LEFM Fracture Mechanics: Why Cracks Kill

Why a scratch doesn't break a bridge, but a crack does

Real materials fail at stresses far below their theoretical atomic strength — often by a factor of 100 or more. In 1921, Alan Griffith explained why: any pre-existing flaw, however small, concentrates stress locally, and once a crack starts to grow it releases stored elastic energy faster than fresh crack surface can absorb it. Linear elastic fracture mechanics (LEFM) is the quantitative theory built on that insight — it's why a windshield chip spreads into a spiderweb overnight while a similar scratch in a block of steel just sits there, and it's the engineering discipline behind every "safe life" or "damage tolerant" design in aerospace and civil structures.

Griffith's energy criterion

A through-crack of half-length a in an infinite plate under remote stress σ releases elastic strain energy as it grows, at a rate G (energy release rate, plane stress):

G = π σ² a / E

// Crack grows when G ≥ Gc = 2γ  (γ = surface energy per unit area)
// → fracture stress:
σf = √(2Eγ / πa)

The key result: fracture stress scales as 1/√a. Double the crack length and the material fails at just 71% of the original fracture stress — small flaws matter disproportionately, which is exactly why fracture mechanics exists as a separate discipline from ordinary strength-of-materials calculations.

The stress intensity factor K

Griffith's energy balance is a global statement; it doesn't describe the stress field near the tip itself. George Irwin (1957) showed that for any crack geometry, the elastic stress field close to the tip always takes the same universal singular form:

σ_ij(r, θ) = K / √(2πr) · f_ij(θ)      // r = distance from crack tip

K = Y · σ · √(π a)      // stress intensity factor
// Y = dimensionless geometry factor (Y=1, infinite plate;
//     tabulated for finite width, holes, edge cracks, etc.)

This is why cracks are so much more dangerous than smooth stress raisers like round holes: the 1/√r term diverges as r → 0, so a perfectly sharp mathematical crack has no finite stress limit at all — the sharper the tip, the higher the local stress, capped in reality only by the material's finite crack-tip radius and by local plastic yielding. For a finite-width plate of width W, the geometry factor picks up a correction, Y ≈ √(sec(πa/W)), that grows sharply as the crack approaches the plate edges — a centre crack that's small relative to W behaves like the infinite-plate case, but the correction becomes essential as a/W grows past about 0.3.

There are three fundamental crack-opening modes — Mode I (tensile opening, the most critical for brittle fracture), Mode II (in-plane shear) and Mode III (out-of-plane tearing) — and each has its own K. The material fails once K reaches its fracture toughness, K_Ic: K_I ≥ K_Ic. K_Ic is a genuine material property (MPa·√m), linked back to Griffith's energy criterion by G_c = K_Ic²/E in plane stress.

Fatigue: the Paris law

Most real-world fractures don't happen from one overload — they happen from millions of small stress cycles growing an existing flaw a tiny amount each time, well below K_Ic. Paris and Erdogan (1963) found empirically that the crack growth rate per cycle follows a power law in the cyclic stress intensity range ΔK = K_max − K_min:

da/dN = C · (ΔK)^m        // C, m = material constants, m ≈ 2–4 for metals

// Region I: ΔK < ΔK_th   → no measurable growth (infinite-life design)
// Region II: linear on log(da/dN) vs log(ΔK)  → steady Paris growth
// Region III: K_max → K_Ic → rapid acceleration to final fracture

// Cycles to failure, integrating from a0 to critical size a_c=(K_Ic/Yσ)²/π:
N_f = [a0^(1-m/2) − a_c^(1-m/2)] / [C(1-m/2)(Y·Δσ·√π)^m]     (m ≠ 2)

This equation is the mathematical foundation of damage-tolerant design: instead of assuming a component is flaw-free, engineers assume a plausible initial flaw size (set by inspection resolution), integrate the Paris law forward to find the predicted crack-growth life, and schedule inspections well inside that window — the standard approach for aircraft fuselages, pressure vessels and bridges, where you can't guarantee a part starts perfect but you can guarantee you'll catch a growing crack before it reaches a_c.

Ductile vs brittle: why fracture toughness varies so much

The K-based field predicts infinite stress exactly at the crack tip, which is physically impossible — in any real ductile material, local yielding caps the stress once it reaches the yield strength, over a region Irwin estimated as r_p = (1/2π)(K_I/σ_ys)². Brittle materials like soda-lime glass (K_Ic ≈ 0.7–0.8 MPa√m) have almost no plastic zone to absorb energy, so they fail suddenly and near the Griffith prediction. Ductile metals — structural steel (K_Ic ≈ 40–100 MPa√m) or aluminium alloys (K_Ic ≈ 26–35 MPa√m) — dissipate enormous extra energy through plastic deformation at the crack tip before final fracture, which is exactly why steel structures can survive cracks that would shatter glass instantly.

Frequently asked questions

Try it live

Every stress field and growth curve above runs live in LEFM Fracture Mechanics. Set crack length, panel width and loading, and watch K and the crack-tip field update instantly, entirely in your browser.