Structure and bonding: packing and blocking
Introduced| English | Meaning |
|---|---|
| engineering materials | The study and selection of materials through relationships between structure, processing and properties. |
| crystal structure | The ordered arrangement of atoms, ions or molecules in a crystalline material. |
| atomic packing factor | the geometric fraction of a unit cell filled by hard-sphere atoms |
| dislocation | A line defect in a crystal lattice whose motion can produce plastic deformation. |
| grain boundary | An interface between crystal grains with different lattice orientations. |
| grain refinement | Reducing grain size, often to increase strength within a material's established Hall-Petch range. |
Spheres, cells and slip
- The running case is engineering materials 工程材料 - the study of what to make the mount of. Metallic bonds give ductile conductors; the crystal structure 晶体结构 sets how easily that ductility flows: BCC iron packs $\pi\sqrt{3}/8 = \mathbf{0.6802}$, FCC aluminium $\pi/(3\sqrt{2}) = \mathbf{0.7405}$.
- The atomic packing factor 致密度 is geometry, not density - atomic mass still decides $ho$.
*Denser packing, finer grains: structure writes the property
Atomic packing factor of BCC (four decimals)?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
pi*sqrt(3)/8.
Atomic packing factor of FCC (four decimals)?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
pi/(3*sqrt(2)).
The atomic packing factor measures:
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Geometry only - mass still decides density.
Walls that strengthen
- A dislocation 位错 is the line defect whose glide IS plastic flow; a grain boundary 晶界 is a wall that blocks it.
- Grain refinement 细晶强化: $\sigma_y = \sigma_0 + k/\sqrt d$ - the bracket steel runs $120 + 0.45/\sqrt{50 imes 10^{-6}} = \mathbf{183.6}$ MPa at 50 µm and $\mathbf{262.3}$ at 10 µm, a gain of 78.7 MPa with no chemistry changed.
Which mechanism, which lever?
Route each structural fact to its property consequence.
Hall-Petch with sigma_0 = 120 MPa and k = 0.45 MPa sqrt(m): yield at d = 50 micrometres (one decimal)?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
120 + 0.45/sqrt(50e-6).
Same steel at d = 10 micrometres (one decimal)?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
120 + 0.45/sqrt(10e-6).
Eliminating all dislocations would make a metal more ductile.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Dislocation glide IS the ductility mechanism - removing them embrittles.
Three structure slips
- APF read as density. It is a sphere-packing ratio.
- Hall-Petch extrapolated far below 10 µm. Other mechanisms take over.
- Dislocations treated as defects to eliminate. They are the mechanism of ductility.
Largest grain size in micrometres that still meets 250 MPa (one decimal)?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
d = (0.45/130)^2.
Strength gain from refining 50 to 10 micrometres (one decimal, MPa)?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
262.3 - 183.6.
Read the certificate the same way
- The mill certificate reports grain size alongside chemistry; the buyer multiplies by Hall-Petch before quoting a strength.
- "Fine-grain condition" on the certificate is the 183.6-vs-262.3 question answered in advance - check which number the supplier means.
Recap
- BCC 0.6802 (iron), FCC 0.7405 (aluminium); FCC's extra 8.9% packing buys the slip planes.
- $\sigma_y = \sigma_0 + k/\sqrt d$: 183.6 → 262.3 MPa for 50 → 10 µm.
- The 250 MPa specification needs $d \leq 12.0$ µm - processing, not alloying.