Fig. E8.8, below, is a chart with axes of \(\mathrm{m} / \mathrm{m}_{o}\) and \(C / C_{o}\) derived in Exercise E8.7 Mild steel (here labelled 'Low carbon steel') lies at the coordinates \((1,1)\). Sketch a trade-off surface - remember that it is simply a lower envelope to the data, with no mathematical function associated with it. Then plot contours of \(Z^{*}\) that are approximately tangent to the trade-off surface for \(\alpha^{*}=1\) and for \(\alpha^{*}=100\). They are plots of the linear relationship

\[\alpha^{*} \frac{m}{m_{o}}=-\frac{1}{\alpha^{*}} \frac{C}{C_{o}}\]

but appear as curves because of the log scales. What selections do these suggest?

Data From E8.8

m/mo (Density/7850)*(314/yield strength) 0.11 10 Metals Copper Phenolics Zn-alloys Epoxies Cast Al-alloys Cast iron Low carbon steel- Medium carbon steel High carbon steel Low alloy steel Age-hardened Al-alloys GFRP Ni-alloys Ti-alloys Wrought Mg-alloys CFRP Composites 10-2 10-2 0.1 = 1 10 C/Co (Density/7850) *(Price/0.55)*(314/yield strength) FIGURE E8.8 Polymers PTFE PEEK MFA 16 100