Segment tip geometry of sheet intrusions, II: Field observations of tip geometries and a model for evolving emplacement mechanisms

Igneous sheet intrusions are segmented across several orders of magnitude, with segment tip geometry commonly considered indicative of the propagation mechanism (brittle or nonbrittle). Proposed propagation mechanisms are inferred to represent host rock mechanical properties during initial magma emplacement; typically, these models do not account for 15 segment sets that show a range of tip geometries within the same lithology. We present a detailed structural characterization of basaltic sill segments and their associated host rock deformation from the Little Minch Sill Complex, Isle of Skye, UK, and a broader comparison with segment geometries in three additional intrusive suites (Utah, USA; and Mull and Orkney, UK). Each separate host lithology shows multiple tip geometries and styles of host 20 rock deformation, from elastic-brittle fracture, to viscous indentation and fluidisation. We attribute this range of host rock deformations to evolving conditions that occur at the tips both during sheet growth and arrest. Non-Technical Summary 25 Magma commonly moves through Earth’s crust within sheet-like plumbing systems. These sheets start as short segments, which grow and link through time. Underground magma plumbing system growth cannot be observed directly: Much of our understanding of sheet growth comes from excavated and long-since cooled remnants of extinct volcanoes. The way


Notice
This manuscript is submitted to EarthArXiv and has not yet been peer reviewed.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 any 5 constructive feedback. that sheets grew is typically inferred from their shape, and the way in which the surrounding rocks were broken (deformation) to allow magma to flow through. Current models invoke growth by a single style of rock deformation. Here, we analyse exposed magmatic segments at four different locations, each of which show numerous sheet shapes, and styles of rock deformation. We attribute these features to local changes in the magma and rock properties 5 during and after sheet growth. Our results have direct implications to the mechanics of active magmatic plumbing systems, as the signals of magma propagation may change over time.

Introduction
Magmatic sheet intrusions, such as sills and dykes, play a fundamental role in magma 10 transport through the Earth's crust. Field observations, seismic surveys, and analogue models have revealed that sheet intrusions are commonly segmented across several orders of magnitude in scale, from centimetres to kilometres [e.g. Pollard et al., 1975;Hansen and Cartwright, 2006;Wyrick et al., 2014]. Segment geometry is typically linked to a mechanism of propagation, by association with host rock deformation. However, the process of intrusion, 15 and therefore the causal link to geometry, cannot be directly observed in nature. Instead, remote sensing methods such as monitoring seismicity and ground deformation are used to infer mechanisms of host rock deformation in the shallow crust [e.g. Biggs et al., 2009;Ágústsdóttir et al., 2016;Magee et al., 2018]. Remote methods are informed by field observations, hence preserved magmatic networks are critical in our understanding of the 20 mechanism(s) of magma propagation, and the likely signatures of emplacement.
Lobate or blunt tips cannot be accommodated elastically, hence LEFM does not hold. This has led to the development of various anelastic propagation models, including viscous 30 indentation and bulldozing [e.g. Spacapan et al., 2017], host rock fluidization [e.g. Schofield et al., 2010], and localised shear-failure and/or ductile flow [e.g. Pollard, 1973;Haug et al., 2017]. These mechanisms are considered representative of the initial host rock lithology and mechanical properties at the time of magma emplacement [e.g. Schofield et al., 2010;2012;Spacapan et al., 2017;Haug et al., 2018].
An issue that is generally omitted in intrusion growth models, is that magma and host rock properties can change during the lifetime of an intrusive event [Poppe et al., 2020]. 5 Modifications to the conditions of emplacement during magma propagation, such as increasing magma viscosity during cooling, have been shown to result in modifications to the intrusion geometry and the associated style of host rock deformation for cryptodomes and laccoliths [e.g. Currier and Marsh, 2015;Mattson et al., 2018;Burchardt et al., 2019]. Such modifications are also likely to occur in segmented sheet intrusions, and since late stage 10 processes have the potential to overprint early structures, segment geometries as observed in the field, may be representative only of the final stage of emplacement, rather than the emplacement process as a whole [Walker et al., in prep].
Here, we use detailed field observations of intrusive segments to develop a new conceptual dynamic model to account for evolving intrusive segment geometries in a 15 dominantly brittle host rock, due to modifications to the conditions of emplacement. We present examples from four field areas where intrusive segments display different tip geometries in the same host rock to highlight that intrusive segment geometry and the style of host rock deformation may vary with changes to the conditions of emplacement. Segments observed in the field provide a snapshot of the final conditions of the arresting intrusion 20 process.

Elastic-Brittle tip-zone Models 5
Dyke and sill segment propagation has been considered traditionally as an elastic-brittle deformation ( Fig. 1a-b). Two models have been proposed: (1) the elastic-splitting model ( Fig. 1a; [Pollard, 1973]), and (2) the Barenblatt-cohesive zone model ( Fig. 1b; [Rubin, 1993]). The Elastic Splitting Model conforms to linear elastic fracture mechanics (LEFM), where fractures have slit-like geometries with sharp wedge-shaped (tapered) tips ( Fig. 1a; 10 e.g. Pollard [1973]), as with hydrofracture propagation. This occurs when the driving fluid pressure Δ (the total fluid pressure ( ) minus the normal stress ( ) acting on the fracture plane) equals or exceeds the tensile strength ( ) of the host rock ( ≥ ). In this model, the host rock bends to accommodate the intrusion thickness; host rock layer thickness above and below the intrusion remains constant. The stress intensity factor ( ) at the crack tip is >0, 15 and anelastic deformation at the tip (the plastic process zone) is small relative to the length of the intrusion [Rubin, 1993]. Elastic splitting is linked to emplacement in brittle host rock at low confining stress and temperatures, or at high strain rates under high confining stress [Pollard, 1973].
The Barenblatt-Cohesive Zone Model builds upon the elastic splitting model to 20 include a cohesive process zone at the intrusion tip (Fig. 1b). Cohesive stresses act to resist dilation and are on the order of the host rock tensile strength, such that the stress intensity ( ) at the crack tip is reduced toward zero ( Fig. 1b; [Rubin, 1993]). During propagation of a bladed magma-filled fracture, the average flow velocity of the magma in the direction of propagation approaches the tip velocity [Rubin, 1995]. In this condition, the pressure gradient 25 ∆ (where is the along-intrusion dimension) varies approximately as 1 2 , hence the magma should lose its pressure within the very narrow tip region, leaving a gap between the viscous magma and the physical fracture tip [Barenblatt, 1962]. This zero-pressure cavity may become filled with exsolved volatiles from the magma, or from inflow of host rock pore fluids [e.g. Pollard, 1973;Lister, 1990;Rubin, 1993Rubin, , 1995. Inflow of pore fluids may cause 30 non-localised inelastic damage in the process zone ahead of the intrusion, which increases the energy required for continued fracture propagation [Rubin, 1993;Gudmundsson, 2011].
However, the pressure gradient arising from low pressure in the cavity also serves to drive magma flow toward the tip . Importantly in the Barenblatt model, if the magma driving pressure cannot facilitate further propagation of the fracture plane, the driving pressure will instead be accommodated by inflation and rounding of the magma front, without an increase in fracture length. 5

Non-brittle tip-zone models
During non-brittle magma propagation, the host rock behaves as a ductile or viscous medium ahead of the propagating tip. Viscosity contrasts influence the style of host rock deformation: when the magma viscosity exceeds that of the host rock, the magma may behave as a viscous indenter, whereas a low viscosity magma intruding into a more viscous host rock may 10 generate a Saffman-Taylor instability, causing the magma front to break down into an array of magma fingers [Saffman and Taylor, 1958]. Two key models of non-brittle magma propagation have been proposed: (1) fluidization, and (2) bulldozing or viscous indentation.
Fluidization is typically attributed to local heating of pore-fluids close to the intrusion, which causes the pore-fluid-pressure to exceed the host rock cohesion, causing 15 disaggregation and flow of the host rock [Kokelaar, 1982;Schofield et al., 2010;2012].
Accordingly, fluidized host rock is most commonly associated with intrusions emplaced into saturated, unconsolidated or poorly consolidated sedimentary materials that have low or zero cohesion. Fluidisation is most commonly associated with shallow-level intrusions [Kokelaar, 1982;Robertson, 1988;Schofield et al., 2010;2012;. Intrusions that propagate via 20 fluidization typically have lobate and/or irregular tip geometries, and are associated with an enveloping zone, or carapace, of fluidized host rock that has lost its internal structure ( Fig.   1c; [Schofield et al., 2010;2012;).
Bulldozing or Viscous Indentation models involve host rock buckling, folding, faulting, and/or ductile flow. This deformation results in a thickening of the host rock ahead 25 of a rounded or blunt intrusion tip, of a magnitude similar to the thickness of the intrusion ( Fig. 1d; [e.g., Pollard, 1973;Merle and Donnadieu, 2000;Spacapan et al., 2017]). Lobate segments with a large width relative to thickness generate large shear and compressive stress concentrations at their tips, which may overcome the shear strength of the adjacent host rock or discontinuities (e.g. bedding planes) to cause bedding plane slip and buckling of strata 30 [Pollard et al., 1975;Rubin, 1993]. Pollard [1973] proposed two end-member models to account for wedge-shaped zones of faulted host rock ahead of rounded segment tips: brittle-shear failure and ductile-shear failure, both of which resemble Hencky-relation slip line solutions for a pressurized hole in an infinite plate [e.g., Nadai, 1950;d'Escatha and Mandel, 1974]. Brittle shear failure generates faults oriented ~30°dependent on the angle of internal friction ( ) for the rockfrom the maximum local compressive stress (σ 1 ) about the intrusion tip (Fig. 1e), while ductile-shear failure produces faults oriented ~45° from the local 5 σ 1 (i.e., in the plane of maximum shear stress, ) ( Fig. 1f; [Pollard, 1973]). Notably, the segment tip geometry controls the location of shear stress concentrations about the tip [e.g., Obert and Duvall, 1967]. Shear stress will concentrate at the segment tip maximum curvature; in the case of a circular tip this is a single concentration distributed around the tip [e.g., Pollard, 1973, Souche et al., 2019, but in the case of a superelliptical or rectangular tip 10 geometry [e.g., Merle and Donnadieu, 2000;Walker et al., in prep], this is not the case. Pollard [1973] suggested that ductile-shear failure, brittle-shear failure, and elastic splitting models form a spectrum of propagation mechanisms that depend on the confining stress, temperature, pore fluid pressure, and strain rate. Changing one or more of these

Geometrical classification of segmented sheet intrusions
We define individual segments by the length of their semi-axes, with the maximum, 25 intermediate, and minimum semi-axes a ≥ b ≫ c; total segment thickness = 2 , width = 2 , and length = 2 . The spatial relationship between adjacent segments is described by their offset and separation [Delaney and Pollard, 1981]. Offset is measured here as the c-axisparallel distance between the centre-line of adjacent segments; separation is the widthparallel distance between the tips of adjacent segments, which may be overlapping, or 30 underlapping. Following Walker et al. [2017] and Healy et al. [2018], we refer to the region between two adjacent segments as the relay zone (termed 'bridge' by Delaney and Pollard [1981]), and more specifically as either an overlap zone, or underlap zone. Where segments have linked across the relay zone, it is termed a breached relay (cf. 'broken bridge': Schofield et al. [2010]).

Field Observations
Our primary data set is derived from a small (~1.1 km 2 ) outcrop of segmented sills at Neist Point, Isle of Skye, UK ( Fig. 2a-b), situated inboard of the eastern North Atlantic volcanic passive margin. The studied sills form part of the Little Minch Sill Complex (LMSC), a ~4000 km 2 sill network emplaced into the Minch Basin between 61-54 Ma, associated with 10 Paleocene magmatic activity during early rifting phases of the North Atlantic ( Fig. 2b; [Gibb and Gibson, 1989;Saunders et al., 1997;Chambers and Pringle, 2001;Fowler et al., 2004]).
Offshore, the LMSC is confined to the Mesozoic sedimentary infill of the Minch Basin, which forms part of a NE-SW striking Mesozoic half-graben system [Roberts and Holdsworth, 1999;Schofield et al., 2016]. No sills crop out in the footwall of the Minch 15 Fault, on the isles of Lewis and Harris. Onshore Skye, the sills predominantly crop out along the northern and north-eastern coastline [Gibb and Gibson, 1989]. Sill geometries along the NE coast of Skye were used by Schofield et al. [2016] to infer that the LMSC propagated eastwards and was fed by a dyke that exploited the basin-bounding Minch fault at depth.
Three thick sills (>10 m thick) crop out in the Neist Point study area and can be traced 20 in section for more than one kilometre ( Fig. 2a-b). Those sills dip at a low-angle toward the ESE, between 5° and 7° (Fig. 2c), and are mildly discordant to bedding, which has an average dip of 4° ESE (Fig. 2d). These thick sills are spatially associated with interconnected and cross-cutting networks of thin sills (≤2 m thick) (Fig. 3).
Our study focusses predominantly on the network of thin sills shown in Figure 3. 25 Here, the sills intrude the Jurassic Leat Shale Formation, which consists of horizontal and interbedded limestones, sandstones, siltstones, and mudstones. The sills also intrude into the base of the Skye Main Lava Series, however due to their inaccessibility we focus only on segments in the sedimentary strata.
The thin sills are exposed in a NE-SW oriented ~20 m high cliff section formed of 3 30 units: A lower sandstone unit interbedded with thin (cm to tens-of-centimetre thick) mudstone beds, this is overlain by a mudstone unit, and an upper sandstone unit ( Fig. 3a). At the level of exposure, the sills occupy ~50% of the total rock volume [Angkasa et al., 2017].
Continuous sheets transgress through the sedimentary sequence as a series of linked offset segments (e.g. segments X-Y-Z on Fig. 3a; Fig. 3b-c). Non-linked offset and collinear segment arrays are also identified (e.g. Fig. 3d); these are the main focus here. 5 Locally, individual segments are parallel to bedding for short distances. In total, we identified 43 segments; 42% of the identified segments are hosted in mudstone, 35% in sandstone, and 14% in limestone; the remaining 9% are thick segments (>2m) with tips that transect multiple beds (Fig. 3e).
Stepping directions between adjacent offset segments are inconsistent across the study area (compare Figs 3a-c and 4a). Breached relays between offset segments are predominantly formed of single tip-toplane linkages, noted by an abandoned tip (cf. 'horn': Nicholson and Pollard [1985]) associated with one segment (Fig. 3b-c). Breached relays also display near-vertical offset of 15 host rock units across the linkage (Fig. 3b-c). Approximately collinear segments, on the other hand, are consistently joined by tip-to-tip links (e.g. Fig. 3d). Relay zones between adjacent segments trend NW-SE and NE-SW (Fig. 3f); prolate amygdales observed at the upper contact of the lower sill in the key study area (Fig. 3a) have long-axes trending NW-SE ( Fig.   3g-h), suggesting that in this location, the primary magma flow direction was oriented NW-20 SE, parallel to 61% of the measured relay trends, and perpendicular to the main section. The studied NE-SW oriented outcrop therefore provides an ideal cross-section of the sill segments, and an excellent opportunity to characterise segment geometry as well as the style and distribution of associated host rock deformation.
Following Pollard et al. [1975], the results are plotted and compared to curves for idealised oval and elliptical geometries ( Fig. 5b-c). The thickness to width ( / ) ratio 5 reflects the overall geometry of the segment, where ⁄ → 0 represents a thin slit, ⁄ ≤ 0.2 represents a sheet, and ⁄ → 1 represents a magma finger with a near-circular crosssection. The tip geometry is characterised as to segment width ratio ( / ); where sharp tapered (wedge-shaped) tips would plot at = 0, rectangular tips would plot at = ∞ [Pollard et al., 1975], and blunt tips with superellipse geometries plot above the 10 oval line.
Of the 43 identified segments, 30 segments could be measured and a total of 47 segment tip geometries were characterised. In some cases, only one segment tip could be measured due to limited exposure or tip-to-plane links between adjacent segments (e.g. segments B-C, Fig. 4a). Most of the measured segments display sheet-like geometries (where 15 0 < ⁄ < 0.2; Fig. 5b), and predominantly plot between the elliptical and oval lines. All overlapping segments have tapered tips and have / values ≤0.01 (Fig. 5b), while underlapping segments display a range of tip geometries from tapered to blunt (superelliptical). Segments with tapered and blunt tip geometries occur within the same host rock lithologies: sandstone ( Fig. 3a-c), mudstone (Fig. 4a), and limestone ( Fig. 4b).

Figure 5c
shows segment geometry and the style of host rock deformation about the tip.
Comparing Figure 5b and 5c shows that segments with tapered tips are associated with deflected bedding around the tapered tip and localised fracturing across all host rock 25 lithologies. Most segments with elliptical to blunt tips are associated with a loss of primary host rock structure.
Figure 5b-c also highlights segment asymmetry denoted by the line length between the data points. Through re-plotting and colour-coding for segments that belong to particular underlapping multi-segment arrays, we find that the segments central to an array are 30 approximately symmetrical about their centre and have oval to blunt tip geometries (Fig. 6).
Segments nearer the periphery of the array are asymmetrical, with tapered to elliptical distal tips and proximal tips that approach blunt geometries (Fig. 6). This is similar to analogue model results of Pollard et al. [1982] of collinear crack development in an elastic medium (Fig 6.d). In their model, tip rounding occurs due to adjacent segments reaching a separation distance at which both segments begin to compete for the available mechanical energy; this competition inhibits Mode I fracture propagation, causing the driving stress to be 5 accommodated via segment inflation and tip rounding.
Notably, Figure 6c shows that some segment tips comprise two parts; a light brown chilled glassy outer tip, and a dark finely crystalline inner tip (see also Fig. 4b). In most cases the finely crystalline inner tips are more rounded than the chilled glassy outer tips. Segment B of Array 3 comprises two finely crystalline segments enveloped by a chilled glassy margin. 10

Host Rock Deformation
The thinly bedded and laminated sedimentary sequence in the Neist Point study area, has laterally continuous units at the scale of outcrop. This is important for identifying the style(s) of host rock deformation associated with each intrusive segment [Spacapan et al., 2017;15 Galland et al., 2019].
Non-linked, offset, overlapping segments are accommodated consistently by rotation of bedding across the relay zone between adjacent segments (Fig. 7a-e), across scales (cmdam). To compare scales, we measure the host rock rotation angle , the angle between undisturbed bedding and the rotated bedding orientation in the relay zone. Overlap zones 20 show a negative linear regression for and the relay zone aspect ratio (calculated as separation/offset) indicates that larger overlaps cause smaller deflections of host rock layering, as would be expected (Fig. 7h). Secondary structures (i.e. those that are only found between adjacent segments) also occur, which include: fractures, faults, and intrusive sheets that cross-cut the relay zone (Fig. 7). The angle of secondary structures (α) is measured from 25 the plane of the intrusive segment to the plane of the structure [after Tentler and Acocella, 2010]. In the measured overlap zones the angle of secondary structures increases linearly with relay zone aspect ratio (Fig. 7i).
Offset underlapping segments also display multiple styles of host rock deformation in the underlap zone. Host rock bending is noted in the underlap zone between adjacent 30 underlapping segments (Fig. 7g). Cross-fractures and minor thrust faults in underlap zones are oriented oblique to sheets with rounded segment tips ( Fig. 7f-g). Localised zones of brecciated sandstone also occur immediately ahead of rounded segment tips.
Underlap zones between approximately collinear segments in sandstone have lost their primary structure (i.e. bedding and laminations; Fig. 3d). In those cases, the segment tips have an overall blunt geometry with local irregularities; small magma fingers extend 5 from the segment tip into the underlap zone towards the adjacent segment (Fig. 3d). Similar deformation and segments with rounded to blunt tip geometries with local-irregularities are also observed between underlapping segments in mudstone (Fig. 8); here the deformed zone is discoloured relative to the rest of the mudstone unit and small magma fingers also extend from the segment tips into this zone (Fig. 8b, d, e). Linked collinear segments are identified 10 by cusps of deformed host rock at the intrusion contact (Fig. 8c); similar to those observed by Pollard et al. [1975] for collinear segments in the Shonkin Sag sill, Montana, emplaced into interbedded sandstone and shale units.

Figure 9
shows an approximately 7 m thick sill segment with a rounded to superelliptical tip. Host rock xenoliths occur close to the upper contact of the thick sill 15 segment (Fig. 9a), however limited accessibility made it impossible to determine whether these xenoliths are the result of stoping, host rock lenses between stacked sills, or remnant relay zones between linked sill segments. A fold and thrust zone occurs at the western tip ( Fig. 9). Host rock deformation is most intense at the sill tip, and the deformation intensity decays over a distance of approximately 10 m. Immediately ahead of the sill tip, minor thrust 20 faults accommodate local horizontal shortening and vertical extension (Fig. 9b-c); material stacking has caused the thickness of the host rock package to increase by ~49% at the sill tip compared to that at a distance of ~20 m from the sill tip. Ahead of the sill tip major thrust faults (those with lengths >5m) dip away from the sill tip, suggestive of material movement towards and over the sill segment. Minor conjugate thrust fault pairs are also observed at the 25 major thrust faults (Fig. d-e). Note that no thrust faults, or significant folding of the host rock, are observed above the intrusion (Fig. 9a).
Through plotting the separation and offset for each measured relay zone akin to methods used for dyke and fault analysis ( Fig. 10; [e.g. Delaney and Pollard, 1981;Fossen and Rotevatn, 2016;Long and Imber, 2011]), we can assess whether segment tip geometry 30 and style of host rock deformation correlate to the spatial relationship between adjacent segments, comparable to the work of Delaney and Pollard [1981]. Collectively, our data spans almost five orders of magnitude for segment offset, ranging from the millimetre to tens-of-metre scale. All observed overlapping sill segments in the Neist Point study area have asymmetrical tapered tips (and tapered abandoned tips at breached overlaps) ( Fig. 7; and A-B on Fig. 10a). Only three of the measured overlap zones are intact (i.e. not breached, e.g. Fig.   7), and together with the breached overlap zones they fit a positive near-linear power-law regression with an R 2 value of 0.83 (Fig. 10a). Underlapping segments, however, display 5 rounded tip geometries at high relay zone aspect ratios (C-D on Fig. 10a), and approaching low relay zone aspect ratios (E on Fig. 10a). Where the underlap is relatively large (e.g. C on Overall, the underlapping segments display a poor positive correlation between separation and offset (R 2 = 0.29 : Fig 10a). The poor correlation is likely due to the fact that underlapping segments may stop propagating at any given separation, which is not primarily 15 controlled by the offset.

Controls on sheet intrusion tip geometry
Intrusive segment tip geometry and the mechanism of magma propagation are commonly 20 inferred to reflect the mechanical properties of the host rock at the time of initial magma emplacement [e.g. Schofield et al., 2010;2012;Spacapan et al., 2017;Vachon and Hieronymus, 2017;Bertelsen et al., 2018;Kjøll et al., 2019;Schmiedel et al., 2017;.
These emplacement models are based upon field observations, which preserve the final stage of intrusion. There is, however, the potential for post-emplacement and/or late-stage syn-25 emplacement overprinting, and the preserved growth mechanism may not be representative of intrusion growth as a whole [see e.g., Spacapan et al., 2017, andHaug et al., 2017]. We can consider the potential for post-emplacement overprinting through detailed field-based textural and structural characterisation of the host rock and intrusion [e.g., Bons et al., 2004]. Few studies have focused on syn-emplacement variations in host rock and magma properties, 30 which may cause a transition in the intrusion tip geometry and emplacement mechanism [Poppe et al., 2020]. Tapered through to superelliptical sill segments outcrop in the Neist Point study area, within no correlation between host rock lithology and emplacement mechanism (Fig. 5b-c). Examples of tapered sill tips are associated with host rock bending, and demonstrate emplacement through elastic brittle processes, at least at the scale of observation (e.g. Fig. 7a-e). Elliptical to superelliptical geometries are associated with 5 several styles of localised host rock deformation ahead of the sill segment tips, including fluidisation and brecciation (Figs 3d, 7f-g, 8), and shear-band formation (Fig. 9).
A similar array of sheet intrusion tip geometries are also observed in a range of host rock types in other locations. The Loch Scridain Sill complex (Isle of Mull, UK) comprises basaltic sill segments with tapered and rounded geometries hosted in subvertically bedded 10 and foliated metasedimentary Moine basement ( Fig. 11a-b) and horizontal basaltic lavas ( Fig. 11c-f; see Supplemental File 1 for geological background). Tapered tips are associated with pre-existing fractures and linked offset overlapping segments, while rounded tips are associated with host rock bending in the Moine basement (Fig. 11a); and chilled tips ( Fig.   11c-d) and densely fractured relay zones (Fig. 11 e-f)  Birsay localities). Offset overlapping segments display tapered tips associated with deflected bedding about sill tips in Utah (Fig. 12a-c) and deflected subvertical joints about dyke tips in Birsay (Fig. 14a, d). In Utah, offset and colinear underlapping segments display rounded to blunt tips, associated with localised host rock brecciation (Figs 12d-g, 13), in some cases thin sheets cut obliquely across the brecciated relay zone, emanating from the blunt segment tips 25 (Figs 12d-g, 13c; a schematic model for their development is shown in Fig. 12h). In Birsay, underlapping segments are associated with localised fracturing in in the tip zone ( Fig. 14b-c,   e). In Mull and Birsay, where the host rock at the segment tip zone is densely fractured, the rounded segment tips also display magma fingers indicative of propagation via viscous fingering (Figs 11f, 14e). 30 Differences in host rock deformation style can reflect the initial host rock shear cohesion and tensile strength [Baer, 1991]. Lithologies with higher shear cohesion are more likely to localise strain into a single fracture, whereas those with low cohesion are not able to withstand elevated shear stresses, and will fail through distributed fracture along grain boundaries; with increased fluid pressure, disaggregation can lead to fluidization. The scale of the zone of brecciation and fluidisation may therefore relate to the scale of existing discontinuity within the material: In a mudstone, this is the grain size, and in intercalated units (mudstone-sandstone sequences for instance), this may be the layer thickness; in the 5 case of lavas and pre-existing sills, this may be the cooling joint network. All of the observed Caledonides; an earlier set with tapered tips and a later set with rounded tips. They suggest 20 that the later set resulted from magma emplacement into a ductile host due to the thermal effects of the first dyke set locally raising the brittle-ductile transition. Wilson et al. [2016] also suggest a two-stage emplacement model for the Trachyte Mesa intrusions, Henry Mountains, Utah similar to models of Hunt [1953] and Corry [1988]: initial elastic-brittle fracture propagation followed by segment inflation, tip-rounding and associated shear failure 25 of the host rock. In this model, shear failure is caused by overburden flexure and bending once the sill reaches a critical diameter in relation to emplacement depth. The examples presented in our study do not resemble cyclic emplacement processes, and the presence of sharp intrusion contacts and brittle fractures throughout the study areas suggests that in these cases the host rock was primarily brittle. We observe variations in sill tip geometry and styles 30 of host rock deformation for segments varying from cm to tens-of-metre thickness, suggesting that the depth of emplacement and segment size do not necessarily control the final (preserved) geometry or local style of deformation at the segment tip.
During magma emplacement, dynamic changes may modify the properties of the magma and host rock ahead of the propagating tip. Magma viscosity may increase due to cooling, crystallization (and generation of crystal mush), or vesiculation and degassing [e.g. Shaw, 1969;Johnson and Pollard, 1973;Hess and Dingwell, 1996;Currier and Marsh, 2015;Chanceaux and Menand, 2016]. Host rock rheology and cohesion may be altered due to local 5 heating and volatile loss into the host rock [e.g. Aarnes et al., 2011a;2011b;Annen, 2011;Currier and Marsh, 2015;Chanceau and Menand, 2016;Mattsson et al., 2018]; and localised pore fluid boiling may cause fracturing, brecciation, or fluidization ahead of segment tips [e.g. Pollard, 1973;Pollard et al., 1975]. In their scaling parameters for comparison between natural and modelled intrusions, Galland et al. [2014] suggest that magma viscosity and host 10 rock cohesion are coupled by the dimensionless ratio , where η is magma viscosity, v is flow velocity, C is cohesion, and T is intrusion thickness. This provides a dynamic ratio between time-dependent viscous stresses in the flowing magma and stresses in the host rock; and indicates that increasing magma viscosity has the same effect as decreasing host rock cohesion: inhibits elastic-brittle propagation and promotes tip rounding. 15 The transition from a tapered to rounded tip geometry would also cause a change in the distribution of stress in the process zone, from a stress singularity at a tapered tip to distributed radial and circumferential stress ahead of an elliptical or oval tip [Pollard, 1973;Souche et al., 2019]. The zones of maximum stress are concentrated at the corners ( ) of superelliptical segments, which would likely propagate as viscous indenters causing shear 20 failure of the host ( Fig. 9; [e.g. Abdelmalak et al., 2012;Guldstrand et al., 2017;Haug et al., 2018;Walker et al., in prep]. As the radius of curvature of the corners decreases (i.e. toward a right angle) the stress becomes increasingly localised, and the stress concentration factor (i.e. the ratio of the maximum stress at the contact to the far-field stress: ∞ ), tends towards infinity [Jaeger et al., 2007;Walker et al., in prep]. Propagation of segments with 25 superelliptical to blunt tips, however, is inefficient [Pollard et al., 1975]. Rounded segment tips at the extremities of intrusions, such as those observed in Skye, Mull, Utah, and Orkney are therefore unlikely to represent the initial stages of intrusion growth (i.e., more proximal to source), and are most likely developed later due to modifications to the conditions of emplacement.

A model for segment evolution
Based on previous propagation models, and our observations of sill segment geometries and associated host rock deformation, we propose a multi-stage model for the evolution of intrusive segment geometry for a basaltic melt in an initially brittle host rock in the shallow crust. Our model accounts for changes to the conditions of emplacement over the lifespan of 5 the segment. We envisage the following stages: Stage I: Emplacement and propagation as Mode I fractures (Fig. 15: Stage I); controlled by periods of driving pressure increase and relaxation following the theory of linear elastic fracture mechanics (LEFM) [e.g. Atkinson, 1987;Cañón-Tapia and Merle, 2006]. Pre-existing planes of weakness (e.g. joints, fractures, bedding, foliation) may be 10 preferentially dilated and intruded during Stage I if the planes are optimally orientated relative to the remote stress state (i.e. oriented normal to the minimum compressive stress) and/or if the magma pressure is sufficiently high to enable dilation of non-optimally oriented planes [Martínez-Poza et al., 2014;Stephens et al 2018]. During Stage I, segments will propagate until the driving pressure is unable to facilitate the next increment of growth; for 15 lateral segment propagation this likely due to local competition for the available mechanical energy between offset and/or collinear adjacent segments [e.g. Pollard et al., 1982], or due to a drop in driving pressure as a function of the increasing segment length for frontal tip propagation. The segment likely accrues the majority of its length and width during this stage. 20 Stage II: Localised modifications to the conditions of emplacement (Fig. 15). This may include modifications to the magma viscosity due to, e.g., volatile loss, degassing, cooling, and crystallization, and/or local modifications to the host rock mechanical properties due heat transfer and/or volatile loss from the magma (increasing the pore fluid pressure ahead of the segment tip). High porosity and/or low cohesion host rock may be fluidized 25 during this stage, whereas cohesive host rock may be fractured or brecciated ahead of the segment tip [Baer, 1991]. Magma degassing and cooling and crystallization cause magma rheology to evolve from viscous to viscoelastic to brittle once chilled [Dingwell, 2006].
During the transition the resistance to flow is increased, which dissipates mechanical energy as viscous drag [Pollard et al., 1975;Rubin, 1993]. The chilled margin may have a higher 30 tensile strength than the host rock, meaning a larger driving pressure would be required to reestablish elastic-brittle propagation [Dingwell, 2006;Currier and Marsh, 2015]. Depending on the cooling rate relative to the rate of increasing driving pressure, magma flow may become localised into the centre of the segment, cause segment inflation or breaching somewhere along the intrusion, or propagation my cease altogether, in which case the primary tapered tip geometry is preserved (Figs 4b, 11d; [Chanceaux and Menand, 2016]).
Stage II is therefore associated with stalled propagation, segment inflation, and a change in 5 tip geometry. We envisage minimal change to the segment width during this stage, as the driving pressure is accommodated by segment inflation without further propagation.
Stage III: Minor non-brittle propagation and/or termination. The transition from Stage II to III is marked by renewed propagation (Fig. 15). Three scenarios are envisaged for Stage III: (A) flow localisation, (B) breakthrough of the chilled margin to enable resumed Mode I 10 propagation, or (C) renewed propagation via anelastic mechanisms (e.g. viscous indentation, bulldozing, or fluidization). Segment tip rounding during Stage II modifies the local tip stress distribution from a tensile stress singularity to a circumferential tension with radial compression and shear stress concentration. When the width exceeds the thickness of a lobate segment (i.e. a segment aspect ratio (thickness/width) <1), the shear stress concentrations at 15 the tip may facilitate propagation via bulldozing or viscous indentation [Pollard et al., 1975;Rubin, 1993;Soushe et al., 2019]. Local host rock fluidization may enable instigation of a Saffman-Taylor instability at the magma -fluidized host rock interface, for renewed propagation via viscous fingering [Saffman and Taylor, 1958;Pollard et al., 1975]. Segment linkage through a locally fluidized zone may result in a 'cusp' of deformed host rock 20 preserved at the intrusion contact, enabling identification of this mechanism Pollard et al. [1975]. Notably, the renewed propagation mechanism will vary on a segment-by-segment basis, dependent on the changes that occur. Multiple tip geometries and styles of host rock deformation could, therefore, occur in the same host rock units and across a single intrusive complex. 25 Any further propagation will be governed by the magma driving pressure. We expect the propagation distance during Stage III to be minor relative to Stage I due to the reduction in stress concentration, and inefficiency of shear propagation of a rounded or blunt segment tip relative to a tapered tip (Mode I fracture) [Inglis, 1913;Pollard et al., 1975;Walker et al., in prep]. Unless the driving pressure increases significantly to generate a break-out and 30 continued Mode I propagation [e.g. Baer, 1991;Currier and Marsh, 2015: Fig. 15, Scenario   B] (in which case the process reverts to Stage I), in our model renewed propagation via nonbrittle mechanisms represents the final stage in the life of the segment. Hence, the preserved tip-zone deformation may not be representative of the initial propagation mechanism that controlled the main phase of segment growth.
Notably, our proposed model provides an evolutionary pathway for magmatic segments that do not link during initial brittle propagation. The timescale of tip geometry evolution is 5 dependent on the rate of change to the host rock and/or magma properties. Our model is consistent with observations of sills and dykes ranging from centimetre to tens of metre thickness and emplaced in the brittle crust (up to ~2km depth), so should therefore be consistent with the development of large intrusive complexes, as imaged in subsurface seismic surveys. Importantly, this model provides an explanation as to how multiple styles of 10 host rock deformation, and seemingly multiple propagation mechanisms, occur in a single field area.
Our model applies to emplacement of low viscosity (basaltic) melts at shallow emplacement depths. We note that intrusion of higher viscosity melts (e.g. rhyolite) or magma emplacement at greater confining stresses can cause the initial tip geometry and/or 15 the mechanism of initial host rock deformation to vary from our model. However, in these cases an evolving pathway also occurs [e.g. Currier and Marsh, 2015;Mattson et al., 2018;Burchardt et al., 2019], which is consistent with our findings here.

20
We present a quantitative geometrical characterization of a segmented sill network in the Little Minch Sill Complex, Isle of Skye, UK and compare these data to three additional intrusive complexes: The Loch Scridain Sill Complex, Isle of Mull, UK; sills in the San Rafael Subvolcanic field, Utah; and dykes in Birsay, Orkney, UK. In each study area intrusive segments display tip geometries varying from tapered through to blunt. Multiple 25 styles of host rock deformation were observed from those consistent with elastic-brittle fracture, associated with tapered sill tips, to those consistent with non-brittle propagation models: fluidization, brecciation, and viscous indentation. Our observations suggest that the emplacement mechanism may evolve over the active magmatic lifespan of the segment. We present a conceptual multi-stage emplacement model to account for the variety of segment 30 geometries and styles of host rock deformation observed: elastic-brittle fracture, followed by localised fluid overpressure at the segment tip, and non-brittle propagation. Non-brittle propagation is likely minor relative to its preservation potential in the rock record: it is representative of the final stages in the life of the segment prior to complete crystallization.

Figure 5.
Geometrical characterisation of sill segment geometries in the Neist Point study area. (a) Schematic diagram to illustrate circular through to rectangular tip geometries and how the macro-tip form radius of curvature (ρtip; red circle) was measured. Note that once tip geometry surpasses an oval, corners develop and the superelliptical geometry can also be defined by two minimum radii of curvature (see green circles in "squared ellipse" example). Following the method of Pollard et al. [1975] the macro-tip forms were plotted to show: (b) segment geometry and host rock lithology; (c) segment geometry and observed style of localised host rock deformation ahead of the tip. Dashed lines for oval and elliptical geometries are shown for comparison to measured geometries. Individual segments are shown as two data points (denoting the geometry of each tip) connected by a line; line length correlates to the segment asymmetry about its c-axis (longer line = more asymmetric); isolated points are where only one tip could be measured. Symbols indicate segment arrangement (underlapping or overlapping). Note that schematic segments below the x-axis in (a) and (b) are to illustrate segment t/w ratio only. (Data available in Supplemental File 2).

Figure 6.
Geometrical characterization of individual segment geometries within an array. (a) Plot to define segment geometries: segment tips are displayed as circles and diamonds (for left and right tips, as observed in outcrop), individual segments are shown as linked symbols and the line length correlates to segment asymmetry about its c-axis (longer line = more asymmetric). Relay zones occur between the diamond and circle tips of adjacent segments (Data available in Supplemental File 2). Note that schematic segments below x-axis are to illustrate segment t/w ratio only. (b-c) Schematic sketches of two of the arrays: Array 1 (Fig. 8); Array 3 ( Fig. 4b), only 4 segments were observed. (d) Illustration to show the generalised asymmetric geometry of collinear segments in an elastic medium (redrawn and modified from analogue model of Pollard et al. [1982]).   Illustrations of high (10:1), unity (1:1), and low (1:10) aspect ratios for underlapping and overlapping segments; separation, S; offset, O. Data in (a) shows relay zone geometry of underlapping (blue squares) and overlapping (orange squares) sill segments in the Neist Point study area. Low and high aspect ratio lines are from Long and Imber [2011]. Note that underlapping segments are plotted with positive values. Best fit power law regressions are fitted to the underlapping (blue line) and overlapping (red line) segments. The sill relay zone data is also plotted with published relay zone data for dykes intruded into shale at Ship Rock, New Mexico (triangles). The data was manually digitized from Delaney and Pollard [1981;their Fig.10] using WebPlotDigitizer, and replotted. Segment sketches A-E were drawn from examples in the Neist Point study area: A-B, overlapping and interacting; C-E, underlapping segments with rounded to blunt tips and minor magma fingers. Sketches of segment tip geometries (F-H) are also redrawn from Delaney and Pollard [1981;their Fig.10]: F, underlapping segments with rounded tips; G, slightly overlapping segments with asymmetrical tips, indicative of interaction; H, overlapping segments with symmetrical tapered tips, indicative of no segment interaction.