Crack Propagation: Griffith's Energy Balance and the Stress Intensity Factor

Why a crack is more dangerous than a hole

Drill a small circular hole in a stressed plate and the stress right at its edge is three times the applied far-field stress — a well-known result from Inglis's 1913 elasticity solution. That factor of three is uncomfortable but survivable. Replace the hole with a sharp crack of the same length and the mathematics changes character completely: for an ideally sharp crack tip, linear elasticity predicts an infinite stress, because the tip has effectively zero radius of curvature. A real material cannot sustain infinite stress, so something else must control whether the crack grows — and that something is energy, not stress at a point.

Griffith's energy balance

A. A. Griffith's 1921 insight was to stop asking whether stress at the tip exceeds a threshold and instead ask a global energy question: does the system lower its total energy by extending the crack a small amount? Extending a crack by da does two things. It relaxes stored elastic strain energy in the material around the crack (energy is released, since the newly cracked surfaces can no longer carry load) — call the amount released per unit new area the strain energy release rate G. And it creates two new free surfaces, which costs energy 2γs per unit area, where γs is the material's surface energy. Griffith's criterion is that the crack grows exactly when the energy released equals or exceeds the energy consumed:

G ≥ 2γs                  Griffith's criterion (energy form)

for a through-crack of length 2a in an infinite plate under
remote stress σ (plane stress):
   G = π σ² a / E

so the critical stress at which the crack becomes unstable is
   σc = sqrt( 2 E γs / (π a) )

That formula already tells you the most important qualitative fact about brittle fracture: the critical stress falls as 1/√a. Doubling a crack's length does not double the danger, it multiplies it by √2, but — critically — it also means there is no safe stress at which an arbitrarily long crack cannot eventually propagate. Griffith's own 1920s experiments on glass fibres, where flaws are essentially unavoidable, first demonstrated that measured strengths track predicted flaw sizes rather than any single material constant.

Stress intensity factor: the practical handle

Griffith's energy view is elegant but awkward to use directly on real, irregular geometries. George Irwin reformulated it in the 1950s in terms of the stress intensity factor K, which characterises the strength of the stress singularity right at the tip rather than the energy of the whole body. For a crack of length a under remote stress σ, in Mode I (the crack faces pulling straight apart — the dominant, most dangerous mode):

K_I = Y · σ · sqrt(π a)        Y = a dimensionless geometry factor (≈1 for a
                               long crack in a wide plate, larger near a free edge)

stress field near the tip, at distance r and angle θ from it:
   σ_yy(r, θ) ≈ K_I / sqrt(2π r) · f(θ)      — the 1/√r singularity

K_I and G are two views of the same physics and are directly related, G = K_I²/E′ (with E′ = E in plane stress, E/(1−ν²) in plane strain), so Griffith's energy criterion and Irwin's stress-intensity criterion predict identical crack growth. A crack propagates when K_I reaches the material's fracture toughness K_IC — a genuine material property, measured in MPa√m, that plays the same role for cracked bodies that yield strength plays for uncracked ones. Glass has K_IC around 0.7-0.8 MPa√m; structural steels are commonly 50-150 MPa√m, which is most of why steel structures tolerate flaws that would shatter glass instantly.

The three modes, and why Mode I dominates

Cracks can be loaded three ways: Mode I (opening, faces pulled straight apart), Mode II (in-plane shear, faces sliding past each other in the crack plane), and Mode III (out-of-plane shear, tearing). Real cracks are usually mixed-mode, but brittle cracks have a strong tendency to curve as they grow so as to locally maximise Mode I and minimise shear — a crack under an oblique load kinks until it is running perpendicular to the maximum principal stress, which is exactly what the erratic, branching paths seen in shattered glass and the meandering front in this simulation are doing.

Beyond ideal brittleness: the plastic zone

Real metals are not perfectly brittle. Right at the crack tip the stress predicted by linear elasticity would exceed the yield stress σy, so a small region actually yields plastically instead of following the elastic 1/√r law. Irwin's estimate of that plastic zone size is

r_p ≈ (1/2π) · (K_I / σy)²        (plane stress; smaller by ≈3× in plane strain,
                                     where the surrounding material constrains yielding)

As long as r_p is small compared with the crack length and the specimen dimensions, linear elastic fracture mechanics (LEFM) still applies with K_IC as the toughness — this is the small-scale yielding condition every K_IC test is designed to satisfy. When the plastic zone is not small (thin sheets of ductile metal, most polymers), the crack tip blunts, absorbs far more energy through plastic flow than the bare surface energy Griffith assumed, and the analysis has to switch to elastic-plastic fracture mechanics — the J-integral and crack-tip-opening displacement — which is why tough ductile alloys resist crack growth so much better than a naive Griffith estimate would suggest.

Fatigue: growth without any single overload

A crack can also grow under cyclic loads that never individually reach K_IC, one microscopic increment per cycle, in the regime engineers call fatigue. The Paris law is the standard empirical model:

da/dN = C · (ΔK)^m            ΔK = K_max − K_min per cycle, C and m fitted per material

Because a grows every cycle, and K_I depends on √a, ΔK itself increases as the crack lengthens, which makes crack growth accelerate — slowly at first, then faster and faster until K_max reaches K_IC and the remaining ligament fails suddenly in one final, brittle burst. This is why fatigue failures so often look sudden even though the crack had, in fact, been growing invisibly for a very long time.

Frequently asked questions

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