No Unique Scaling Law for Igneous Dikes

In linear elastic fracture mechanics (LEFM), veins, dikes, and sills grow in length when the stress intensity factor KI ${K}_{I}$ at the tip reaches a critical value: the host rock fracture toughness KIc ${K}_{Ic}$. This criterion is applied broadly in LEFM models for crack growth and it is often assumed that the pressure inside the crack is uniform. When applied to intrusion length versus thickness scaling, a significant issue arises in that derived KIC=300to3000MPam ${K}_{IC}=300\,\mathrm{t}\mathrm{o}\,3000\,\mathrm{M}\mathrm{P}\mathrm{a}\,\sqrt{m}$, which is about 100-1,000 times that of measured KIc ${K}_{Ic}$ values for rocks at upper crustal depths. The same scaling relationships applied to comparatively short mineral vein data gives KIc<10MPam ${K}_{Ic}< 10\,\mathrm{M}\mathrm{P}\mathrm{a}\,\sqrt{m}$, approaching the expected range. Here we propose that intrusions preserve non-equilibrated pressures as cracks controlled by kinetics, and therefore cannot be treated in continuum with fracture-controlled constant pressure (equilibrium) structures such as veins, or many types of scaled analogue model. Early stages of dike growth (inflation) give rise to increasing length and thickness, but magma pressure gradients within intrusions may serve to drive late-stage lengthening at the expense of maximum thickness (relaxation). For cracks in 2D, we find that intrusion scaling in non-equilibrium growth is controlled by the magma injection rate and initial dike scaling, effective (2D) host rock modulus, magma viscosity and cooling rate, which are different for all individual intrusions and sets of intrusions. A solidified intrusion can therefore achieve its final dimensions via many routes, with relaxation acting as a potentially significant factor, hence there is no unique scaling law for dike intrusions.