Failure and selection: show the arithmetic
Introduced| English | Meaning |
|---|---|
| fatigue/fəˈtiːɡ/ | Damage and failure under repeated loading, which can occur below the monotonic yield stress. |
| corrosion/kəˈrəʊʒn/ | Deterioration through chemical or electrochemical reactions with the environment. |
| material index | the property combination that maximises performance for one load case |
Fatigue: below yield, above death
- Fatigue 疲劳 kills below yield under repetition: Basquin's $\sigma_a = \sigma_f'(2N)^b$ - with $\sigma_f' = 900$ MPa, $b = -0.10$: $\mathbf{226.1}$ MPa at $10^6$ reversals, $\mathbf{142.6}$ at $10^8$.
- A mean stress eats the amplitude budget: Goodman's $\sigma_a/\sigma_e + \sigma_m/\sigma_{uts} = 1$ cuts 250 to $\mathbf{175.0}$ MPa at 150 mean.
*Fatigue prices the cycles; the index ranks the candidates
Basquin with sigma_f = 900 MPa and b = -0.10: amplitude at 1e6 reversals (one decimal)?
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900 x 10^-0.6.
Amplitude at 1e8 reversals (one decimal)?
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900 x 10^-0.8.
Goodman with sigma_e = 250, sigma_uts = 500, mean 150: allowable amplitude (one decimal)?
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250 (1 - 150/500).
Service amplitude 160 MPa against a Goodman budget of 175: the verdict is:
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(175-160)/175.
The quiet ledger
- Corrosion 腐蚀 is electrochemical loss, budgeted up front: 0.12 mm/yr × 25 seasons $= \mathbf{3.0}$ mm of wall.
- The certificate carries Basquin, Goodman and the drop-test budget side by side.
Which failure mode, which tool?
Match each service condition to its failure analysis.
Wall loss at 0.12 mm/yr over 25 seasons (one decimal)?
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Multiply it out.
The index ranks, constraints eliminate
- The material index 材料指数 for a stiffness-critical part is $\sqrt E/ ho$: steel $\mathbf{1.80}$, aluminium $\mathbf{3.10}$, nylon $\mathbf{1.23}$, CFRP $\mathbf{7.37}$ GPa$^{1/2}$·m³/Mg.
- The mount ships steel (impact joules, corrosion budget); the carry-handle ships CFRP (2.4× aluminium at stiffness-per-mass) - one method, two right answers.
Stiffness index sqrt(E)/rho of aluminium with E = 70 GPa, rho = 2.70 Mg/m3 (two decimals)?
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sqrt(70)/2.70.
Same index for the laminate with E1 = 139.2 GPa, rho = 1.60 (two decimals)?
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sqrt(139.2)/1.60.
A part loaded below its yield strength cannot fail by fatigue.
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Fatigue lives entirely below yield - that is its definition.
Three failure slips
- Designing fatigue to yield. Fatigue lives below it.
- Ignoring the mean stress. Goodman is not optional.
- Comparing indices across load cases. Each case has its own exponent.
Recap
- Basquin 226.1 → 142.6 MPa over $10^6 o 10^8$ reversals; Goodman 175.0 at 150 mean.
- Corrosion 3.0 mm per 25 seasons, bought as wall.
- $\sqrt E/ ho$: CFRP 7.37 > aluminium 3.10 > steel 1.80 > nylon 1.23.