This is the 2D statistical counterpart to the 3D box-of-capsules visualizer, computed independently rather than flattened from its 3D scene. Instead of rendering capsules as spheres in space, it plots the population's actual Weibull weakest-link fracture statistics directly: a strength-density histogram of the batch's sampled critical stresses on top, and a linearized Weibull probability plot — the classic reliability-engineering diagnostic — underneath.
Each capsule's critical stress: σ_c = σ0 · (−ln(1−U))^(1/m) (inverse-CDF sampling, U~Uniform(0,1))
Rupture condition: σ_c ≤ σ_applied (irreversible once triggered)
Population CDF: P(rupture | σ) = 1 − exp[−(σ/σ0)^m]
Linearization (probability plot): ln(−ln(1−F)) = m·ln(σ) − m·ln(σ0) ⇒ a straight line of slope m
Sorting the N sampled thresholds and assigning each its median rank F_i = (i−0.3)/(N+0.4) turns the whole batch into points on that linearized axis. A well-behaved Weibull population falls on a straight line whose slope recovers the modulus m and whose x-intercept recovers ln(σ0) — exactly how real fracture-strength data from tensile tests on brittle shells is analyzed. The bottom panel plots the batch's own points against that line live, and a least-squares fit (shown in green) is computed every time the batch regenerates so you can see how closely a finite sample of ~400 capsules recovers the true σ0, m used to generate it.
- σapplied slider — sweeps a vertical cursor across both panels; capsules to its left (lower critical stress) have already ruptured, irreversibly, exactly as in the 3D scene.
- Auto-ramp — sweeps the load 0 → 40 MPa automatically so the histogram's ruptured region and the probability-plot cursor advance together.
- σ0 / m sliders — regenerate a fresh batch from a Weibull distribution with the chosen mean strength and scatter; watch the fit line and the analytic line converge as you compare a tight, well-manufactured batch (high m) against a wide, defect-prone one (low m).
- The Weibull prediction readout is the closed-form P(rupture|σ) at the current load; the simulated ruptured fraction should track it, with the residual finite-sample scatter itself being the same statistic every materials engineer has to account for when sizing a real microcapsule loading (White et al., Nature 2001).