Material Selection

Intuition

Mechanical engineering education suffers from a chronic confusion between description and selection.[fn::This distinction is not academic. Most failures I have seen in early-stage design reviews trace to an engineer reaching for "what they know" — usually mild steel or 6061 aluminum — rather than what the objective function demands. The metallurgy note (Metallurgy) tells you what steel is; this note tells you how to decide whether steel is the right answer at all.] The metallurgy note is a textbook: it catalogues phase diagrams, heat treatments, and the microstructural mechanics of what a given alloy is. Useful, but it does not answer the question an engineer is actually paid to ask: "given an objective and a constraint, which material should I pick?" That question belongs to selection methodology, and the honest answer is almost never the material you first reached for.

The central insight of Ashby's program is that the design problem and the material problem are coupled through a small number of dimensionless-ish indices — functions of material properties that rank candidates independent of geometry.[fn::Strictly, the indices are not always dimensionless; M = E^{1/2}/ρ has units of (Pa^{1/2})/(kg/m^3). The ranking is what matters, since all candidates carry the same units. The pedant who insists on dimensional homogeneity before ranking has missed the point — which is, predictably, a mistake I have made.] Plot properties on log-log axes, overlay the index as a straight line of a particular slope, and the ranking becomes a visual scan: materials above the line are better; below, worse. This is the Ashby chart, and it is the single most useful tool in the materials engineer's kit — provided it is treated as a screening device, not an oracle.

This note is the tool counterpart to the textbook that meche_metallurgy.org already is. Where metallurgy describes, this note decides.

First principles

The objective-constraint decomposition

Every material selection problem — every one, without exception I have been able to find in the literature — can be decomposed into:

1. An objective: minimize mass, minimize cost, maximize stiffness, maximize some utility. 2. A constraint: fixed length, fixed stiffness, fixed deflection, fixed load, fixed cost. 3. A free variable: usually a cross-sectional dimension A or second moment I, free to vary so as to satisfy the constraint at minimum objective. 4. A material, which the engineer gets to choose.

The trick — and it is a trick worth stating plainly because the textbooks often bury it — is to eliminate the free variable using the constraint, substitute into the objective, and factor the result into a product of (geometric terms) × (a material index M). The geometry cancels or becomes a constant; the material index alone ranks candidates.

Derivation: tie rod, minimize mass, stiffness-constrained

A tie rod of length L, cross-section A, density ρ, Young's modulus E. Objective: minimize mass m = ρAL. Constraint: axial stiffness k = EA/L ≥ k_0 (some required stiffness). Free variable: A.

Eliminate A from the constraint: A = k_0 L / E. Substitute:

m = ρ (k_0 L / E) L = (k_0 L^2) · (ρ / E).

The bracketed geometric term is fixed; the material ranking is by minimizing ρ/E, i.e. maximizing E/ρ. This is the specific stiffness, the material index for a light, stiff tie rod:

M_tie = E / ρ.

Note the form: it is a ratio, not a logarithm or a power, because the loading is pure tension — no bending moment distribution to introduce shape-dependent exponents. The simplicity is misleading; beams are where the real structure appears.[fn::The tie rod is the case undergraduates remember because it is the only one where the index is a clean ratio. Engineers who never revisit the derivation retain a false intuition that "specific stiffness" is the universal figure of merit. It is not; it is the figure of merit for axial loading only. The beam index below is the one that actually governs most real structures.]

Derivation: beam, minimize mass, bending-stiffness-constrained

A cantilever of length L, square cross-section b × b, second moment I = b^4/12, density ρ, modulus E. End load F; tip deflection δ = FL^3 / (3EI). Constraint: δ ≤ δ_0. Objective: minimize m = ρ b^2 L. Free variable: b.

From the constraint: EI ≥ FL^3 / (3 δ_0), so I ≥ FL^3 / (3 E δ_0). With I = b^4/12:

b^4 = 12 FL^3 / (3 E δ_0) = 4 FL^3 / (E δ_0), b^2 = (4 FL^3 / (E δ_0))^{1/2}.

Substitute into m:

m = ρ L (4 FL^3 / (E δ_0))^{1/2} = (const) · ρ / E^{1/2}.

So minimize ρ / E^{1/2}, i.e. maximize the beam stiffness index:

M_beam = E^{1/2} / ρ.

The square root appears because bending stiffness scales as E·I, and I scales with the fourth power of the linear dimension while mass scales with the second — the exponent (1/2) falls out of that 4-to-2 ratio. Change the loading (tie, beam, plate, column) and you change the exponent. *The index is not a property of the material; it is a property of the material-and-the-loading-case.*[fn::This is the single most underappreciated point in the methodology. Engineers who memorize "E/ρ is the specific stiffness" then apply it to a beam in bending have committed a real error, not a pedantic one: E^{1/2}/ρ and E/ρ rank CFRP, Al, Ti, and steel in different orders at the margins. See the worked example below for the numerical consequence.]

Derivation: beam, minimize mass, bending-strength-constrained

Same beam, but now the constraint is strength: the maximum bending stress σ must not exceed the yield σ_y. For a rectangular section, σ_max = (M · c) / I where c = b/2 and M = FL (worst-case moment at the root). So σ_max = FL·(b/2) / (b^4/12) = 6FL / b^3 ≤ σ_y.

Solve for b^3 = 6FL / σ_y, so b^2 = (6FL / σ_y)^{2/3}. Mass:

m = ρ L b^2 = ρ L (6FL)^{2/3} · σ_y^{-2/3} = (const) · ρ / σ_y^{2/3}.

So the strength index for a light, strong beam is

M_strength-beam = σ_y^{2/3} / ρ,

while for a tie rod (pure tension, σ = F/A ≤ σ_y → A = F/σ_y → m = ρ L F / σ_y) it is simply

M_strength-tie = σ_y / ρ,

the specific strength. The exponent again tracks the loading: tension gives the clean ratio; bending introduces the 2/3 because strength goes as section modulus (~b^3) while mass goes as ~b^2. Plates in bending give 1/2; columns in buckling give 1/2 for the Euler case. The pattern: the more the load is carried by geometric stiffening (bending, buckling) rather than area (tension), the smaller the exponent on the property, and the less the highest-property material dominates.

Ashby chart construction

Given an index, say M = E^{1/2}/ρ, the chart is built as follows:

• Plot log E (vertical) against log ρ (horizontal), one point per material.
• Take the log of the index: log M = (1/2) log E − log ρ. Setting M constant gives log E = 2 log ρ + 2 log M, a line of slope 2 on the chart.
• Materials lying above the line have higher M (better); below, lower. Slide the line up to find the best candidates, down to relax the constraint.

The slope is set by the exponent on the denominator property — slope = (exponent of numerator)/(exponent of denominator). For E/ρ (tie), slope 1. For E^{1/2}/ρ (beam), slope 2. For σ_y^{2/3}/ρ (strength beam), slope 3/2. For σ_y/ρ (strength tie), slope 1. The chart literally encodes the loading case in its slope; a different problem is a different chart.

Trade-off maps and penalty functions

Single-index selection is the toy version. Real designs have multiple objectives — light and cheap and manufacturable and impact-tough — that no single index captures. The penalty-function method formalizes this:

Define a value function V = (material's contribution to objective 1) + α · (contribution to objective 2) + β · (contribution to objective 3) + …

where α, β are exchange rates the designer sets — e.g. "one kilogram of extra mass is worth $X in cost," derived from a system-level trade study.[fn::The exchange rates are the hard part. They are never given; they must be argued for from mission/system context. A satellite and a bicycle frame have wildly different $/kg, and the wrong rate silently inverts the ranking between CFRP and aluminum. I have yet to see a design doc that justified its exchange rate rather than asserted it.] On the chart, the penalty function appears as a family of parallel lines, one per exchange rate; the optimal material is the one whose point is tangent to the lowest attainable penalty line. This is exactly Lagrangian duality in disguise, but graphical.

The CES EduPack / Granta Selector methodology codifies this: it provides a limit stage (apply constraints — must have ρ < 5000 kg/m^3, must have σ_y > 200 MPa, must be recyclable) to screen out infeasible candidates, then a ranking stage (apply the index) to order the survivors, then a documentation stage (read the case studies, the supplier data, the failure modes) to make the final call. The discipline is in not skipping the third stage: the chart is a sieve, not a verdict.

Worked example

Problem. Select a material for a light, stiff cantilever beam, length L = 1 m, end load F = 500 N, tip deflection constrained δ ≤ 2 mm. Square cross-section b × b is free to vary. Compare CFRP (unidirectional, fiber direction), 6061-T6 aluminum, and Ti-6Al-4V, computing the actual mass of each at equal stiffness.

Index. From the derivation above, M_beam = E^{1/2}/ρ — maximize. But to make the comparison concrete we need actual stiffness, so we'll compute b and m for each candidate at δ = 2 mm exactly.

Constraint. δ = FL^3 / (3EI) = 2 × 10^{-3} m. So EI = FL^3 / (3 δ) = (500)(1) / (3 · 2 × 10^{-3}) = 500 / (6 × 10^{-3}) = 8.333 × 10^4 N·m^2. With I = b^4/12: b^4 = 12 · 8.333 × 10^4 / E = 10^6 / E. So b = (10^6 / E)^{1/4} m, and m = ρ · b^2 · L = ρ · (10^6 / E)^{1/2} · 1 = ρ · 10^3 / E^{1/2} (in SI units).

Now plug in (carrying units):

Check the indices directly:

The ranking by the index (CFRP > Al > Ti) matches the ranking by actual mass (4.35 < 10.28 < 13.12 kg). The index works because the geometric term canceled — which is the entire point of deriving it.[fn::The cancellation is also why the index is robust: it does not depend on F, L, or δ. Change the load to 5000 N and the absolute masses scale by sqrt(10) (~3.16×), but the ranking is unchanged. This invariance is the practical value of the method — you can rank materials before you've finalized the geometry.]

Counter-observation. Note that Ti-6Al-4V — a prestige aerospace alloy — is worse here than ordinary 6061 aluminum, by 28% mass, despite costing ~10× more per kg. The reason: titanium's high density (4430 vs 2700) outweighs its modulus advantage. This is exactly the kind of finding the index produces, and exactly the kind of finding engineers miss when they "reach for the fancy material." A reviewer who sees "Ti-6Al-4V cantilever" in a mass-critical design has grounds to question it.[fn::The exception: titanium's corrosion resistance, fatigue behavior, and high-temperature retention may justify it in service environments where Al or CFRP fail. The index says nothing about these because they are not in the objective or the constraint. This is the canonical demonstration that selection is not ranking — the index is a sieve, the final call is the designer's.]

Cost as a tie-breaker. If we now add cost as a second objective with exchange rate α = $200/kg (typical aerospace-systems value of mass), the penalty functions are:

• CFRP: 4.35 × 200 + material cost. CFRP ~$40/kg → 870 + 174 = $1044 (material only, ignoring layup).
• Al: 10.28 × 200 + 10.28 × $4 = 2056 + 41 = $2097.
• Ti: 13.12 × 200 + 13.12 × $80 = 2624 + 1050 = $3674.

Now CFRP wins by an even larger margin because the mass penalty dominates the material cost. But lower the exchange rate to α = $20/kg (consumer product):

• CFRP: 87 + 174 = $261.
• Al: 206 + 41 = $247.
• Ti: 262 + 1050 = $1312.

Aluminum now wins. The exchange rate inverts the ranking between CFRP and aluminum — a single scalar, asserted by the designer, deciding the answer. This is why the penalty function is powerful and dangerous: the methodology gives you a clean answer, but only to the problem you posed, with the rates you chose.[fn::A common failure: the exchange rate is set by analogy to a prior project without examining whether the analogy holds. A satellite's $/kg has no business in a bicycle frame trade study. I would treat any single number quoted as "the mass penalty" with suspicion unless its derivation is shown.]

Pitfalls

1. Optimizing a single index and ignoring manufacturability. The index M = E^{1/2}/ρ puts CFRP first, but a CFRP cantilever requires layup, cure cycles, and tooling that an aluminum extrusion does not. If the build quantity is one, the tooling cost dominates the mass savings; the index is silent on this. Selection methodology assumes you will then apply a documentation stage — the failure mode is stopping at the ranking stage and treating it as the answer.

2. Treating Ashby charts as exact data rather than screening tools. The chart's property bands are ranges: 6061-T6 is one point in a band that spans the whole 2xxx–7xxx family and overlaps with steels in places. A material that looks superior on the chart may, in its specific temper and form, be inferior to the band-average. The chart narrows the search; it does not close it.[fn::The bands also hide anisotropy. The CFRP datum used above is the fiber-direction modulus; the transverse modulus is ~10 GPa, off the bottom of the chart. A beam loaded off-axis behaves nothing like the chart suggests. Anyone using a single chart point for an anisotropic material has lied to themselves, however unintentionally.]

3. Forgetting that the index depends on the loading case. E/ρ is the tie-rod index; E^{1/2}/ρ is the beam index; E^{1/3}/ρ is the plate index; σ_y^{2/3}/ρ is the beam-strength index. Applying the wrong one — say E/ρ to a beam — systematically mis-ranks materials, often in favor of the densest high-modulus candidate (steel, tungsten). This is the most common technical error in selection work I have seen.[fn::There is a related subtler error: choosing the index for the dominant failure mode without checking whether the constraint you used is actually the binding one. A beam may be stiffness-governed at long spans and strength-governed at short spans; using the stiffness index throughout over-designs the short members and under-designs the long ones. The honest move is to check both and take the more restrictive.]

4. Ignoring cost as a constraint. The index ranks by mass or stiffness, not by dollars. A "minimize mass" optimization that ignores cost will happily specify beryllium or carbon-carbon. Real engineering has budgets. Either fold cost into a penalty function (preferred, when the exchange rate is defensible) or apply it as a hard limit in the screening stage ("cost < $X per part").[fn::Hard limits are easier to defend than exchange rates, but they are blunt — a part that exceeds the cost ceiling by 1% is rejected even if it halves mass. Penalty functions are smoother but require the rate. Neither is right; choose by how well you understand the system.]

5. Conflating material properties with design properties. The chart lists material modulus and material strength. A real part has stress concentrations, residual stresses, and size effects (Weibull) that reduce the allowable by safety factors of 1.5–4. The index does not know about your knuckle geometry. Apply knockdowns after ranking, or — better — use a property chart of design allowables where one exists.

6. Selection by precedent ("we always use 17-4 PH"). This is not a pitfall of the methodology but of not using it. The methodology's value is precisely that it surfaces alternatives the precedent-based engineer would not consider. The cost of running an Ashby analysis on a part is minutes; the cost of not running it can be a 30% mass penalty that survives the entire product lifecycle because no one questioned it.

References

• Ashby, M.F. (2005). /Materials Selection in Mechanical Design/, 3rd ed. Oxford: Butterworth-Heinemann. — the canonical reference; all derivations above follow Chs. 5–7.
• Ashby, M.F., Shercliff, H.R., Cebon, D. (2013). /Materials: Engineering, Science, Processing and Design/, 3rd ed. — undergraduate companion, lighter on derivations, heavier on worked cases.
• Granta Design / CES EduPack documentation. — the software implementation of the limit-rank-document methodology; the Selector module is the practical tool.
• Callister, W.D. & Rethwisch, D.G. (latest). /Materials Science and Engineering: An Introduction/. — property data and the metallurgical context; cross-reference with Metallurgy.
• Dieter, G.E. & Schmidt, L.C. (2009). /Engineering Design/, 4th ed. McGraw-Hill. — the broader design-process context, where material selection sits as one decision among many.
• Ashby, M.F. (2004). "Materials Selection in Mechanical Design," /Philosophical Magazine Letters/ — shorter expositions of the index method exist; the book is still the source.
• Metallurgy — the textbook companion to this tool note.
• Polymers — for the lower-modulus, lower-density end of the charts.
• Composites — CFRP, GFRP; the high-M materials that dominate the upper-left of E^{1/2}/ρ charts.
• XRD — characterization, relevant when verifying the material that the Selector ranked first is actually what you received.
• Tribology — surface properties are not on any Ashby chart I know of; a wear-critical selection needs a separate analysis entirely.

Related

• Metallurgy
• Polymers
• Composites
• XRD
• Tribology