From a wish to a number: the performance index
In guide 1 you learned the Ashby translation: any part carries a function (hold a load, spring back, conduct heat), a set of constraints it must not violate (do not break, do not stretch past a limit, survive to 300 degrees C), an objective you want to push (usually be as light or as cheap as possible), and one or more free variables you are allowed to adjust (the section area, the wall thickness, and — the one we care about — the material itself). The magic step of materials selection is that for a huge class of problems, when you write the objective and the constraints down and eliminate the geometry you were free to change, the material properties fall out as one tidy group. Maximize that group and you have the best material, no trial and error. That group is the performance index.
The cleanest example is the lightest stiff tie — a straight rod of fixed length that must not stretch more than a set amount when you pull it. Its stiffness is S = E A / L, where E is Young's modulus, A the cross-section, L the length. The length is fixed by the job and the material is what we are choosing, so the only knob left is A. Fatten the rod and it is stiffer but heavier; slim it and it is lighter but floppier. We want the lightest rod that still meets the stiffness target — and when you do the algebra, the answer is: pick the material with the largest E divided by density. That ratio, E over rho, is the specific stiffness, and it is the performance index for this job.
- State the function and its constraint: a tie of fixed length L must reach at least a target stiffness S*, and stiffness of a rod is S = E A / L.
- Name the free variable and the objective: we may adjust the cross-section area A, and we want to minimize the mass m = A x L x rho.
- Use the constraint to kill the free variable: from S* = E A / L we get A = S* x L / E — the thinnest rod that still meets the target.
- Substitute A back into the mass: m = (S* x L / E) x L x rho = S* x L^2 x (rho / E).
- Read off the material group: S* and L are fixed by the job, so mass is smallest when rho/E is smallest — that is, when E/rho is largest. The index is E/rho, the specific stiffness.
The twist that surprises everyone: geometry sets the exponent
Here is the part that catches beginners off guard. The performance index for stiffness is not always E/rho — it depends on how the part is loaded. Run the same elimination for a light stiff beam bent by a load, and because a beam's bending stiffness depends on the section's second moment (roughly the width to the fourth power) rather than plain area, the algebra spits out a different group: maximize E^(1/2)/rho. For a flat panel loaded in bending it is E^(1/3)/rho. The exponent on E keeps dropping — 1 for a tie, 1/2 for a beam, 1/3 for a panel — and each drop rewards low density more heavily.
Why does thinning the exponent favor light materials? Because in bending you are allowed to make the section deep. A featherweight material can be spread into a fat, hollow, or tall section that is enormously stiff for its mass, whereas a dense material forced into the same deep shape would weigh far too much. In simple tension you cannot play that game — the rod just has to have enough area, full stop — so density and stiffness trade one-for-one and the exponent stays at 1. The moment bending enters, low density gets a leverage bonus, and the whole ranking of materials can flip.
Watch it happen with real numbers. For a tie (E/rho), steel gives 200/7.9 = about 25, aluminium 70/2.7 = about 26, magnesium 45/1.74 = about 26 — all three metals essentially tie, an honest and famous result: you cannot save weight in a simple tension member by switching among common metals. Now bend a beam and use E^(1/2)/rho: steel drops to sqrt(200)/7.9 = about 1.8, aluminium rises to about 3.1, wood along the grain (E about 10, rho about 0.5) leaps to about 6.3, and carbon-fibre composite (E about 100, rho about 1.6) reaches about 6.3 as well. The same steel that matched aluminium as a tie is now three to four times worse as a beam. This is why aircraft, bicycles, and diving boards are not made of solid steel — and why the specific stiffness you compute is meaningless until you know the loading.
The Ashby chart: the material universe on one page
An index tells you what to maximize, but it does not tell you which materials sit where. That is the job of the material property chart, Ashby's beautiful invention: plot one property against another — most famously modulus E up the side against density rho across the bottom — and drop every material onto it. The Ashby chart uses logarithmic axes on both sides, and that choice is not cosmetic. Real materials span an absurd range: modulus runs from soft rubber near 0.001 GPa to diamond near 1000 GPa, six orders of magnitude, and density from balsa-light foams to dense tungsten. Only a log scale can fit rubber and diamond on the same sheet without one of them vanishing into the margin.
On such a chart every material family gathers into its own cloud — drawn as a balloon or bubble because a family is a range, not a point (steels alone span a fair spread of strength). Metals cluster high and to the right: stiff, but heavy. Ceramics sit highest of all and a bit to the left: stiffer and lighter than metals, but brittle. Polymers hang in the middle-lower band: light but floppy. Foams drop to the bottom-left corner, light and soft. And composites reach into the prized top-left region — high stiffness at low density — which is exactly why they matter. The chart is a map, and once you can read it, the whole field of engineering materials becomes one glance.
E (GPa, log scale)
1000 +----------------------------------------------
| ( CERAMICS: SiC, Al2O3 )
| ( CFRP ) ( WC )
100 + ( wood, along ( steel )
| grain ) ( Ti ) ( Cu )
10 + ( Al ) ( Mg )
| ( polymers: PMMA, nylon )
1 + ( foams )
| / ( rubber )
0.1 + / guideline: slope 1 -> E/rho = const
| / (slide it up-left; last bubble wins)
0.01 +--/----+--------+--------+--------+----------
0.1 0.3 1 3 10 rho (g/cm^3, log)
slope 1 -> tie index E / rho
slope 2 -> beam index E^(1/2) / rho
slope 3 -> panel index E^(1/3) / rhoSelection lines: reading the index straight off the chart
The genius of putting the index on a log-log chart is that it turns into a straight line you can slide. Take E/rho = M and take logs of both sides: log E = log rho + log M. That is a straight line of slope 1, and every material lying on it has the exact same specific stiffness M. Materials above the line beat M; materials below it fall short. So to find the best material, you lay a slope-1 ruler on the chart and slide it up and to the left, raising M as you go, until only a handful of bubbles remain above it. Those survivors — typically woods, then carbon composites, at the very top — are the best light stiff materials, ranked by how far above the last line they sit.
The slope encodes which index you are using, and this is the whole trick of the chart. Because the beam index E^(1/2)/rho rearranges to log E = 2 log rho + constant, its selection line has slope 2; the panel index E^(1/3)/rho gives slope 3. So you do not redraw anything to switch jobs — you just tilt the ruler. Slope 1 for a tie, slope 2 for a beam, slope 3 for a panel, laid on the very same modulus-density chart, each sweeping toward a different champion. The steeper the line, the more sharply it rewards low density, which is precisely why the beam and panel rulers sweep past the heavy metals and settle on wood, balsa, and foams.
Honest limits: one index is never the whole story
The charts hide a deep truth worth saying out loud: stiffness and strength do not respond to the same tricks. Young's modulus is set by the stiffness of the atomic bonds themselves, so it barely moves with heat treatment or cold work — you cannot quench or age your way up the modulus chart, and a material family's E/rho is essentially locked. Strength is different: dislocations, grain refinement, precipitation, and work hardening can multiply a metal's yield by a large factor, so its position on a strength chart really can be pushed. That asymmetry is a recurring theme of this whole ladder, and it means the two charts must be read with different expectations — you engineer strength, but you mostly just choose stiffness.
And a single index is only ever a first cut, because a real part must survive every constraint at once, not just the one you optimized. The lightest stiff beam might be so brittle it shatters on the first knock, so you also test it on a fracture-toughness-versus-strength chart to screen for damage tolerance; a strong-but-weathering polymer fails a service-life screen. The disciplined recipe is to plot each constraint on its own chart, draw the box or line each one demands, and keep only the materials that clear all of them — then rank the survivors by the objective index. The chart never makes the decision for you; it makes the tradeoffs visible so your judgment has something honest to stand on. The strength-toughness tension you met earlier is right there on the axes.
Two last cautions before guide 3. First, a composite's chart bubble usually quotes its best, along-the-fibre value; across the fibres it is far weaker, so a single point flatters an anisotropic material — read composites with that asterisk in mind. Second, mass is not the only objective. Swap 'be light' for 'be cheap' and the objective becomes minimize cost, so density in the index is replaced by (cost per kg) times density, and the winning index for a cheap stiff beam becomes E^(1/2) divided by (Cm times rho). That single substitution is the bridge to the next guide, where material cost, manufacturability, and sustainability enter the selection as constraints and objectives in their own right — because the cheapest, most makeable, greenest material is often not the one the stiffness chart alone would crown.