Materials Selection Techniques: Objectives and Criteria

In the previous section, we saw that Performance Indices of the form $P = y^{a}/x^{b}$ yielded a set of lines on the Ashby diagram, with each line corresponding to a different value of $P$, and materials data points residing on each line having identical performance.

This means that if we want to optimize our selection of a materials using a performance index, we need find the materials which possess the maximum or minimum values of $P$ on an Ashby diagram.

Let's try this analysis with a new performance index, $P = E^{1/2}/\rho$, where $E$ is elastic modulus and $\rho$ is density. This is a performance index that we'd use if we were to optimize the materials selection for a light, but stiff beam. I haven't (and won't) tell you exactly how we derive these performance indices, but in this case it is essentially a statics analysis of a beam in bending. If you take the advanced Materials Selection course, we'll show you how to derive these performance indices. For this course, we'll simply use the performance indices. However, if you are interested, you can refer to the appendix of performance indices derived by Ashby and others, which lists different indices for different design objectives and limitations.

Selection for a Light, Stiff Beam

We want to select for a light stiff beam with performance index $P = E^{1/2}/\rho$. This means that, ideally, we'd have a very large values of $E$ and a very small value of $\rho$. Let's use what we learned from the previous section to find the slope of the line we'd observe on a logarithmic $E$-$\rho$ plot:

$$ \begin{align} P &= \frac{E^{1/2}}{\rho}\\ \log{P} &= \frac{1}{2}\log{E} -\rho\\ \log{E} &= 2\rho+2\log{P}\\ \end{align} $$

This is in the form of $y = mx + b$. This means that if I draw line with a slope of 2 on a log-log graph (meaning if I go one decade along the $x$-axis I must go two decades along the $y$-axis), all the points on the line with have equivalent performance $P$. We show this in Figure 14.6.1 with four different lines. Test each to prove this to yourself — compute the values of $P$ at each point on one of the lines on the graph. Do they have the same value?

If you computed the values for each line, you'd also find something expected - as you move to the upper left in the graph, the value of $P$ gets larger. Indeed, $P$ for Line (d.) is $10\times$ larger than $P$ for Line (b.). Materials on Line (d.) like wood perform $10\times$ better when used as light, stiff beams than the metals, polymers, and foams on Line (b.).

This analysis also implies that if you want to find the material with the best performance, you simply find the material that has the line with the largest $P$ passing through it. In this case, that is wood, specifically one towards the lower-density side of the bubble.

Selection Criteria

In the section above, we found a material that has the best performance based on a specific performance index. However, in some cases, we may need to apply a selection criterion, a properties requirement that is necessary for a successful design. Let's show two examples of criteria below, using the same $E$-$\rho$ chart we've been working with.

The first criterion is that $E^{1/2}/\rho > 3 \sqrt{\text{GPa}}/(\text{Mg/m}^3)$. We might have this requirement due to geometric constraints or due to constraints regarding current market designs (i.e., we need to have better performance than what's currently on the market). This analysis is very straightforward. All it means is that we cannot accept materials with a Performance Index $E^{1/2}/\rho$ less than > $3 \sqrt{\text{GPa}}/(\text{Mg/m}^3)$. So, we find that line on the graph and eliminate all materials that don't pass the criteria.

To find the line all I do is pick values that are easy to compute and draw the line. I'm going to choose $E = 9 \text{ GPa}$ and $\rho = 1 \text{ Mg/m}^3$:

$$P = \frac{(9 \text{ GPa})^{1/2}}{1 \text{Mg/m}^3} = 3 \sqrt{\text{ GPa}}/(\text{Mg/m}^3)$$

We've drawn that line in Figure 14.6.2, shading in gray the materials that are unacceptable for selection, which includes all foams, elastomers, and polymers, as well as most metals and ceramics, and a good fraction of woods and composites.

We can add as many other criteria as we like. Here, we'll simply require that the material also has to have a minimum elastic modulus of $E > 50 \text{ GPa}$. That line is also included in Figure 14.6.2. It's only effect bound that of the previous criteria is to eliminate all woods and a small subset of composites. We have our final search region defined as the non-excluded region in the upper left-hand side of the graph.