Navigating Ashby Diagrams

Very frequently, we need to consider the balance of two different materials properties. We need a material to be strong, but lightweight. Or, we need it to be thermally conductive but low cost. Or, we need it to have a low environmental footprint in terms of $\ce{CO2}$ production, but we want it to highly optically transparent.

Striking the right balance between these materials properties is typically where we begin in the study of Materials Selection. And since we often need to consider two properties as once, perhaps it's best if we plot the materials properties ranges for many materials on two axes? That's what Michael Ashby thought in the early 1990s, when he came up with the Ashby Diagram.

We've seen a few of these in class before, but we'll spend some more time on their utility here. Figure 14.5.1 shows our first Ashby Diagram, which has density, in units of Mg/m$^3$ (Mg are megagrams), on the $x$-axis, and Young's modulus, in units of GPa, on the $y$-axis. Both axis are plotted on logarithmic scales. This is customary in Ashby diagrams for a few reasons. The first is that the values of materials properties span many orders of magnitude, so viewing them on a log scale is useful. The second reason will be clear below.

Within the area of the graph we see "bubbles" showing general ranges for the properties for specific classifications of materials. Metals are typically quite density and have high elastic moduli, while foams have low density and low elastic moduli. Elastomers have intermediate densities, but are very "compliant", while composites have similar densities to elastomers but are very "stiff".

This graph shows the first real benefit of plotting the data in this way! If I wanted to quickly understand which materials were dense and stiff, or lightweight and compliant, I can quickly compare them using this graph. However, there's an added feature here that makes these graphs very useful for systematic materials selection.

Let's say I needed to select the material for which sound traveled through the fastest. It turns out that a good first approximation for the speed of sound in a solid is

The speed of sound is simply the square root of the ratio of the elastic modulus and the density. So, the higher the modulus and the lower the density, the larger the speed of sound! This implies that if we want to select the material that will transmit sound the fastest, we can look at the upper left-hand corner of the graph in Figure 14.5.1. Well, there are not many materials up there (an opportunity for materials designers, perhaps?), but we can see quickly that some of the best materials if we want to maximize $c_{\text{sound}}$ could be wood, composites, ceramics, or low-density metals.

Comparing the Speed of Sound for Multiple Materials

Perhaps we can be a bit more precise? Eq. 14.5.1 is very precise. We could pick two points on the graph and compare them and simply see which material has the larger value of $c_{\mathrm{sound}}$. Let's try. I'll choose a polymer with $\rho \approx 1\text{Mg/m}^3$ and $E \approx 1 \text{GPa}$ and a metal at $\rho \approx 10\text{Mg/m}^3$ and $E \approx 10 \text{GPa}$. (I know there are no materials exactly at those points, but humor me.) Let's convert to base SI values so we can compute the speed of sound in m/s. First, for the polymer:

Alright, and for the metal:

Interesting! These two points have the same speed of sound. If we were selecting a material based on just the speed of sound, the performance of the polymer and the metal would be identical. In fact, I could draw a line between the two points and label it as $c = 1000\text{ m/s}$ and all the materials it passes through would have the same performance, or here, speed of sound. This actually arises due to the form of the equation for the speed of sound — which we'll refer to as a Performance Index $P$, and the logarithmic graph.

Overlaying Performance Indices on Ashby Diagrams

Let's work by example, here. Let's look at our performance index $P = c_{\text{solid}} = \sqrt{\frac{E}{\rho}}$, the performance index being the value that we want to maximize for during our selection process.

Let's take our equation Eq. 14.5.1 and take the logarithim base 10 of each side:

$$\log{c_{\text{solid}}} = \log{\sqrt{\frac{E}{\rho}}}$$

Using our logarithmic identities (Section 16.3) we find that

$$2\log{c_{\text{solid}}} = \log{E}-\log{\rho}$$

In our diagram we plotted $E$ on the $y$-axis and $\rho$ on the $x$-axis, so let's rearrange accordingly:

This equation simply has the form of

$$y = mx + b$$

But each term has a logarithmic operator. This means that if I plot $\rho$ vs $E$ on a logarithmic axis, every increase in of $\rho$ by a factor of 10 yields an increase in $E$ by a factor of 10. This is a linear behavior on a log-log plot.

Summary

This might seem confusing at first, but you'll get it quickly with practice. Just remember:

• Performance Indices can by overlayed as lines on Ashby Diagrams.
• All points on the Performance Index's line have identical performance.
• Many different values of the performance index can be plotted as once (i.e., many values of $c_{\text{solid}}$, like in Figure 14.5.1, which has four different values of $P = c_{\mathrm{solid}}$ plotted as parallel "contour" lines.
• We can use these lines to help us select the best performing materials!