Material Selection Index Toughness⚠ unverified
Mechanical / Materials · Compute the toughness-limited material selection index
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Kic | Kic | Pa*m**0.5 | 1.0 | Plane-stra |
| rho | ρ | kg/m**3 | 1.0 | Mass density of the material |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | M | Pa*m**0.5 | Toughness-limited selection index, in (Pa*m**0.5) per (kg/m**3) |
The science & history
Understanding the Parameters
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Fracture toughness $K_{Ic}$ — the material's resistance to crack propagation (Fracture Toughness), the critical stress-intensity at which a sharp crack runs. It is the damage-tolerance numerator.
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Density $\rho$ — the weight penalty, as in every specific-property index.
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Index $M$ — highest value = lightest damage-tolerant material. Which exponent on $K_{Ic}$ is "correct" depends on the failure scenario (yield-before-break vs leak-before-break); this card uses the simple first-power form. Ashby's fracture-limited design also uses indices like $K_{Ic}/\rho$ and $K_{Ic}^{2/3}/\rho$ for different geometries.
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Trade-off with strength — high-strength materials often have lower toughness; the fracture-mechanics ratio $K_{Ic}/S_y$ (Fracture Toughness To Yield Ratio) captures that tension separately.
Derivation (Approaching a Proof)
Fracture-limited selection follows the same Ashby template as the strength index: express the mass of a component that must not fracture, in terms of a fixed functional requirement times a material group, then minimise. For a tie that must tolerate a crack of fixed relative size, the fracture condition $\sigma \le K_{Ic}/\sqrt{\pi a}$ (constant $a$) plays the role of the strength limit, so the failure stress scales with $K_{Ic}$ and the mass scales as $\rho/K_{Ic}$. Minimising mass then maximises
$$M = \frac{K_{Ic}}{\rho}.$$
Other geometries (fixed crack length rather than fixed relative size, or leak-before-break vessels) shift the exponent on $K_{Ic}$, giving the related indices $K_{Ic}^{1/2}/\rho$ and $K_{Ic}^{2/3}/\rho$.
Dimensional check. $M = \dfrac{K_{Ic}}{\rho} = \dfrac{\text{Pa}\cdot\text{m}^{0.5}}{\text{kg}/\text{m}^3} = \text{Pa}\cdot\text{m}^{3.5}/\text{kg}$ — confirming the registry label drops the density term.
History and Development
Toughness–density selection is part of Ashby's treatment of fracture-limited design, plotted on the fracture-toughness–density (and toughness–strength) Ashby Charts. The recognition that strength and toughness usually trade off — and that both, weighted by density, matter for safe lightweight structures — is central to aerospace and pressure-vessel material choice, where flaw tolerance is as important as strength.
Related Concepts: Fracture Toughness To Yield Ratio, Fracture Toughness, Stress Intensity Factor, Material Selection Index Strength, Ashby Chart Index, Ashby Charts
Notes: Higher $K_{Ic}/\rho$ = lighter damage-tolerant material. Registry unit label drops the density division (true unit Pa·m$^{3.5}$/kg). Exponent on $K_{Ic}$ varies with geometry (yield-before-break / leak-before-break). Strength–toughness trade-off: see Fracture Toughness To Yield Ratio.