Chapter 1 · Materials & the Environment

Ashby's Maps

How a log–log chart turns a mechanics derivation into a straight line, what to do when one line is not enough, and how the same mathematics carries an eco-cost axis.

Every engineering component is a compromise between what a material can do and what it costs us: in mass, in money and, increasingly, in damage to the environment. Michael Ashby's contribution was to make that compromise visible. Plot two properties of every candidate material on logarithmic axes, and the best choice for a given job sits on one side of a straight line whose slope comes from mechanics.

This chapter rebuilds that method from first principles, extends it to several objectives through a value function, and swaps price for an eco-cost to obtain the eco-Ashby chart. It ends where two-dimensional charts run out of room, which is where Chapter 2 begins.

1Ashby's method, as code

Take a flat panel of fixed length L and width b, loaded in bending. It must not deflect more than a set amount, so its bending stiffness S is fixed. Its thickness t is free. We want the lightest panel.

m = \rho\, L\, b\, t, \qquad S = C\,\frac{E\, b\, t^3}{L^3}.

Eliminate the free variable t using the stiffness constraint, and the mass becomes

m = \underbrace{\left(\frac{12 S}{C b}\right)^{1/3} b\,L^{2}}_{\text{fixed by the design}} \;\cdot\; \underbrace{\frac{\rho}{E^{1/3}}}_{\text{material}}

Every term except the last is set by the design. The material enters only through \rho/E^{1/3}, so the lightest panel is the one that maximises the material index M = E^{1/3}/\rho.

Take logarithms of any index of the form M = E^{a}/\rho:

\log E = \tfrac{1}{a}\,\log \rho + \tfrac{1}{a}\,\log M.

On log–log axes, all materials with the same M lie on a straight line of slope 1/a. Sliding that line upward raises M; the materials that remain above it are the better choices. The mechanics sets the slope, and the chart does the ranking.

Loading case	a	Index M	Guideline slope
Tie, axial, stiffness-limited	1	E/\rho	1
Beam or shaft, bending/torsion, stiffness-limited	1/2	E^{1/2}/\rho	2
Panel, bending, stiffness-limited	1/3	E^{1/3}/\rho	3

Chart Reset line

Figure 1.1A live Ashby chart. Choose a property pair and a loading case, then drag the ochre guideline up or down. Materials below the line fade out; the material with the highest index is starred. The 22 materials and their densities and moduli are the CES/Granta set used throughout the book. Strengths and prices are approximate handbook values added for teaching.

Try the panel case on the stiffness–density chart. Woods and foams climb the ranking because the cube root rewards low density far more than high stiffness.

2Why one index is rarely enough

A single index answers a single question: the lightest panel, the cheapest tie, the stiffest beam per unit mass. Real components ask several at once. An aircraft panel must be light and affordable; a car door must be stiff, cheap and recyclable. Each requirement has its own index, and the indices disagree. Beryllium makes a superb light panel and a ruinous cheap one.

When objectives conflict there is no single best material, only a set of best compromises. Finding that set, and then choosing a point on it, is the subject of the next two sections.

3Trade-off surfaces and the Pareto set

Plot each candidate by two performance metrics that we want to minimise, say a mass index P_1 and a cost index P_2. A candidate is dominated when another candidate is better on both metrics at once. Dominated candidates can be discarded without knowing how much we value mass against money.

What remains is the non-dominated set, the Pareto front or, in Ashby's vocabulary, the trade-off surface. Moving along it, every gain on one metric costs something on the other. The front tells us which materials are worth arguing about. It does not tell us which one to pick.

4Value functions and exchange constants

To pick a single point we need a price for each property. The value function adds the penalised properties with weights:

V = \sum_i \alpha_i\, P_i = \alpha_1 P_1 + \alpha_2 P_2 + \dots

The exchange constant \alpha_i is what one unit of property P_i is worth. If P_2 is a cost, then \alpha_2 = 1 and \alpha_1 is the money we would pay to save one kilogram. The lowest V wins. Geometrically, lines of constant V are straight on linear axes and curved on log axes; the optimum is where the lowest such contour touches the trade-off surface.

Sector	Exchange constant α₁ (£ per kg saved)
Family car (mass and fuel)	≈ 1
Civil aircraft	100 – 500
Space vehicle	3 000 – 10 000

The spread is four orders of magnitude. The same material data, valued by a car maker and by a satellite builder, lead to different materials.

5Worked application: co-minimising mass and cost

Return to the stiffness-limited panel. Its mass index is P_1 = \rho/E^{1/3} and its material cost index is P_2 = C_m P_1, with C_m the price per kilogram. The value function is V = \alpha_1 P_1 + P_2. The six candidates are those of Ashby's own example.

Exchange constant α₁

£0.50/kg

Figure 1.2Choosing along the trade-off surface. Top: mass index against cost index on log axes. Hollow markers are dominated; the dashed curve is the constant-V contour through the current winner. Bottom: the winner across the full range of exchange constants. It changes only at a few discrete crossover points.

At \alpha_1 = £0.5/kg, a value typical of commodity products, nodular cast iron wins with V = 0.98: cheap, stiff enough, and heavy enough not to matter. At \alpha_1 = £500/kg, the value of mass in aerospace, beryllium wins with V = 205 despite costing a thousand times more per kilogram. These figures match Ashby's published table to rounding.

Between the two extremes the winner changes only three times: from cast iron to the aluminium alloy near £0.55/kg, to the aluminium–SiC composite in a narrow window around £140–170/kg, and to beryllium beyond that. Each crossover is an exchange constant at which two materials have equal value. Knowing those crossovers is often more useful than knowing the winner at a single guessed constant.

6Eco-costs and the eco-Ashby chart

Price is one axis among many. The eco-cost method assigns an environmental burden a monetary value: the marginal cost of preventing that burden so that it stays within what the Earth can carry. For greenhouse gases the prevention cost is of the order of €0.13–0.15 per kilogram of CO₂-equivalent. Unlike a classical life-cycle assessment, which reports a dozen impact categories in different units, eco-costs reduce to one number in euros, and that number can go on an Ashby axis.

Eco-costs are tabulated per kilogram. A component is sized by volume, so multiply by density:

\text{eco-cost}_V = \text{eco-cost}_{kg}\;\rho \quad [\text{EUR/m}^3].

The same derivations as in Section 1 now give eco-material indices, to be minimised:

I_{E} = \frac{\text{eco-cost}_V}{E^{1/2}} \;\;\text{(stiff beam)},\qquad I_{\sigma,\text{bend}} = \frac{\text{eco-cost}_V}{\sigma_y^{2/3}} \;\;\text{(strong beam)},\qquad I_{\sigma} = \frac{\text{eco-cost}_V}{\sigma_y}\;\;\text{(tie)}.

At product level the companion measure is the eco-efficient value ratio, \text{EVR} = \text{eco-costs}/\text{value}: the environmental cost per euro a customer is willing to pay.

Figure 1.3An eco-Ashby chart. Eco-cost per cubic metre against modulus (guideline slope 1/2) or yield strength (slope 2/3). The best materials sit to the lower right; drag the guideline to filter. Right: the six materials ranked by the selected eco-index. All values are illustrative order-of-magnitude teaching figures, not Idemat or CES data.

Wood and concrete lead both rankings, because their eco-cost per kilogram is tiny and their density moderate. Aluminium, PET and GFRP trail by one to two orders of magnitude. This is the eco-Ashby chart's main lesson: per unit of function, the ordering of materials can differ completely from the ordering per kilogram.

7Preliminary conclusion

Three tools, one lineage:

• Charts reduce a mechanics derivation to a slope on log axes, but they show one index and two properties at a time.
• Value functions combine any number of objectives into one score, at the price of the picture.
• Eco-Ashby charts keep the mathematics and replace price with a defined environmental prevention cost.

All three still need a person to choose the axes. When a decision hinges on three or more properties, the chart runs out of dimensions. Chapter 2 replaces the chart with a learned, continuous map and turns selection into a search.