Exercise set

Polymer and Composite Materials Design Exercises

Solved polymer and composite exercises for mixture rules, fiber volume, laminate balance, adhesive and bolted joints, moisture, creep, voids and impact.

Branch: Materials Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

These exercises focus on polymers and fiber composites: mixture rules, fiber volume, laminate balance, adhesive and bolted load introduction, moisture uptake, creep, cure margin, permeability, voids, impact knockdowns and validation evidence. Ceramic fracture and thermal-shock practice is handled in a separate specialist exercise set.

Use the calculations as material-system screens. Real release needs controlled material grade, batch, cure or molding route, fiber architecture, surface preparation, inspection evidence, environmental conditioning, repair rules and approved allowables.

How to use these exercises

Work through the set as a design-release sequence, not as isolated arithmetic. Exercises 1 to 4 establish whether the assumed fiber fraction, density data and ply mix are internally consistent. Exercises 5 to 8 then check how load is introduced into the composite through adhesive and bolted features, where local failure can dominate global laminate strength. Exercises 9 to 17 add the environmental, process and damage variables that normally decide whether a polymer or composite result can be released. Exercise 18 combines those gates into a hold-or-release decision.

For each calculation, write down the material system, direction, unit basis and service state before using the equation. A stiffness or allowable from a dry coupon, a quasi-isotropic laminate, a wet hot environment, a repaired panel and a bolted lug are not interchangeable. The engineering comment below each exercise is the minimum interpretation step: it tells which assumption must be verified before the numeric result can support a design decision.

Release Evidence Notes

Polymer and composite evidence should state material grade, fiber and matrix system, layup, cure route, void limit, moisture state, temperature, surface preparation, joining method, inspection method, damage state, direction-specific allowable and release authority. The record should also identify whether the value comes from coupon testing, handbook data, supplier data, a project-specific qualification program or a conservative preliminary estimate.

The evidence package must keep manufacturing and service conditions together. A high fiber volume fraction is not useful if the associated void content, resin-rich zones, ply waviness or incomplete cure are outside the qualified process window. Likewise, a passing dry room-temperature allowable does not release a wet hot part, a bonded repair, a pressurized barrier or a damage-tolerant aerospace structure unless the relevant knockdowns and inspection assumptions are included.

Engineering Boundary Notes

The exercises use first-pass effective properties and simple stress screens. They do not replace laminate analysis, coupon allowables, durability testing, repair manuals or process qualification. Treat the results as gates for review: pass results justify continuing the design route, while failed results identify the next change in layup, material, cure, joint detail, inspection method or environmental derating.

Directionality is the main boundary. A unidirectional stiffness estimate cannot be reused for transverse loading, shear, bearing, compression after impact or fatigue without the appropriate test basis. Time and environment are the second boundary. Polymers and polymer-matrix composites can change behavior with moisture, solvent exposure, temperature, creep duration, ultraviolet exposure and thermal cycling, so a release calculation must name the exposure state.

Common Release Mistakes

Common mistakes include using fiber-direction stiffness for off-axis load, ignoring moisture and creep, accepting adhesive averages without edge effects, treating void content as harmless, and releasing a composite joint without bearing, net-section and edge-distance checks. Another frequent error is mixing mass fraction and volume fraction, which can shift stiffness and density estimates enough to hide a wrong architecture.

Do not use a single favorable test result as a material allowable. Release values need statistical basis, batch traceability, test direction, conditioning state and failure mode review. Also avoid treating non-destructive inspection as a substitute for process control: inspection may find voids, delamination or impact damage, but it does not prove cure, surface chemistry, fiber alignment or long-term environmental durability unless those checks are explicitly qualified.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Composite architecture	1, 2, 3, 4, 13	Modulus, volume fraction, balance and knockdowns	Use, revise layup or reject assumption.
Interfaces and joints	5, 6, 7, 8, 18	Adhesive shear, bearing, net section, edge distance and repair overlap	Accept or redesign load introduction.
Polymer environment and defects	9, 10, 11, 12, 14, 15, 16, 17	Moisture, creep, Tg, cure, void, permeation, impact and validation	Condition, derate, inspect or hold release.

Exercise 1: Composite Longitudinal Modulus

A unidirectional composite has $V_f=0.58$ , fiber modulus 230 GPa and matrix modulus 3.2 GPa. Estimate longitudinal modulus.

Solution

E_1=V_fE_f+(1-V_f)E_m=0.58(230)+0.42(3.2)=134.7\ \text{GPa}

Engineering Comment

This is a fiber-direction screen and says little about transverse, shear or impact behavior.

Plausibility Check

The result is much closer to fiber modulus because fiber volume is high.

Exercise 2: Transverse Modulus

Use inverse rule of mixtures with the same data. Estimate transverse modulus.

Solution

E_2=\left(\dfrac{0.58}{230}+\dfrac{0.42}{3.2}\right)^{-1}=7.48\ \text{GPa}

Engineering Comment

Matrix-dominated stiffness is far lower than fiber-direction stiffness.

Plausibility Check

The result is closer to matrix modulus than fiber modulus.

Exercise 3: Fiber Volume from Mass Fraction

Fiber mass fraction is 0.68. Fiber density is 1.80 g/cm3 and matrix density is 1.20 g/cm3. Estimate fiber volume fraction.

Solution

V_f=\dfrac{0.68/1.80}{0.68/1.80+0.32/1.20}=0.586

Engineering Comment

Volume fraction, not mass fraction, controls most mixture-rule stiffness estimates.

Plausibility Check

The volume fraction is lower than mass fraction because fiber density is higher.

Exercise 4: Laminate Balance

A laminate has 10 plies: four 0-degree, two 90-degree and four +/-45-degree plies. Find 0-degree ply fraction.

Solution

f_0=\dfrac{4}{10}=40\%

Engineering Comment

Ply fractions should match load path, buckling, bearing and damage-tolerance needs.

Plausibility Check

Four out of ten plies is directly 40 percent.

Exercise 5: Adhesive Average Shear

A bonded lap joint carries 12 kN over a 25 mm by 50 mm bond area. Find average shear stress.

Solution

\tau=\dfrac{12000}{25(50)}=9.6\ \text{MPa}

Engineering Comment

Average shear hides peel and edge stress concentrations.

Plausibility Check

12,000 N over 1250 mm2 gives about 10 MPa.

Exercise 6: Bolted-Joint Bearing Stress

A composite lug carries 8 kN through a 6 mm bolt in a 3 mm thick laminate. Find bearing stress.

Solution

\sigma_b=\dfrac{8000}{6(3)}=444\ \text{MPa}

Engineering Comment

Bearing allowable depends on layup, hole quality, clamping and moisture state.

Plausibility Check

The bearing area is only 18 mm2, so stress is high.

Exercise 7: Net-Section Stress

The lug width is 24 mm, hole diameter 6 mm and thickness 3 mm under 8 kN. Find net-section stress.

Solution

\sigma_n=\dfrac{8000}{(24-6)(3)}=148\ \text{MPa}

Engineering Comment

Net-section failure may control even when bearing appears acceptable.

Plausibility Check

The net area is 54 mm2, so stress is much lower than bearing stress.

Exercise 8: Edge-Distance Ratio

Bolt center is 11 mm from the laminate edge and bolt diameter is 6 mm. Find edge-distance ratio.

Solution

e/d=\dfrac{11}{6}=1.83

Engineering Comment

Low edge distance increases tear-out risk and may invalidate simple bearing checks.

Plausibility Check

11 mm is slightly less than two bolt diameters.

Exercise 9: Moisture Uptake

A polymer part increases from 82.0 g to 83.4 g after conditioning. Find moisture uptake percent.

Solution

M=\dfrac{83.4-82.0}{82.0}\times100=1.71\%

Engineering Comment

Moisture can change dimensions, modulus, Tg and dielectric behavior.

Plausibility Check

The mass gain is 1.4 g on 82 g, a little under 2 percent.

Exercise 10: Creep Strain Increase

Initial strain is 0.80 percent and strain after 1000 h is 1.12 percent. Find creep strain increase.

Solution

\Delta\varepsilon=1.12-0.80=0.32\%

Engineering Comment

Creep decisions should reference stress, temperature, humidity and service duration.

Plausibility Check

The final strain is higher, so creep increment is positive.

Exercise 11: Glass-Transition Margin

Qualified wet Tg is 105 C and maximum service temperature is 82 C. Required margin is 20 C. Check.

Solution

M=105-82=23^\circ\text{C}

The margin passes.

Engineering Comment

Dry Tg is not enough when the part sees wet or solvent exposure.

Plausibility Check

23 C is slightly above the 20 C requirement.

Exercise 12: Cure Degree Gate

DSC estimates cure degree of 94 percent. Release threshold is 95 percent. Decide.

Solution

94\%<95\%

The cure gate fails.

Engineering Comment

Incomplete cure can reduce Tg, strength and chemical resistance.

Plausibility Check

The result is directly below the threshold.

Exercise 13: Environmental Knockdown Allowable

Dry open-hole allowable is 260 MPa. Moisture knockdown is 0.85 and hot-temperature knockdown is 0.90. Find design allowable.

Solution

\sigma_{allow}=260(0.85)(0.90)=199\ \text{MPa}

Engineering Comment

Knockdowns should come from qualified data, not convenience factors.

Plausibility Check

Two reductions lower the allowable below 260 MPa.

Exercise 14: Void Content from Density

Theoretical laminate density is 1.56 g/cm3 and measured density is 1.50 g/cm3. Estimate void fraction.

Solution

V_v=\dfrac{1.56-1.50}{1.56}\times100=3.85\%

Engineering Comment

Void content can reduce compression, shear and fatigue performance.

Plausibility Check

The density deficit is about 4 percent of theoretical density.

Exercise 15: Barrier Permeation

A polymer membrane passes 0.18 g/day through 0.020 m2. Limit is 12 g/(m2 day). Check flux.

Solution

J=\dfrac{0.18}{0.020}=9.0\ \text{g/(m}^2\text{ day)}

The membrane passes.

Engineering Comment

Permeation depends strongly on temperature, pressure difference and conditioning.

Plausibility Check

0.18 g over one fiftieth of a square metre scales to 9 g/m2 day.

Exercise 16: Compression-After-Impact Gate

CAI allowable is 155 MPa and design compression stress is 132 MPa. Find margin.

Solution

M=\dfrac{155-132}{132}=17.4\%

Engineering Comment

Impact damage tolerance requires inspection capability and allowable damage limits.

Plausibility Check

Allowable exceeds demand, so margin is positive.

Exercise 17: Scarf Repair Overlap

A scarf repair uses 20:1 taper through a 2.5 mm laminate. Find scarf length.

Solution

L=20(2.5)=50\ \text{mm}

Engineering Comment

Repair release also needs surface prep, cure, ply restoration and inspection evidence.

Plausibility Check

Twenty times a few millimetres gives a few centimetres.

Exercise 18: Polymer Composite Release Gate

Release requires cure pass, void content below 3 percent and Tg margin above 20 C. Results are fail, 3.85 percent and 23 C. Decide.

Solution

\text{cure}=\text{fail},\qquad 3.85\%>3\%,\qquad 23>20

The release fails because cure and void content fail.

Engineering Comment

A good Tg margin cannot rescue incomplete cure and excessive voids.

Plausibility Check

Two required checks fail, so the hold decision is clear.

Validation Package Checklist

material grade, batch, layup and cure route are controlled;
direction-specific properties are used for the load path;
moisture, temperature, creep and permeability conditions match service;
bond, bolt and repair details include surface and inspection evidence;
void, impact and NDE limits are documented;
knockdowns are tied to qualified wet, hot, aged, damaged or repaired states;
coupon, element or subcomponent test evidence is traced to the same architecture;
failure modes are named, including bearing, net section, tear-out, delamination, creep rupture, permeation and impact damage;
release decision states accept, condition, inspect, redesign or hold.

A complete validation package should make the decision reproducible. Another engineer should be able to see which material state was calculated, which state was tested, which state will be installed, and where the margin is carried. If those four states do not match, the page result is only a screening calculation.

REF