Microcapsule Rupture: Weibull Fracture Statistics

Applied Stress

Bulk load σ_applied 0.0 MPa

Capsule Population

Mean shell strength σ₀ 18.0 MPa

Weibull modulus m (scatter) 4.0

Live Readout

Ruptured

0 / 0 (0%)

Weibull prediction

0.0%

Healing agent released

0.00 mm³

How it works

Each microcapsule is a thin polymer (urea-formaldehyde) shell of radius r and wall thickness t, holding liquid healing agent under a matrix that transmits an external load. Treating the shell as a thin pressurized sphere, the hoop (membrane) stress in the wall scales as:

σ_wall ≈ σ_applied · (r / 2t)      (thin-shell stress concentration)
Capsule ruptures when σ_wall ≥ σ_f  (wall fracture strength)

Real capsule populations are never uniform — small random variations in wall thickness and flaw size mean each capsule has its own critical rupture stress. This is modeled with Weibull weakest-link statistics, the standard fracture-strength distribution for brittle shells:

P(rupture | σ) = 1 − exp[ −(σ / σ0)^m ]
σ0 = characteristic (mean-scale) strength
m  = Weibull modulus — larger m ⇒ tighter, more uniform batch

σ_applied slider — ramps the bulk mechanical load transmitted through the surrounding polymer matrix; capsules whose individually-sampled threshold is exceeded rupture irreversibly and release their payload.
Auto-ramp — sweeps the load 0 → 40 MPa automatically so you can watch the population fail as a smooth statistical front rather than one click at a time.
σ0 / m sliders — regenerate a fresh capsule batch drawn from a Weibull distribution with the chosen mean strength and scatter, letting you compare a tight, well-manufactured batch (high m) against a wide, defect-prone one (low m).
The dashed Weibull prediction readout is the closed-form P(rupture|σ) evaluated at the current load — the simulated fraction ruptured should track it, converging as the sample size grows, exactly the logic engineers use to size microcapsule loading in self-healing composites (White et al., Nature 2001).

Intact capsule Ruptured shell Released healing agent

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