FR39 · Crystal defects

Edge Dislocation Glide: When Does Shear Tear a Dipole Apart?

How much shear does it take to pull apart two edge dislocations that attract each other? A shear above the passing stress, the peak pull one dislocation exerts on the other. For a pair on glide planes four atomic spacings apart in a metal with Poisson ratio 0.3, that is about 1.4 percent of the shear modulus. Halving the spacing doubles it, so narrow dipoles are the hardest to break.

Shear a crystal holding edge dislocations. Watch their stress fields, see a dipole break above its passing stress, and drag a dislocation onto another plane.

Applied shear stress now $\tau/\mu=0.0177$

Passing stress of the pair $\tau_{\mathrm{p}}/\mu=0.0142$

Offset between neighbouring dislocations $\Delta x=1.20\,b$

Angle of the pair to the glide plane $\varphi=73.3^{\circ}$

Shear a crystal that holds edge dislocations

Arrangement

Amplitude of the cyclic shear

0.025

\tau_0/\mu

Spacing of the glide planes

4

h/b

Poisson ratio

0.3

\nu

Loading period

8000

T\mu M

Drag a dislocation along its dashed glide plane, or onto its partner's plane to make the pair annihilate.

Playback speed

Model time per second of animation

640

\dot t\,\mu M

Model, method and sources

$\sigma_{xy}=\frac{\mu b}{2\pi(1-\nu)}\,\frac{x(x^2-y^2)}{(x^2+y^2)^2},\qquad v=\pm M\,b\,(\tau+\sigma_{xy}),\qquad \tau_{\mathrm{p}}=\frac{\mu b}{8\pi(1-\nu)h}$

Straight edge dislocations in an isotropic crystal carry the Volterra displacement and stress fields of linear elasticity. Each one glides along its slip plane at a speed proportional to the Peach–Koehler force from the applied shear and the other dislocations; climb is switched off. The atoms are drawn at the displaced lattice sites, with the field cut off inside one Burgers vector of a core, where elasticity does not hold. Lengths are in b, stresses in the shear modulus and time in one over the shear modulus times the mobility. A dislocation stops 21.5 b from the centre so it stays in view; a pair held at that stop has already broken.

Niu, Zhu, Dai & Xiang (2018), arXiv:1707.05161v2 · KIT IAM, Dislocations: properties (lecture 4c, 2024)

Derive the passing stress of a dipole

Model assumptions

Each dislocation is a straight edge dislocation in an isotropic crystal, described by the Volterra displacement and stress fields of linear elasticity. The atoms are drawn at their displaced positions, so the inserted half-plane above or below each core is visible. A dislocation glides along its slip plane at a speed proportional to the Peach–Koehler force of the applied shear plus the stress of the other dislocations; climb, the core structure and lattice friction are left out. The passing stress is the largest stress one dislocation of the pair can exert on the other, derived from the same field. Everything shown is computed from these formulas; no measured data are shown.

Applied shear and pair offset over time

Glide force between the two neighbours (elastic theory)

More numbers and how they are computed

Plastic slip carried by the dislocations: $0$
Model time: $0$
Loading cycles completed: $0$
Dislocations in the crystal: $0$
Shear amplitude over passing stress: $0$
Plane spacing and Poisson ratio: $0$

The interaction curve is the closed-form glide stress of one edge dislocation on a parallel plane, drawn for the pair nearest each other; it is a reference, not a measurement. The offset chart records the simulated positions. The exported file lists the settings and the full position history.

Physics tutorial

A crystal deforms one row of atoms at a time

BackgroundA perfect crystal would need an enormous stress to slide one half over the other all at once. Real metals yield at a far smaller stress, because they hold line defects, dislocations, that let the slip happen one atomic row at a time.

Why it mattersDislocations carry long-range stress fields, so they push and pull on each other. Those interactions decide whether a metal work-hardens, how dislocation walls form during recovery, and how much stress a pair of trapped dislocations can resist.

Start with the essentials

Focus question: How much shear does it take to pull apart two edge dislocations that attract each other?
One-sentence intuition: Two edge dislocations of opposite sign on parallel glide planes $h$ apart settle at 45 degrees, where their mutual glide force vanishes. As the offset grows, the attraction first rises and then falls; its peak, the passing stress $\tau_{\mathrm p}=\mu b/[8\pi(1-\nu)h]$ , is the largest applied shear the pair can resist.

Core mathematical model

Shear stress around an edge dislocation

\sigma_{xy}=\frac{\mu b}{2\pi(1-\nu)}\,\frac{x\,(x^2-y^2)}{(x^2+y^2)^2}

The field falls off as one over distance and changes sign on the lines $x=0$ and $x=\pm y$ , which gives the four-lobed pattern in the scene.

Peach–Koehler glide force

f_{\mathrm g}=\pm b\,\bigl(\tau+\sigma_{xy}^{\mathrm{others}}\bigr),\qquad v=M f_{\mathrm g}

Only the shear stress resolved on the glide plane pushes an edge dislocation along it. The sign follows the Burgers vector, so the same shear drives the two members of a dipole in opposite directions.

Common difficulties

A larger dipole is weaker, not stronger

Typical misconceptionDislocations farther apart must be harder to separate, since more distance has to be covered.

Better mental modelThe stress field weakens as one over distance, so the peak pull between the planes scales as one over their spacing. Doubling the spacing halves the passing stress.

Run the experiment

01
Tear a dipole apart
Start with A dipole torn apart and watch the offset chart against the dashed passing-stress lines.
What to observe: Each time the applied shear climbs past the passing stress, the pair snaps through the vertical and the two dislocations glide far apart. They return when the shear reverses.
02
Halve the spacing
Choose A narrow dipole holds, which keeps the same shear amplitude with planes two Burgers vectors apart.
What to observe: The passing stress doubles past the applied amplitude, and the pair only rocks around its 45 degree position.
03
Make the pair annihilate
Drag one dislocation onto the glide plane of its partner.
What to observe: The two glide together and vanish, and the lattice around them becomes perfect again.

Edge Dislocation Glide: When Does Shear Tear a Dipole Apart?

A crystal deforms one row of atoms at a time

Core mathematical model

Shear stress around an edge dislocation

Peach–Koehler glide force

Common difficulties

A larger dipole is weaker, not stronger

Run the experiment

Tear a dipole apart

Halve the spacing

Make the pair annihilate

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