Polymers, ceramics and composites: the catches
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| composite/ˈkɒmpəzɪt/ | a designed combination of phases at a scale visible to the eye |
| rule of mixtures | property averaging by volume fraction, exact axially and inverse transversely |
| ceramic | a stiff, brittle, strongly-bonded solid with statistical strength |
| viscoelasticity | the time- and temperature-dependent stiffness of polymers |
| glass transition | the softening temperature of an amorphous polymer, nylon 6 near 55 C |
| Weibull modulus | the shape parameter m pricing the scatter of brittle strength |
One laminate, two stiffnesses
- A composite 复合材料 trades properties by design; the rule of mixtures 混合定律 is the receipt: along the fibres an average, across them the inverse.
- Carbon-epoxy at $V_f = 0.60$: $E_1 = \mathbf{139.2}$ GPa, $E_2 = \mathbf{7.36}$ GPa - an 18.9× anisotropy.
*Fibres are one-way promises; ceramics are statistics
Rule of mixtures along fibres: E1 with Vf = 0.60, Ef = 230 GPa, Em = 3.0 GPa (one decimal)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Vf Ef + Vm Em.
Transverse modulus E2 for the same laminate (two decimals)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Ef Em / (Vf Em + Vm Ef).
A laminate datasheet should quote modulus as:
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Anisotropy is the material.
The polymer clock
- Viscoelasticity 黏弹性 makes stiffness depend on time and temperature; the glass transition 玻璃化转变 of nylon 6 sits at $\mathbf{55°}$C - too close to a 45°C festival booth.
- The certification thermal-cycles the nylon candidate and re-reads the spring rate.
Glass transition of nylon 6 in degrees C (nearest integer)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
The polymer clock.
A viscoelastic mount at room temperature predicts its stiffness at 45 C.
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Near Tg the modulus can halve - thermal-cycle and re-read.
Brittle by statistics
- A ceramic 陶瓷 is strong in compression and statistical in tension: survival $\exp[-(\sigma/\sigma_0)^m]$, the Weibull modulus 威布尔模数 $m$ pricing the scatter.
- With $m = 10$, $\sigma_0 = 300$ MPa: $\mathbf{98.3\%}$ survive 200 MPa, $\mathbf{36.8\%}$ survive 300, only $\mathbf{0.2\%}$ survive 360 - twenty per cent over the characteristic strength erases almost everyone.
Which family, which catch?
Match each material behaviour to its family and design rule.
Weibull survival at 200 MPa with m = 10, sigma_0 = 300 (three decimals)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
exp(-(2/3)^10).
Survival at the characteristic strength 300 MPa (three decimals)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
exp(-1).
Survival at 360 MPa (three decimals)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
exp(-1.2^10).
Three non-metal slips
- E₁ quoted where the load crosses the fibres. The 19-fold error.
- Rule of mixtures for strength. Fibres fail statistically, not in unison.
- Ceramics designed to a mean. Design to a survival probability.
Recap
- $E_1 = 139.2$, $E_2 = 7.36$ GPa: quote a modulus per direction, never one number.
- Nylon's $T_g$ = 55°C: check the sun before trusting the datasheet.
- Survival $e^{-(\sigma/\sigma_0)^m}$: 0.983 / 0.368 / 0.002 at 200 / 300 / 360 MPa.