Exercise set

Materials Processing and Manufacturing Routes Exercises

Worked materials engineering exercises for processing and manufacturing routes covering material yield, forming strain, heat-treatment acceptance, welding heat input, sintering shrinkage, additive anisotropy, CT porosity, coating thickness, process capability, rheology, repair limits, and validation evidence.

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

These exercises practise materials processing and manufacturing route selection as engineering decision-making. They cover material yield, forming strain, heat-treatment acceptance, welding heat input, sintering shrinkage, additive-manufacturing anisotropy, CT porosity, coating thickness effects, process capability, rheology, repair limits, and validation evidence.

The goal is not only to calculate a manufacturing number. The goal is to decide whether a route can repeatedly produce the material state, geometry, surface integrity, defect population, and inspection evidence assumed by design.

Assume simplified screening models unless an exercise states otherwise. Real process qualification should also check supplier capability, material lot, product orientation, thermal history, residual stress, surface condition, inspection access, process windows, sampling strategy, repair rules, and change control.

How to Use These Exercises

For each exercise, define:

the manufacturing route and the material state it is expected to create;
the critical-to-quality characteristic being controlled;
the failure mode or production risk connected to that characteristic;
the evidence needed to accept, reject, repair, or requalify the part;
the process change that would invalidate the current evidence.

The common mistake is treating manufacturing as execution after design. For materials engineering, manufacturing is part of the design basis.

Use the exercises as production gates: accept or reject a route, adjust a process window, quarantine a lot, require repair disposition, requalify an inspection plan, challenge capability evidence, or stop release when material state, defect population, or change-control evidence no longer matches the design assumption.

Exercise 1: Material Yield from Stock to Finished Part

A machined component is produced from a forged blank with mass:

m_{blank}=12.5\ \text{kg}

The final component mass is:

m_{part}=7.8\ \text{kg}

Calculate the material yield and the removed mass.

Solution

Material yield is:

\displaystyle Y=\frac{m_{part}}{m_{blank}}

\displaystyle Y=\frac{7.8}{12.5}=0.624

Convert:

Y=62.4\%

Removed mass is:

m_{removed}=m_{blank}-m_{part}=12.5-7.8=4.7\ \text{kg}

Engineering Comment

Low material yield may be acceptable for a fatigue-critical forged part if the route delivers grain flow, defect control, and inspection access. It becomes questionable if the same performance could be achieved with a near-net route that preserves quality while reducing machining time, waste, and cost.

Exercise 2: Thickness Strain in Sheet Forming

A sheet-metal part starts with thickness:

t_0=2.00\ \text{mm}

After forming, a critical region measures:

t_f=1.72\ \text{mm}

Calculate percent thinning and decide whether it passes a maximum allowed thinning of 12 percent.

Solution

Percent thinning is:

\displaystyle \%T=\frac{t_0-t_f}{t_0}\times100

Substitute:

\displaystyle \%T=\frac{2.00-1.72}{2.00}\times100=14\%

Compare:

14\%>12\%

The forming screen fails.

Engineering Comment

The response should review local strain path, blank holder force, lubrication, die radius, edge condition, material anisotropy, and forming simulation. A tensile coupon can pass while the formed geometry fails locally because strain is concentrated by tooling and geometry.

Exercise 3: Heat-Treatment Hardness Acceptance

A heat-treated gear requires surface hardness:

58\leq HRC\leq62

Five measured values are:

Location	Hardness
1	60 HRC
2	61 HRC
3	57 HRC
4	59 HRC
5	62 HRC

Decide whether the part passes the hardness screen.

Solution

The acceptance interval is 58 to 62 HRC.

Locations 1, 2, 4, and 5 pass. Location 3 fails because:

57<58

The part fails the hardness screen.

Engineering Comment

A single low location may indicate nonuniform quench, inadequate case depth, decarburization, fixture shadowing, grinding damage, or measurement error. The next step is not only retesting; it is understanding whether the local material state still supports tooth contact fatigue, bending fatigue, wear, and distortion requirements.

Exercise 4: Welding Heat Input

A welding procedure uses:

V=24\ \text{V}

I=180\ \text{A}

Travel speed is:

S=6.0\ \text{mm/s}

Estimate heat input per unit length using:

\displaystyle Q=\frac{VI}{S}

Express the result in kJ/mm.

Solution

Compute:

\displaystyle Q=\frac{(24)(180)}{6.0}=720\ \text{J/mm}

Convert:

Q=0.720\ \text{kJ/mm}

Engineering Comment

Heat input affects cooling rate, heat-affected-zone hardness, distortion, residual stress, toughness, and corrosion sensitivity. The number should be checked against a qualified welding procedure, including joint thickness, preheat, interpass temperature, filler material, shielding, and inspection method.

Exercise 5: Sintering Shrinkage

A ceramic green part has diameter:

D_0=42.0\ \text{mm}

After sintering, the diameter is:

D_f=35.7\ \text{mm}

Calculate linear shrinkage percentage.

Solution

Linear shrinkage is:

\displaystyle S_L=\frac{D_0-D_f}{D_0}\times100

Substitute:

\displaystyle S_L=\frac{42.0-35.7}{42.0}\times100=15.0\%

Engineering Comment

Shrinkage must be predictable across geometry, lot, furnace load, binder removal, atmosphere, and powder condition. If shrinkage varies spatially, the part can distort, crack, or miss critical dimensions even when average shrinkage looks correct.

Exercise 6: Additive-Manufacturing Anisotropy

An additively manufactured alloy has tensile strength parallel to the build plane:

\sigma_{xy}=620\ \text{MPa}

and tensile strength in the build direction:

\sigma_z=515\ \text{MPa}

Calculate the build-direction strength ratio:

\displaystyle R_z=\frac{\sigma_z}{\sigma_{xy}}

Solution

Substitute:

\displaystyle R_z=\frac{515}{620}=0.831

Convert:

R_z=83.1\%

Engineering Comment

The build direction is about 17 percent weaker in this simplified comparison. This may affect orientation, support strategy, heat treatment, machining stock, inspection plan, and whether design allowables must be direction-specific.

Exercise 7: CT Porosity Volume Fraction

An x-ray CT analysis of a small casting region finds total pore volume:

V_p=18\ \text{mm}^3

within inspected material volume:

V_i=4{,}500\ \text{mm}^3

Calculate porosity volume fraction as a percentage.

Solution

Porosity fraction is:

\displaystyle \phi=\frac{V_p}{V_i}

\displaystyle \phi=\frac{18}{4{,}500}=0.004

Convert:

\phi=0.4\%

Engineering Comment

Porosity percentage alone is incomplete. A small number of pores near a fatigue-critical fillet can be worse than the same total pore volume distributed in low-stress material. Acceptance should include pore size, shape, location, clustering, inspection resolution, and stress field.

Exercise 8: Coating Thickness and Shaft Fit

A cylindrical shaft has machined diameter before coating:

D_0=24.960\ \text{mm}

The coating adds thickness:

h=18\ \mu\text{m}

on each side. The maximum allowed finished diameter is:

D_{max}=25.000\ \text{mm}

Calculate the coated diameter and decide whether the shaft passes the dimensional screen.

Solution

Convert coating thickness:

18\ \mu\text{m}=0.018\ \text{mm}

Coating adds to both sides of the diameter:

D_f=D_0+2h

D_f=24.960+2(0.018)=24.996\ \text{mm}

Compare:

24.996\ \text{mm}<25.000\ \text{mm}

The shaft passes the maximum-diameter screen with margin:

M=25.000-24.996=0.004\ \text{mm}

Engineering Comment

The margin is only 4 micrometres. The process plan should check coating-thickness variation, measurement uncertainty, masking, edge buildup, surface roughness, and whether post-coating finishing is allowed without damaging corrosion or wear performance.

Exercise 9: Process Capability for a Critical Diameter

A process produces a critical diameter with specification limits:

LSL=19.95\ \text{mm}

USL=20.05\ \text{mm}

The measured process mean is:

\mu=20.01\ \text{mm}

and standard deviation is:

s=0.012\ \text{mm}

Calculate the process capability index:

\displaystyle C_{pk}=\min\left(\frac{USL-\mu}{3s},\frac{\mu-LSL}{3s}\right)

Solution

Upper capability term:

\displaystyle \frac{USL-\mu}{3s}=\frac{20.05-20.01}{3(0.012)}=\frac{0.04}{0.036}=1.11

Lower capability term:

\displaystyle \frac{\mu-LSL}{3s}=\frac{20.01-19.95}{3(0.012)}=\frac{0.06}{0.036}=1.67

Therefore:

C_{pk}=1.11

Engineering Comment

The process is closer to the upper limit than the lower limit. If the project requires Cpk of 1.33 or higher for production release, this process is not ready. Corrective action may include centering the mean, reducing variation, improving measurement capability, or revising the route.

Exercise 10: Resin Flow Time from Viscosity Screen

A simplified resin-transfer process uses the screening relation:

\displaystyle t_f=\frac{\mu L^2}{K\Delta P}

where fill time is proportional to viscosity. A baseline resin with viscosity:

\mu_1=0.45\ \text{Pa}\cdot\text{s}

fills the mold in:

t_1=210\ \text{s}

If a colder resin batch has viscosity:

\mu_2=0.72\ \text{Pa}\cdot\text{s}

estimate the new fill time assuming geometry, permeability, and pressure are unchanged.

Solution

Because all other terms are unchanged:

\displaystyle \frac{t_2}{t_1}=\frac{\mu_2}{\mu_1}

Therefore:

\displaystyle t_2=t_1\frac{\mu_2}{\mu_1}

\displaystyle t_2=210\frac{0.72}{0.45}=336\ \text{s}

Engineering Comment

Higher viscosity increases fill time and may raise risk of dry fiber, voids, incomplete wet-out, cure advancement, or excessive pressure. The process window should define resin temperature, viscosity limit, injection pressure, venting, and acceptance evidence.

Exercise 11: Repair Limit for Weld Undercut Length

A weld inspection procedure permits total undercut length per 300 mm inspection segment of at most:

L_{max}=12\ \text{mm}

Measured undercut indications in one segment are:

Indication	Length
A	3 mm
B	5 mm
C	6 mm

Calculate total undercut length and decide whether repair is required.

Solution

Total undercut length is:

L_{tot}=3+5+6=14\ \text{mm}

Compare:

14\ \text{mm}>12\ \text{mm}

The segment fails the undercut limit and requires disposition.

Engineering Comment

Disposition may be repair, engineering acceptance, additional inspection, or rejection depending on stress level, fatigue consequence, weld class, geometry, and repair risk. A repair can introduce new heat input and residual stress, so repair rules need their own acceptance criteria.

Exercise 12: Production-Release Evidence Completion

A production-release package for a safety-critical additively manufactured bracket requires ten evidence items:

approved build orientation;
qualified parameter set;
powder lot traceability;
heat-treatment record;
machining and surface-finish record;
CT inspection acceptance;
tensile coupon results from matching orientation;
fatigue-critical surface inspection;
repair procedure;
change-control plan.

Eight items are complete at release review.

Calculate the completion percentage and decide whether the package can be released if all ten items are mandatory.

Solution

Completion fraction is:

\displaystyle f=\frac{8}{10}=0.80

Convert:

f=80\%

Because all ten evidence items are mandatory, the package cannot be released even though 80 percent is complete.

Engineering Comment

Mandatory evidence items are not interchangeable. Missing repair rules or change control can create lifecycle risk even when build data and inspection results look strong. Production release should wait until the complete process basis is documented and controlled.

Review Checklist

When reviewing a material-process route, ask:

Does the route create the material state assumed by design?
Are route-specific defects identified before production starts?
Are coupons and inspections representative of part geometry, orientation, heat treatment, surface condition, and critical locations?
Are process parameters, tooling, suppliers, powders, fillers, resins, coatings, and repairs under change control?
Are acceptance limits tied to fatigue, fracture, corrosion, leakage, wear, biocompatibility, electrical performance, or dimensional function?
Does inspection detect the defects that matter, rather than only convenient defects?
Are capability statistics supported by a stable process, representative sampling, and a measurement system that is fit for the tolerance?
Are nonconformances used to improve process control instead of being absorbed as production noise?
Does release evidence define what process, supplier, material-lot, tooling, or repair change would require requalification?

Good materials processing is controlled transformation. The manufacturing route is successful only when it repeatedly produces the intended material state with evidence strong enough for the failure consequence.

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