X-ray diffraction, SEM and TEM, DSC and TGA, tensile, hardness, fatigue and fracture testing, ASTM methods, polymer, ceramic and thin-film processing, and materials databases.

Materials Characterization & Processing

## Why it matters

A tensile test is the most common mechanical test in the world. Its results (modulus, yield strength, tensile strength, elongation) appear on every material certificate and data sheet, and designers use them directly. Hardness tests are faster and nearly non-destructive, so they are the everyday check on heat treatment and incoming material. Being able to compute these numbers yourself from raw data, and to spot a bad test, is a core skill.

## From load and extension to stress and strain

A load frame pulls a specimen of known cross-section `A0` and records force `F`; an extensometer measures the change in a gauge length `L0`.

```text
engineering stress  sigma = F / A0        (N / mm^2 = MPa)
engineering strain  e     = delta_L / L0  (dimensionless; x100 for %)
```

"Engineering" means we divide by the **original** area and length. True stress and strain use the instantaneous values and matter for forming simulations, but certificates report engineering values.

## The properties

- **Young's modulus E:** the slope of the initial straight part. Fit a line only to the clearly elastic region. Grip slack and specimen seating make the first few points unreliable, and a crosshead-displacement "strain" without an extensometer underestimates E badly.
- **0.2% offset yield strength:** many metals have no sharp yield point. Draw a line with slope E starting at 0.2% strain (0.002) and find where it crosses the curve. That stress is the yield strength `Rp0.2` (or `sigma_y`).
- **Ultimate tensile strength (UTS):** the maximum engineering stress. After it, the specimen necks and engineering stress falls.
- **Elongation at break** and **reduction of area:** measures of ductility, taken from the broken specimen fitted back together, or from the strain at fracture minus the elastic recovery.

### Worked example

A round steel specimen has `A0 = 20.0 mm^2` (about 5.05 mm diameter) and `L0 = 25.0 mm`.

1. At 4,000 N the extension is 0.0250 mm: `sigma = 4000 / 20 = 200 MPa` and `e = 0.025 / 25 = 0.0010`, so the slope is `200 / 0.0010 = 200,000 MPa` (200 GPa), as expected for steel. In real data, fit a line through many elastic points rather than trusting one.
2. The 0.2% offset line `sigma = 200,000 (e - 0.002)` crosses the curve at about 350 MPa: the yield strength.
3. The peak force is 9,000 N: `UTS = 9000 / 20 = 450 MPa`.

The yield-to-tensile ratio, 350/450 = 0.78, is a quick indicator of how much strain hardening the material has left.

## Hardness

Hardness tests press an indenter into the surface and measure the resulting dent (Brinell, Vickers, Knoop) or the depth (Rockwell).

- **Vickers (HV):** a square diamond pyramid. Measure both diagonals of the dent and average them as `d` (in mm). With the force `F` in kgf, `HV = 1.8544 * F / d^2`. Example: 10 kgf with diagonals 0.300 and 0.304 mm gives `d = 0.302`, `HV = 1.8544 * 10 / 0.0912 = 203`, reported as **203 HV10**.
- **Rockwell (HRC, HRB...):** a depth-based number read straight from the machine; fast and common on the shop floor. The scale letter matters: HRC and HRB are not interchangeable.
- **Brinell (HBW):** a large ball and high load, which averages over coarse microstructures such as castings.

Hardness correlates with tensile strength for many steels, and published conversion tables exist, but conversions are approximate and material-specific. Do not use them to certify a part.

## Specimens and methods

Specimen shape, gauge length, test speed and how you compute each property are set by the test standard (for metals, ASTM E8/E8M or ISO 6892-1; for Vickers, ASTM E92 or ISO 6507-1). Results from different gauge lengths or speeds are not directly comparable, which is why test reports always name the method. The next lessons cover finding and following these methods.

## Common mistakes

- **Fitting the modulus through grip-seating data** or using crosshead displacement as strain.
- **Using the actual (necked) area** and calling it engineering stress.
- **Reporting UTS as yield** for a material with no clear yield point.
- **Indenting too close** to an edge or another indent, or on a rough or curved surface, which distorts hardness.
- **Mixing scales**: an HRC value from a soft material, or a converted value reported as measured.

## Safety and training

Load frames store a lot of energy, and specimens can break suddenly and throw fragments. Hardness testers and specimen preparation (cutting, grinding, polishing) have their own hazards. You operate them only after your lab's training, with guards and eye protection, under its procedures. In this course you analyse data.

## Recap

- Stress = F / A0, strain = delta_L / L0. E is the elastic slope, yield is the 0.2% offset crossing, UTS is the peak.
- Vickers hardness = 1.8544 F / d^2 (kgf, mm); always report the scale and load.
- The standard method defines specimens, speeds and calculations, so name it in every report.

## Try it

Open the **Do** tab to compute modulus, offset yield, UTS and Vickers hardness from a synthetic steel test with numpy.


Turn load-extension data into stress and strain, then compute modulus, 0.2% offset yield, tensile strength and Vickers hardness.

Tensile and hardness testing

Understanding Material Strength, Ductility and Toughness

Rockwell hardness test simply explained (comparison with Brinell and Vickers)

## Why it matters

Most mechanical failures in service are not single overloads. They are **fatigue**: a crack starts at a stress concentration and grows a little with every load cycle until the remaining section breaks. Axles, aircraft skins, welds, springs, medical implants and bicycle frames are all designed around fatigue. **Fracture toughness** tells you how big a crack a part can tolerate before it breaks suddenly. Together they connect lab testing to safe-life and damage-tolerant design.

## Stress cycles

A cyclic load has a maximum `sigma_max` and a minimum `sigma_min`:

```text
stress amplitude  sigma_a = (sigma_max - sigma_min) / 2
mean stress       sigma_m = (sigma_max + sigma_min) / 2
stress ratio      R       = sigma_min / sigma_max
```

A "fully reversed" test (R = -1) has zero mean stress. Rotating-beam and axial tests at R = -1 are the classic source of S-N data.

## The S-N curve

Specimens are tested at several stress amplitudes until they fail, and amplitude is plotted against cycles to failure on a log scale. In the high-cycle region the curve is close to a straight line in log-log coordinates, described by **Basquin's equation**:

```text
sigma_a = sigma_f' * (2N)^b
```

`2N` is reversals (two per cycle), `sigma_f'` is the fatigue strength coefficient, and `b` is the fatigue strength exponent, typically between about -0.05 and -0.12 for metals. Because the line is straight in log-log space, you fit it with ordinary linear regression on `log10(sigma_a)` vs `log10(2N)`.

Many steels show a **fatigue (endurance) limit**: below a certain amplitude, roughly 10^6 to 10^7 cycles and beyond, they appear not to fail. Aluminium alloys do not show a clear limit, so designers quote a fatigue strength at a stated life, for example 5 × 10^8 cycles.

**Fatigue data scatter.** At one amplitude, lives can vary by a factor of 10 or more. Real design curves are statistical (for example, a 95% survival curve), not a line through the averages. Surface finish, size, notches, environment and residual stress all shift the curve, which is why test reports describe the specimen and surface in detail.

### Worked example

A steel tested at R = -1 gives these results:

| sigma_a (MPa) | 600 | 550 | 500 | 450 | 400 | 350 |
|---|---|---|---|---|---|---|
| N (cycles) | 2.4e3 | 6.4e3 | 1.9e4 | 6.3e4 | 2.3e5 | 1.0e6 |

A regression of `log10(sigma_a)` on `log10(2N)` gives `b` of about -0.089 and `sigma_f'` of about 1,280 MPa. The predicted life at 420 MPa is `N = 0.5 * (420 / 1280)^(1 / -0.089)`, about 1.3 × 10^5 cycles. Treat it as a median estimate, not a guarantee.

## Mean stress

A tensile mean stress shortens life. The simple **Goodman** correction converts a cycle with amplitude `sigma_a` and mean `sigma_m` into an equivalent fully reversed amplitude:

```text
sigma_ar = sigma_a / (1 - sigma_m / S_u)
```

With `sigma_a = 200 MPa`, `sigma_m = 150 MPa` and `S_u = 600 MPa`, `sigma_ar = 200 / 0.75 = 267 MPa`. Goodman is conservative for many ductile metals; others (Gerber, Morrow, SWT) exist and your employer's design method decides which one you use.

## Fracture toughness

Linear-elastic fracture mechanics describes the stress field at a crack tip with the **stress intensity factor**:

```text
K = Y * sigma * sqrt(pi * a)
```

`a` is the crack length (for an edge crack) or half-length (for a centre crack), and `Y` is a geometry factor (about 1.12 for a small edge crack in a wide plate). Fracture happens when `K` reaches the material's **plane-strain fracture toughness** `K_IC`, measured on pre-cracked specimens by a standard method such as ASTM E399. Rearranged, the **critical crack size** is

```text
a_c = (1 / pi) * (K_IC / (Y * sigma))^2
```

**Illustrative example.** A high-strength aluminium alloy with `K_IC = 25 MPa·sqrt(m)` carries 300 MPa with a small edge crack (`Y = 1.12`): `a_c = (1/pi) * (25 / 336)^2 = 0.00176 m`, about 1.8 mm. Inspections must reliably find cracks well below that size. A steel with twice the toughness tolerates a crack four times longer at the same stress, because `a_c` scales with `K_IC^2`.

Crack growth between inspection intervals is modelled with the **Paris law**, `da/dN = C (delta_K)^m`, using constants measured in crack-growth tests. That is the basis of damage-tolerant design in aerospace.

## Common mistakes

- **Fitting cycles instead of reversals** and then plugging the constants into the other form.
- **Ignoring scatter** and designing to the mean curve.
- **Using smooth-specimen data for a notched or welded part**: notches and welds need their own data or methods.
- **Using a non-valid K_IC**: thin specimens can give apparent toughness that is too high; the standard's validity checks matter.

Real fatigue and fracture-critical designs follow your industry's codes and are reviewed by qualified, often licensed, engineers.

## Recap

- S-N data follow Basquin's power law; fit it in log-log space and respect scatter.
- Mean stress matters: Goodman converts to an equivalent fully reversed amplitude.
- `K = Y sigma sqrt(pi a)`; the critical crack size scales with `(K_IC / sigma)^2`.

## Try it

Open the **Do** tab to fit Basquin constants to S-N data, predict a life and compute a critical crack size.


Read and fit an S-N curve, correct for mean stress, and use fracture toughness to estimate a critical crack size.

Fatigue and fracture testing

Understanding Fatigue Failure and S-N Curves

FRACTURE TOUGHNESS and Crack Modes  in Under 10 Minutes!

Go to https://www.astm.org and search for E8/E8M. Open the standard's page (no purchase needed) and note its full title, the current year suffix and what the scope says it covers.

Check whether your school or public library gives access to ASTM Compass or ISO standards. Write down how you would get a controlled copy at work (ask your lab's quality or document control lead).

From the free summary, list the referenced standards it mentions for machine force verification and extensometers, and look up their titles too.

Search iso.org for ISO 6892-1 and write two sentences comparing it with E8/E8M (scope and units). Note which one a European customer would probably call up.

On paper, write a one-page test plan: material and product form, standard and edition, number of specimens, the properties to report (modulus, 0.2% offset yield, UTS, elongation), and which equipment needs current calibration.

Draft a report template with the fields a test report needs: specimen IDs, dimensions, standard and edition, test speed, temperature, results per specimen, mean and standard deviation, deviations, and operator and date.

Add a 'deviations' line with one realistic example (for example, one specimen broke outside the gauge length) and write how you would record it instead of dropping it.

## Why it matters

A number like "yield strength 350 MPa" means little without the method behind it. Specimen shape, gauge length, test speed, temperature, conditioning and how each property is calculated all change the result. **Standard test methods** fix those choices so that a lab in Ohio and a lab in Osaka get comparable numbers, customers can write specifications, and auditors can check your work. Materials scientists read, follow and cite standards every week.

## Who writes standards

- **ASTM International** (https://www.astm.org) publishes over 12,000 standards written by volunteer committees of producers, users and academics. Metals testing lives mostly in committee E28, plastics in D20.
- **ISO** (https://www.iso.org) publishes international standards, adopted nationally (for example as EN ISO in Europe).
- Industry and government bodies add their own: SAE/AMS for aerospace materials, ASME for pressure equipment, military specifications, and customer specifications that call up ASTM or ISO methods.

Standards are copyrighted and most are sold, not free. Many universities and employers subscribe; check your library before buying. ASTM's site shows each standard's title, scope and summary free, which is often enough to choose the right one. Checked on 2026-10-23.

## Reading a designation

Take **ASTM E8/E8M-24**, the tension test for metals:

- `E` is the category (miscellaneous subjects, which includes most metals test methods); `D` is used for plastics and many other materials.
- `8` is the number.
- `/E8M` means the standard covers both inch-pound and SI units in one document.
- `-24` is the year of the last revision. A trailing year in parentheses, like `(2019)`, means it was reapproved without changes; a superscript epsilon marks an editorial change.

Always record the exact edition you used. Contracts and specifications often call up a specific year.

### Common methods in a materials lab

| method | what it covers |
|---|---|
| ASTM E8/E8M, ISO 6892-1 | Tension testing of metals at room temperature |
| ASTM E18 | Rockwell hardness of metals |
| ASTM E92, ASTM E384 | Vickers and Knoop hardness; microindentation hardness |
| ASTM E466 | Force-controlled constant-amplitude axial fatigue of metals |
| ASTM E399 | Plane-strain fracture toughness (K_IC) of metals |
| ASTM D638 | Tensile properties of plastics |
| ASTM D3418, ASTM E1356 | DSC transition temperatures of polymers; glass transition by DSC |
| ASTM E1131 | Compositional analysis by thermogravimetry |

Use this list only to know where to look. The documents themselves define the details.

## The structure of a test method

Most ASTM methods follow the same outline, so once you have read one you can navigate any of them:

1. **Scope**: what materials and conditions the method covers, and what it does not.
2. **Referenced documents**: other standards you also need (for example, E4 for force verification of the machine, E83 for extensometers).
3. **Terminology**, **summary** and **significance and use**.
4. **Apparatus**: required equipment and its calibration.
5. **Test specimens**: dimensions, tolerances, how to take them from the product.
6. **Procedure**: speeds, temperatures, conditioning, the exact sequence.
7. **Calculation** of each property.
8. **Report**: the information you must include.
9. **Precision and bias**: how much results vary within and between labs.

The **report** section is your checklist. The **precision and bias** section tells you whether a difference between two results is real or within normal lab scatter.

## Following a method in practice

- Work from the **current controlled copy** your lab holds, not an old PDF.
- Many labs write a short **work instruction** that applies the standard to their own machines. The standard wins when they conflict, unless the customer specification says otherwise.
- Any **deviation** (a different specimen size, a missed conditioning time) must be recorded in the report, not quietly absorbed.
- Calibration matters: load cells, extensometers and thermocouples are verified to their own standards at set intervals.
- Accredited labs work to **ISO/IEC 17025**, which audits exactly this: methods, calibration, records and competence.

## Common mistakes

- **Citing a standard without the year**, or using a superseded edition.
- **Using the wrong specimen geometry** and comparing results with data from another geometry.
- **Skipping conditioning** for plastics, whose properties depend on temperature and humidity.
- **Reporting fewer specimens** than the method requires, or leaving out failed tests without a stated reason.

## Safety and training

Running tests means operating machines, furnaces and chemicals covered by your lab's procedures. Follow your institution's training and supervision before running any method for real. The activity below is a paper exercise.

## Recap

- Standards fix the choices that change results; always cite designation and year.
- ASTM and ISO methods share a structure: scope, apparatus, specimens, procedure, calculation, report, precision.
- Record deviations, keep calibrations current, and use the report section as your checklist.

## Try it

Open the **Do** tab to find a method on the ASTM website and draft a test plan and report outline that follow it.


Find the right ASTM or ISO test method, read its structure, and plan a test and report that follow it.

ASTM methods

ISO and ASTM Standards Explained Simply

Compute properties from tensile and hardness tests, read S-N curves and fracture toughness data, and find and follow ASTM test methods.