.
Significant strain at fracture, typically $0.10$ to $0.40$ ( $10\%$ to $40\%$ ) for engineering metals; up to $0.6$ for gold and lead.
Large area under the curve — the energy per unit volume absorbed before fracture is large. Ductile materials are tough.
Necking after UTS: the cross-section thins locally; engineering stress (which uses the original cross-section) drops; true stress (using the actual reduced cross-section) keeps rising until fracture.

Ductile materials are ideal for:

Drawing into wires — copper for electrical cables, steel for piano wires, gold for jewellery filaments. The wire-drawing process exploits plastic deformation directly.
Rolling into sheets — aluminium foil, car body panels, ship plate.
Pressing and forging — engine blocks, machine parts.
Absorbing impacts — modern cars have "crumple zones" designed to deform plastically in a crash, dissipating kinetic energy through plastic work rather than transferring it to the occupants. The Insurance Institute for Highway Safety publishes "energy-absorbed-per-volume" data for crash structures, with values around $5$ – $20$ MJ m $^{-3}$ .

Why Are Real Metals Ductile?

In a perfect crystal, deformation would require simultaneously breaking all the bonds between two adjacent atomic planes — needing a stress of $E/10$ or so ( $\sim 20$ GPa for steel), $\sim 100$ times higher than the observed yield stress.

Real metals contain dislocations — line defects in the crystal lattice where an extra half-plane of atoms is inserted. When stress is applied, dislocations move through the lattice, breaking only one line of bonds at a time as they advance. The required stress is therefore $\sim 100$ times smaller than the theoretical bond-strength limit.

Strengthening a metal generally amounts to pinning its dislocations — preventing them from moving. Strategies:

Cold working (bending, rolling) creates dense tangles of dislocations that interfere with each other.
Alloying introduces foreign atoms that physically block dislocation motion (carbon in iron creates steel).
Grain-boundary strengthening uses many small grains (fine-grained metal) so that dislocations cannot travel far before hitting a grain boundary.
Heat treatment alters the dislocation density and grain size.

These strategies underpin every metallurgical advance from the Bronze Age to today's titanium aerospace alloys.

Brittle Materials

Brittle materials — glass, ceramics, cast iron, concrete, most rocks, hardened steel, diamond — show little or no plastic deformation before fracture. Characteristic features:

Nearly straight line all the way from origin to fracture; little to no curvature.
Very small strain at fracture — typically $< 0.5\%$ ( $0.005$ ) for engineering brittle materials, sometimes as low as $0.0001$ for glass.
Small area under the curve — brittle materials store little elastic energy per volume before fracture, so they are not tough.
Can still be very stiff and very strong — high Young modulus and high UTS — they simply do not yield before breaking.

Common Exam Mistake: Confusing "brittle" with "weak". Diamond ( $E \sim 1050$ GPa, $\sigma_{\text{UTS}} \sim 1$ – $10$ GPa depending on orientation) is extraordinarily brittle yet also among the strongest materials known. Tungsten carbide ( $E \sim 600$ GPa, $\sigma_{\text{UTS}} \sim 1$ GPa) is similarly brittle and strong. Brittle means "no plastic deformation"; weak means "low failure stress". The two are independent.

Brittle materials fail suddenly from a tiny surface flaw or crack that concentrates stress at its tip. A scored glass rod snaps cleanly along the scratch because the scratch acts as a stress concentrator — the stress at the crack tip is enormously larger than the bulk stress. Griffith's analysis of crack propagation (1920) showed that the fracture stress scales as $\sigma_{\text{frac}} \propto 1/\sqrt{a}$ where $a$ is the crack length: even a fine scratch can dramatically weaken brittle materials.

Engineering strategies for making brittle materials usable:

Toughened (tempered) glass — the surface is deliberately put in residual compression by rapid cooling. Cracks cannot propagate into a compressive region. Car-window glass is tempered for this reason; the residual stresses cause it to shatter into small, blunt cubes rather than long sharp shards when it eventually breaks.
Reinforced concrete — embedded steel rebar carries tensile loads (concrete cracks at $\sim 3$ MPa in tension) while concrete handles compressive loads ( $\sim 30$ MPa).
Fibre-reinforced composites (GRP, CFRP) — small fibres in a polymer matrix; the matrix prevents crack propagation and shares the load between fibres.
Pre-stressing — concrete or steel cables are tensioned during installation so that the operational load merely reduces the compression, never inducing tension.

Worked Example 1 — Energy Stored in a Brittle Material at Fracture

A brittle material has a straight-line stress-strain curve from the origin to fracture at $\sigma_f = 150$ MPa, $\varepsilon_f = 0.002$ . Find the energy per unit volume stored at the moment of fracture, and compare with a ductile material with $\sigma_f = 400$ MPa, $\varepsilon_f = 0.30$ .

For the brittle material (straight-line / triangular region):

$u_{\text{brittle}} = \tfrac{1}{2}\sigma_f \varepsilon_f = \tfrac{1}{2} \times 150 \times 10^6 \times 0.002 = 1.5 \times 10^5\;\text{J m}^{-3}$

For the ductile material (approximate the area under the curve as a rectangle of height $\sigma_f$ and base $\varepsilon_f$ , since the plastic plateau is nearly flat):

$u_{\text{ductile}} \approx \sigma_f \varepsilon_f = 400 \times 10^6 \times 0.30 = 1.2 \times 10^8\;\text{J m}^{-3}$

The ductile material absorbs about $800 \times$ more energy per unit volume before fracture. This is precisely why structures designed to absorb impact (car crumple zones, military body armour, climbing rope) prefer ductile or polymeric materials over brittle ones, even when the brittle material has higher Young modulus and higher peak stress.

Polymeric Materials

Polymers are long-chain molecules — natural (rubber, proteins, cellulose) or synthetic (polyethylene, nylon, PVC, Kevlar). They show behaviour intermediate between the crystalline regularity of metals and the amorphous fragility of glass, with the additional feature of strain-dependent stiffness due to chain coiling/uncoiling.

Polymers split into two broad families:

Thermoplastics — e.g. polyethylene, nylon, PVC

Flexible and tough at room temperature.
Plastic deformation occurs by uncoiling and sliding of polymer chains past each other.
Often show a clear yield followed by cold-drawing — a near-flat plastic region in which chains align in the direction of stress and the material elongates dramatically.
Can be remelted and reshaped — they have no permanent cross-links.
Engineering range $E \sim 0.1$ – $5$ GPa, $\sigma_f \sim 10$ – $100$ MPa, $\varepsilon_f \sim 0.1$ – $1.0$ .

Elastomers — e.g. natural rubber, silicone, neoprene

Extremely large elastic strains — up to $\varepsilon = 5$ ( $500\%$ , six times natural length) or more for rubber.
Non-linear stress-strain curve: soft at small strains (chains uncoil), much stiffer at large strains (chains fully extended and bond-stretching dominates).
Significant hysteresis — the unloading curve falls below the loading curve; the enclosed area is the energy dissipated as heat per cycle.
Stretching is entropic in origin: the random coiled state is more probable than the aligned stretched state, so the polymer naturally restores itself to the coiled state on release. This is fundamentally different from the bond-stretching elasticity of metals.

Polymeric Stress-Strain Sketch

flowchart LR
  A[Origin] --> B[Initial low stiffness: chains uncoil]
  B --> C[Strain-stiffening: chains fully extended]
  C --> D[Final steep rise: bond-stretching dominates]
  D --> E[Fracture at very high strain]

A rubber band can stretch to several times its natural length yet still recover. Its Young modulus, computed as $\sigma/\varepsilon$ , varies enormously with strain — from $\sim 0.01$ GPa near zero strain to $\sim 0.1$ GPa near the strain-stiffening regime. Unlike a Hookean material, no single Young modulus value characterises rubber's stiffness across all strains; it must be quoted at a specified strain or as a tangent modulus.

Hysteresis in Rubber

When a rubber sample is loaded and then unloaded, the unloading curve lies below the loading curve. The area between them is the energy dissipated as heat per loading-unloading cycle. This is why:

A stretched rubber band warms detectably. Try it: hold a thick elastic band against your lip, stretch and release rapidly, and you can feel the temperature change.
Car tyres heat up on motorway driving — the cyclic compression and expansion as the tyre rolls dissipates energy via hysteresis. About $20\%$ of fuel energy is lost to tyre rolling resistance.
Rubber is the material of choice for vibration damping and shock isolation: engine mounts, washing-machine pads, anti-seismic building bearings. The hysteresis ensures that vibrational energy is converted to heat rather than transmitted onward.

For an ideal elastic material (no hysteresis), the loading and unloading curves would coincide, and all the work done in stretching would be recoverable on release. Steel approaches this ideal below its elastic limit; rubber departs from it significantly.

Comparing the Three Material Classes

Feature	Ductile (copper)	Brittle (glass)	Polymeric (rubber)
Young modulus $E$	Medium-high (100–200 GPa)	High (70 GPa)	Very low (0.01–0.1 GPa)
Strain at failure $\varepsilon_f$	Large (10–40%)	Tiny ( $<$ 1%)	Enormous (100–500%+)
Stress-strain shape	Linear, then yield, plateau, UTS, necking, fracture	Almost straight, sudden fracture	Non-linear throughout: soft → stiff → fracture
Plastic deformation?	Yes, large	Almost none	Limited in elastomers; significant in thermoplastics
Toughness (area under curve)	Very large (1–500 MJ m $^{-3}$ )	Small ( $\sim 0.1$ MJ m $^{-3}$ )	Medium-large
Hysteresis?	Below elastic limit: minimal	Below fracture: minimal	Significant, in every cycle
Engineering uses	Wires, sheets, structural beams, crumple zones	Windows, optical fibres, kitchen ceramics	Tyres, seals, vibration damping, climbing rope

Worked Example 2 — Identifying Materials From Stress-Strain Curves

Three samples are tested in a tensile machine. Their stress-strain curves show:

A: straight line from origin to fracture at $\sigma = 200$ MPa, $\varepsilon = 0.001$ .
B: linear to $\sigma = 250$ MPa at $\varepsilon = 0.0015$ , then a clear yield, then plastic region to $\sigma = 450$ MPa at $\varepsilon = 0.22$ , fracture at $\sigma = 400$ MPa, $\varepsilon = 0.28$ .
C: non-linear throughout, stretches to $\varepsilon = 4.0$ , fractures at $\sigma = 20$ MPa.

Identify the likely class of each material.

A: Brittle. Small $\varepsilon_f$ , straight line all the way, no plastic deformation. Young modulus $E = \sigma/\varepsilon = 200 \times 10^6 / 0.001 = 2.0 \times 10^{11}$ Pa $= 200$ GPa — this is in the range of stiff materials like cast iron or a hard ceramic. (Glass would be $E \sim 70$ GPa, lower; this is more likely a high-strength brittle metal alloy.)
B: Ductile metal. Clear yield, long plastic region between $\varepsilon = 0.0015$ and $\varepsilon = 0.22$ , UTS at $0.22$ , then necking-driven drop to fracture at $0.28$ . Young modulus $E = 250 \times 10^6 / 0.0015 = 1.7 \times 10^{11}$ Pa $\approx 170$ GPa — consistent with mild steel or stainless steel.
C: Polymeric / elastomer. Non-linear curve, huge strain at fracture ( $\varepsilon_f = 4$ ), low stress ( $\sigma_f = 20$ MPa). Almost certainly rubber or a similar elastomer.

Worked Example 3 — Unloading Curve and Permanent Set

A mild-steel sample is stretched to the point where $\varepsilon = 0.050$ , $\sigma = 300$ MPa. The yield strain is $\varepsilon_{\text{yield}} \approx 0.002$ . The sample is then unloaded. Compute the elastic recovery and permanent set.

On unloading, the sample follows a straight line of slope $E$ (the Young modulus, $\sim 200$ GPa) back toward the strain axis:

Stress-Strain Graphs and Material Behaviour

Stress-Strain Graphs and Material Behaviour

The Generic Stress-Strain Graph for a Ductile Metal

Elastic vs Plastic Deformation

Ductile Materials

Why Are Real Metals Ductile?

Brittle Materials

Worked Example 1 — Energy Stored in a Brittle Material at Fracture

Polymeric Materials

Thermoplastics — e.g. polyethylene, nylon, PVC

Elastomers — e.g. natural rubber, silicone, neoprene

Polymeric Stress-Strain Sketch

Hysteresis in Rubber

Comparing the Three Material Classes

Worked Example 2 — Identifying Materials From Stress-Strain Curves

Worked Example 3 — Unloading Curve and Permanent Set

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