Segment tip geometry of sheet intrusions, I: Theory and numerical models for the role of tip shape in controlling propagation pathways

Richard J. Walker*, Tara L. Stephens, Catherine Greenfield, Simon P. A. Gill, David Healy, Sam Poppe

*Corresponding author for this work

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Inferences about sheet intrusion emplacement mechanisms have been built largely on field observations of intrusiontip zones: magmatic systems that did not grow beyond their observed state. Here we use finite element simulationof elliptical to superelliptical crack tips, representing observed natural sill segments, to show the effect of silltip 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. Super elliptical tips concentrate maximum tensile stress pσmaxqand shear stress out-of-plane of the sill, which may promote non-coplanar growth, vertical thickening, or coplanar viscous indentation. We find that σmax“Pep1`2? aκmaxq, where Peis 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.
Original languageEnglish
Pages (from-to)189–201
Number of pages13
Issue number2
Publication statusPublished - 18 Oct 2021

Bibliographical note

This study was conducted without allocated funding. Catherine Greenfield is a Daphne Jackson Trust Research Fellow funded by NERC and the University of Leicester. Simon Gill is supported by a Royal Society Apex Award. Sam Poppe was supported by a frs-FNRS postdoctoral fellowship while at UlB and currently by a ULAM scholarship (NAWA Poland) and NCN Poland grant 2020/37/K/ST10/02447 at SRC PAS. The au- thors thank Tim Davis and Steffi Burchardt for discussions that benefitted the formulation of this paper, and thank the three anonymous reviewers and Editor—Jamie Farquharson—for review commentary that helped to improve the clarity of this paper.

Data Availability Statement

Supporting datasets are available via the University of Leicester’s Figshare portal at doi: 10.25392/, including:
• Supplement 1. Calculations and calculator for de- termining κmax based on Equation 7, 8, 9 and 10.
• Supplement 2. Full results for numerical models of elliptical to superelliptical tips, extracted from Comsol Multiphysics.
• Supplement 3. Comsol Multiphysics model file.


  • Igneous sill
  • Igneous dike
  • Superellipse
  • Curvature
  • Stress


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