Question 3 of 7: Tensile Test of a Magnesium-Alloy Bar — Full Mechanical-Property Workup

Question 3: Tensile Test of a Magnesium-Alloy Bar — Full Mechanical-Property Workup (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. Original diameter $d_0=12$ mm, original gauge length $L_0=30.00$ mm ($A_0=\tfrac{\pi}{4}d_0^2=113.10$ mm$^2$). Load–gauge-length pairs:

After unloading the fractured halves: gauge length $32.61$ mm, diameter $11.74$ mm.

Find. (i) 0.2%-offset yield strength. (ii) Tensile strength (UTS). (iii) Modulus of elasticity $E$. (iv) $\%$ elongation. (v) $\%$ reduction in area. (vi) Engineering stress at fracture. (vii) True stress at fracture.

Approach

Convert every load/length pair to engineering stress $\sigma=P/A_0$ and engineering strain $\varepsilon=(L-L_0)/L_0$, plot the curve, fit $E$ to the initial linear (elastic) points, then apply the standard offset-yield construction, read UTS as the peak engineering stress, and use the unloaded post-fracture gauge length/diameter (which reflects elastic springback) for $\%$EL/$\%$RA, since the source gives that measurement separately from the under-load fracture-instant row.

1. (iii) Modulus of elasticity. Converting the first three non-zero loads to stress/strain: $$\sigma(5\,\text{kN})=44.21\ \text{MPa},\ \varepsilon=0.000987;\quad \sigma(10\,\text{kN})=88.42\ \text{MPa},\ \varepsilon=0.001973;\quad \sigma(15\,\text{kN})=132.63\ \text{MPa},\ \varepsilon=0.002960$$ All three give the identical ratio $\sigma/\varepsilon\approx44{,}807$ MPa, confirming these three points are elastic; at $20$ kN the ratio drops sharply (to $\approx35{,}368$ MPa), showing the material has already begun to yield by that point: $$\boxed{E\approx44.8\ \text{GPa}}$$ This is consistent with a magnesium alloy (handbook $E_{\text{Mg}}\approx45$ GPa), a useful check on both the data and the material identification.
2. (i) 0.2%-offset yield strength. The offset line $\sigma=E(\varepsilon-0.002)$ is intersected against the measured curve, piecewise-linear between data points. It lies below the curve at the $20$ kN point and above it by the $25$ kN point, so the crossing falls on the $20$–$25$ kN segment; linear interpolation on that segment gives: $$\boxed{\sigma_{0.2\%}\approx180.6\ \text{MPa (at 20.4 kN, }\varepsilon\approx0.00603\text{)}}$$
3. (ii) Tensile strength. The engineering stress is highest at the maximum recorded load ($27$ kN, $L=31.50$ mm): $$\sigma_{UTS}=\frac{P_{max}}{A_0}=\frac{27{,}000}{113.10}$$ $$\boxed{\sigma_{UTS}\approx238.7\ \text{MPa}}$$ The two data points beyond this (load falling back to $26.5$ then $25$ kN while the gauge length keeps growing) are the post-necking descending branch of the engineering curve, not higher stresses.
4. (iv) $\%$ Elongation. Using the unloaded post-fracture gauge length (springback measurement, $32.61$ mm), not the under-load fracture-instant row ($32.79$ mm): $$\%EL=\frac{L_f-L_0}{L_0}\times100=\frac{32.61-30.00}{30.00}\times100$$ $$\boxed{\%EL\approx8.70\%}$$
5. (v) $\%$ Reduction in area. Using the unloaded post-fracture diameter ($11.74$ mm), $A_f=\tfrac{\pi}{4}(11.74)^2=108.25$ mm$^2$: $$\%RA=\frac{A_0-A_f}{A_0}\times100=\frac{113.10-108.25}{113.10}\times100$$ $$\boxed{\%RA\approx4.29\%}$$ The modest $\%$RA (compared with $\%$EL) is consistent with magnesium's limited room-temperature ductility (few active slip systems in its HCP structure).
6. (vi) Engineering stress at fracture. The load at the instant of separation was $25$ kN (table's "fracture" row), still divided by the original area $A_0$: $$\sigma_{f,\text{eng}}=\frac{25{,}000}{113.10}$$ $$\boxed{\sigma_{f,\text{eng}}\approx221.0\ \text{MPa}}$$
7. (vii) True stress at fracture. True stress uses the actual (necked-down) cross-section carrying the load at fracture. The best available measurement of that area is the post-fracture diameter, $A_f=108.25$ mm$^2$ (the standard simplification used when only a post-test diameter is reported, flagged below): $$\sigma_{f,\text{true}}=\frac{P_f}{A_f}=\frac{25{,}000}{108.25}$$ $$\boxed{\sigma_{f,\text{true}}\approx231.0\ \text{MPa}}$$