No abstract

Inferences about sheet intrusion emplacement mechanisms have been built largely on field observations of intrusion tip zones: magmatic systems that did not grow beyond their observed state. Here we use finite element simulation of elliptical to superelliptical crack tips, representing observed natural sill segments, to show the effect of sill tip shape in controlling local stress concentrations, and the potential propagation pathways. Stress concentration magnitude and distribution is strongly affected by the position and magnitude of maximum tip curvature κmax. Elliptical tips concentrate stress in-plane with the sill, promoting coplanar growth. Superelliptical tips concentrate maximum tensile stress (σmax) and shear stress out-of-plane of the sill, which may promote non-coplanar growth, vertical thickening, or coplanar viscous indentation. We find that σmax =  Pe(1+ 2(√[aκmax]), where Pe is magma excess pressure and a is sill half length. At short length-scales, blunted tips locally generate large tensile stresses; at longer length-scales, elliptical-tipped sills become more efficient at concentrating stress than blunt sills.

This manuscript is submitted to EarthArXiv as a pre-print and has not yet been peer-reviewed. Please note that following peer-review, subsequent versions of this paper may have slightly different content. If accepted for publication, the final version of this pre-print will also be made available. Please feel free to contact the corresponding author directly. We welcome constructive feedback. Title: Segment tip geometry of sheet intrusions, I: Theory and numerical models for the role of tip shape in controlling propagation pathways.

Previous studies have suggested a link between rates of sea-level variation and eruptions globally [McGuire et al., 1997], with Satow and coauthors [2021] presenting the first detailed comparison between sea-level change and eruptive history for a single island-volcano. They use robust, high-resolution ages for volcanic deposits at Santorini, combined with a 2D numerical model to correlate sea-level reduction with volcanism. Lowering sea level reduces overburden pressure and is predicted to increase tensile stress in the magma chamber roof, leading to diking and eventually eruption. Having independently reproduced their results, we disagree with the numerical model for three main reasons: (1) predictions of stress distribution and magnitudes caused by sea level change are solely dependent on the size and boundary conditions of the 2D model; (2) minor changes to the model dimensions, dimensionality (2D to 3D), and/or addition of a mantle analogue, removes correlation between sea level and eruptions; and (3) crustal loading conditions at the volcano absent from the model are more significant than sea level change.

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=3000.25emnormaltnormalo0.25em30000.25emnormalMnormalPnormala0.25emm ${K}_{IC}=300\,\mathrm{t}\mathrm{o}\,3000\,\mathrm{M}\mathrm{P}\mathrm{a}\,\sqrt{m}$, 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<100.25emnormalMnormalPnormala0.25emm ${K}_{Ic}< 10\,\mathrm{M}\mathrm{P}\mathrm{a}\,\sqrt{m}$, 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.