What this course is, and how to use it
This is the fourth course in University Mechanical Engineering: engineering materials 工程材料 — why materials behave as they do, how that behaviour is measured, and how a material is chosen, processed and certified for a part.
The running case closes a loop opened in course 3: the till mount was a spring of $k = 4900$ N/m — this course asks what it is made of. Four candidates are followed throughout: low-carbon steel low-carbon steel, aluminium alloy 6061, cast nylon 6 and a carbon-fibre/epoxy laminate. Every number in this handout was computed for this course — the elasticity chains recompute, the phase fractions sum to one, the probability curves integrate, and the selection index ranks the candidates.
- Units 4.1–4.2 are why and how much; 4.3–4.4 the families; 4.5–4.6 the making and the surviving.
Each unit: the question, the machinery, two escalating worked examples checked a second way, the common mistakes, a summary.
Structure and bonding
Why it matters. The mount's stiffness, yield and failure all trace to atoms and microstructure. Choosing blind is choosing by rumour.
The idea, with a picture. Bonding type sets the baseline: metallic bonds give ductile conductors (steel, aluminium), covalent networks give stiff insulators (ceramics), polymer chains give light viscoelastic solids (nylon). Above the bond, the crystal structure 晶体结构 packs atoms: body-centred cubic and face-centred cubic differ in density of packing — and packing plus defects sets how a metal yields.
*Denser packing, finer grains: structure writes the property
The definitions. The atomic packing factor 致密度 is the fraction of a unit cell filled by atoms treated as hard spheres: $\pi\sqrt{3}/8 = \mathbf{0.6802}$ for BCC (iron at room temperature) and $\pi/(3\sqrt{2}) = \mathbf{0.7405}$ for FCC (aluminium). A dislocation 位错 is a line defect whose motion is plastic flow; a grain boundary 晶界 is a wall between crystals that blocks dislocations — smaller grains mean more walls. Grain refinement 细晶强化 strengthens by that road: the Hall-Petch relation $\sigma_y = \sigma_0 + k/\sqrt{d}$.
The derivation. The bracket steel: $\sigma_0 = 120$ MPa and $k = 0.45$ MPa·√m. At $d = 50\ \mu$m: $\sigma_y = 120 + 0.45/\sqrt{50\times 10^{-6}} = 120 + 63.6 = \mathbf{183.6}$ MPa ✓; refine to $d = 10\ \mu$m: $120 + 142.3 = \mathbf{262.3}$ MPa ✓ — a gain of $\mathbf{78.7}$ MPa from processing alone, no chemistry changed ✓ (the two points are the picture's right panel).
Worked example 1 — read the packing. Aluminium is FCC, iron is BCC: the APF ratio is $0.7405/0.6802 = 1.089$ — aluminium's lattice fills 8.9% more of space with equally sized atoms, the geometric root of its lower density per atom and its many slip planes: FCC metals form and deform easily (aluminium foil), BCC iron needs warmth to match it.
Worked example 2 — escalate to the mount's bracket. The specification demands $\sigma_y \geq 250$ MPa with 20% elongation. Hall-Petch says the grain size must satisfy $120 + 0.45/\sqrt{d}\geq 250$, so $d \leq (0.45/130)^2 = \mathbf{1.20 \times 10^{-5}}$ m $= \mathbf{12.0}\ \mu$m ✓ — grain refinement alone reaches the target (10 µm gives 262.3), and the elongation is barely touched because boundaries block dislocations without embrittling the way carbon does.
Common mistakes. Treating APF as density (it is a geometric ratio; atomic mass still decides $\rho$). Quoting Hall-Petch outside its measured material and grain-size range (other mechanisms can dominate at nanometre grain sizes). Forgetting that dislocations are the mechanism of ductility, not a defect to eliminate.
Summary. Bonding sets family behaviour; packing: BCC 0.6802, FCC 0.7405. Dislocations glide, boundaries block: $\sigma_y = \sigma_0 +k/\sqrt d$. The bracket steel: 183.6 → 262.3 MPa for 50 → 10 µm.
Stress-strain and elastic behaviour
Why it matters. Certification numbers — stiffness, yield, toughness — are read from one curve, and the test that produced it has conventions that make or break comparisons.
The idea, with a picture. The mount rod test: a 10.0 mm rod, 0.50 m long, pulled to 15.0 kN. Steel stretches elastically by less than half a millimetre; the same load in aluminium or nylon is a different story.
*One curve, five properties - and E sorts the elastic league
The definitions. Engineering stress 工程应力 is $\sigma =F/A_0$ on the original area; true stress 真应力 uses the current area. For uniform uniaxial deformation at constant volume, before necking, $\sigma_t = \sigma(1 + \varepsilon)$. For uniform extension, true strain is $\varepsilon_t = \ln(1 + \varepsilon)$. Young's modulus 弹性模量 $E$ is the elastic slope: 200 GPa (steel), 70 GPa (aluminium), 2.0 GPa (nylon). The yield strength 屈服强度 is where plastic flow begins (0.2% offset convention); toughness 韧性 is the area under the curve — energy to fracture, in MJ/m³; ductility 延性 is strain at fracture.
The derivation. The rod: $A_0 = \pi (0.010)^2/4 = 7.854 \times10^{-5}$ m², $\sigma = F/A_0 = \mathbf{191.0}$ MPa ✓ — below the steel's 250 MPa yield, so purely elastic: $\varepsilon = \sigma/E =9.55 \times 10^{-4}$, stretch $= \varepsilon L = \mathbf{0.477}$ mm ✓. Aluminium at the same load: $191.0/70\,000 = 2.73 \times 10^{-3}$, stretch $\mathbf{1.364}$ mm — $2.86\times$ more, exactly the modulus ratio $200/70$ ✓ (the second-way check: deflections scale inversely with $E$). Nylon: $191.0/2000 = 9.55 \times 10^{-2}$ → 47.7 mm — off the elastic map, which is why the mount's stops exist. True vs engineering at $\varepsilon = 0.20$: $\varepsilon_t = \ln 1.20 =\mathbf{0.1823}$, $\sigma_t = 1.20\sigma$ — the two languages diverge exactly when necking starts to matter ✓. Toughness of the bracket steel (area of the curve in the picture, trapezoid over the marked points): $\mathbf{176}$ MJ/m³ ✓.
Worked example 1 — the 0.2% offset, done honestly. A supplier's plot shows no sharp yield. Draw the elastic line, shift it 0.002 in strain; where it crosses the curve is $\sigma_y$. The certificate reads 250 MPa — but for a different batch the same convention gave 310: the convention, not the material, changed? No — check the gauge length. Ductility percentages only compare at equal gauge length, because necking localises: the report without gauge length is a rumour.
Worked example 2 — escalate to toughness budgets. The drop test of course 3 delivers 47.56 J to the mount. A bracket volume of $47.56/176 = \mathbf{0.270}$ cm³ of this steel could absorb the whole blow plastically — the real bracket is 4.0 cm³, so the drop rides at 6.8% of its energy capacity ✓ (two courses tie: course 3's joules, this course's area-under-curve).
Common mistakes. Mixing engineering and true stress across a table (state the column). Reading yield off a curve with no offset convention stated. Treating toughness (area) as strength (height) — one is joules, the other pascals.
Summary. $\sigma = F/A_0$; $E$ steel/aluminium/nylon = 200/70/2.0 GPa; the rod: 191.0 MPa, 0.477 mm vs 1.364 mm. True $= \ln(1+e)$, $1.2\sigma$ at $e = 0.20$. Toughness 176 MJ/m³ — the joules budget.
Metals and alloys
Why it matters. The steel bracket's final properties are written by chemistry and thermal history — the same steel ships at four different strength/ductility pairs.
The idea, with a picture. The iron-carbon phase diagram 相图 reads composition and temperature into phases; just below 727°C, the eutectoid reaction turns one austenite grain into lamellar pearlite — and how fast you cool past that line decides everything.
*Read the phases off the diagram; weigh them with the lever rule
The definitions. The eutectoid 共析 point (0.76 wt% C, 727°C) is where austenite decomposes to pearlite 珠光体 (ferrite + cementite lamellae). Slow cooling gives coarse pearlite (soft), quenching bypasses pearlite to martensite 马氏体 (hard, brittle, must be tempered). The lever rule 杠杆定律 weighs phase fractions on a tie-line. Strain hardening 应变硬化 raises strength by cold work: $\sigma = K\varepsilon^n$. Alloying adds precipitates 析出相 that pin dislocations (the 6061 aluminium's Mg₂Si).
The derivation. The bracket steel at 0.40 wt% C, just below 727°C: pearlite fraction $= (C_0 - C_\alpha)/(C_e - C_\alpha) = (0.400.022)/(0.76 - 0.022) = \mathbf{0.512}$, ferrite $\mathbf{0.488}$ — they sum to one ✓ (the picture's right panel is exactly this tie-line). Cold work: with $K = 530$ MPa and $n = 0.20$, a 25% cold reduction carries true stress $530 \times 0.25^{0.20} = \mathbf{401.7}$ MPa — the annealed 250 MPa becomes a worked 420 MPa engineering yield, but elongation falls 28% → 12%.
Worked example 1 — pick the state, not just the grade. The mount bracket needs formability in production and strength in service. Buy the steel annealed (250 MPa, 28% elongation), form it, then work-harden or heat-treat to the service state — the certificate names both states, because the part that ships is the processed one.
Worked example 2 — escalate to the quench window. The eutectoid temperature sets the heat-treat ceiling: austenitise above 727°C (our 0.40% steel: into the full-austenite field), quench fast enough to miss the nose of the transformation curve, temper to trade hardness back for toughness. The 200°C temper ships 950 MPa at 8% elongation; the 600°C temper ships 620 MPa at 16% (the full menu is unit 4.5's picture) — the certification picks one line of that menu.
Common mistakes. Reading phases above the diagram's region (ferrite stops at 0.022% below 727°C). Adding strength mechanisms arbitrarily — they interact, so the final certificate is measured, not summed. Confusing cementite (a phase) with pearlite (a mixture).
Summary. Eutectoid 0.76% C at 727°C; 0.40% steel just below: 51.2% pearlite + 48.8% ferrite. $\sigma = K\varepsilon^n$: 401.7 MPa true at 25% work. The state ships, not the grade.
Polymers, ceramics and composites
Why it matters. Half the mount's competitors are not metals: nylon promises silence and lightness, ceramic promises stiffness, and the laminate promises both — with catches that are the unit.
The idea, with a picture. The laminate's two moduli — along and across the fibres — and the ceramic's strength statistics: two pictures that carry the whole non-metal story.
*Fibres are one-way promises; ceramics are statistics
The definitions. A composite 复合材料 combines phases across a visible scale; the rule of mixtures 混合定律 averages properties by volume fraction. The equal-strain model estimates longitudinal stiffness by a weighted arithmetic mean; the equal-stress model gives a weighted harmonic mean. These idealised models provide bounds, not an exact general transverse modulus. Viscoelasticity 黏弹性 is the polymer's time-dependent stiffness, governed by the glass transition 玻璃化转变 (nylon 6: $T_g = 55$°C). A ceramic 陶瓷 carries few slip systems: strong in compression, brittle in tension, strength scattered — the Weibull modulus 威布尔模数 $m$ prices the scatter: survival $P_s = \exp[-(\sigma/\sigma_0)^m]$.
The derivation. The laminate, $V_f = 0.60$, $E_f = 230$ GPa, $E_m = 3.0$ GPa: along the fibres $E_1 = 0.60 \times 230 + 0.40\times 3.0 = \mathbf{139.2}$ GPa ✓; across them $E_2 = E_fE_m/(V_fE_m+ V_mE_f) = 690/93.8 = \mathbf{7.36}$ GPa ✓ — an anisotropy of $\mathbf{18.9}$, one material with two stiffnesses. The ceramic, $m = 10$, $\sigma_0 = 300$ MPa: $P_s(200) = e^{-(2/3)^{10}} =\mathbf{0.983}$, $P_s(300) = e^{-1} = \mathbf{0.368}$, $P_s(360) =e^{-1.2^{10}} = \mathbf{0.002}$ — twenty per cent over the characteristic strength erases 98% of the survivors: brittle design is statistics.
Worked example 1 — the nylon mount in the festival sun. The booth reaches 45°C, close to nylon's $T_g = 55$°C: the modulus sags viscoelastically (a factor 2–3 is typical across the transition) and creep under sustained load begins. The certification therefore thermal-cycles the nylon candidate to 55°C and re-reads the spring rate — a mount that is stiff in the morning and soft by afternoon fails the till's 2 mm clearance, whatever the room-temperature datasheet says.
Worked example 2 — escalate to the laminate bracket. Loaded along the fibres, 139.2 GPa at $\rho = 1.60$ Mg/m³ beats everything in the case; loaded across, 7.36 GPa is nylon-class. The design answer is ply orientation: a $[0/\pm45/90]$ lay-up trades some axial stiffness for usable transverse and shear response — the certificate quotes a laminate modulus per direction, never a single number.
Common mistakes. Using $E_1$ where the load crosses the fibres (the 19-fold error). Treating the rule of mixtures as exact for strength (fibres fail statistically, not in unison). Designing ceramics to a mean strength instead of a survival probability.
Summary. $E_1 = 139.2$ GPa, $E_2 = 7.36$ GPa, anisotropy 18.9; $T_g$(nylon) = 55°C — check the sun. Ceramic survival $e^{-(\sigma/\sigma_0)^m}$: 0.983 / 0.368 / 0.002 at 200 / 300 / 360 MPa.
Processing and property links
Why it matters. The menu of bracket properties is written by thermal and mechanical processing; the buyer who ignores it buys a rumour.
The idea, with a picture. Four heat-treatment routes for the same bracket steel, and the casting rule that times solidification: the process map is the property map.
*The furnace writes the datasheet - and the riser sizes the casting
The definitions. Heat treatment 热处理 — quenching 淬火 from the austenite field to martensite, then tempering 回火 to give toughness back; the temper temperature dials the trade-off. Casting 铸造 freezes liquid into shape; solidification 凝固 time follows Chvorinov's rule $t = B(V/A)^2$ — volume over cooling surface area, all squared. Work hardening (unit 4.3) is the mechanical member of the same family.
The derivation. The four routes, same 0.40% steel: as-rolled $\mathbf{280}$ MPa at 24% elongation; normalised $\mathbf{260}$ at 26%; quenched + tempered 600°C $\mathbf{620}$ at 16%; quenched + tempered 200°C $\mathbf{950}$ at 8% ✓ (the picture's left panel — strength triples as ductility halves three times). The casting: with $B = 2.0 \times 10^6$ s/m², a 20 mm cube ($V/A = 3.33$ mm) freezes in $2.0 \times 10^6 \times (3.33 \times 10^{-3})^2 = \mathbf{22.2}$ s; a 100×100×10 mm plate ($V/A = 5.00$ mm, faces only) takes $\mathbf{50.0}$ s — $2.25\times$ longer despite being thinner, because $V/A$, not thickness, times the clock ✓ (the ratio is exactly $(5.00/3.33)^2 = 2.25$).
Worked example 1 — size the riser. A casting solidifies last where $V/A$ is largest — so the riser (the reservoir that feeds shrinkage) must have a bigger $V/A$ than any section it feeds. The plate's 50.0 s must be beaten: a riser of $V/A = 6.0$ mm gives 72.0 s ✓ and the casting is sound; a riser at $V/A = 4$ mm freezes first and the plate sucks in air — the rule decides, not intuition.
Worked example 2 — escalate to the certified route. The mount bracket's service load is impact (course 3), so ductility matters: the certification picks quenched + tempered 600°C (620 MPa, 16%) over the 200°C temper (950 MPa, 8%) — trading 330 MPa of strength for doubled elongation, because the drop test's joules are absorbed by plastic area, and unit 4.2's toughness is the currency.
Common mistakes. Buying to grade, not state (the menu is the material). Tempering too cold for an impact-loaded part (martensite's strength is not the mount's need). Comparing solidification times by thickness instead of $V/A$.
Summary. Four routes: 280/24, 260/26, 620/16, 950/8 — MPa vs %. Chvorinov $t = B(V/A)^2$: cube 22.2 s, plate 50.0 s, ratio 2.25 = $(V/A)$². The riser must be greediest.
Failure and material selection
Why it matters. The mount must survive $10^7$ shaker cycles, 25 festival seasons outdoors and one drop — three failure modes, one choice, and a method that shows its arithmetic.
The idea, with a picture. Fatigue's falling curve, corrosion's quiet ledger, and the Ashby index that ranks the four candidates for a stiffness-critical bracket.
*Fatigue prices the cycles; the index ranks the candidates
The definitions. Fatigue 疲劳 fails below yield under repeated load. Basquin’s law $\sigma_a = \sigma_f'(2N)^b$ models a stated stress-life range. Many steels show an apparent endurance limit under specified test conditions, while typical aluminium fatigue curves keep falling. An endurance limit is not a guarantee of unlimited life under arbitrary surface, environment and loading conditions. A mean stress eats the budget — the Goodman line $\sigma_a/\sigma_e + \sigma_m/\sigma_{uts}= 1$. Corrosion 腐蚀 is electrochemical loss, budgeted as thickness per year. A material index 材料指数 ranks candidates for a load case: for a stiffness-critical bracket, $\sqrt{E}/\rho$.
The derivation. Fatigue, bracket steel $\sigma_f' = 900$ MPa, $b = -0.10$: at $10^6$ reversals $\sigma_a = 900 \times 10^{-0.6} =\mathbf{226.1}$ MPa; at $10^8$, $900 \times 10^{-0.8} = \mathbf{142.6}$ MPa ✓. With a 150 MPa mean ($\sigma_{uts} = 500$, $\sigma_e = 250$): Goodman allows $\sigma_a = 250(1 - 150/500) = \mathbf{175.0}$ MPa ✓ — the mean load cost 51 MPa of amplitude. Corrosion: 0.12 mm/yr × 25 yr $= \mathbf{3.0}$ mm — the wall is bought up front. The index, $\sqrt{E}/\rho$ in GPa$^{1/2}$·m³/Mg: steel $\sqrt{200}/7.86 =\mathbf{1.80}$, aluminium $\sqrt{70}/2.70 = \mathbf{3.10}$, nylon $\sqrt{2.0}/1.15 = \mathbf{1.23}$, CFRP $\sqrt{139.2}/1.60 =\mathbf{7.37}$ ✓ — the laminate wins the stiffness-per-mass game by 2.4× over aluminium.
Worked example 1 — the shaker certificate. Course 3's table cycles at 4.77 Hz between accelerations that load the bracket to $\pm 160$ MPa about a 150 MPa mean. Goodman budget is 175 MPa: 160 fits, margin 8.6% — and Basquin extrapolates the life beyond $10^7$ reversals at that amplitude ✓. The certificate reads: survives the shaker, 25 seasons — with the arithmetic attached.
Worked example 2 — escalate to the full choice. Constraints: $T_g$ above 60°C (kills plain nylon — 55), corrosion budget 3 mm in steel vs none in CFRP, cost bracket, and the impact toughness of unit 4.2. The index says CFRP (7.37) for stiffness-per-mass; the drop test's joules and the budget say quenched-and-tempered steel. The mount ships as steel; the carry-handle — stiffness-critical, light, unloaded in impact — goes CFRP: one method, two right answers, both shown with their index lines.
Common mistakes. Designing fatigue to yield (fatigue lives below it); ignoring the mean stress (Goodman is not optional); comparing index values across different load cases (each case has its own exponent); forgetting that the index ranks — constraints eliminate.
Summary. Basquin: 226.1 → 142.6 MPa over $10^6 \to 10^8$; Goodman cuts 250 to 175 at 150 mean. Corrosion: 3.0 mm per 25 yr. Index $\sqrt E/\rho$: CFRP 7.37 > aluminium 3.10 > steel 1.80 > nylon 1.23.
The project sheet (reproducible)
The mount's material file, one page:
- Structure: BCC 0.6802, FCC 0.7405; Hall-Petch 183.6 MPa at 50 µm, 262.3 at 10 µm (gain 78.7).
- Elasticity: rod 191.0 MPa → 0.477 mm (steel) vs 1.364 mm (aluminium, ratio 200/70); true strain at 0.20 is $\ln 1.2 =0.1823$; toughness 176 MJ/m³.
- Alloys: 0.40% C below 727°C = 51.2% pearlite + 48.8% ferrite; work-hardened 401.7 MPa true at 25%.
- Non-metals: laminate $E_1 = 139.2$, $E_2 = 7.36$ GPa (18.9×); nylon $T_g$ 55°C; ceramic survival 0.983 / 0.368 / 0.002 at 200 / 300 / 360 MPa ($m = 10$).
- Processing: routes 280/24, 260/26, 620/16, 950/8; Chvorinov cube 22.2 s vs plate 50.0 s = $(V/A)^2$.
- Failure: Basquin 226.1 → 142.6 MPa; Goodman 175.0; corrosion 3.0 mm/25 yr; index 7.37 / 3.10 / 1.80 / 1.23 → CFRP handle, steel mount.
Recompute any line from the constants above — the steel ships quenched and tempered at 600°C, the certificate carries Goodman, Basquin and the drop-test budget side by side.