Exercises

All materials in an Ashby have the same performance index. Convince yourself that this is true for any two points on a single contour line in Figure 14.7.1. These lines represent the speed of sound $c = (E/\rho)^{1/2}$. Prove this mathematically (again using $\LaTeX$, if you like, enclosing your syntax in $ symbols), and be aware of units (it is best to convert everything to SI).

Materials selection is about finding the best-performing material(s) for an application. Explain how you might graphically use these contour lines to find the best-performing material based on a performance index. You can explain what you'd do within the context of the figure in Figure 14.7.1, describing how you'd select the material with the largest $c$.

In acoustic engineering, you often want to select materials in which the speed of sound matches in order to reduce acoustic impedance. Are there any ceramic materials that acoustically match wood materials? Explain, based on data in Figure 14.7.1.

• When selecting a material from a performance index, we will maximize (or minimize) the index using an Ashby diagram. In the video, we showed an example for a light, stiff beam, and it seemed that ceramics were comparable to woods. You've probably seem wood beam, but you probably haven't seen ceramic beams... why? What does this imply about the selection process we outlined?
• All the indices we've shown have two independent variables (e.g. $c = (E/\rho)^{1/2}$). Do you think we ever have to do selection with three variables? What would the Ashby diagram look like? Would we still use contour lines?