Toughness-controlled magma intrusion ($R=(24V/\pi {\kappa }_f{)}^{1/4}$)

With the axisymmetric clamped-plate deflection and fixed front curvature ${\kappa }_f>0$, a magma intrusion hasThese relations describe the advancing fracture branch, with roof bending rather than hydrostatic pressure controlling the interior load.

Let $V_T$ be total magma volume shared between a reservoir and an advancing toughness-controlled magma intrusion, $p_r=E(V_T-V)$, and conduit conductance $\beta >0$. With $A=\sqrt{8\pi /3}B{\kappa }_f^{3/2}$,Equilibria satisfy $\sqrt{V}(V_T-V)=A/E$. Two positive equilibria exist when $A/E<2V_T^{3/2}/(3\sqrt{3})$; the smaller is unstable and the larger is stable under this volume equation. An advancing solution between them approaches the larger. The singular pressure as $V\to 0$ prevents this late-time model from nucleating an intrusion by itself.