How the structure of metals, ceramics, polymers and composites sets their properties: crystal structure, defects and diffusion, mechanical, thermal and electrical properties, failure, phase diagrams, heat treatment and materials selection.

Materials Science Fundamentals

## Why it matters

The tensile test is the most common mechanical test in materials engineering. A carefully machined sample is pulled in a machine until it breaks, while force and elongation are recorded. Converted to stress and strain, the record becomes a **stress-strain curve**, and nearly every number on a material datasheet (modulus, yield strength, tensile strength, elongation) is read from it. Being able to read one is a basic skill for engineers and materials scientists alike.

## From the test to the curve

**Engineering stress** is force divided by the sample's *original* cross-sectional area, σ = F/A0, and **engineering strain** is the change in gauge length divided by the original length, ε = ΔL/L0. Test standards such as ASTM E8 (metals) and ASTM D638 (plastics) fix the sample shape and speed so labs get comparable results.

## Reading the curve for a ductile metal

Walking along a typical curve from the origin:

1. **Elastic region.** A straight line: stress is proportional to strain. Its slope is **Young's modulus** E. Unload here and the sample returns to its original length.
2. **Yield.** The curve bends over as planes of atoms start to slip (dislocation motion). Many metals have no sharp yield point, so engineers use the **0.2 % offset yield strength**: draw a line parallel to the elastic line, starting at a strain of 0.002, and read the stress where it crosses the curve.
3. **Strain hardening.** Stress keeps rising as dislocations multiply and tangle, so more force is needed for more deformation.
4. **Ultimate tensile strength (UTS).** The highest point on the engineering curve. Beyond it the sample starts to **neck**: deformation concentrates in one region, the local area shrinks, and the force needed falls.
5. **Fracture.** The sample breaks. The **elongation at fracture** (often written as % elongation) measures ductility. It is the *plastic* strain at fracture: the total strain minus the elastic part that springs back when the sample breaks.

Two other properties come from the area under the curve: **resilience** (energy absorbed elastically, the area under the elastic part) and **toughness** (energy absorbed up to fracture, the total area). Ductile metals are tough because they deform a lot before breaking.

## Worked example

A test on an aluminium alloy gives these points (strain, stress in MPa):

```text
(0.001, 70)  (0.002, 140)  (0.003, 210)  (0.004, 280)  ...
(0.0065, 285)  ...  (0.080, 310)  ...  (0.120, 280) fracture
```

**Modulus.** The early points lie on a line through the origin with slope 70 MPa / 0.001 = 70,000 MPa = 70 GPa, a normal value for aluminium.

**0.2 % offset yield.** The offset line is σ = 70,000 × (ε − 0.002). At ε = 0.0065 it gives 315 MPa, already above the measured 285 MPa, while at ε = 0.006 it gives 280 MPa, below the curve. So the offset line crosses the curve between these points, at about 285 MPa: the yield strength.

**UTS.** The highest stress is 310 MPa, reached at a strain of 0.08.

**Elongation at fracture.** Total strain at fracture is 0.120. The elastic strain that springs back is 280 / 70,000 = 0.004. Plastic strain is 0.116, so elongation ≈ 11.6 %.

## How different materials look

- **Brittle ceramics and glass:** a straight line to fracture, with almost no plastic region. Their tensile strength is very sensitive to surface flaws, so it scatters widely between samples.
- **Ductile metals:** a long plastic region, a clear UTS and necking.
- **Polymers:** strongly depend on temperature and test speed. Some neck and draw a long way (polyethylene), others are brittle (polystyrene), and elastomers stretch hundreds of percent with a curved, non-linear response.

## True stress and strain

Engineering stress uses the original area, so it drops after necking even though the material in the neck is still getting stronger. **True stress** (force over the actual area) keeps rising. Engineering values are used for design data; true values are used for modelling forming processes.

## Common mistakes

- **Reading yield where the curve first looks curved.** Use the 0.2 % offset method unless a standard says otherwise.
- **Quoting total strain as elongation.** Subtract the elastic recovery.
- **Treating datasheet values as guaranteed.** Typical values vary with grade, temper, product form and direction; designs use specified minimum values from the relevant material standard.

## Recap

- Elastic slope = E; 0.2 % offset gives yield strength; the peak gives UTS.
- Elongation at fracture is plastic strain and measures ductility.
- The area under the curve measures toughness.

## Try it

On the Do tab, you will analyse a full set of tensile test data in Python. The code finds UTS, offset yield and elongation; you write the modulus fit.


Read a stress-strain curve for modulus, yield strength, tensile strength, ductility and toughness.

Mechanical properties

Understanding Material Strength, Ductility and Toughness

## Why it matters

Mechanical properties decide whether a part breaks; thermal and electrical properties decide whether it works at all. A heat sink must move heat; a building's insulation must stop it. Wires must conduct, and the insulation around them must not. Parts that heat up must expand without cracking or jamming. This lesson compares how the four classes of materials conduct heat and electricity and how much they expand, and links each behaviour back to bonding and structure.

## Thermal conductivity

Heat flows through a solid by two routes: **free electrons** (in metals) and **lattice vibrations**, called phonons (in everything). Fourier's law gives the heat flow rate through a slab:

```text
q = k · A · ΔT / L        (watts)
```

where k is the **thermal conductivity** in W/(m·K), A the area, ΔT the temperature difference across it and L its thickness.

Typical room-temperature values (approximate):

| material | k (W/m·K) |
|---|---|
| copper | about 400 |
| aluminium | about 240 (pure), 120–200 (alloys) |
| carbon steel | about 50 |
| stainless steel | about 15 |
| alumina (ceramic) | about 30 |
| glass | about 1 |
| concrete | about 1–2 |
| polyethylene | about 0.4 |
| foam insulation | about 0.03 |
| still air | about 0.026 |

Metals conduct best because free electrons carry heat. Alloying scatters electrons, which is why stainless steel conducts about a quarter as well as plain carbon steel. Polymers and glasses have disordered structures that scatter phonons, so they conduct poorly. Foams and fibre insulation work by trapping still air.

**Worked example.** A 1 m² single pane of glass, 4 mm thick, has 20 °C on the inside surface and 0 °C on the outside surface. q = 1 × 1 × 20 / 0.004 = 5,000 W. In reality the air films on each side add much more resistance than the glass itself, so actual losses are far lower, but the calculation shows why thin glass alone is a poor insulator and why double glazing traps a layer of gas.

## Thermal expansion

Most materials expand when heated:

```text
ΔL = α · L · ΔT
```

with α the coefficient of linear thermal expansion (per K, often quoted in 10⁻⁶/K). Approximate values: aluminium 23, copper 17, steel 12, soda-lime glass 9, borosilicate glass 3.3, alumina about 8, polyethylene 100–200, Invar (an iron-nickel alloy) about 1.5.

**Worked example.** A 2.0 m aluminium rail warms by 40 K: ΔL = 23 × 10⁻⁶ × 2.0 × 40 = 1.84 mm. A steel rail of the same length would grow only 0.96 mm. When two materials with different α are joined (a metal frame around glass, a polymer part screwed to steel), the mismatch causes stress as the temperature changes. Borosilicate glass resists cracking when heated quickly partly because its expansion is so low.

Weak bonds mean large α: polymers expand roughly ten times as much as metals. Strong bonds in ceramics mean small α, but ceramics are also brittle, so **thermal shock** (a sudden temperature change across the part) can still crack them.

## Electrical conductivity

Electrical resistivity ρe (Ω·m) and conductivity σe = 1/ρe span more than twenty orders of magnitude, more than almost any other physical property:

| material | resistivity (Ω·m, approximate) |
|---|---|
| copper | 1.7 × 10⁻⁸ |
| aluminium | 2.7 × 10⁻⁸ |
| stainless steel | about 7 × 10⁻⁷ |
| pure silicon | about 2 × 10³ |
| glass | 10¹⁰ or more |
| polyethylene | about 10¹⁵ |

**Band theory** explains the spread. In metals, the highest occupied electron energy band is only partly filled, so electrons move freely: **conductors**. In **insulators** the filled valence band is separated from the empty conduction band by a wide gap (several eV), so almost no electrons can cross. **Semiconductors** have a small gap (about 1.1 eV for silicon), so some electrons are thermally excited across it, and adding tiny amounts of dopant (phosphorus or boron in silicon) controls the conductivity precisely.

Temperature affects them in opposite ways: a metal's resistance **rises** with temperature (vibrating atoms scatter electrons), while a semiconductor's resistance **falls** (more electrons are excited across the gap).

Copper is the standard conductor for wiring. Aluminium conducts about 60 % as well by volume but is much lighter, so overhead power lines use aluminium conductors, often with a steel core for strength.

## Linking the two

In metals the same free electrons carry heat and charge, so good electrical conductors are usually good thermal conductors (the Wiedemann-Franz relationship). Diamond is a notable exception: an electrical insulator that conducts heat superbly through its very stiff lattice.

## Common mistakes

- **Confusing conductivity with resistivity**, which are reciprocals.
- **Ignoring units.** k in W/(m·K) and thickness in metres, not millimetres.
- **Assuming a strong material conducts well.** Stainless steel is strong but a modest conductor of heat and electricity.

## Recap

- Metals conduct heat and electricity through free electrons; polymers and glasses insulate.
- ΔL = α·L·ΔT; polymers expand most, ceramics least.
- Band gaps separate conductors, semiconductors and insulators.
- Metal resistance rises with temperature; semiconductor resistance falls.


Compare thermal conductivity, thermal expansion and electrical conductivity across the classes of materials.

Thermal and electrical properties

Thermal Conductivity

Conductivity and Semiconductors

## Why it matters

Parts rarely fail because a load was simply bigger than the tensile strength. Far more often they fail by a mechanism that a static strength check misses: a crack that grows a little with every load cycle (**fatigue**), slow stretching at high temperature (**creep**), or a sudden **fracture** that starts at a small flaw. Failure analysis engineers read broken parts like a crime scene, and the features on a fracture surface usually reveal which mechanism did it. This lesson describes those features in words so you can recognise them in photographs and real parts.

## Ductile and brittle fracture

**Ductile fracture** happens after a lot of plastic deformation. A round tensile sample of a ductile metal necks, small voids form and join in the neck, and the final break makes the classic **cup-and-cone** shape: a fibrous, dull, grey centre with a shear lip around the edge. Under a microscope the surface is covered in tiny **dimples**. The part visibly stretches or bends first, which gives warning.

**Brittle fracture** happens with little or no plastic deformation, at high speed, along flat surfaces roughly perpendicular to the main tensile stress. The surface is often bright and granular (in metals, **cleavage facets** that glint in the light) and may show **chevron** or **river** marks that point back toward the origin of the crack. There is little or no warning.

Some steels (with BCC crystal structure) change from ductile to brittle behaviour as the temperature drops: the **ductile-to-brittle transition**. Several welded Liberty ships of the Second World War cracked in cold water, some breaking completely in two, partly because their steel was below its transition temperature. Impact tests such as the Charpy test measure the energy absorbed at different temperatures to find the transition. FCC metals such as aluminium and austenitic stainless steel do not show this transition.

## Fracture mechanics in one equation

Brittle fracture starts at flaws: scratches, pores, weld defects, small cracks. Stress concentrates at the tip of a crack, and fracture mechanics captures this with the **stress intensity factor**:

```text
K = Y · σ · √(π · a)
```

where σ is the applied stress, a the crack length (half the length for an internal crack), and Y a geometry factor near 1. Fracture occurs when K reaches the material's **fracture toughness** K_IC.

**Worked example.** A plate with Y = 1 carries 200 MPa and contains a 2 mm crack:

```text
K = 1 × 200 × √(π × 0.002) = 200 × 0.0793 ≈ 15.9 MPa·√m
```

Typical K_IC values: glass about 0.7–0.8, alumina about 3–5, high-strength aluminium alloys about 25, and tough steels 50 or more MPa·√m. So this crack would break glass at a tiny fraction of that stress but would not grow in a tough steel. Inspection programmes (for aircraft, pipelines, pressure vessels) set inspection intervals so cracks are found long before they reach the critical size.

## Fatigue

Under repeated loading, a crack can start and grow at stresses well below the yield strength. Fatigue causes a large share of mechanical failures in service: shafts, axles, springs, bolts, welded joints, aircraft skins.

A fatigue fracture surface usually shows three zones:

1. an **origin**, often at a surface feature such as a sharp corner, keyway, scratch, weld toe or corrosion pit;
2. a smooth, flat region with curved **beach marks** (also called clamshell marks) spreading out from the origin, recording pauses and changes in loading; under high magnification, finer **striations** mark individual cycles;
3. a rough **final fracture** zone where the remaining section broke in one go. A small final zone means the stress was low; a large one means it was high.

Fatigue behaviour is shown on an **S-N curve**: stress amplitude against the number of cycles to failure. Many steels have an **endurance limit**, a stress below which they appear to last indefinitely (often roughly 35–60 % of tensile strength under ideal laboratory conditions), whereas aluminium alloys have no true limit and are rated at a set number of cycles. Smooth surfaces, generous radii, compressive surface stresses (shot peening) and avoiding corrosion all improve fatigue life.

## Creep

At high temperature, roughly above 0.4 of the absolute melting temperature, materials slowly and permanently stretch under a constant load. Creep curves show a short primary stage, a long steady secondary stage (whose rate designers use), and a tertiary stage of accelerating strain and voids leading to **rupture**. Creep limits turbine blades, boiler tubes and furnace parts, and it explains why lead pipes sag and why plastic shelves droop over time at room temperature: for polymers and lead, room temperature is already "hot". Creep fractures often show many small voids along grain boundaries and visible stretching near the break.

## Common mistakes

- **Assuming a part that passed a static test is safe.** Fatigue can fail it at a fraction of the yield stress.
- **Polishing or cleaning away evidence.** Fracture surfaces must be protected for failure analysis.
- **Treating handbook endurance limits as design values.** Real parts need corrections for size, surface finish, environment and reliability, from design standards.

## Recap

- Ductile fracture: necking, cup-and-cone, dimples, warning. Brittle: flat, shiny, chevrons, sudden.
- K = Y·σ·√(πa); fracture when K reaches K_IC.
- Fatigue: origin, beach marks, final fracture zone.
- Creep: slow strain at high homologous temperature, ending in rupture.


Recognise ductile and brittle fracture, fatigue and creep from the features of a broken part.

Failure: fatigue, creep, fracture

Understanding Fatigue Failure and S-N Curves

How materials can fail due to Creep

Read a stress-strain curve, compare thermal and electrical behaviour across material classes, and recognise how materials fail by fatigue, creep and fracture.