A Frank-Read source is a dislocation segment of length L pinned at both ends (by precipitates, jogs, or intersections with other dislocations threading the slip plane). Under a resolved shear stress τ, the line feels an outward glide force per unit length τb (Peach-Koehler), opposed by its own line tension T ≈ ½Gb², which resists curvature and tries to shorten the line back to straight:
Force balance: τ b = T κ = T / R
Equilibrium radius: R = T / (τ b) = G b / (2τ)
Critical stress: τ꜀ = G b / L (reached when R shrinks to L/2, a semicircle)
Below τ꜀ the segment bows to a stable arc and stops — the dynamics simulated here integrate the actual local curvature at every point of the line (via the circumradius of neighbouring points) and let the force balance find that equilibrium itself, rather than drawing a fixed shape.
Once τ exceeds τ꜀, no circular arc pinned at both ends can satisfy the force balance — the line has no equilibrium and keeps bowing past a semicircle, the ends curling around the pinning points into a spiral. In a real crystal the two trailing arms eventually meet behind the pins, annihilate where they touch, and pinch off a closed loop that expands outward under the same τ while the original segment reforms between the pins to repeat the cycle. This simulation reproduces that force balance exactly for the growing arc, and represents the pinch-off itself (a full 3D self-intersection event) with a simplified trigger — once the line's arc length passes roughly one loop's worth, a new independent loop is spawned at the same driving stress and the source segment resets.
- τ, G, L — set the physical regime; τ꜀ = Gb/L updates live (b = 0.25 nm, a typical metallic Burgers vector).
- Simulation speed — scales simulated time only, not the physics.
- Reset source — clears emitted loops and reseeds the segment.
This is the dominant mechanism by which crystals generate new dislocations under load (dislocation multiplication) — it is why plastic deformation increases dislocation density and drives work hardening, first proposed by Frank & Read in 1950.