Ashby Chart Index⚠ unverified
Mechanical / Materials · Compute a generalized Ashby material selection index
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| E | E | Pa | 1.0 | Young's modulus of the material |
| rho | ρ | kg/m**3 | 1.0 | Mass density of the material |
| n | n | — | 1.0 | Exponent applied to the modulus, dimensionless. Common values are ``1`` (tie), ``1/2`` (beam), and ``1/3`` (plate) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | M | meter ** 3 * pascal / kilogram | Ashby selection index, in (Pa**n) per (kg/m**3) |
The science & history
Understanding the Parameters
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Exponent $n$ — the heart of the index. It comes from how the stiffness constraint couples to the free geometric variable (area or thickness). $n = 1$: only area is free (tie). $n = 1/2$: beam depth is free, and stiffness $\propto$ (second moment) makes $E^{1/2}$ the right power. $n = 1/3$: plate thickness is free, giving $E^{1/3}$.
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Young's modulus $E$ — raised to the power $n$; the more geometry can compensate (smaller $n$), the less raw modulus matters and the more density dominates — which is why low-density materials win bending applications so decisively.
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Density $\rho$ — always in the denominator to the first power: mass is proportional to $\rho$ at fixed performance, so it is penalised linearly regardless of loading mode.
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Using the index — compute $M$ for each candidate at the appropriate $n$ and pick the largest; on an Ashby chart this is a straight guideline of slope $1/n$ swept toward the top-left.
Derivation (Approaching a Proof)
For a stiffness-limited component, write the performance as (functional requirement) $\times$ (geometric term) $\times$ (material group), then minimise mass. Two cases:
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Tie (area $A$ free, length $L$ fixed): stiffness $\propto EA$, mass $\propto \rho A$; eliminating $A$ gives mass $\propto \rho/E$, so maximise $E/\rho$ — i.e. $n = 1$.
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Beam (square section side $b$ free): bending stiffness $\propto E b^4$, mass $\propto \rho b^2$; eliminating $b$ gives mass $\propto \rho/E^{1/2}$, so maximise $E^{1/2}/\rho$ — i.e. $n = 1/2$.
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Plate (thickness $t$ free, in-plane size fixed): bending stiffness $\propto E t^3$, mass $\propto \rho t$; eliminating $t$ gives mass $\propto \rho/E^{1/3}$, so $n = 1/3$.
Each case yields the same template $M = E^{n}/\rho$ with the geometry setting $n$. The general index simply parameterises the family.
Dimensional check. $M = \dfrac{E^{n}}{\rho} = \dfrac{\text{Pa}^{n}}{\text{kg}/\text{m}^3} = \text{Pa}^{n}\cdot\text{m}^3/\text{kg}$ — dimensionally $n$-dependent, confirming the registry's fixed label holds only at $n = 1$.
History and Development
The parameterised index is the general form of Michael Ashby's material-selection method (Materials Selection in Mechanical Design, 1992). Ashby's key insight was that separating the design objective into function, geometry, and material lets the material be optimised independently, producing indices like $E^{1/2}/\rho$ that explain why wood and composites outperform steel in bending-limited, weight-critical structures. Plotted on log–log Ashby Charts, each index is a family of parallel selection lines.
Related Concepts: Specific Modulus, Specific Strength, Ashby Charts, Material Selection Index Stiffness, Material Selection Index Strength, Material Selection Index Toughness
Notes: $n = 1$ tie, $1/2$ beam, $1/3$ plate (stiffness-limited). Units of $M$ depend on $n$ — the fixed Pa·m³/kg label is exact only at $n=1$; compare materials at the same $n$. Strength-limited design uses $S^{n}/\rho$ analogues (with different exponents).