Fracture Toughness Calculator

Linear elastic fracture mechanics describes the stress field near a crack tip using the stress intensity factor $$K$$. For a mode I (opening) crack under tensile loading:

$$K_I = Y \sigma \sqrt{\pi a}$$

The stress field near the crack tip follows: $$\sigma_{ij}(r,\theta) = \frac{K_I}{\sqrt{2\pi r}} f_{ij}(\theta)$$ showing the characteristic $$1/\sqrt{r}$$ singularity. The geometry factor $$Y$$ depends on the configuration:

• Infinite plate, center crack: $$Y = 1.0$$
• Edge crack in semi-infinite plate: $$Y = 1.12$$
• Penny-shaped embedded crack: $$Y = 2/\pi \approx 0.637$$
• Single-edge notch bend (SENB): $$Y$$ depends on $$a/W$$ ratio (typically 1.0–2.5)

Fracture occurs when $$K_I \geq K_{Ic}$$, where $$K_{Ic}$$ is the plane-strain fracture toughness — a material property measured under standardized conditions (ASTM E399). The critical crack size at a given stress is: $$a_c = \frac{1}{\pi}\left(\frac{K_{Ic}}{Y\sigma}\right)^2$$ And the critical stress for a given crack is: $$\sigma_c = \frac{K_{Ic}}{Y\sqrt{\pi a}}$$

Valid application of LEFM requires that the plastic zone at the crack tip be small compared to the crack length and specimen dimensions (small-scale yielding). For ductile materials with large plastic zones, elastic-plastic methods (J-integral, CTOD) may be more appropriate.

If $$K_I < K_{Ic}$$ (safety ratio > 1), the crack is stable under current loading — but growth may still occur through fatigue or stress corrosion. A safety ratio of 2 or more is typical for critical structures. The critical crack length tells you how large a crack can grow before sudden fracture — essential for inspection interval planning. The critical stress indicates the maximum load the cracked component can sustain. Typical $$K_{Ic}$$ values: aluminum alloys 20–40 MPa√m, structural steels 50–150 MPa√m, titanium alloys 50–100 MPa√m, ceramics 1–5 MPa√m.