Stress-strain: read the certificate
Introduzido| English | Significado |
|---|---|
| yield strength | the stress at which plastic flow begins, fixed by the 0.2% offset convention |
| Young's modulus | the elastic proportionality between stress and strain |
| engineering stress | load divided by the original cross-sectional area, the reporting convention |
| true stress | load divided by the current area, sigma(1+e) with true strain ln(1+e) |
| toughness | the energy per volume to fracture, the area under the stress-strain curve |
| ductility/dʌkˈtɪlɪti/ | the strain at fracture, the property heat treatment trades away for strength |
One curve, five properties
- The certificate is read from one plot: elastic slope, yield strength 屈服强度, ultimate, fracture, and the area - each with a convention attached.
- The rod test: 10.0 mm bar, 0.50 m long, 15.0 kN gives $\sigma = F/A_0 = \mathbf{191.0}$ MPa - elastic in steel.
*Young's modulus does the sorting
A 10.0 mm rod at 15.0 kN: engineering stress in MPa (one decimal)?
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F over pi d²/4.
Yield strength by the certificate convention is read at:
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The offset construction, always stated.
Two languages of stress
- Engineering stress 工程应力 is $F/A_0$ on the original area; true stress 真应力 is $F/A$ now: $\sigma_t = \sigma(1+e)$, $\varepsilon_t = \ln(1+e)$ - at $e = 0.20$: $0.1823$ and $1.20\sigma$.
- Young's modulus 弹性模量 sorts the league: 200 / 70 / 2.0 GPa → the same rod stretches 0.477 / 1.364 / 47.7 mm - deflections scale as $1/E$.
Which property answers it?
Route each design question to its curve quantity.
Elastic stretch of its 0.50 m steel length in mm (three decimals)?
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sigma L / E with E = 200 GPa.
Same rod in aluminium (E = 70 GPa), stretch in mm (three decimals)?
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Ratio 200/70 applied to 0.477.
True strain at 0.20 engineering strain (four decimals)?
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ln(1.2).
True stress as a multiple of engineering stress at that point (two decimals)?
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sigma_t = sigma(1+e).
Toughness is area, strength is height
- Toughness 韧性 is the area under the curve - $\mathbf{176}$ MJ/m³ for the bracket steel; ductility 延性 is strain at fracture.
- The drop test's 47.56 J needs only 0.270 cm³ of this steel at full plastic absorption - the real 4.0 cm³ bracket rides at 6.8%.
Bracket volume in cm³ that could absorb the 47.56 J drop at 176 MJ/m³ (three decimals)?
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J divided by MJ/m³ gives m³; convert.
Toughness and strength have the same units.
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Toughness is energy per volume; strength is stress.
Three testing slips
- Yield with no convention. State the 0.2% offset or the number means nothing.
- Ductility without gauge length. Necking localises; only equal gauges compare.
- Toughness read as strength. Joules per cubic metre are not pascals.
Recap
- $\sigma = F/A_0 = 191.0$ MPa; steel stretches 0.477 mm, aluminium 2.86x more.
- True strain $\ln 1.2 = 0.1823$; true stress $1.2\sigma$ at 20% engineering.
- Toughness 176 MJ/m³ - the joules budget of the drop test.