Dislocations: Burgers Vector, Stress Fields & the Hall-Petch Relation

The edge dislocation: an extra half-plane of atoms

An edge dislocation is a line defect formed when an extra half-plane of atoms is wedged into an otherwise regular crystal lattice. The edge of that half-plane — the dislocation core — is where the lattice is most distorted: atoms directly above are compressed, atoms below are stretched. Real metals are riddled with them; an annealed metal typically carries 10¹⁰–10¹² metres of dislocation line per cubic metre.

The Burgers vector: quantifying the distortion

The Burgers vector b is found by tracing a closed loop around the dislocation core and comparing it with the same loop in a perfect crystal — the gap that fails to close is b. For an edge dislocation, b lies perpendicular to the dislocation line, in the slip direction, and its magnitude equals one lattice spacing (roughly 0.25 nm in typical metals). Gliding a dislocation only requires breaking and reforming one row of bonds at a time — like moving a ruck across a carpet rather than sliding the whole carpet — which is why real crystals deform at a small fraction of their theoretical strength.

The stress field: an exact elastic solution

The elastic stress field around an edge dislocation has an exact closed form — the Volterra solution — derived from linear elasticity theory:

σ_xx = −Dy(3x²+y²)/r⁴
σ_yy =  Dy(x²−y²)/r⁴
τ_xy =  Dx(x²−y²)/r⁴
D = μb / [2π(1−ν)]     μ = shear modulus,  ν = Poisson's ratio

The field is tensile above the slip plane and compressive below, decaying as 1/r — long-ranged enough that dislocations feel and interact with each other from far away, which is exactly what causes them to tangle and pile up during deformation. A dislocation only starts to glide once the applied shear stress τ exceeds the Peierls stress — roughly 1.2% of the shear modulus for a typical metal; below that threshold, the core stays pinned in place.

Hall-Petch: why finer grains mean higher strength

Grain boundaries are the biggest obstacle a gliding dislocation ever meets — it cannot cross directly into a differently-oriented crystal, so it piles up and generates a back-stress that resists further slip. The Hall-Petch relation captures the resulting strengthening quantitatively:

σ_y = σ_0 + k/√d      d = average grain diameter

Nanocrystalline: d < 5 µm    Fine-grained: d < 50 µm
Medium: d < 150 µm            Coarse: d > 150 µm

Each material — mild steel, copper, aluminium — has its own σ₀ (single-crystal-like baseline strength) and k (grain-boundary strengthening coefficient) fitted to the same square-root law. This single idea underlies why cold-rolled, fine-grained sheet steel is measurably stronger than the same alloy annealed into large grains, without a single change in composition.

Frequently asked questions

Try it live

Everything above runs in your browser — open Crystal Dislocations and drag the shear-stress and grain-size sliders to watch the core glide, the stress-field heatmap update, and the Hall-Petch marker climb the curve. Nothing is installed, nothing is uploaded.