Fracture Stability Map: Dendrite Crack Growth 2D

Cell Conditions

Plating current density 2.0 mA/cm²

Initial flaw size a₀ 5.0 µm

Electrolyte fracture toughness K_IC 0.9 MPa·√m

Applied stack pressure 1.0 MPa

Crack State

Idle

Stress intensity K

0.00 MPa·√m

Crack length a

5.0 µm

Wedging stress σ

0 MPa

Time to short

stable

How to read the map

This is the 2D companion to the 3D crack-rendering simulator — same fracture-mechanics equations, computed independently and shown as a stability map instead of a rendered crack:

K = σ·√(π·a)          (Griffith stress-intensity factor)
σ ≈ k_p·i − k_P·P_stack (net electrochemical wedging stress)
Grows while:  K > K_IC
Critical flaw: a_c = (K_IC / σ)² / π

The top panel plots K as a function of crack length a (log scale) for the current wedging stress σ — a rising curve, since K scales with √a. The dashed line is K_IC. Wherever the curve sits above that line (shaded), the crack grows; wherever it sits below, it is mechanically arrested. The glowing dot is the cell's actual current state — it always sits exactly on the curve, and slides right as the crack lengthens.
The curve crossing K_IC marks the critical flaw length a_c — any flaw longer than that, at this σ, runs away on its own.
The bottom panel is a real-time strip chart of crack length a(t), so you can watch the same growth law unfold over time instead of only in phase space.
Raising stack pressure or lowering current density pushes σ down, which lowers the whole K(a) curve and pushes a_c out — the same suppression mechanism as pressurized solid-state cell packaging (Kasemchainan et al., 2019).

K(a) curve at current σ K_IC (fracture toughness) Current state / trail