Fracture Toughness To Yield Ratio⚠ unverified
Mechanical / Materials · Compute the fracture toughness to yield strength ratio
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Kic | Kic | Pa*m**0.5 | 1.0 | Plane-stra |
| Sy | Sy | Pa | 1.0 | Yield strength of the material |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Kic/Sy | m**0.5 | Fracture toughness index, in square-root metres (m**0.5) |
The science & history
Understanding the Parameters
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Fracture toughness $K_{Ic}$ — resistance to crack propagation (Fracture Toughness): the critical stress-intensity at which a crack runs unstably. High $K_{Ic}$ = damage tolerant.
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Yield strength $S_y$ — the flow strength. There is a fundamental trade-off: processing that raises $S_y$ (hardening, cold work) usually lowers $K_{Ic}$, so this ratio, not either alone, measures true flaw tolerance.
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The ratio squared — $(K_{Ic}/S_y)^2$ is proportional to both the critical crack length at yield-level stress and the plastic-zone size at the crack tip. A material with a large ratio tolerates large cracks and develops a big, blunting plastic zone; a small ratio means tiny critical flaws and brittle behaviour.
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Design meaning — it separates "leak-before-break" materials (large ratio: a crack grows through-wall and leaks detectably before it becomes unstable) from "break-before-leak" (small ratio: sudden fracture).
Derivation (Approaching a Proof)
Linear-elastic fracture mechanics gives the stress-intensity at a crack of length $a$ under remote stress $\sigma$ as $K = Y\sigma\sqrt{\pi a}$ (Stress Intensity Factor). Fracture occurs when $K = K_{Ic}$. Ask: how large a crack can the material tolerate while the net stress is still at yield, $\sigma = S_y$? Setting $K_{Ic} = Y\,S_y\sqrt{\pi a_c}$ and solving,
$$a_c = \frac{1}{\pi Y^2}\left(\frac{K_{Ic}}{S_y}\right)^2 \;\propto\; \left(\frac{K_{Ic}}{S_y}\right)^2.$$
Likewise the crack-tip plastic-zone radius is $r_p = \tfrac{1}{2\pi}(K_{Ic}/S_y)^2$ (plane stress). Both key length scales of ductile/brittle behaviour are set by this ratio squared — which is why $K_{Ic}/S_y$ is the material figure of merit for flaw tolerance.
Dimensional check. $\dfrac{K_{Ic}}{S_y} = \dfrac{\text{Pa}\cdot\text{m}^{0.5}}{\text{Pa}} = \text{m}^{0.5}$ — as the output label states; squaring gives a length, confirming the critical-crack interpretation.
History and Development
The toughness/yield ratio crystallised out of linear-elastic fracture mechanics (Irwin, 1950s), building on Griffith's energy theory (1920). The realisation that a small flaw plus low toughness causes catastrophic fracture — driving failures from Liberty-ship hulls to Comet aircraft — made $K_{Ic}/S_y$ and the critical-crack size it defines central to damage-tolerant design in aerospace, pressure vessels, and pipelines. It is a standard axis pairing on fracture-mechanics selection charts.
Related Concepts: Fracture Toughness, Stress Intensity Factor, Material Selection Index Toughness, Material Selection Index Strength, Ashby Charts, Static Failure Theories
Notes: $(K_{Ic}/S_y)^2 \propto$ critical crack length and plastic-zone size. Large ratio = damage tolerant / leak-before-break; small = brittle. Captures the strength–toughness trade-off that neither property shows alone.