Whether the dike can move at all is decided by fracture mechanics + buoyancy, exactly as in the 3D version. But which way it moves at every step is decided separately, by the local stress field: a dike always propagates perpendicular to the local minimum compressive stress σ₃ (equivalently, parallel to the local σ₁ axis) — real dikes bend, curve and get deflected around magma chambers for exactly this reason.
Magnitude (can it move?):
ΔP(h) = P₀ + (ρ̄_rock − ρ_magma)·g·h_risen
nucleates only if ΔP·√(π·L₀) > K_IC
Direction (which way does it move?):
σ(x,y) = σ_regional(x,y) + Σ σ_source_i(x,y) (2×2 stress tensors, superposed)
θ(x,y) = ½·atan2(2σxy, σxx − σyy) (principal-axis angle, σ₁ direction)
step direction = θ(x,y), resolved to point upward
- Regional stress — the far-field trend. At 0° obliquity σ₃ is horizontal and the dike simply climbs straight up, same as the 3D model's default. Tilt it and the whole trend leans.
- Stress magnitude — how strongly the regional trend resists being overridden by a nearby source. Weak regional stress lets a placed chamber dominate and bend the path sharply; strong regional stress keeps the dike nearly straight even close to a source.
- Stress sources — each placed chamber is treated as a small pressurized (Mogi-like) body: it puts the crust immediately around it into radial compression and tangential extension, so σ₃ near it runs tangential and the dike is pushed to trace around it rather than through it — the same mechanism that produces real radial dike swarms around volcanic centers.
- The buoyancy/toughness magnitude criterion and the stress-direction criterion are evaluated independently every step, then combined: the dike only advances (magnitude test passes) and, when it does, it advances along the locally computed σ₁ direction (direction test).
This is a simplified 2×2-tensor superposition for teaching purposes — real boundary-element dike models integrate over finite-size sources and 3-D elastic layering — but the governing rule (propagation ⟂ σ₃, tensors from independent sources add) is the real one structural geologists use to explain dike deflection and arrest.