Exercise set

Materials Characterization, Testing, and NDE Exercises

Worked materials engineering exercises for characterization, testing, and non-destructive evaluation covering tensile properties, modulus, hardness maps, CT resolution, ultrasonic thickness, XRD, XRF screening, defect margins, measurement uncertainty, sampling, and validation evidence.

Branch: Materials Engineering
Content: Exercise set
Updated: Jun 20, 2026
Revision: v1.1.0 · reviewed

These exercises practise materials characterization, mechanical testing, and non-destructive evaluation as engineering evidence. They cover tensile-property extraction, modulus estimation, hardness mapping, CT resolution, ultrasonic thickness, XRD spacing, XRF screening, defect detection margin, measurement uncertainty, sampling evidence, and validation readiness.

The goal is not only to calculate a test result. The goal is to decide whether the evidence is strong enough to accept a material, release a process, inspect a critical defect, or update a design basis.

Assume simplified screening models unless an exercise states otherwise. Real test and inspection decisions should also check product form, specimen orientation, surface condition, calibration, fixture alignment, temperature, environment, data processing, acceptance criteria, inspector qualification, and traceability.

How to Use These Exercises

For each exercise, define:

the engineering decision supported by the measurement;
the material state and product form represented by the specimen;
the critical property, defect, or phase being assessed;
the measurement uncertainty or detection limit;
the release action if the result is marginal or incomplete.

The common mistake is collecting data without connecting it to a decision. A test is engineering evidence only when it answers a requirement, risk-control, manufacturing, or service-life question.

Use the exercises as release controls: accept or reject a property value, hold a material lot, qualify an inspection method, challenge a detection claim, require confirmatory testing, update a process window, or refuse production release when calibration, uncertainty, sampling, or operator evidence is incomplete.

Exercise 1: Engineering Stress from Tensile Load

A tensile specimen has original cross-sectional area:

A_0=50\ \text{mm}^2

During a universal testing machine test, the measured load is:

F=18{,}000\ \text{N}

Calculate the engineering stress.

Solution

Engineering stress is:

\displaystyle \sigma_e=\frac{F}{A_0}

Convert area:

50\ \text{mm}^2=50\times10^{-6}\ \text{m}^2

Substitute:

\displaystyle \sigma_e=\frac{18{,}000}{50\times10^{-6}}=360{,}000{,}000\ \text{Pa}

Therefore:

\sigma_e=360\ \text{MPa}

Engineering Comment

This value is only meaningful if the specimen geometry, grip alignment, strain rate, temperature, surface condition, and product orientation represent the material state used in design. A tensile coupon from the wrong orientation can overstate strength for rolled plate, forged stock, additive builds, and composites.

Exercise 2: Elastic Modulus from Stress-Strain Data

In the linear elastic region of a tensile test, two data points are:

Stress	Strain
80 MPa	0.00040
180 MPa	0.00090

Estimate the elastic modulus from these points.

Solution

Elastic modulus is the slope:

\displaystyle E=\frac{\Delta\sigma}{\Delta\epsilon}

Stress change is:

\Delta\sigma=180-80=100\ \text{MPa}

Strain change is:

\Delta\epsilon=0.00090-0.00040=0.00050

Therefore:

\displaystyle E=\frac{100\ \text{MPa}}{0.00050}=200{,}000\ \text{MPa}

Convert:

E=200\ \text{GPa}

Engineering Comment

The result is plausible for steel-like materials, but modulus extraction is sensitive to extensometer setup, grip seating, machine compliance, strain range, and data filtering. If stiffness is a design-critical requirement, the test method should state how the linear region was selected.

Exercise 3: Ductility from Elongation

A tensile specimen has gauge length before testing:

L_0=50\ \text{mm}

After fracture and rejoining the specimen halves, the final gauge length is:

L_f=61\ \text{mm}

Calculate percent elongation.

Solution

Percent elongation is:

\displaystyle \%EL=\frac{L_f-L_0}{L_0}\times100

Substitute:

\displaystyle \%EL=\frac{61-50}{50}\times100=22\%

Engineering Comment

Ductility supports damage tolerance and forming capability, but it is not proof of fracture toughness or fatigue resistance. The fracture location, necking behavior, surface defects, and specimen orientation should be recorded because they affect interpretation.

Exercise 4: Hardness Map Acceptance

A heat-treated component requires hardness between:

H_{min}=280\ \text{HV}

and:

H_{max}=340\ \text{HV}

A local hardness map gives:

Location	Hardness
A	305 HV
B	318 HV
C	276 HV
D	331 HV
E	288 HV

Decide whether the component passes the hardness requirement.

Solution

Compare each value with the acceptance range:

A: 305 HV, pass;
B: 318 HV, pass;
C: 276 HV, fail because it is below 280 HV;
D: 331 HV, pass;
E: 288 HV, pass.

Because one measured location is outside the acceptance range, the component fails the hardness screen.

Engineering Comment

The engineering response should investigate whether location C represents a heat-treatment shadow, surface preparation issue, decarburization, local repair, measurement error, or real process drift. Repeating the test without root-cause review can hide a material-state problem.

Exercise 5: Ultrasonic Thickness and Corrosion Rate

An ultrasonic thickness baseline measured a pipe wall at:

t_1=8.20\ \text{mm}

Four years later, the same location measures:

t_2=7.72\ \text{mm}

Estimate the apparent wall-loss rate.

Solution

Wall loss is:

\Delta t=t_1-t_2

\Delta t=8.20-7.72=0.48\ \text{mm}

The rate is:

\displaystyle r=\frac{\Delta t}{\Delta T}

\displaystyle r=\frac{0.48}{4}=0.12\ \text{mm/year}

Engineering Comment

This rate is meaningful only if both readings use comparable calibration, location control, surface preparation, couplant, probe frequency, and measurement grid. If the degradation is pitting rather than uniform thinning, a single spot reading can underestimate local risk.

Exercise 6: Detection Margin for a Critical Crack

A fracture analysis shows that a surface crack becomes critical at:

a_c=1.6\ \text{mm}

The selected inspection method is qualified to detect cracks of:

a_d=0.7\ \text{mm}

Calculate the detection margin.

Solution

Detection margin is:

M_a=a_c-a_d

M_a=1.6-0.7=0.9\ \text{mm}

The inspection method has a 0.9 mm margin between qualified detection size and critical crack size.

Engineering Comment

This is acceptable only if the 0.7 mm detection claim was qualified for the actual material, crack orientation, surface condition, geometry, access, and inspector procedure. A detection claim from flat reference coupons may not transfer to curved weld toes, rough castings, composite laminates, or additive parts.

Exercise 7: CT Voxel Size and Minimum Resolvable Feature

An x-ray CT scan has voxel size:

v=18\ \mu\text{m}

The inspection procedure requires a defect to span at least four voxels across its smallest dimension before it can be measured reliably.

Estimate the minimum reliably measurable defect size.

Solution

Minimum measurable size is:

d_{min}=4v

d_{min}=4(18)=72\ \mu\text{m}

The minimum reliably measurable defect size is about 72 micrometres.

Engineering Comment

Voxel size is not the same as validated detection capability. Material density, part thickness, beam hardening, reconstruction settings, thresholding, noise, and defect contrast can make real detection worse than the simple voxel calculation suggests.

Exercise 8: Bragg Spacing from X-Ray Diffraction

An XRD peak is measured at:

2\theta=44^\circ

using x-ray wavelength:

\lambda=0.154\ \text{nm}

Assume first-order diffraction:

n=1

Use Bragg’s law:

n\lambda=2d\sin\theta

Estimate the lattice-plane spacing (d).

Solution

First compute:

\displaystyle \theta=\frac{2\theta}{2}=22^\circ

Rearrange Bragg’s law:

\displaystyle d=\frac{n\lambda}{2\sin\theta}

Substitute:

\displaystyle d=\frac{(1)(0.154)}{2\sin(22^\circ)}

Using:

\sin(22^\circ)\approx0.375

therefore:

\displaystyle d=\frac{0.154}{2(0.375)}=0.205\ \text{nm}

Engineering Comment

XRD can support phase identification, residual-stress studies, or process verification, but a single peak does not fully characterize a material. Calibration, surface condition, peak overlap, texture, phase mixtures, and stress state affect interpretation.

Exercise 9: XRF Alloy Screening

An incoming stainless-steel lot is screened by XRF. The specification requires chromium content between:

Cr_{min}=17.0\%

and:

Cr_{max}=19.0\%

The measured chromium content is:

Cr_{meas}=16.6\%

The measurement uncertainty is:

U=0.3\%

Use a conservative guard-band rule: accept only if the measured value minus uncertainty is at or above the lower limit and the measured value plus uncertainty is at or below the upper limit.

Decide whether the lot passes this chromium screen.

Solution

Lower guarded value:

Cr_{low}=Cr_{meas}-U=16.6-0.3=16.3\%

Upper guarded value:

Cr_{high}=Cr_{meas}+U=16.6+0.3=16.9\%

The lower guarded value is below the minimum:

16.3\%<17.0\%

The lot fails the conservative chromium screen.

Engineering Comment

XRF is useful for material identification and sorting, but it does not prove heat treatment, mechanical properties, passivation quality, surface contamination absence, or corrosion performance. A failed screen should trigger segregation, confirmatory analysis, supplier review, or disposition by engineering authority.

Exercise 10: Measurement Uncertainty Against an Acceptance Limit

A machined material coupon must have thickness at least:

t_{min}=2.00\ \text{mm}

A measurement gives:

t_{meas}=2.04\ \text{mm}

with expanded uncertainty:

U=0.06\ \text{mm}

Using a conservative rule, accept only if:

t_{meas}-U\geq t_{min}

Decide whether the measurement supports acceptance.

Solution

Compute the guarded thickness:

t_g=t_{meas}-U

t_g=2.04-0.06=1.98\ \text{mm}

Compare with the limit:

1.98\ \text{mm}<2.00\ \text{mm}

The measurement does not support acceptance under the conservative rule.

Engineering Comment

The part may or may not actually be below the limit, but the evidence is insufficient for guarded acceptance. The team should review measurement method, calibration, surface condition, repeatability, and whether another qualified method can reduce uncertainty.

Exercise 11: Sampling Pass Fraction

A process qualification builds 24 test coupons from different production locations and orientations. The requirement is that at least 95 percent of coupons meet the acceptance criteria.

The number of passing coupons is:

N_p=22

The total number tested is:

N=24

Calculate the observed pass fraction and decide whether the process meets this simple criterion.

Solution

Pass fraction is:

\displaystyle f_p=\frac{N_p}{N}

\displaystyle f_p=\frac{22}{24}=0.917

Convert to percentage:

f_p=91.7\%

Compare with the requirement:

91.7\%<95\%

The process fails the simple pass-fraction criterion.

Engineering Comment

The failed coupons should be more informative than the average result. Their location, orientation, fracture appearance, process history, defect population, and measurement records can reveal whether the issue is random scatter, a local process variable, an inspection method problem, or an uncontrolled failure mode.

Exercise 12: Validation Evidence Completion

A method-qualification plan requires eight evidence items:

material and geometry represented by reference samples;
defect type represented by known flaws;
critical defect size defined by engineering analysis;
detection threshold demonstrated below the critical size;
calibration and setup procedure documented;
operator competency recorded;
repeatability checked;
acceptance and reporting criteria released.

At design freeze, six evidence items are complete.

Calculate the completion fraction. If the project requires all eight items before production release, decide whether the inspection method is ready.

Solution

Completion fraction is:

\displaystyle f=\frac{6}{8}=0.75

Convert:

f=75\%

Because two required items are incomplete, the method is not ready for production release.

Engineering Comment

For critical inspection methods, partial completion is not enough. Missing calibration, repeatability, operator competency, or reporting criteria can invalidate otherwise strong laboratory evidence. The release decision should be tied to completed evidence, not optimism about future procedure updates.

Review Checklist

When reviewing characterization, testing, or NDE evidence, ask:

What engineering decision does the measurement support?
Does the specimen or inspected part represent the real product form, orientation, surface condition, and process route?
Are acceptance limits tied to the failure mode, not only workmanship language?
Is the detection limit below the critical defect size with documented margin?
Are calibration, uncertainty, repeatability, and operator effects included?
Are nonconforming results investigated rather than averaged away?
Are guard bands and decision rules stated before the data are interpreted?
Can the result be traced to material lot, process records, instrument settings, procedure revision, and raw data?
Is validation evidence representative of the real geometry, defect orientation, surface state, access condition, and production variability?

Good characterization turns invisible assumptions into reviewable evidence. The strongest test plan is the one that lets engineers make a defensible accept, reject, repair, redesign, or revalidate decision.

REF