Michael Ashby's material-selection charts turned an abstract optimisation — choosing the best material for a function — into something an engineer could see: two properties on log–log axes, a guideline of the right slope, and the winning family lights up. This is Part 1 of a two-part note. It follows that idea through two more generations — a multi-objective value function for when one chart is not enough, and an eco-cost axis that turns the same method toward sustainability — and stops exactly where the two-dimensional chart itself starts to run out of room. Part 2, “Ashby's Maps, Renewed,” picks up from there with a generative, learned alternative.

1

Ashby's method, as code

Ashby developed a simple, powerful method for selecting materials in engineering design. Plot two material properties against each other on log–log axes — Young's modulus E against density \rho, say — one bubble per material, grouped by family. For a given design goal, mechanics gives a material index: a combination of properties to maximise or minimise, derived by writing the objective, writing the constraint, and eliminating the free geometric variable that connects them.

1.1

Why one index is rarely enough

Consider a panel of length \ell, width b and free thickness t, loaded in bending, with the objective of minimising mass m=\ell b t \rho subject to a stiffness constraint S = C_1 E I/\ell^3 \ge S^\ast with I=bt^3/12. Eliminating the free variable t between these two equations gives:

m \;\propto\; \underbrace{\left(\frac{12\,S^\ast b^2}{C_1}\right)^{1/3}\ell^2}_{\text{fixed by the design}} \;\times\; \underbrace{\frac{\rho}{E^{1/3}}}_{\text{material index}}

Everything in the first bracket is fixed by the design; only \rho/E^{1/3} depends on the material choice. That is the material index: to minimise mass at a given stiffness, pick the material with the smallest \rho/E^{1/3}. On log–log axes, lines of constant index are straight, so sliding a guideline across the chart instantly reveals the winning family — one of the most quietly effective ideas in engineering design.

That last sentence is worth deriving properly, because it is the whole reason the Ashby chart works at all. Any index of the form M = E^a/\rho — density always to the first power, the stiffness exponent a set by the geometry of the loading case — traces a straight line on log–log axes. Taking logs of E^a=M\rho:

E^a = M\rho \quad\Longrightarrow\quad a\log E = \log M + \log\rho \quad\Longrightarrow\quad \log E = \underbrace{\frac{1}{a}}_{\text{slope}}\log\rho \;+\; \frac{1}{a}\log M

So on a chart of \log E against \log\rho, every value of M draws a line of the same slope 1/a; changing M only slides that line up or down, parallel to itself. Since the goal is to maximise M, the favourable direction is always up and to the left — and the last material the line touches before leaving the data cloud is the answer. Three cases recur often enough to have their own guideline slope on the classic stiffness–density chart:

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

These are exactly the three guideline families printed on Ashby's own stiffness–density chart, and the algebra generalises without change to any pair of properties: replace E and \rho with any two log-scaled axes and the same construction — a straight guideline whose slope is fixed by the case, whose position encodes \log M — still applies. This is the same construction used, non-interactively, by teaching tools such as the Cambridge Engineering Department's material-selection charts, whose full “Select chart” list the figure below reproduces: eleven property pairs, all drawn from the same 22-material set used throughout this article.

Try it. Pick a chart from the dropdown below. Stiffness–density and strength–density carry the named engineering cases derived above (their guideline slopes are checked directly against this article's own results table); every other pair still gets the same draggable log–log guideline — just without an invented textbook name, since M=Y^a/X is a valid index for any two properties, named case or not. Materials below the line dim out of contention as you drag; the survivor, pushed as far up-left as the data allow, is starred. Recycle fraction–cost is the one exception — a bounded 0–1 fraction isn't a power law, so it's a plain scatter instead.

But real components rarely answer to one objective. A tie that must also be cheap and light, a heat exchanger that must resist both thermal shock and corrosion — the moment a second performance metric P_2 matters as much as the first, P_1, a single index and a single chart no longer say which material is best.

1.2

Trade-off surfaces and the Pareto set

With two performance metrics P_1,P_2 — different units, in conflict — plot every candidate on a P_1–P_2 chart. A solution is dominated if another beats it on both metrics at once; the non-dominated solutions trace the trade-off (or Pareto) surface — the only candidates worth considering further.

Dominance, made visible: of ten illustrative candidates, only the six on the trade-off surface are ever worth choosing — every dominated point is beaten on both axes by some other candidate (M7, M4, M1, M3).

Applied to a practical material selection problem — choosing a material that is both stiff (P_1=\rho/E, to minimise) and well damped (P_2=1/\zeta, to minimise) — the same dominance rule turns a table of alloys into a legible trade-off between aluminium at one end and lead at the other:

Stiffness vs. damping: aluminium alloys anchor the stiff, poorly-damped end and lead alloys the compliant, well-damped end, with cast irons and magnesium occupying the middle of the front.

A trade-off surface narrows the field to good compromises but does not by itself choose one. Three strategies do: judgement (inspect and pick intuitively), constrain all-but-one objective and optimise the remaining one, or combine every objective into a single composite score — a value function.

1.3

Value functions and exchange constants

V = a_1 P_1 + a_2 P_2 + \cdots + a_i P_i + \cdots

The coefficients a_i are exchange constants: a_i = \partial V/\partial P_i at fixed P_{j\ne i} — how much one unit of P_i is worth. The best material minimises V. With two objectives, lines of constant V are parallel straight lines of slope -a_1/a_2, and the optimum sits where such a line is tangent to the trade-off surface. When cost itself is one of the objectives (P_1=C, a_1=1), substituting a new material for an incumbent is worthwhile exactly when

\frac{\Delta P}{\Delta C} \le -\frac{1}{a}.

Exchange constants come from technical/economic modelling, historical price data etc. Published values for weight-saving in transport span four orders of magnitude: about £1/kg in a family car (fuel saving), £100–500/kg in a civil aircraft (payload), and £3,000–10,000/kg in a space vehicle. The exchange constant, not the material property alone, determines what “the best material” means for a given application.

1.4

Worked application: co-minimising mass and cost

Return to the panel and add cost as a second objective. From the mass performance equation, define P_1=\rho/E^{1/3} (mass index) and P_2=C_m\rho/E^{1/3}=C_m P_1 (cost index, where C_m is cost per kilogram):

P_1 = \frac{\rho}{E^{1/3}}, \qquad P_2 = C_m\frac{\rho}{E^{1/3}} = C_m\,P_1

Since cost is one of the objectives, a_2=1, so V=a_1P_1+P_2. Recomputing this directly from six real material datasheets, at the two exchange constants (Ashby is from UK) used in Ashby's paper — a_1=£0.5/kg (family-car-like) and a_1=£500/kg (aerospace-like) — reproduces the paper's own tables to within rounding:

Stiffness-constrained panel — recomputed from raw data (cf. Ashby's Table 4).

Material	ρ (Mg/m³)	E (GPa)	Cm (£/kg)	V, a1=0.5	V, a1=500
Cast iron, nodular	7.30	175	0.25	0.98	652.9
Low-alloy steel (4340)	7.85	210	0.45	1.25	660.9
Al 6061-T6	2.85	70	0.95	1.00	346.4
Al-6061-20%SiC, PM	2.77	102	25.0	15.12	311.2
Ti-6-4, B265 grade 5	4.43	115	20.0	18.68	473.7
Beryllium, SR-200	1.84	305	250.0	68.47	205.0

Mass index P_1 vs. cost index P_2. The shallow line (weight cheap) is tangent to nodular (ductile) cast iron; the steep line (weight expensive) is tangent to beryllium.

At low a_1, nodular cast iron wins — cheap and conventional, exactly right when cost outweighs weight. At high a_1, the ranking inverts and beryllium takes over — mechanically remarkable and beside the point until weight becomes expensive enough to pay for it. Sweeping a_1 continuously, rather than checking two hand-picked points, exposes every crossover:

Rank-1 material as the exchange constant a_1 sweeps continuously from £0.1/kg to £1000/kg, for both the stiffness- and strength-constrained panel. The optimum changes discretely, at a handful of crossover points — the direct numerical analogue of sliding a value-function line across the chart.

Beyond two objectives the graphical tangent-line construction stops being practical; the paper's own answer is to rank candidates directly by V=\sum_i a_i P_i, no picture required This is exactly the limit Part 2 removes: a latent space needs no picture at any number of objectives, because it never relied on one..

1.5

Eco-costs and the eco-Ashby chart

An eco-cost is the amount of money that would have to be spent today, with best available technology, to prevent an environmental burden from exceeding what the Earth can carry in the long run. It is a marginal prevention cost, not a cost anyone has actually paid. Avoiding 1000 kg of CO₂ emissions by building extra offshore wind capacity to displace coal power costs on the order of €150, giving an eco-cost for climate change of roughly:

\text{eco-cost}_{\text{CO}_2} \approx 0.13\text{--}0.15\ \text{\euro{}/kg CO}_2\text{-eq}

The same logic applies to every impact category — acidification, eutrophication, toxicity, resource depletion, land use — each with its own prevention cost, summed into a single monetary eco-cost per kilogram. Expressing environmental impact this way, rather than in physical units or dimensionless “points”, buys comparability across categories, plain communication to engineers who already think in cost, and transparency: every number traces back to an explicit, auditable prevention technology, unlike the less transparent weighting schemes used in classical Life Cycle AssessmentBecause eco-costs cover the full environmental profile rather than only carbon, they avoid “carbon tunnel vision”: a material with a small carbon footprint but severe toxicity or depletion impacts does not automatically look attractive.. Eco-costs become a business tool through the Eco-efficient Value Ratio (EVR):

\text{EVR} = \frac{\text{eco-costs}}{\text{value (price paid by the customer)}}

A lower EVR means the same customer value at a smaller environmental burden. The eco-costs framework extends the Ashby method by replacing (or complementing) the price axis with the eco-cost per unit volume, \text{eco-cost}_V = \text{eco-cost per kg}\times\rho, so that a chart plots eco-cost against a mechanical property instead of against price. Three eco-material indices follow the same derivation as any Ashby index:

I_\sigma = \frac{\text{eco-cost}_V}{\sigma_y}, \qquad I_{\sigma,\text{bend}} = \frac{\text{eco-cost}_V}{\sigma_y^{2/3}}, \qquad I_{E} = \frac{\text{eco-cost}_V}{E^{1/2}}

A lower index means a more eco-efficient choice for that structural function: the same mechanical performance for a smaller environmental prevention cost.

Illustrative material data and derived eco-material indices (order-of-magnitude values, for teaching only).

Material	ρ (kg/m³)	E (GPa)	σy (MPa)	eco-cost (€/kg)	IE	Iσ,bend
Structural wood	500	10.0	40	0.05	7.9	2.1
Concrete	2400	30.0	30	0.02	8.8	5.0
Mild steel	7850	210.0	250	0.40	216.7	79.1
Aluminium alloy	2700	70.0	100	1.00	322.7	125.3
PET plastic	1350	2.8	55	0.60	484.1	56.0
GFRP composite	1800	20.0	200	1.50	603.7	78.9

Two things stand out, typical of real eco-Ashby analyses: low-tech, low-processing materials such as wood and concrete carry very low eco-costs per kilogram, often outweighing their modest mechanical properties; metals and engineered composites, although mechanically excellent, carry a much higher environmental prevention cost per unit of stiffness or strength delivered.

Eco-Ashby chart: eco-cost per unit volume vs. Young's modulus. The guideline of slope 1/2 traces constant I_E.

Eco-Ashby chart: eco-cost per unit volume vs. yield strength, guideline of slope 2/3 for I_{\sigma,\text{bend}}.

The same six materials, ranked directly by each eco-material index — the eco-Ashby chart's tabular twin, exactly as Step 4 above falls back to a ranked table once judgement by eye stops scaling.

This is the added value of eco-Ashby charts: they let a designer ask not just “which material is technically best?” but “which material achieves this technical performance at the lowest environmental prevention cost?” — using the same intuitive, visual method engineers already trust.

2

Preliminary conclusion

Three generations of the same idea sit side by side in this note. The chart compresses a mechanics derivation into a slope on two log axes — unbeatable for one index, incapable of more than two properties at a time. The value function removes this limitations by aggregating any number of objectives into one weighted sum, at the cost of losing visual representation: beyond two dimensions, Ashby's own method falls back to a table and a ranking, as §1.4 showed directly. The eco-Ashby chart does not change the mathematics at all — it changes what one of the axes is allowed to mean, substituting a rigorously defined environmental prevention cost for price, and in doing so turns “which material is best?” into “best for whom, and at what true cost?” without asking the designer to learn a new method.

What all three share is a hard limit: every one of them still needs a human to pick the axes, and the moment a design question genuinely depends on three, four, or six properties at once — a spring that must be light and strong and cheap, a heat exchanger balancing thermal shock against corrosion — the chart runs out of dimensions and the value function runs out of picture. That is precisely where this note stops being about charts and starts being about search.

Part 2, “Ashby's Maps, Renewed,” takes the same 22-material database this article has used throughout and trains a variational autoencoder on it — a continuous, differentiable stand-in for the discrete chart, searched by gradient ascent rather than read by eye. The two baseline cases worked by hand in §1.1 above, the stiffness-limited bar and the deflection-limited beam, are the first thing Part 2 checks the learned model against.