An edge dislocation is an extra half-plane of atoms inserted into a crystal lattice. Under shear stress τ it glides along the slip plane — atoms near the core switch bonds one at a time, like pushing a ripple through a rug. This enables plastic deformation at stresses hundreds of times lower than the theoretical shear strength of a perfect crystal.
σ_xy = Gb / [2π(1−ν)] · x(x²−y²) / (x²+y²)²
Glide velocity: v = M · (τ − τ_PN) [τ_PN ≈ 2G·exp(−2πw/b), w ≈ 1.5b]
→ τ_PN ≈ 13 MPa at G = 80 GPa
Burgers vector magnitude: |b| = a₀/√2 (FCC, ⟨110⟩ slip)The concept of a dislocation was independently proposed by Taylor, Orowan, and Polanyi in 1934 to explain why real metals yield at stresses 10,000× lower than theory predicts. It took until 1956 — when the electron microscope improved sufficiently — for dislocations to be directly observed for the first time.
What is an edge dislocation in a crystal?
An edge dislocation is a line defect in a crystal lattice formed by inserting an extra half-plane of atoms. The boundary of that extra half-plane — where it terminates inside the crystal — is the dislocation line. Atoms near the core are compressed above and stretched below, creating a strong local stress field.
What is the Burgers vector and why is it important?
The Burgers vector b is a lattice vector that quantifies the magnitude and direction of the displacement mismatch caused by a dislocation. For an edge dislocation b is perpendicular to the dislocation line. Its magnitude equals the lattice spacing and determines how far the crystal shears each time a dislocation crosses a slip plane.
Why does dislocation glide require far less stress than the theoretical shear strength?
The theoretical shear strength of a perfect crystal is roughly G/30, because every bond in a plane must break simultaneously. With a dislocation present, only bonds at the core switch one at a time — like moving a rug by pushing a ripple across it. This sequential bond-switching reduces the required stress by several orders of magnitude.
The Volterra solution gives the shear stress component σ_xy = Gb/[2π(1−ν)] · x(x²−y²)/(x²+y²)², where G is the shear modulus, b the Burgers vector magnitude, ν Poisson's ratio, and (x,y) coordinates from the dislocation core. The field is long-range (falls off as 1/r) and has quadrupole symmetry.
The slip plane is the crystallographic plane on which a dislocation moves. For edge dislocations the slip plane contains both the dislocation line and the Burgers vector. Plastic flow is easiest on close-packed planes because atoms are most densely packed there, reducing the Peierls–Nabarro stress needed to move the dislocation.
The Peierls–Nabarro stress τ_PN = 2G·exp(−2πw/b) is the minimum shear stress needed to move a dislocation through the lattice at 0 K, where w is the dislocation core width. Wider cores (covalent vs ionic materials) have lower τ_PN, explaining why metals with metallic bonding deform easily while ceramics are brittle.
Taylor hardening gives Δσ = αGb√ρ, where ρ is dislocation density and α ≈ 0.3. As a metal is cold-worked, dislocations multiply and their stress fields overlap, impeding each other's motion. This raises the flow stress — a mechanism called work hardening or strain hardening.
Glide (conservative motion) is movement of a dislocation within its slip plane under shear stress — the subject of this simulation. Climb is non-conservative motion perpendicular to the slip plane, requiring diffusion of vacancies or interstitials at elevated temperature. Cross-slip allows a screw dislocation to switch to another slip plane with the same Burgers vector.
Grain boundaries are regions of severe lattice mismatch between crystallites of different orientation. A gliding dislocation cannot pass directly through a grain boundary; it piles up against it, creating a back-stress that opposes further dislocation motion. This is the basis of the Hall–Petch relation: smaller grains → more boundaries → higher yield strength.
Metal forming processes — rolling, forging, drawing, stamping — all rely on dislocation glide for permanent shape change. Precipitation hardening (age hardening in aluminium alloys) pins dislocations with nanometre-scale precipitates. Annealing removes stored dislocations by allowing climb and recovery. Understanding glide is central to designing high-strength, high-toughness structural alloys.
This simulation models the glide of an edge dislocation through a two-dimensional crystal lattice under an applied shear stress. An edge dislocation is a line defect formed by inserting an extra half-plane of atoms into the lattice; under sufficient shear stress it moves along the slip plane as atoms at the core switch bonds one at a time, enabling permanent plastic deformation. Users can observe how the dislocation core distorts the surrounding lattice, visualise the quadrupole shear-stress field predicted by the Volterra solution, and explore how parameters such as shear modulus, Poisson ratio, and applied stress control glide velocity.
Dislocation glide is the primary mechanism by which metals are shaped by rolling, forging, and drawing, and understanding it underpins the design of high-strength structural alloys used in aerospace, automotive, and civil engineering. The theory was independently proposed by Taylor, Orowan, and Polanyi in 1934 to explain why real metals yield at stresses thousands of times lower than those predicted for perfect crystals.
An edge dislocation is a linear crystal defect created by inserting an extra half-plane of atoms into an otherwise perfect lattice. The lower boundary of that half-plane — where it terminates inside the solid — is the dislocation line. Atoms directly above the core are compressed, while those below are in tension, producing a long-range elastic stress field that decays as 1/r from the core.
The Burgers vector b quantifies the magnitude and direction of the lattice displacement mismatch caused by a dislocation. It is determined by tracing a closed circuit (Burgers circuit) around the dislocation in a perfect crystal and comparing it with the same circuit in the defect crystal; the closure failure is the Burgers vector. For an edge dislocation b is perpendicular to the dislocation line, and its magnitude equals the lattice spacing on the slip direction.
The theoretical shear strength of a defect-free crystal is approximately G/30 (where G is the shear modulus), because all atomic bonds across a slip plane must be broken simultaneously. A dislocation acts like a ripple being pushed across a carpet: only the bonds at the dislocation core switch at any instant, drastically reducing the energy barrier. This sequential bond-switching lowers the required stress by two to four orders of magnitude compared with the theoretical limit.
The Volterra solution gives the shear stress component as σxy = Gb / [2π(1−ν)] · x(x²−y²) / (x²+y²)², where G is the shear modulus, b the Burgers vector magnitude, ν Poisson’s ratio, and (x, y) the coordinates measured from the dislocation core. The field has quadrupole symmetry: it is positive in two opposite quadrants and negative in the other two. The simulation colour-maps this field in real time as the dislocation glides.
The Peierls–Nabarro stress τPN ≈ 2G·exp(−2πw/b) is the minimum shear stress required to move a dislocation through the lattice at absolute zero, where w is the dislocation core half-width. Materials with narrow cores (such as covalent ceramics) have high τPN and are brittle, while metals with wide cores and metallic bonding have much lower τPN and deform readily. The simulation uses τPN ≈ 1.2% of G as a pinning threshold.
The slip plane is the crystallographic plane on which the dislocation moves; for an edge dislocation it contains both the dislocation line and the Burgers vector. Dislocation glide is easiest on close-packed planes because atoms are most densely packed, maximising the core width and minimising the Peierls–Nabarro stress. In face-centred cubic (FCC) metals the primary slip system is {111}<110>, giving 12 independent slip systems that enable extensive plastic flow before fracture.
As a metal is deformed, dislocation sources (notably Frank–Read sources) generate new dislocations, rapidly increasing the dislocation density ρ from roughly 1010 m−2 in an annealed metal to 1015 m−2 after heavy cold work. The overlapping stress fields of neighbouring dislocations impede each other’s motion. Taylor hardening quantifies this as Δσ = αGb√ρ (where α ≈ 0.3), predicting the measured square-root increase in flow stress with plastic strain.
Metal forming operations — rolling sheet steel, forging crankshafts, drawing wire, deep-drawing car body panels — all rely entirely on dislocation glide to permanently reshape material without fracture. Precipitation hardening (age hardening) in aluminium alloys for aircraft structures works by dispersing nanometre-scale precipitates that pin dislocations, raising strength by a factor of three or more. Conversely, annealing heat treatments restore ductility by allowing dislocations to rearrange and annihilate through climb and recovery.
The dislocation concept was independently proposed in 1934 by Geoffrey Taylor, Egon Orowan, and Michael Polanyi to reconcile the enormous gap between theoretical and measured yield strengths of metals. Despite its explanatory power, the dislocation remained a theoretical construct for over two decades. In 1956, Peter Hirsch and colleagues at the Cavendish Laboratory in Cambridge first observed moving dislocations directly using transmission electron microscopy, a breakthrough that validated the theory and launched the modern field of physical metallurgy.
Glide (conservative motion) is the movement of a dislocation within its slip plane driven by shear stress — the mechanism shown in this simulation. Climb is non-conservative motion perpendicular to the slip plane and requires the diffusion of vacancies or interstitial atoms to or from the dislocation core; it is thermally activated and significant only at elevated temperatures. Cross-slip is a mechanism specific to screw dislocations (which have b parallel to the line), allowing them to switch from one slip plane to another of the same zone, bypassing obstacles and contributing to dynamic recovery.
Active research areas include atomistic and molecular-dynamics simulations of dislocation cores at the angstrom scale to capture quantum mechanical effects missed by continuum elasticity; dislocation behaviour in high-entropy alloys (HEAs) where chemical disorder creates a fluctuating energy landscape for glide; radiation damage in nuclear reactor steels where dislocation loops formed by neutron cascades cause hardening and embrittlement; and machine-learning interatomic potentials that allow billion-atom dislocation simulations at near-DFT accuracy, opening new windows on deformation mechanisms in complex alloys.