Question 4 of 8: Ti BCC→HCP Volume Change; the Charpy Impact Test

Question 4: Ti BCC→HCP Volume Change; the Charpy Impact Test (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. BCC Ti: $a_{bcc}=0.332$ nm (2 atoms/cell). HCP Ti: $a_{hcp}=0.2978$ nm, $c_{hcp}=0.4735$ nm (6 atoms/cell).

Find. (a) %volume change on cooling through the transus. (b) Description of the Charpy test and its relation/advantages relative to a tensile test.

Approach

The number of atoms is conserved across the transformation but the two unit cells contain a different number of atoms (2 for BCC vs. 6 for HCP), so unit-cell volumes cannot be compared directly — the fair comparison is volume per atom in each structure.

1. Volume per atom, BCC. $$V_{cell,bcc}=a_{bcc}^3=(0.332)^3=0.03659\ \text{nm}^3,\qquad v_{bcc}=\frac{V_{cell,bcc}}{2}=0.018297\ \text{nm}^3/\text{atom}.$$
2. Volume per atom, HCP. The hexagonal cell volume is $V=\left(\tfrac{3\sqrt3}{2}\right)a^2c$: $$V_{cell,hcp}=\left(\frac{3\sqrt3}{2}\right)(0.2978)^2(0.4735)=0.10909\ \text{nm}^3,\qquad v_{hcp}=\frac{V_{cell,hcp}}{6}=0.018182\ \text{nm}^3/\text{atom}.$$
3. Percent volume change. $$\%\Delta V=\frac{v_{hcp}-v_{bcc}}{v_{bcc}}\times100 =\frac{0.018182-0.018297}{0.018297}\times100$$ $$\boxed{\%\Delta V\approx-0.62\%}$$ — a small contraction on cooling through 882°C (the low-temperature HCP $\alpha$-phase packs marginally more efficiently than the high-temperature BCC $\beta$-phase), consistent with titanium's well-documented small negative transformation volume change.
4. (b) The Charpy impact test. A notched bar specimen (standard V-notch or keyhole-notch geometry) is struck by a swinging pendulum hammer released from a fixed height; the pendulum's rise on the far side after fracturing the specimen is converted to the energy absorbed in fracture (read directly off the test machine's calibrated scale). Testing a series of identical specimens over a range of temperatures produces the characteristic ductile-to-brittle transition curve for BCC metals (a sharp drop in absorbed energy over a fairly narrow temperature band) — the single most important piece of information the test provides, since it identifies the lowest safe service temperature for a structure.
5. Relation to the tensile stress-strain curve, and the impact test's advantages. There is no direct, general quantitative relation between Charpy energy and the tensile stress-strain curve (e.g. Charpy energy is not simply the tensile toughness, the area under $\sigma$-$\varepsilon$, measured a different way) — the two tests load the material in fundamentally different ways: tensile testing applies a slow, uniaxial, unnotched load, while Charpy applies a load at very high strain rate to a sharply notched specimen, producing a triaxial stress state at the notch root. That difference is precisely the test's value: high strain rate and a sharp notch both promote brittle behaviour, so the Charpy test reveals a material's susceptibility to brittle, low-energy fracture under impact/shock loading and locates the ductile-brittle transition temperature — information a slow, unnotched tensile test cannot provide at all, even though the tensile test gives more precise quantitative strength and ductility numbers under the conditions it does test.