🪨 Crack Propagation & Fracture Mechanics
A real per-vertex stress field (Irwin's near-tip approximation, σ ∝ 1/√r) rendered on a 3D plate under tension. Grow a pre-existing crack, cross the Griffith critical stress σc = √(2Eγ/πa), and watch unstable propagation and a Voronoi brittle shatter unfold.
Drag to orbit · Scroll to zoom · Colour = stress concentration factor at that point
Glass — brittle, low toughness
Glass has a true surface energy of only about γ ≈ 1 J/m², so even a microscopic flaw (a few micrometres) is enough to drop the critical stress far below the material's theoretical strength. Increase the applied stress above σc, or press Fracture!, to watch the crack accelerate and the plate shatter.
Griffith's Energy-Balance Criterion
The idea — A crack grows only if doing so releases more
elastic strain energy than it costs to create the two new crack-face surfaces (energy
2γ per unit area). Balancing the energy released, πσ²a²/E per unit thickness for a
centre crack of half-length a, against the surface cost 4γa gives the critical
stress σc = √(2Eγ/(πa)).
Why longer cracks are worse — σc falls as 1/√a, so
doubling a flaw's length lowers the stress needed to fracture it by a factor of √2.
This is exactly why a small scratch shatters a glass pane that would otherwise
survive far higher loads.
Stress intensity factor — Real design work uses
K = σ√(πa), the amplitude of the 1/√r stress singularity at the tip, and compares it
to the material's fracture toughness Kc = √(2Eγ) = √(E·Gc). Failure occurs when
K ≥ Kc — identical to σ ≥ σc, just expressed in the units engineers actually measure.
About Crack Propagation & Fracture Mechanics
This simulation renders a real per-vertex stress field on a 3D plate under uniform tension, using Irwin's near-tip approximation σ ∝ K/√(2πr) to show the classic stress-concentration region ahead of a crack tip. A pre-existing centre crack of adjustable half-length a sits in the plate; once the applied stress crosses the Griffith critical stress σc = √(2Eγ/πa) for the current crack length and material, the crack tip actually advances through the mesh, the stress field recomputes live, and growth becomes unstable — accelerating exactly as real fast fracture does. Pushing stress far past critical, or pressing Fracture!, triggers a Voronoi-tessellation-based shatter that breaks the plate into flying fragments.
Fracture mechanics underpins why brittle materials like glass and ceramics fail at stresses far below their theoretical strength, and it gives engineers the stress-intensity-factor / fracture-toughness framework (K vs Kc) used to certify everything from aircraft fuselages to pressure vessels.
Frequently Asked Questions
What is Griffith's fracture criterion?
Griffith's criterion is an energy balance: a crack propagates only when the elastic strain energy released by extending it exceeds the energy needed to create the new crack-surface area, 2γ per unit area of new surface (two faces), where γ is the material's surface energy. Balancing these two energies for a centre crack of half-length a in a plate under stress σ gives the critical stress σc = √(2Eγ/(πa)).
How do I use this simulation?
Pick a material preset (brittle glass or tougher metal), set the initial crack half-length and applied stress with the sliders, and watch the live readouts for critical stress σc, the current stress ratio, and the stress intensity factor K. If the applied stress exceeds σc the crack tip will visibly advance through the mesh; press Fracture! to force an immediate unstable-growth-and-shatter demonstration, or Reset to restore the original crack.
Why does a small scratch shatter glass so easily?
Because σc falls as 1/√a, a longer flaw dramatically lowers the stress needed to fracture the material. Glass has a very low surface energy (γ ≈ 1 J/m²) and essentially no plastic deformation to blunt a crack tip, so even a microscopic surface scratch a few micrometres long can cut its practical strength from the theoretical GPa range down to tens of megapascals — exactly what this simulation's σc readout demonstrates as you lengthen the initial crack.
What is the stress intensity factor and fracture toughness?
The stress intensity factor K = σ√(πa) is the amplitude of the 1/√r stress singularity at a crack tip — it packages the applied stress and crack geometry into a single number. Fracture toughness Kc = √(2Eγ) = √(E·Gc) is a material property, independent of the crack or the load. Failure occurs when K reaches Kc, which is mathematically identical to σ reaching σc but is the form used throughout real structural engineering because K and Kc can be measured directly and compared for any crack shape or loading, not just the idealised centre crack Griffith solved.
Why does crack growth become unstable ("fast fracture")?
Because σc decreases as the crack grows (σc ∝ 1/√a) while the applied stress stays fixed, the excess stress above critical keeps increasing as the crack lengthens. That positive feedback loop is why once a crack exceeds the critical length at a given stress, it doesn't creep forward slowly — it accelerates toward the speed of sound in the material within microseconds, which is exactly what this simulation's growth model reproduces as the crack half-length climbs past σc's crossing point.
Why are metals so much tougher than glass or ceramics?
Metals can deform plastically at a crack tip: dislocations glide and blunt the sharp tip, spreading the load over a larger region and consuming a great deal of energy as heat and lattice damage rather than just breaking bonds. This raises their effective fracture energy (often modelled as an "effective γ" of hundreds to thousands of J/m², versus about 1 J/m² for the true surface energy of glass) even though the true surface energy of the metal itself is similar in magnitude. Ceramics and glasses, lacking easy dislocation motion at room temperature, stay brittle and rely on their true surface energy alone.
What is a Voronoi tessellation and why use it for shatter effects?
A Voronoi tessellation partitions a surface into regions, each containing all the points closest to one particular seed point. Real brittle shatter patterns — glass panes, ceramic tiles, rock fragmentation — closely resemble Voronoi cells because fracture surfaces tend to propagate along paths that divide the material into roughly convex regions around flaw or impact sites. This simulation scatters seed points (biased toward the crack line) and assigns every triangle of the plate mesh to its nearest seed, producing a physically-motivated fragment pattern rather than an arbitrary grid break.
How is this used in real engineering?
Aircraft, pressure vessels, pipelines and bridges are designed using fracture mechanics inspection limits: engineers measure or assume a maximum undetected flaw size, compute the stress intensity factor at the design load, and require it stay safely below the material's certified fracture toughness Kc. This "damage tolerant design" philosophy, which grew directly out of Griffith's 1920s glass-fibre experiments and the 1950s Comet airliner fatigue-crack disasters, is why aircraft undergo scheduled crack inspections rather than being designed to never crack at all.
Who developed fracture mechanics and when?
A. A. Griffith published the energy-balance theory of brittle fracture in 1920, testing it on glass fibres. His theory under-predicted the toughness of ductile metals until G. R. Irwin extended it in the 1950s to include the near-tip stress field (the K used in this simulation) and an effective surface energy that accounts for plastic work, founding modern linear elastic fracture mechanics (LEFM). Irwin's work followed directly from investigations into the 1954 de Havilland Comet crashes, caused by fatigue cracks growing from square window cutouts.
What is an active research frontier in fracture mechanics?
Phase-field fracture modelling — simulating crack initiation and propagation as a smooth, evolving damage field rather than tracking an explicit crack-tip geometry — is a major active area, because it can naturally handle crack branching, merging and complex 3D fracture surfaces that are very difficult for classical LEFM. It's used to study everything from fracking-induced rock fracture networks to fatigue in additively-manufactured (3D-printed) metal parts, where internal porosity creates unpredictable flaw populations that classical single-crack Griffith analysis cannot capture.