Toughness-controlled magma intrusion
= Toughness-controlled magma intrusion
{title2=$R=(24V/\pi\kappa_f)^{1/4}$}
With the <axisymmetric clamped-plate deflection> and fixed front <curvature> $\kappa_f>0$, a <magma intrusion> has
$$
R=\left(\frac{24V}{\pi\kappa_f}\right)^{1/4},\qquad
p=\sqrt{\frac{8\pi}{3}}B\kappa_f^{3/2}V^{-1/2}.
$$
These relations describe the advancing fracture branch, with roof bending rather than hydrostatic pressure controlling the interior load.