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

Alloys are mixtures, and a mixture can exist as one phase or several, depending on composition and temperature. A **phase diagram** is a map that tells you, for any alloy composition and temperature at equilibrium, which phases are present, what each contains, and how much of each there is. Metallurgists use phase diagrams to choose casting and heat-treating temperatures, to predict microstructures, and to understand why, for example, solder melts at a low temperature or why steel can be hardened.

## Phases and components

A **component** is a pure substance (copper, nickel, iron, carbon). A **phase** is a region of uniform structure and composition: a liquid, or a solid solution with one crystal structure, or a compound. A binary phase diagram plots temperature (vertical) against composition of two components (horizontal, usually in weight percent).

## The isomorphous diagram: copper-nickel

Copper and nickel are both FCC with similar atom sizes, so they dissolve in each other completely in the solid state. Their diagram has just three regions:

- **liquid (L)** at the top;
- **solid solution (α)** at the bottom, for every composition;
- a lens-shaped **L + α** region between them, bounded by the **liquidus** (above it, all liquid) and the **solidus** (below it, all solid).

Unlike a pure metal, an alloy melts over a range of temperatures.

## Reading a two-phase point: tie line and lever rule

Take a Cu-Ni alloy with C0 = 35 wt% Ni held at 1,250 °C. That point falls in the L + α region.

**Step 1: compositions (tie line).** Draw a horizontal line through the point to the liquidus and solidus. Where it meets the liquidus gives the liquid's composition, CL ≈ 31.5 wt% Ni; where it meets the solidus gives the solid's composition, Cα ≈ 42.5 wt% Ni.

**Step 2: amounts (lever rule).** The fraction of each phase is the length of the *opposite* arm of the tie line divided by the whole tie line:

```text
W_L = (Cα − C0) / (Cα − CL) = (42.5 − 35) / (42.5 − 31.5) = 7.5 / 11 = 0.68
W_α = (C0 − CL) / (Cα − CL) = (35 − 31.5) / 11 = 0.32
```

So at 1,250 °C the alloy is 68 % liquid (containing 31.5 % Ni) and 32 % solid (containing 42.5 % Ni) by weight. In a single-phase region there is no lever rule: one phase, with the alloy's own composition.

## The eutectic diagram: lead-tin

When two components have limited solubility in each other, the diagram has two solid phases (α on one side, β on the other) and a special **eutectic point** where liquid transforms directly into both solids at one temperature:

```text
L  →  α + β        (Pb-Sn: 61.9 wt% Sn, 183 °C)
```

The eutectic alloy melts at a single, lowest temperature, which is why traditional tin-lead solder used compositions near 60-40. (Lead-free solders are now required in much electronics; they use other eutectic-like systems such as tin-silver-copper.) Cooling a eutectic alloy gives a fine layered mixture of α and β called the eutectic microstructure.

## The iron-carbon diagram

The most important phase diagram in engineering covers iron with up to 6.70 wt% carbon (the compound cementite, Fe₃C). Key phases:

- **ferrite (α):** BCC iron, which dissolves at most 0.022 wt% C;
- **austenite (γ):** FCC iron, which dissolves up to 2.14 wt% C;
- **cementite (Fe₃C):** hard, brittle iron carbide.

The key reaction for steels is the **eutectoid**, at 0.76 wt% C and 727 °C:

```text
γ (austenite)  →  α (ferrite) + Fe₃C (cementite)
```

Slowly cooled, eutectoid steel forms **pearlite**, fine alternating layers of ferrite and cementite. Steels with less carbon (hypoeutectoid) form some ferrite first, then pearlite; those with more (hypereutectoid) form cementite first. There is also a eutectic at 4.30 wt% C and 1,147 °C, relevant to cast irons. Steels are iron-carbon alloys up to about 2.14 wt% C; above that are cast irons.

**Worked example.** A 0.40 wt% C steel cooled slowly to just below 727 °C contains pearlite and the ferrite that formed earlier. The lever rule between 0.022 and 0.76 wt% C gives the fraction of pearlite:

```text
W_pearlite = (0.40 − 0.022) / (0.76 − 0.022) = 0.378 / 0.738 ≈ 0.51
```

About half pearlite and half ferrite: that mix of soft ferrite and harder pearlite is typical of medium-carbon structural and machine steels.

## Limits of phase diagrams

Phase diagrams show **equilibrium**, which needs slow cooling. Fast cooling can produce phases not on the diagram at all, most importantly martensite in quenched steel. That is the subject of the next lesson.

## Common mistakes

- **Using the near arm in the lever rule.** The fraction of a phase uses the arm on the *opposite* side.
- **Confusing wt% with at%.** Most engineering diagrams use weight percent.
- **Expecting equilibrium phases after rapid cooling.**

## Recap

- Phase diagrams map phases against temperature and composition at equilibrium.
- Tie lines give phase compositions; the lever rule gives phase amounts.
- Eutectic: L → α + β at one temperature (Pb-Sn at 61.9 % Sn, 183 °C).
- Fe-C eutectoid: γ → α + Fe₃C at 0.76 % C, 727 °C, forming pearlite.


Read a binary phase diagram with tie lines and the lever rule, including the iron-carbon diagram.

Phase diagrams

Intro to Phase Diagrams {Texas A&M: Intro to Materials}

Fe-C phase diagram

## Why it matters

A spring, a file, a chisel and a car body panel can all be made of steel, yet one is springy, one is hard enough to cut other steel, and one is soft enough to press into shape. Much of that difference comes from **heat treatment**: heating and cooling a metal in a controlled way to change its microstructure without changing its shape or composition. This lesson explains the main treatments for steel, and briefly for aluminium, at the level an engineer needs to read a drawing note such as "quench and temper to 30–35 HRC" and understand what it means.

Heat treatment is industrial work with furnaces at hundreds of degrees, hot metal and quench tanks of oil or water, which can cause burns, fires and steam explosions. This lesson is for understanding only. Do not try it at home; it is done in properly equipped shops or labs, with training and supervision.

## The starting point: austenite

From the iron-carbon diagram, heating a steel above roughly 727–912 °C (depending on carbon content) turns it fully or partly into **austenite**, FCC iron that can dissolve carbon. Every steel heat treatment starts here (austenitising) and differs in how the steel is cooled.

## Slow cooling: annealing and normalizing

- **Full annealing:** heat into the austenite region, then cool very slowly, usually in the furnace. The result is coarse pearlite and ferrite: soft, ductile and easy to machine or form. Annealing also removes the effects of cold work.
- **Normalizing:** heat a little higher, then cool in still air. Faster cooling gives finer pearlite, so the steel is somewhat stronger and harder than annealed, with a more uniform grain structure. It is often used to refine the structure after forging or welding.
- **Stress relieving:** heating well below the transformation temperature to relax residual stresses from welding or machining, without changing the microstructure much.

## Fast cooling: quenching to martensite

If austenite is cooled very rapidly, by plunging it into water, brine or oil, the carbon atoms do not have time to diffuse out to form ferrite and cementite. Instead the FCC lattice shears into a distorted body-centred tetragonal structure supersaturated with carbon: **martensite**. Martensite is very hard and strong, but also brittle and full of internal stress. Its hardness depends mainly on carbon content: low-carbon steels (below about 0.25 % C) cannot be hardened much by quenching, which is one reason they weld easily.

Quenching has risks. Water quenches are severe and can distort or crack parts, especially with sharp corners or sudden changes in section. Oil and polymer quenchants are gentler. **Hardenability** is a steel's ability to form martensite deep below the surface rather than only at the skin. Alloying elements such as chromium, molybdenum and nickel slow the transformation to pearlite, increasing hardenability so thicker sections harden with a gentler quench. Hardenability is measured by the Jominy end-quench test.

Time-temperature-transformation (TTT) and continuous-cooling-transformation (CCT) diagrams show how fast a given steel must cool to avoid pearlite and bainite (another, tougher microstructure) and form martensite.

## Tempering

Freshly quenched martensite is usually too brittle to use. **Tempering** reheats it to a temperature below 727 °C, often somewhere between about 200 and 650 °C, for a set time. Carbon comes out as tiny carbide particles, internal stresses relax, and the result, **tempered martensite**, trades some hardness for much better toughness.

The tempering temperature sets the balance:

- low tempering (around 200 °C): high hardness and wear resistance, for cutting tools and bearings;
- higher tempering (around 500–650 °C): lower hardness but much greater toughness, for shafts, bolts, axles and springs.

The combination is called **quench and temper (Q&T)**. A drawing might specify a steel grade, "Q&T to 28–34 HRC", where HRC is the Rockwell C hardness scale. Standards such as SAE/AISI grade data give the tempering temperatures that achieve a given hardness.

## Surface hardening

Many parts need a hard, wear-resistant surface over a tough core: gear teeth, camshafts, bearing races.

- **Carburizing:** diffuse carbon into the surface of a low-carbon steel at around 900–950 °C (the diffusion lesson computed how deep it goes), then quench and temper. The surface becomes high-carbon martensite; the core stays tough.
- **Nitriding:** diffuse nitrogen at lower temperatures to form hard nitrides, with little distortion.
- **Induction or flame hardening:** heat only the surface of a medium-carbon steel quickly, then quench it.

## Aluminium: precipitation hardening

Heat-treatable aluminium alloys such as 6061 and 7075 are strengthened differently, because aluminium does not form martensite:

1. **Solution treatment:** heat to dissolve the alloying elements into a single solid solution.
2. **Quench:** cool rapidly to trap them in a supersaturated solution.
3. **Ageing:** hold at a moderate temperature (artificial ageing) or at room temperature (natural ageing) so that tiny precipitates form and block dislocations.

The temper designation "T6" means solution treated and artificially aged. Overageing (too long or too hot) makes the precipitates coarsen and the alloy weaker. Welding can locally undo the treatment, so welded heat-treated aluminium is weaker near the weld.

## Common mistakes

- **Expecting every steel to harden.** Low-carbon steel barely responds to quenching.
- **Skipping tempering.** As-quenched martensite can crack in service, or even on the bench.
- **Assuming harder is better.** Higher hardness usually means lower toughness and ductility.

## Recap

- Annealing and normalizing cool slowly: soft, ductile pearlite and ferrite.
- Quenching forms martensite: hard, brittle, stressed. Tempering restores toughness.
- Hardenability (alloying) controls how deep martensite forms.
- Aluminium alloys are hardened by solution treatment, quench and ageing (T6).


Explain annealing, quenching and tempering of steel and precipitation hardening of aluminium.

Heat treatment

Understanding Steels and Heat Treatment

Intro to heat treatment of steel (hardening and tempering)

Write the design brief in four lines: the function (carry books in bending), the constraints (for example: must not sag more than about 5 mm, must cost under a budget you choose, must not be damaged by room humidity), the objective (minimise mass, or minimise cost), and the free variable (the material and the shelf thickness).

Pick the matching performance index from the lesson's table. For a stiff, light beam it is E^(1/2)/ρ; for a stiff, light panel (a wide, thin shelf) it is E^(1/3)/ρ. Write down which one fits your shelf and why.

List at least five candidate materials with their E (GPa) and ρ (Mg/m³). Use the lesson's table and add at least one more, such as plywood (E about 9, ρ about 0.6), glass (E about 70, ρ about 2.5), polycarbonate (E about 2.3, ρ about 1.2) or MDF (E about 3.5, ρ about 0.75). Note where each value came from.

Calculate the index for every candidate and rank them from best to worst.

Sketch an Ashby-style chart on paper: log scale for E on the vertical axis (0.1 to 1,000 GPa) and for ρ on the horizontal axis (0.1 to 10 Mg/m³). Mark each candidate as a dot, then draw a straight line of slope 2 (for the beam index) through your best candidate. Check that the materials above the line are the ones your calculation ranked highest.

Strike out candidates that fail a constraint: for example glass (brittle, dangerous if it breaks), or anything over your budget. Write one line for each reason.

Do a safe stiffness comparison: place a ruler flat on a table with about 20 cm sticking over the edge, hold the rest down firmly with a stack of books, and hang the bag of coins at the tip. Measure how far the tip drops with a second ruler. Repeat with a ruler of a different material (wood, plastic or metal) with the same overhang and the same bag. Keep loads light and your feet clear of anything that could drop.

Compare the deflections with what the E values predict (deflection is proportional to 1/(E·I), so a ruler of a different thickness will also change the result) and note any surprises.

Write your final choice and a three-sentence justification that mentions the index ranking, at least one constraint, and one factor the index does not capture (cost, appearance, manufacturing or sustainability).

## Why it matters

There are tens of thousands of engineering materials. Picking one by habit ("we always use steel") or by a single property ("it's the strongest") often gives heavy, expensive or failure-prone designs. Michael Ashby's method, taught in engineering schools worldwide, makes selection systematic: translate the design into requirements, turn those into a **performance index**, and compare materials on charts that plot one property against another. This lesson walks through the method so you can apply it with a calculator and paper.

## Step 1: translate the design

Write down four things:

- **Function:** what does the part do? (Carry a bending load, hold pressure, conduct heat.)
- **Constraints:** non-negotiable requirements. (Must not yield; must not deflect more than 5 mm; must survive 80 °C; must cost less than a set amount.)
- **Objective:** what to minimise or maximise. (Minimise mass, minimise cost.)
- **Free variables:** what you are allowed to change. (Usually the material, and often one dimension such as thickness.)

## Step 2: derive a performance index

Combine the objective with the constraint, and eliminate the free dimension. The material properties that remain form the index.

**Example: a light, stiff tie** (a rod in tension, fixed length L, must not stretch more than a set amount; free variable: its area A). Stiffness is E·A/L, so the required area is A = k·L/E, and mass is m = A·L·ρ = k·L²·(ρ/E). To minimise mass, maximise **E/ρ**.

**Example: a light, stiff beam** (fixed length, square section of free size, must not deflect more than a set amount). Working through the bending stiffness, which depends on E·I with I proportional to the section size to the fourth power, gives an index of **E^(1/2)/ρ**.

Common indices to minimise mass:

| part and requirement | maximise |
|---|---|
| tie, stiffness | E / ρ |
| beam, stiffness | E^(1/2) / ρ |
| panel (plate), stiffness | E^(1/3) / ρ |
| tie, strength | σf / ρ |
| beam, strength | σf^(2/3) / ρ |

Here σf is the failure strength (usually yield strength for metals). For minimum cost instead of mass, replace ρ with the cost per unit volume.

## Step 3: compare materials

**Worked example: a light, stiff beam.** Approximate properties (E in GPa, ρ in Mg/m³, which is the same number as g/cm³):

| material | E | ρ | E/ρ | E^(1/2)/ρ |
|---|---|---|---|---|
| steel | 200 | 7.8 | 25.6 | 1.8 |
| aluminium alloy | 70 | 2.7 | 25.9 | 3.1 |
| titanium alloy | 110 | 4.5 | 24.4 | 2.3 |
| CFRP (quasi-isotropic laminate) | 100 | 1.6 | 62.5 | 6.3 |
| softwood (along grain) | 11 | 0.55 | 20 | 6.0 |

For a stiff tie, steel, aluminium and titanium are almost identical (E/ρ ≈ 25). For a stiff beam, the square root changes everything: low-density materials win, and wood and carbon-fibre composite beat steel by a factor of more than three. That is why light aircraft wing spars, bicycle frames and racing car chassis have moved from steel to aluminium, then to composites, and why wood has been used for so long in bending members.

## Ashby charts

An **Ashby chart** plots two properties on log-log axes, for example Young's modulus against density, with every material family appearing as a bubble. On such a chart, a performance index becomes a straight line: for E^(1/2)/ρ the lines have slope 2. Slide a line of the right slope across the chart; materials above the line are better than those below. Constraints (such as a maximum cost or a minimum service temperature) are drawn as boxes that remove candidates. Free and commercial software (for example Ansys Granta EduPack, used in many universities) contains these charts and databases.

## Step 4: screen, rank, then check the details

The index gives a short list, not a final answer. Finally consider what it cannot capture:

- **cost** and availability;
- **manufacturing:** can it be shaped, joined and finished in the quantities needed?
- **environment:** corrosion, UV, temperature, moisture (wood and some composites change with humidity);
- **failure modes:** toughness, fatigue, fire behaviour;
- **sustainability:** embodied energy and CO₂, recyclability;
- **codes and standards** that limit which materials may be used.

For any part that carries safety-critical loads, the chosen material and its design values must come from certified data and the governing standard, and the design must be reviewed by a qualified engineer (a licensed professional engineer where required).

## Common mistakes

- **Comparing raw properties instead of an index.** Steel's high E looks impressive until you divide by its density.
- **Using the wrong index.** A tie and a beam reward different properties.
- **Ignoring units.** Use the same units for every candidate; the absolute value of an index matters less than the ranking.

## Recap

- Function, constraints, objective, free variables.
- Eliminate the free variable to get a performance index.
- E/ρ for stiff ties, E^(1/2)/ρ for stiff beams, E^(1/3)/ρ for stiff panels.
- Ashby charts turn indices into lines; then check cost, processing and environment.

## Try it

On the Do tab, select a material for a simple household part on paper, using an index and a hand-sketched Ashby-style chart.


Choose a material for a job with performance indices and an Ashby-style chart.

Materials selection

How to select materials using Ashby plots and performance indexes

Visual Materials Selection -- Lesson 2

Use phase diagrams to predict microstructure, see how heat treatment changes steel, and choose a material for a job with performance indices and Ashby charts.