Material Selection Index Stiffness⚠ unverified
Mechanical / Materials · Compute the stiffness-limited 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 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | M | kg/m**3 | Stiffness-limited selection index, in pascals per (kg/m**3) |
The science & history
Understanding the Parameters
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Young's modulus $E$ — the stiffness numerator; for an axial (tie) stiffness requirement, mass scales as $\rho/E$, so a high $E/\rho$ minimises mass.
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Density $\rho$ — the mass penalty. As with Specific Modulus, $E/\rho$ is nearly constant ($\approx 25$ MJ/kg) across steel, aluminium, and titanium, so this index rarely separates the common structural metals for a tie.
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Index $M$ — a pure material property; the largest value wins. For bending-limited members the correct index is $E^{1/2}/\rho$ (beam) or $E^{1/3}/\rho$ (plate) instead — see Ashby Chart Index.
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Unit caveat — the true unit is energy per mass (J/kg), the same as specific modulus; the registry's kg/m³ label is a bug.
Derivation (Approaching a Proof)
For an axial member of fixed length $L$ meeting a stiffness target $S = EA/L$, the required area is $A = SL/E$ and the mass is
$$m = \rho A L = (S L^2)\,\frac{\rho}{E}.$$
With $S$ and $L$ fixed, mass is minimised by maximising $E/\rho$. This is identical to the derivation for Specific Modulus — the two cards compute the same index.
Dimensional check. $M = \dfrac{E}{\rho} = \dfrac{\text{Pa}}{\text{kg}/\text{m}^3} = \dfrac{\text{J}}{\text{kg}}$ — energy per mass, confirming the registry's kg/m³ label is incorrect.
History and Development
The stiffness index $E/\rho$ is the first and most-cited of Ashby's material-selection indices (1980s), introduced to answer "what is the lightest material for a stiff strut?" Plotted on the modulus–density Ashby Charts it is a slope-1 guideline. Its near-constancy across metals is the standard motivating example for composites in stiffness-critical, weight-sensitive design.
Related Concepts: Specific Modulus, Material Selection Index Strength, Ashby Chart Index, Ashby Charts, Material Selection Index Toughness, Material Selection Index Stiffness Thermal
Notes: Duplicate of Specific Modulus ($E/\rho$). Output unit should be J/kg (registry mislabels kg/m³). Axial (tie) index — bending uses $E^{1/2}/\rho$, $E^{1/3}/\rho$ (Ashby Chart Index).