Dislocation theory

Plastic deformation of a metal is slip: one part of the crystal shifts over another along a crystal plane. If the whole plane slid rigidly, every bond across it would stretch at the same moment. The shear stress would then peak near $G/2\pi$, where $G$ is the shear modulus ($\mu$ in the figure). Real metals yield at $10^{-3}$ to $10^{-5}$ of that (Theoretical shear strength).

The figure is a raft of soap bubbles, the model in which Bragg and Nye first watched slip (Bragg and Nye 1947). Each bubble pushes and pulls on its neighbours like a bonded atom. Load a perfect raft. Nothing gives until the stress comes close to the dashed ideal line, and then new defects form at the corners. Now load the same raft with one dislocation. The blue half glides over the grey half at a few hundredths of the shear modulus. The first run stays on the chart for comparison.

The difference is the ⊥, where an extra half-plane of bubbles ends at the slip plane. Only the bonds at its edge break and re-form at each moment, so the stress falls by orders of magnitude. After the line has crossed, the blue half has moved by one bubble spacing, the same slip that a rigid shift gives. Reset the raft and load it again to watch the step form at the edge.

A dislocation is this line defect, and its motion carries plastic deformation in crystalline metals (Dieter 1988). Dislocation theory is the physical layer underneath crystal plasticity. Crystal plasticity averages the glide into a continuum law per grain. Discrete dislocation dynamics simulates the lines one by one. At the bottom, molecular dynamics and density functional theory resolve the dislocation core. This note is the overview. Each idea below is worked out, with its own figures, in the note it links to.

How the idea arrived

The theory at a glance

The theory is a chain of six ideas, and each one needs the ideas before it. Each stage borrows the figure of the note that works its idea out.

1 · What a dislocation is

A dislocation is a line that carries slip. Its Burgers vector $\mathbf b$ measures that slip. Take a circuit of equal atom-to-atom steps round the dislocation: it misses its start, and the gap is $\mathbf b$. Move the circuit off the line and it closes again. The angle between $\mathbf b$ and the line sets the character: edge, screw or mixed. A loop in its slip plane shows all three, because $\mathbf b$ stays fixed while the line turns. $\mathbf b$ is a lattice translation, so the Bravais lattice fixes the slip systems.

2 · How it pushes and is pushed

Away from its core the line is elastic (Dislocation elasticity). Its shear stress falls as $1/r$, so lines feel each other far away. Bring a second dislocation close and the two interact strongly. A stress pushes on a line with the Peach–Koehler force $\mathbf f=(\boldsymbol\sigma\cdot\mathbf b)\times\boldsymbol\xi$. The line’s energy, about $\tfrac12Gb^2$ per length, makes it a string under tension. Pinned at two points, it bows into an arc and stops. Above the critical stress no arc can balance the push.

3 · What the core adds

Within a few atoms of the line elasticity fails, and the Peierls–Nabarro model adds the lattice energy. That energy sets the core width and the friction on a moving line. Many crystals lower it further by splitting the line into partial dislocations joined by a ribbon of stacking fault. The figure shows the split in a raft that lies on a fixed second layer. Lower the fault energy and the ribbon widens. Raise it and the ribbon narrows. Push the pair and both partials glide together.

4 · How lines move

The area a line sweeps has slipped by $\mathbf b$, so glide is plastic strain. The Orowan equation gives its rate, $\dot\gamma=\rho_{\mathrm m}bv$, from the mobile density $\rho_{\mathrm m}$ and the line velocity $v$. Follow loops in one plane from the start through the middle to the end. The swept fraction saturates at 1 when the plane has slipped by one full $\mathbf b$. The velocity comes from a mobility law. Lines also leave their plane by cross-slip and climb, and some crystals shear by deformation twinning instead.

5 · How many lines behave

Above about $Gb/L$ a pinned segment of length $L$ emits loop after loop (Dislocation multiplication). The segment bows, runs away past the semicircle and wraps round both pins. Its arms meet and annihilate, which frees a loop and restores the segment. The density grows by five or six orders of magnitude. The lines then block each other, and the flow stress rises as $\tau=\alpha Gb\sqrt\rho$ (Taylor relation). Junctions set how one slip system hardens another, and the strain hardening stages follow. Non-uniform slip stores geometrically necessary dislocations, which bring a length into gradient plasticity. Under cyclic load the lines gather into persistent slip bands.

6 · What makes a metal strong

Obstacles in the way of the lines make a metal strong: other dislocations, solute atoms, precipitates and grain boundaries (Strengthening mechanisms). A line held between obstacles a distance $L$ apart passes at about $Gb/L$. Load the raft with a hard particle: the line waits at the particle until the stress forces it past. Load a free dislocation for comparison. Two grains add a boundary, whose resistance is the subject of slip transmission at grain boundaries. Steels add phase changes on top (Iron phases and hardening). How people see all of this in real metals is in Observing dislocations.

Play with the raft

The raft of the opening figure has more experiments than the hook shows. An opposite pair of ⊥ and ⊤ attracts and annihilates. A square crystal has bumpier rows, so its dislocation needs much more stress to move. In tension the horizontal slip plane feels no shear, so slip starts on the inclined planes. Two grains meet at a wall of ⊥, and shear moves the whole wall. Under “more controls”, the drag slider sets the linear mobility law $v=F/B$. Click a bubble to pop it (a vacancy) or to swell it (a solute atom).

Simulating dislocations

Everything above is the physics. People compute it with a ladder of models (Dislocation plasticity across scales). Each rung tracks a coarser object than the one below and takes a law from it:

Two notes say what the averaging between rungs loses: Crystal plasticity as averaged dislocation dynamics and Continuous density fields cannot represent topology change. CADD couples atoms and lines in one run instead of handing a law up. The discrete–continuous model couples lines and a continuum solver the same way. Machine learning now enters at every hand-over: interatomic potentials, learned mobility laws, faster line dynamics and learned closures for the densities.

The geometric theory of dislocations sits beside the ladder rather than on it. It reads the dislocation density as the torsion of the crystal lattice, and residual stress as a metric that does not fit in flat space.

The textbooks behind this map are Dieter’s Mechanical metallurgy, Part 2 (Dieter 1988), and Hull and Bacon’s Introduction to dislocations (Hull and Bacon 2011).

Gaps

• The notes on observation, twinning, hardening stages, persistent slip bands and slip transmission are landscape notes, cited mostly from abstracts. None has a literature digest yet. Continuum dislocation dynamics now has digests for field dislocation mechanics only; its density theories are still cited from abstracts.
• Not yet in the vault at all:
  • thermally activated glide and kink pairs in BCC metals;
  • non-Schmid effects;
  • dislocations in HCP metals;
  • dislocation–solute interaction (Cottrell atmospheres, yield points).

Links:

• Dislocation plasticity across scales — the simulation half of this map: the models, and the hand-overs between them.
• Crystal plasticity — the continuum theory built on the elasticity, motion and hardening of dislocations.
• Bravais lattices — the lattice translations that a perfect dislocation’s Burgers vector must be.
• Phase field dislocation dynamics — the field-based rival to discrete dislocation dynamics: core and reactions from the γ-surface, at the cost of a small domain.
• Molecular dynamics — calibrates the mobility law and serves as ground truth for line simulations. Deep reads: mobility from molecular dynamics (Bertin 2024) · dislocations in iron (Bataev 2026)
• Computational material engineering — where these simulations sit among the other multiscale methods, and the software.
• Mechanical metallurgy · Creep · Alloying — the engineering uses.

References: