This paper examines the development of a subvolcanic magmatic breccia located along the contact of a granitic intrusion using fractal analysis and thermal-elastic modeling. The breccia grades from clast-supported, angular clasts adjacent to unfractured host rock to isolated, rounded clasts supported by the granitic matrix adjacent to the intrusion. Field observations point to an explosive breccia mechanism, and clast size distribution analysis yields fractal dimensions (Ds > 3) that exceed the minimum value known to result from explosion (Ds > 2.5). Field observations, clast size distribution data, clast circularity data, and boundary roughness fractal dimension data suggest that the clast sizes and shapes reflect post-brecciation modification by partial melting and thermal fracture. Numerical modeling is employed to explore the possible thermal-elastic effects on the size distribution of clasts. Instantaneous immersion is assumed for metasedimentary clasts of a fractal size distribution in a superheated granitic matrix for different matrix volume percentages. Thermal analysis is restricted to conductive heat transfer corrected for latent heat. Partial melting of metasedimentary clasts is an effective secondary modification process that was probably responsible for markedly altering the clast size distribution for clast populations adjacent to the intrusion. Diabase clasts experienced late-stage fracture due to the instantaneous thermal pulse in which angular clasts with high surface area to volume ratios were preferentially fractured, although this process does not appear to have markedly influenced the clast size distribution.
 Subvolcanic systems, much like eruptive sequences, can preserve a long and varied history of instantaneous magmatic events [e.g., Johnson et al., 2002; Metcalf, 2004; Kemp et al., 2008; Marianelli et al., 2006]. They typically contain a wide variety of intrusive phases and are potentially a rich source of information about the evolution of volcanic root zones and the tops of upper-crustal magma reservoirs. The detailed intrusive relationships in these complexes and the intimate timing relationships between the various intrusions and deformational structures are preserved in part due to rapid quenching of some units, and in part due to the sequential series of intrusions that produce a magmatic stratigraphy. The study of magma reservoir walls is important because they reflect a variety of processes associated with inflation and deflation of the reservoir during recharge and evacuation events.
 We describe the Cadillac Mountain intrusive complex (Figure 1), a subvolcanic igneous system that preserves a spectacular concentric magmatic breccia zone known as the Shatter Zone [Gilman et al., 1988]. We explored possible mechanisms of Shatter-Zone formation by combining geological observations with fractal analytical methods, including clast size distribution, boundary roughness fractal dimension, and clast circularity analysis. The fractal data imply that an explosion caused wall rock brecciation, but the data distributions are affected by secondary modification of clast sizes and shapes by thermally driven processes. Thermal-elastic modeling was employed to investigate the possibility of clast shape and size modification resulting from partial melting and thermal fracturing.
2. Geologic Setting
2.1. Cadillac Mountain Intrusive Complex
 The Cadillac Mountain intrusive complex of Mount Desert Island (Figure 1) is part of the Coastal Maine Magmatic Province [Hogan and Sinha, 1989], a group of over 100 plutons of bimodal granitic-gabbroic composition with ages ranging from Late Silurian to Early Carboniferous. The complex is roughly circular in map view covering an approximate area of 14 km × 20 km, and consists of the Cadillac Mountain Granite and the younger Somesville granite suite to the west, both of which are inferred to be underlain by composite gabbro-diorite sheets exposed to the west [Hodge et al., 1982; Wiebe, 1994]. The Cranberry Island volcanic series, located in the south, features bimodal geochemistry that correlates well with the intrusive complex, leading some authors to consider the two to be genetically related [e.g., Seaman et al., 1995, 1999; Wiebe et al., 1997].
 The complex was emplaced into the Ellsworth Schist, Bar Harbor Formation, and the Cranberry Island volcanic series. This study focuses on the contact between Cadillac Mountain Granite and Bar Harbor Formation on the eastern side of Mount Desert Island. The Bar Harbor Formation is a metasedimentary rock which underwent regional diagenesis followed by contact metamorphism along the perimeter of the Cadillac Mountain Granite [Metzger, 1959]. The unit is approximately 610 m thick and stratified with dominant rock types including pelitic clays, sandstones, conglomerates, calc-silicates, and volcanic tuff, with the dominant sediment sources including 1) Ellsworth Schist, 2) an underlying amphibolite unit, and 3) sediments from local volcanic activity. The Bar Harbor Formation probably formed as a subaqueous fan [Metzger and Bickford, 1972; Metzger, 1979].
 The perimeter of the Cadillac Mountain Granite is defined by the Shatter Zone, an aureole of fragmented country rock in a granitic matrix that varies in apparent width from 450 to 1000 m [Chapman, 1962; Gilman et al., 1988]. The rock fragments in the eastern Shatter Zone consist of: (1) Bar Harbor Formation, (2) Devonian diabase dikes, and (3) large felsic volcanic xenoliths near the gradational contact with the Cadillac Mountain Granite. Clast sizes range from millimeter to meter scale [Gilman et al., 1988]. The matrix of the Shatter Zone is a fine-grained biotite leucogranite that was locally intruded by late-stage pegmatite dikes, both of which differ compositionally from the dry, A-type Cadillac Mountain Granite. The average matrix grain size gradually increases from the very fine grained outer margin to the coarse grained Cadillac Mountain Granite [Wiebe et al., 1997].
3. Background on Quantitative Methods of Breccia Classification
where N(≥ r) is the count of clasts with a radius greater than or equal to r, and k is a curve fitting parameter equal to N(≥ 1). Ds is the fractal dimension for clast distribution, and it is considered to be a measure of fracture resistance relative to the mechanism or process of fragmentation. Ds is proportional to the magnitude and rate of stress loading, the inherent strength properties of the rock, and the possibility of repeated fracturing events, and therefore can be used to determine the mechanisms responsible for rock fragmentation [Sammis et al., 1987; Jebrak, 1997]. It is also sensitive to any secondary mechanisms that may alter the original distribution [e.g., Farris and Paterson, 2007]. The negative slope signifies an increase in clast frequency with decreased size.
 To compare clast frequency with size, the volume or area of a non-spherical clast must be represented by an equivalent value for radius. The radius is calculated by defining a circle with an area identical to that of the clast of interest [e.g.,Brittain, 2001; Bjork et al., 2009]. This is a common method used to display CSD data from 2D measurements [Barnett, 2004; Farris and Paterson, 2007; Bjork et al., 2009]. Data bias caused by progressively approaching the detection limits of the observational method can produce a non-fractal trend for the finer clast sizes; therefore, it is necessary to place a minimum radius limit on counted clasts [Blenkinsop, 1991; Clark and James, 2003; Barnett, 2004].
 Although CSD data are typically collected in 2D sections, they can be expressed in 3D given assumptions of homogeneity and isotropy. If an object is fractal in 2D, it is also fractal in 3D, and
This conversion is validated by the fact that Ds refers to the line, surface, or space that dissects the fractal object. An increase in Euclidean dimension requires the same increase in Ds [e.g., Sammis et al., 1987]. This conversion is justified for a breccia comprised of a homogeneous, isotropic material, but error can be introduced if this assumption is used on anisotropic materials [e.g., Barnett, 2004; Farris and Paterson, 2007]. To reduce this potential error, outcrops with 3D exposures can be used. Clast distribution can also be expressed as a function of clast frequency versus mass [e.g., Hartmann, 1969; Blenkinsop, 1991], or percent sample by weight versus diameter [e.g., Schoutens, 1979], both of which are directly related to a 3D distribution. These results also pertain to power law distributions and therefore their slopes are proportional and can be converted to Ds [Blenkinsop, 1991; Perfect, 1997].
3.2. The Fractal Dimension-Brecciation Mechanism Link for Clast Size Distribution
 Clast size distribution data are a function of the self-similar manner by which fractures propagate through a medium, and the fractal dimension Dsis influenced by the intensity of fracturing. Understanding the brecciation mechanism will provide important information on the mechanical response of rigid wall rock during pressure fluctuations in the magma reservoir. There are several potential mechanisms that could have formed the Shatter Zone, and three possible end-members will be discussed: 1) pre-eruptive magma emplacement causing hydraulic brecciation of wall rocks, 2) caldera subsidence producing an abrasive collapse breccia along concentric ring faults, or 3) the rapid volume expansion of volatiles during eruption leading to explosive fracture of the wall rocks. These three mechanisms are fundamentally different and will therefore produce different breccias with unique Ds values. Hydraulic fracture and explosion should lead to minimum and maximum Ds values, whereas abrasive breccias tend to result in intermediate Ds for a single rock type without repeated fragmentation [Jebrak, 1997; Clark and James, 2003; Barnett, 2004; Sammis et al., 1987], although some overlap in Ds for these mechanisms has been noted [Turcotte, 1986; Blenkinsop, 1991].
 Abrasive breccias form by sliding along shear fractures leading to abrasive fragmentation along the fracture walls [Goodman, 1980; Jebrak, 1997]. Like hydraulic breccias, abrasive breccias can also form incrementally, but approximately simple-shear kinematics and relatively moderate stress loading produce more complex fragmentation patterns [e.g.,Sammis et al., 1987; Blenkinsop, 1991]. Continued grinding, plucking, and reduction of clast size results in the development of fault gouge, and rotation and flow of elongate clasts leads to a preferred alignment [e.g., Jebrak, 1997]. Ds values for abrasive breccias can range widely depending on strain rate, the normal stress on fracture planes, and the number and duration of abrasion events [Sammis et al., 1986, 1987; Blenkinsop, 1991]. However, they typically have Ds values of 2–2.7 [Sammis et al., 1987; Blenkinsop, 1991]. In addition, clasts typically show evidence of imbrications and preferred orientation [Lipman, 1984].
 Circularity is a measure of the compactness of a shape, and can be used to determine the relative degree of post-breccia clast boundary modification for originally intact rock [e.g.,Clark, 1990; Dellino and Volpe, 1996]. Circularity is the ratio of clast cross-sectional area A versus the area of a circle with perimeter p equal to the clast:
Very angular, elongate shapes approach 0 and very compact shapes approach 1.
4. Field Relations in the Shatter Zone
 The Shatter Zone is characterized by a spatial gradient in rock fragmentation and brecciation (Figures 1 and 4), with the degree of fragmentation and relative volume of the granite intrusion increasing toward the contact with the Cadillac Mountain Granite. This gradient is completely exposed along the eastern coast of Mount Desert Island and intermittently visible on the island interior. The Shatter Zone can be divided into three breccia types based on differences in appearance and degree of fragmentation and their distributions are indicated in inset B of Figure 1.
4.1. Type 1
 The width of Shatter Zone subtypes can vary greatly; maximum thickness of Type 1 is approximately 100 m, but it is also absent in some locations (Figure 1, outcrop 1, and Table 1). The transition from solid, coherent Bar Harbor Formation to Type 1 Shatter Zone is gradual. Granite veinlets (typically <1 cm wide) intruded parallel to primary bedding structures, and are interconnected by less common fractures across bedding layers (Figures 3a and 3b). Type 1 outcrops have an average of 16.1% matrix by area. Approximately 50 m into the Shatter Zone from unfractured Bar Harbor Formation, pockets of brecciated material appear between the more common bed-parallel vein structures. All matrix material is fine grained and there is no evidence for late stage fracture after the initial brecciation event. Few diabase dikes are present, and no diabase clasts exist at this stage in the available outcrops shown inFigure 1b.
Table 1. Average CSD Values
Type 3 Bar Harbor
Type 3 Diabase Dike
4.2. Type 2
 The maximum width of Type 2 is approximately 450 m (Figure 1, outcrops 2–4, and Table 1). Type 2 outcrop fabrics are more chaotic with less preservation of the original bedding structures (Figures 3c and 3d). As with Type 1, all matrix material is fine grained and there is no evidence for late-stage fracture. Type 2 outcrops have an average of 34.4% matrix by area. Diabase dikes are more frequent, and some are fractured and fragmented by pegmatite veins. Pockets in the Type 2 Shatter Zone show a greater degree of fragmentation divided by sections with weakly intact bed fabric.
4.3. Type 3
 The maximum width of Type 3 is approximately 450 m and it lies adjacent to the Cadillac Mountain Granite (Figure 1, outcrops 5–6, and Table 1). The matrix is generally fine grained with pegmatitic material present in late stage cracks. Type 3 outcrops have an average of 75% matrix by area. Type 3 clasts are generally sub-equant and completely supported by the granite matrix (Figures 3e and 3f). The clasts in the Type 3 breccia outcrops are dominantly diabase, contrasting with Types 1 and 2. Many clasts appear well rounded with concentrations of biotite along their rims. There are several occurrences of magma contamination, probably by disaggregated clast material. Evidence for late stage cracking in diabase is not uncommon. Many of the Bar Harbor Formation clasts appear to have recrystallized, and the most abundant appear to be calc-silicate rock. Magmatic fabric dominates in Type 3, with no clast preferred orientation. There are no large scale flow textures in the matrix, but occasional local biotite-rich flow bands may have formed from minor clast rotation and settling.
 There is a fourth type within the Shatter Zone located along the contact with the Cadillac Mountain Granite. Isolated wall rock fragments in Type 4 average ca. 10 m in diameter, are commonly too large to define their boundaries in the limited outcrops, and are sparse compared to Type 3. Schlieren textures trailing above and along the sides of the large fragments suggest that they may have sank into the magma chamber. The large sizes and sparse occurrence of Type 4 wall rock fragments preclude their incorporation in the fractal analysis presented below.
5. Fractal Analysis Methods
 Data from 12,732 clasts were used to identify the brecciation mechanism and assess possible modifications of clast size and shape. Clast data were calculated from image mosaics collected from outcrops representative of Types 1, 2, and 3 of the Shatter Zone (Figures 1 and 3). Grids were overlain on flat outcrops with individual boxes of 30 × 25 cm. High resolution images of each box were stitched to create the image mosaics. Clasts were manually outlined from each mosaic in a drafting program to differentiate between clast and matrix, producing a black (clast) and white (matrix) image. Manual outlining was required because the gray scale separation between clasts and matrix was commonly too small to accurately distinguish them using image analysis software [e.g., Sudhakar et al., 2006]. The outlines were analyzed with NIH ImageJ for clast count, area, circularity, and boundary shape. CSD, boundary roughness, and circularity were calculated from these output data. The number of clasts used for boundary roughness was limited compared to CSD because each randomly chosen outline had to be analyzed individually.
 For CSD, the equivalent radius was used to plot logarithmic size versus cumulate frequency, with a clast radius interval of 100.02 cm. Ds values were calculated using equation (1). Cumulate frequency was standardized to the total area covered for each outcrop location to allow better comparison between locations with greater or smaller outcrop representation. Only clasts greater than 1 mm radius were plotted because of difficulty in distinguishing between smaller clasts and the granite matrix. The plotted cumulate frequency values are averages weighted by outcrop area for multiple studied outcrops with the standard error of regression included for each type.
 Boundary roughness fractal dimension data were produced by 42 measurements of clast outline width and area for 428 clast outlines. Measurements were made using NIH ImageJ. The log of outline width was plotted with respect to the log of outline area, and Dr was calculated from the slope using equation (2).
 Clast circularity data were produced in NIH ImageJ using equation (3). Clast frequency plots were produced using a 0.05 circularity interval for 0.1–1 cm, 1–10 cm, and >10 cm clast radius intervals.
6. Fractal Analysis Data
6.1. Clast Size Distribution (CSD) Data
 A total sample size of 14 imaged outcrops yielding 12,732 clasts with size ranges spanning 3 orders of magnitude (0.1–40 cm) were used for CSD analysis (Table 1). Types 1–3 of the Shatter Zone are represented and results are shown in Figure 4. Type 1 (Figure 4a) Shatter Zone shows a bifractal distribution, with two power law distributions divided by a slope breakpoint. The position of the breakpoint is calculated by fitting power law trendlines to both distributions and adjusting the dividing breakpoint position every 100.02 cm to find the smallest average R2 value [e.g., Stankiewicz and De Wit, 2005]. An average Ds value of 3.027 ± 0.252 characterizes clasts with radii >1.25 cm, with an average Ds value of 1.5 ± 0.088 for clasts with radii <1.25 cm. The R2 values for these two averaged distributions are 0.9868 above and 0.9778 below the breakpoint. The CSD curve for Type 1 represents 1,519 clasts from four outcrop grids with a size range of 0.1–23.4 cm. Type 1 has 16.1% average matrix component by area.
 Type 2 (Figure 4b) also shows a bifractal distribution. An average Ds value of 3.166 ± 0.108 characterizes clasts with radii >1.52 cm, with an average Ds value of 1.875 ± 0.138 for clasts with radii <1.52 cm. The R2 values for these two distributions are 0.9932 above and 0.9865 below the breakpoint. The CSD curve for Type 2 represents 5,538 outlined clasts from four outcrop grids with a clast size range of 0.1–13.02 cm. Type 2 has 34.4% average matrix component by area.
 Clast size populations in Type 3 Shatter Zone are divided by rock type due to the marked increase in diabase dike clast abundance (Figure 4c). The diabase clasts have an average Ds value of 2.62 ± 0.058 and R2value of 0.9865 for the clast size range of 0.41–11.82 cm. There is a gradual decrease in slope with decreasing clast size which will be discussed below. Clasts of the Bar Harbor Formation show a CSD trend that can be characterized in two ways: a bi-fractal distribution with a breakpoint at 1.6 cm and average Dsvalues of 1.66 ± 0.166 above and 4.06 ± 0.905 below, or a non-fractal, exponential curve with an exponential decay coefficient of 1.013 ± 0.166. The bifractal R2 values are 0.9867 above and 0.9492 below the breakpoint. The R2 value for the exponential curve distribution is 0.9915. The CSD curves for Type 3 represent 5,675 outlined clasts from seven outcrop grids, less than 20% of which are Bar Harbor Formation clasts. Diabase size ranges are 0.1–11.8 cm and Bar Harbor Formation size ranges are 0.1–4.24 cm. Type 3 has 75% average matrix component by area.
6.2. Roughness Fractal Dimension Data
 A total of 433 clasts from Type 1, 2, and 3 Shatter Zone, with an additional outcrop (named 2.5) located between types 2 and 3 (Figure 1), were used for Dr analysis (Figure 5a). The Dr data set comes from Hawkins and Johnson . Values of Dr vary only slightly with most values approaching 1 (1.04–1.12). Type 1 Shatter Zone has the highest average Dr value of 1.125 but with the highest standard deviation of 0.062 from 123 clasts. Type 2 has an average Dr of 1.052 with a standard deviation of 0.037 from 79 clasts, and Type 2.5 has an average Dr of 1.047 with a standard deviation of 0.035 from 66 clasts. Type 3 has an average Dr of 1.040 with a standard deviation of 0.019 from 56 clasts.
6.3. Clast Circularity Data
 Circularity data were collected for 8,724 outlined clasts over three orders of magnitude for Shatter Zone Types 1–3 (Figures 5b–5d). Many of the same clasts used for CSD were used to calculate circularity. Type 1 clasts have the lowest average circularity value of 0.48, and the circularity averages for the 0.1–1 cm, 1–10 cm, and >10 cm intervals are 0.54, 0.36, and 0.19, with population sizes of 925, 431, and 5, respectively. Type 2 clasts have an average circularity of 0.58, and the circularity averages for the 0.1–1 cm, 1–10 cm, and >10 cm intervals are 0.61, 0.47, and 0.27, with population sizes of 2,155, 595, and 9, respectively. Type 3 clasts have the highest degree of circularity with an average of 0.76, and the circularity averages for the 0.1–1 cm, 1–10 cm, and >10 cm intervals are 0.77, 0.67, and 0.63, with population sizes of 3,945, 644, and 14, respectively.
6.4. Summary of Data
 All three types of Shatter Zone yield a value of Ds above 2.5 for distributions of radius greater than 1.25 cm. Below this clast size, Ds varies but is always lower than the Ds of the coarser size range. The breakpoint in the bifractal slope occurs over a small and consistent size range in Types 1 and 2. Data from Types 1 and 2 come dominantly from Bar Harbor Formation clasts, whereas data from Type 3 are split between diabase dike and Bar Harbor Formation clasts. The gradual change in the Type 3 diabase slope is probably due to small clast sample bias. Granitic matrix becomes more abundant with proximity to the Cadillac Mountain Granite interface. Circularity increases with proximity to the granite reservoir and with decreasing clast size for all Types. Roughness fractal dimension values decrease with proximity to the granite reservoir [Hawkins and Johnson, 2004].
6.5. Fractal Analysis Discussion
6.5.1. Initial Fragmentation
 The Dsvalues for coarse size distributions are above 3 for Types 1 and 2, implying a high rate stress-loading mechanism with a large degree of fragmentation [e.g.,Jebrak, 1997; Barnett, 2004]. The two most viable possibilities as postulated previously are collapse abrasion and reservoir explosion. Hydraulic fracture is not a probable mechanism. Shear is the fundamental mechanism that produces an abrasive breccia. The clasts of Type 1 and 2 are preserved approximately in their original positions relative to stratigraphic layering and show no shear or transport fabric. Additionally, the preserved outcrops suggest a 450–1000 m thick breccia with fragmentation intensity increasing with proximity to the intrusive contact and a single intrusion event. This gradient is consistent with a rapid, localized source of energy, as would be expected from the rapid volume expansion of volatiles in the magma reservoir during a volcanic eruption. Given that the calculated Ds values for clasts of Type 1 and 2 breccia are within the lower end of the explosion range [Schoutens, 1979; Turcotte, 1986] and that they are adjacent to a previously active subvolcanic complex, we suggest that chamber explosion is the most probable cause of brecciation. The relatively higher error for Type 1 is due to the heterogeneity present between well brecciated pockets and poorly fractured bedding structures, which becomes less prevalent in Type 2 and is absent in Type 3.
 The Type 3 breccia represents a much more advanced stage of fragmentation and magma injection, and the original positions of clasts relative to a reference frame fixed to pre-breccia stratigraphy are no longer preserved. The curve patterns and relatively high Ds error values of Type 3 Bar Harbor Formation CSD data suggest that this breccia should be defined as nonfractal. Nevertheless, diabase dike clasts in Type 3 have a Ds value consistent with an explosive breccia, and there is no apparent shear fabric; therefore, we also interpret Type 3 Shatter Zone as an explosion breccia preserved at a more advanced stage, consistent with its proximity to the energy source. We therefore consider the spatial gradient in breccia development from Stages 1 to 3 to represent a temporal evolutionary sequence. This would require that the Type 3 breccias began as Type 1 breccias, and evolved through Type 2 to arrive at their present degree of development. Although we cannot prove that the spatial brecciation gradient equates to a temporal gradient, this interpretation is the most consistent with all of the geological observations.
 The development of a subvolcanic explosion breccia is limited to shallow crustal depths where fluid is abundant and reservoir pressure can rapidly overcome wall rock lithostatic pressures [e.g., Lorenz and Kurszlaukis, 2007]. Hydraulic fracture would be a more likely mechanism for brecciation at depths below 8–10 km due to greatly reduced water content and higher confining pressures [e.g., Walther, 1990; Bons and van Milligen, 2001]. Abrasion breccias commonly occur in ring complexes and signify large displacements associated with caldera collapse [e.g., Lipman, 1984; Johnson et al., 2002], but no direct evidence for abrasion exists in the Shatter Zone.
6.5.2. Clast Modification Evidence
 The changes in slope for Type 1 and 2 clasts may represent one or more scale-dependent processes that affected the fractal properties of CSD, particularly for relatively small clasts. The divergence from a fractal distribution in Type 3 Bar Harbor Formation clast populations may suggest brecciation by a mechanism completely different from the one that produced Type 1 and 2, but we consider this to be unlikely based on field observations that indicate a single fragmentation event, presumably by a single explosive eruption. Although there is no field evidence to support it, we cannot rule out the possibility that abrasion was partly responsible for initial Ds ≈ 3 during the early stages of brecciation, or that the combination of explosion followed by limited fragmentation led to an initial bi-fractal distribution. However, boundary roughness and circularity data indicate increased clast modification with proximity to the Cadillac Mountain Granite and with decreased clast size. Values of Dr decrease and circularity values increase with proximity to the Cadillac Mountain Granite, suggesting that some process has altered the original clast boundaries, which may be best represented in Type 1.
 All evidence considered, we postulate that the Cadillac Mountain Granite thermal perturbation caused the originally fractal CSD trends to evolve into bi-fractal, and finally non-fractal trends, and this evolution is displayed by the breccia gradient from Type 1 to Type 3. Two mechanisms that could have an effect on CSD, boundary roughness, and circulation data are: 1) thermal disintegration of clasts by partial melt and assimilation into the surrounding magma; and 2) fluid-assisted thermal fracture of clasts during magma intrusion. Thermal disintegration (Figure 6) requires supersolidus temperatures combined with magma flow to physically disaggregate the clasts [e.g., Braun and Kriegsman, 2001]. Partial melt can play an effective role in the volumetric removal of clasts and has been linked to alteration of clast size distribution trends and reduced Ds values in other studies [Farris and Paterson, 2007]. Thermal fracture results from the rock's elastic response to steep thermal gradients. The response is dependent on the rock's thermal expansivity, and fragmentation is assisted by the presence of fluids that fill and expand inside these fractures [e.g., Clarke et al., 1998].
7. Methods: Clast Melt Model
7.1. Model Setup and Important Parameters
 To assess the possible roles of melting and thermal fracture, transient thermal-elastic equations were solved using the finite element method (COMSOL Multiphysics). We used outcrop outline geometries to create 2D thermal-elastic models for Type 2 and Type 3 breccias (Figures 7a and 7b) in order to show the 2D pattern of heat transfer into clasts with observed fractal distributions. We used a 3D thermal-elastic model (Figure 7c) to quantify the characteristic migration rate of a partial melting boundary into a spherical clast. All models were used to examine the instantaneous immersion of Bar Harbor Formation metasedimentary clasts in a hot magma matrix of infinite extent. We assumed that the area of the clast that achieved supersolidus temperatures was eventually able to disaggregate and be dispersed by matrix magma flow and chemical diffusion into the surrounding magma, altering clast sizes with time [Marko et al., 2005; Clarke, 2007]. Field observations suggest local flow around clasts, consistent with this assumption. These mechanisms would allow transient evolution of clast size distribution, boundary roughness, and clast circularity with progressive heat conduction. We assumed a kinematically static interface between the clast and its granitic matrix; therefore, the rate of heat transfer was entirely dependent on the thermal diffusivities of the granitic magma and the metasedimentary clast, and our results represent a maximum-size-reduction end-member for conduction-controlled heat transfer. Results from the 3D clast model are used to calculate possible changes in size distribution assuming that all clasts are spherical, which diverges from the true clast geometries but provides a first order constraint on thermally modified size distributions.
 Geometry is the dominant factor that determines the pattern of conductive heat transfer [Jaeger, 1961, 1964], and the rate of transfer increases with an increase in surface area to volume ratio and thermal gradient [Jaeger, 1961; Turcotte and Schubert, 1982; Bowers et al., 1990; Furlong et al., 1991; Stuwe, 2002]. The 2D outcrop models contain 75% matrix by volume for Type 3 and 34% for Type 2, and their geometries represent clast geometries in outcrop. The 2D outcrop models use periodic boundary conditions, and nodes are defined by an advancing triangular mesh that fined toward the clast-matrix boundaries. Time stepping was used to collect thermal data for clast sizes that cover 3 orders of magnitude. The 3D clast model assumes that all clasts are spherical and are surrounded by the same average matrix volume percent for each breccia type. Symmetry boundaries are used to extend the spherical surface geometry and an insulation boundary terminates the extent of the matrix. Nodes are defined by a tetragonal mesh that fines toward the clast-matrix boundary, and the model was run for several clast sizes to confirm a characteristic migration rate of the partial melting boundary. Different model runs are defined by the parameters inTable 2.
Table 2. Physical Constants for Thermal Solutions
850 J/kg °K
950 J/Kg °K
k Bar Harbor Formation
3 W/m °K
ρBar Harbor Formation
κBar Harbor Formation
L, Latent heat
 Initial temperatures for clast and magma were required to solve the conductive heating equation. The relatively sparse occurrence of orthopyroxene implies that the rocks were heated to the lower temperature end of orthopyroxene hornfels facies, so initial clast temperature was set to Tclast = 650°C for Type 3 and Tclast = 550°C for Type 2, which is outside of the orthopyroxene hornfels facies [e.g., Spear et al., 1999; Milord et al., 2001; Blatt et al., 2006; Kriegsman and Alvarez-Valero, 2010]. Type 3 magmatic temperatures were modeled at Tintrusion = 900°C and Type 2 at Tintrusion = 800°C. Wiebe et al. suggested that superheated temperatures resulted from many gabbro-diorite sheets that underplated the granite reservoir; we therefore choose these temperatures to constrain the greatest potential for clast melt and thermal fracture. The intrusion temperature for the Type 1 3D clast is 750°C, the clast temperature is 450°C, and initial temperature values for Types 2 and 3 are identical to those used for the 2D models. Initial matrix temperatures reflect magma cooling as it is injected further away from the chamber into the Shatter Zone, and initial clast temperatures reflect the thermal gradient produced by the intrusive complex.
 The solidus for the Bar Harbor Formation is approximately 720°C, calculated using an optimization algorithm in PerpleX [Connolly, 2009] and the thermodynamic database provided by Holland and Powell . Chemical data required for the thermodynamic calculations came from mineral abundance data from Metzger , and results were compared to melt data from similar metasedimentary and metapelite rocks from the Ballachulish aureole [Pattison and Harte, 1988] and from metapelite P-T-t paths [Spear et al., 1999]. We assume a single fragmentation-emplacement event in the Shatter Zone as there was no compelling field evidence for multiple events.
For this equation, latent heat was released within the range of dT = 100°C above Tgranite solidus = 700°C, when most crystallization occurs. We chose not to address the nonlinear effect of magma crystallization rates, so latent heat was evenly dispersed during the entire duration within dT. Microstructural evidence suggests that the Bar Harbor Formation in Type 3 Shatter Zone was metamorphosed to orthopyroxene hornfels facies prior to brecciation, so we ignore endothermic metamorphic reactions that may counterbalance latent heat production [Kerrick, 1991].
7.2. Use of Dimensionless Variables
 We use dimensionless variables to compare heat diffusion patterns from the 3D clast model, thus removing the solution's dependence on clast size, thermal gradient, and thermal diffusivity [Bejan, 1993]. The following dimensionless variables were used:
where τ and R are dimensionless time and clast radius, respectively. κ is the diffusivity coefficient of the heated material. R0 is equal to clast radius at t = 0. Because τ is normalized time with respect to the diffusivity and clast radius ratio, migration of a phase change through any size clast would show similar results facilitating comparison of clast sizes spanning more than three orders of magnitude. Dimensionless time and radius values were used to determine characteristic temperature versus time curves for points located in the center, edge, and a distance 1.1 times the clast radius into the magma (Figure 7d). Dimensionless values were also used to determine the characteristic rate at which the partial melt isotherm migrates into a heated clast (Figure 7e). Dimensionless model results were used to produce hypothetical clast size distribution (CSD) curves using the same methods as for our outcrop-based CSD data. Equivalent radius values were calculated from the area below the solidus temperature for every clast (collected using NIH ImageJ) and data were plotted with a radius interval of 100.3 (Figure 8, discussed further below).
7.3. Clast Melt Results
Figure 7d displays model results from the transient temperature path for points within the spherical clast model for Types 1–3. Results from the 16% magma by volume model with an intrusion temperature of 750°C show that clast centers (solid black curve) do not reach 720°C for Tc = 450°C. Results from the 34% magma-by-volume model with intrusion temperature of 800°C show that clast centers do not achieve 720°C for Tc = 550°C. For a magmatic breccia with 75% matrix, clast centers achieve 720°C by τ = 0.2 (ca. 15 s for 1 cm radius clast) for Ti = 900°C, and the steady state temperature is 840°C. The migration rate of the 720°C isotherm for Type 3 clasts is shown in Figure 7e.
 Results (Figure 7f and Animation S1 in the auxiliary material) from the Type 3 outcrop model show trends for a 900°C intruding magma. Clasts with radii below 1 cm reach 720°C throughout within 4 s, and all clasts with radii below 5 cm reach 720°C throughout before 300 s. For the total clast population, the area above 720°C at 300 s is equal to 35% of the initial bulk clast area. Approximately 50% of clast area reaches the 720°C solidus by 900 s, and all clast area achieves 720°C by 5800 s. Steady state temperatures are achieved within approximately 2 days.
 Results (Figure 7g and Animation S2) from the Type 2 outcrop model show thermal evolution for an 800°C intruding magma with initial clast temperature of 550°C. Contrary to the 3D spherical model, clasts with radii below ca. 1.25 cm achieve core temperatures above the 720°C solidus temperature for a finite amount of time (Figure 7h) if they are isolated enough from the large clasts, but their temperature eventually drops as the outcrop approaches a steady state temperature below 720°C. Small clasts with radaii less than ca. 1.25 cm maintain core temperatures below their solidus if they are directly adjacent to large (>2 cm radius) clasts, and clasts above ca. 1.25 cm radius maintain core temperatures below their solidus regardless of their proximity to large clasts for this outcrop model.
7.4. Clast Melt Discussion
 Based on thermal modeling results and the calculated solidus temperature of 720°C, all Bar Harbor Formation clasts with the average bulk mineralogy [Metzger, 1959] achieved supersolidus temperatures throughout with 75% magma matrix by volume. Small, noncircular clasts were the first to fully reach supersolidus temperatures, and eventually only larger clasts remained (Figure 7f). An order of magnitude change in clast radius will cause two orders of magnitude change in the amount of time for the clast to achieve supersolidus temperatures. Although the thermal model ignores advection of heat due to flow of the matrix magma relative to the clasts, magma flow would have occurred during the fragmentation and intrusion process. Flow of magma around the partially melting clasts early in the breccia development could have facilitated disaggregation, though “ghosting” of some clast margins suggests that diffusion, or local mixing of melt and intruding magma, may have occurred without marked flow of the matrix magma later in the history. Small clasts (<1 cm radius) reached solidus temperatures within seconds after emplacement, and so may have been disaggregated by flowing magma. Large clasts (>10 cm radius), on the other hand, may have lost some marginal material by flow-driven disaggregation, but largely retained much of their original volume as they heated above the solidus temperature in a static magma bath.
 Some clasts will have had higher or lower solidus temperatures than the average one used here because of the heterogeneous composition of the Bar Harbor Formation. For example, volcanic tuff, quartzite, or calc-silicate layers would not melt as rapidly as the pelitic layers, and in some cases may not have undergone any partial melting. This would result in these types of clasts being preferentially preserved, which is generally consistent with outcrop observations in Type 3 Shatter Zone.
 The potential for whole-clast partial melting in Type 2 is limited to small (ca. 1.25 cm radius), isolated clasts for a finite time (Figures 7g and 8h). The time interval in which melt may occur in small clasts is dependent on clast size and proximity to larger clasts that are less affected by the intrusion, with observed time intervals ranging from a few seconds to a few minutes. Melting in small, isolated clasts initiates once they reach 720°C and ceases when the large clast abundance eventually cools the matrix to below 720°C. Angular corners of large clasts are susceptible to partial melting, which can lead to an increase in circularity, decrease in Dr, and a possible decrease in clast size which would affect CSD trends. Larger (>2 cm radius), more circular clasts are otherwise unaffected by the intrusion and would preserve the original CSD. Partial melt of small clasts alters the original CSD for clast sizes below ca. 1.25 cm, which is somewhat close to our empirically determined Type 2 CSD breakpoint of ca. 1.5 cm.
 The initial temperature and volume fraction of wall rock and magma determine the steady state temperature reached at infinite time. As seen in Figure 7d, a lower magma matrix percent by volume would cause less clast melting. Several initial intrusion and clast temperatures are used to approximate potential magma temperatures and wall rock temperatures with distance from the reservoir. Type 1 and 2 Shatter Zone experienced a relatively weaker thermal pulse and therefore would have experienced less partial melting than Type 3. No clast cores reach solidus temperatures for Type 1 and 2 average matrix percents in our spherical clast models, but we show with the 2D Type 2 model that partial melting limited to small clasts does occur in magma channelways with higher than average matrix percentages. This result illustrates the importance of clast proximity in thermal evolution for the more clast-rich breccias. We are unable to replicate this possibility with the 3D spherical clast model because it assumes that all clasts experience the same average exposure to the intruded magma. The initial Type 3 Shatter Zone possibly had a matrix volume percent similar to that of Type 2, but continued melt and disaggregation reduced the area percentage of clasts.
 The trend of CSD evolution with partial clast melt based on results from the 3D clast model (Figures 7c–7e) is shown in Figure 8. Assuming Ti = 900°C and Tc = 650°C, Ds decreases over time with the loss of small clasts and generally no change in large clast populations. Starting with a fractal distribution at time zero, as small clasts quickly disaggregate, a trend develops that could be interpreted as bifractal. Eventually, the trend evolves to one that is better defined by an exponential size distribution. This model can explain the progression of Ds from Type 2 to Type 3 in Bar Harbor Formation clasts. Type 3 Bar Harbor Formation size distributions have the most evolved slope and the greatest thermal exposure. Owing to heterogeneous compositions, we cannot assume that the Bar Harbor Formation clasts precisely followed this migrating CSD trend. The CSD for Bar Harbor Formation clasts therefore reflects partial melting of most of the clasts plus the remnant clasts that did not attain melting temperatures.
8. Thermal-Induced Fracture
 Field observations, specifically the late-stage fractures in diabase clasts, suggest that fracturing occurred after the main fragmentation event (Figure 9). Clasts exposed to magma undergo rapid heating leading to a transient, non-uniform temperature distribution. Most materials tend to volumetrically expand when heated [e.g.,Manson, 1953; Clarke et al., 1998], and non-uniform volume expansion in a constricted solid leads to stress concentration within sharp thermal gradients, potentially causing fracture propagation. Stresses associated with heating are therefore inextricably linked to the thermal gradient. In a clast of irregular shape subjected to surface heating, temperatures in the interior and in the root zones of the corners are lower than those at the surfaces. Tensile stresses thus develop in these regions of non-uniform expansion as the materials there are stretched by the hotter surface materials, causing potential corner break-off. Thermal fracture could be significant for CSD because: 1) it would produce new clasts from the breakdown of old ones in a fractal distribution pattern different from explosion-derived populations [e.g.,Glazner and Bartley, 2006]; and 2) it would increase the surface area to volume ratio of clasts, making them more likely to undergo partial melting and thus further modifying the CSD.
8.1. Model Setup
 A single 2D clast was used to evaluate the stress distributions during heating, using the same methodologies as described in Section 7.1. Figure 10ashows the clast embedded in a magma matrix, and the finite element mesh used in the solution. Models were run for 650°C clasts in 900°C, low-viscosity magma (∼106 Pa s, CIPW Norm calculation based on bulk rock chemistry data from Wiebe et al. ). The thermal-mechanical parameters used for diabase and Bar Harbor Formation clasts are listed inTable 3. The clast and surrounding matrix were considered to be elastically isotropic. The magma surrounding the expanding clast would have flowed viscously to accommodate expansion, therefore thermal expansion of the matrix was set to zero and the Young's modulus was set to three orders of magnitude lower than a typical rock to approximate this behavior. The models were run with a confining pressure consistent with approximately 5 km depth (ca. 0.13 GPa). Our model does not take into account the dynamic nature of the stress field during fracture nucleation and propagation; instead we model a clast that can potentially fracture based on tensile stress buildup without actually producing fractures. We limit our observations to a 2D clast model and use them for discussion purposes below.
Table 3. Physical Constants for Thermal-Mechanical Solutions
8.2. Thermal-Induced Fracture Results and Discussion
 We follow the convention of tensile stress being positive and compressive stress negative for Figure 10 and Animation S3. The first principal stress relates to the maximum value of stress, and the second principal stress relates to the minimum value of stress for a 2D model. Figure 10c shows transient development of the first principal stress distribution in the diabase clast. The surface plot displays the first principle stress field, with tensile stress plotted as red and compressive stress plotted as blue. Arrows designate the second principle stress direction, or the assumed direction of fracture propagation. The green overlay denotes where 20 MPa, the approximate tensile strength of diabase, has been reached. Results display two stages of tensile stress buildup at different spatial scales (Figure 10b): the first initiates early local tensile stress buildup in the roots of sharp corners, while the second occurs after the gradual migration of the thermal gradient to the core of the clast, where large tensile stresses are caused by volumetric expansion of the heated outer perimeter. Eventually thermal stresses relax as the steady state temperature is approached. The sharp thermal gradients produced by instantaneous immersion of the clast causes a tensile stress maximum (Figure 10b), but the portion of the clast in which tensile stresses exceed the tensile strength is miniscule until the thermal gradient migrates into the corner root zones. The maximum tensile stress value reaches a transient low at about 6000 s, when the corner root zones are approaching steady state temperature and the thermal gradient has slackened, but before the sharp thermal gradient approaches the core of the clast. Arrows within the maximum tensile stress locations show that tensile fracture causes corner break-off. For this clast there is a preferred cuspate-shaped fracture pattern that would produce additional corners. We interpret the cuspate-lobate morphology of this clast to result from this process. Eventually the number of corners would be reduced by successive fragmentation and circularity would increase.
 Progressive heating causes tensile stress migration from the roots of corners to the center of the clast. The entire perimeter of the clast is now expanding, causing a second phase of maximum tensile stress at the center of the clast (Figure 10b). At time 10000 s, the tensile stress at the center of the clast is well established and exceeds the tensile strength of diabase, and second principal stress arrows indicate that larger fractures could nucleate from the center and break the clast parallel to its short axis. At this time of thermal exposure, originally elongate clasts may be preferentially broken into more equant fragments, each new fracture surface providing a fresh face to repeat the process. The potential for thermal fracture declines as the thermal gradient in the clast is reduced.
Figure 10d shows results for a Bar Harbor Formation clast under the same conditions. We assume that no thermal fracture can occur in a partially molten clast, therefore we only consider stress buildup within the clast portion that lies at temperatures below the Bar Harbor Formation solidus. Tensile stress concentrates early in corner root zones and exceeds 15 MPa, the approximate tensile strength of Bar Harbor Formation (spatially denoted in green), and migrates inward more rapidly than the solidus. The solidus temperature is therefore on the lower end of the thermal gradient responsible for stress generation and ensures that tensile stresses can build to failure in the cooler portions of the clast. As with the diabase clast, tensile stresses initiate in the small areas of corner root zones and migrate with the thermal gradient to the clast interior area, until the steady state temperature is approached and stresses relax.
 This thermal fracture model provides an explanation for late-stage fracture for both rock types, but it does not take into account the anisotropic, heterogeneous nature of the Bar Harbor Formation. Rock anisotropy and heterogeneity create a more complex thermal stress distribution and affect fracture propagation pathways.Clarke et al.  discussed the three different possibilities for fracture development in anisotropic xenoliths. The first is determined by the thermal gradient and this is used in the above model (thermal gradient cracking). The second mechanism for crack formation in a layered material becomes important when one layer has a differing thermal expansion coefficient than its adjacent counterpart (thermal expansion mismatch cracking). Third, a material may have an anisotropic fabric that leads to preferred crack propagation in one direction (thermal anisotropy cracking). The Bar Harbor Formation can exhibit all three of these fracture types due to its compositional layering, and it may be more susceptible to thermal fracture than suggested by the thermal gradient cracking model.
 If thermal fracture was prevalent, each clast would fracture according to a new size distribution: for thermal fracture, Ds = 2.156 (calculated from data in Glazner and Bartley ). The new value of Ds is the combination of the explosive fracture event and the secondary thermal fracture event, and repeated fracture events always increase Ds by a fractional amount [Jebrak, 1997]. For Bar Harbor Formation clasts, the added component of thermal fracture would produce greater surface area and therefore more rapid clast melting, enhancing the ability for CSD trends to diverge from initial fractal distributions (Figure 8). Field evidence suggests that not all diabase clasts preserve evidence for thermal fracture; therefore, late-stage thermal fracture may not have had a significant effect on diabase dike CSD results.
9. Summary and Conclusions
 Fractal analysis has led to important new insights into the fragmentation mechanisms of geological materials, with examples in environments of abrasion [e.g., Sammis et al., 1986, 1987; Blenkinsop, 1991], chemical weathering [e.g., Bérubé and Jebrak, 1999], hydraulic fracture [e.g., Clark and James, 2003] and explosion [Schoutens, 1979]. However, the application of fractal analysis to magmatic breccias has met with considerable challenges related to post-fragmentation modification of the clast size distributions by thermal processes such as partial melting and thermal fracture. Complexities related to the partial or complete disintegration of relatively small clast populations, and decrease in boundary roughness, have been attributed to thermal processes [e.g.,Jebrak, 1997; Farris and Paterson, 2007], but there is a lack of numerical modeling that would provide more robust predictions of how these processes may alter clast size distributions.
 In this paper we employ novel thermal-elastic modeling to explore the role of partial melting and thermal fracture on modifying clasts. Our results show that these processes are particularly effective agents for change in clast size distribution, boundary roughness, and circularity. Small clasts are more susceptible to partial melting in a relatively short amount of time and Ds values quickly become perturbed with their removal. Partial melt and thermal fracture are most efficient for angular clasts with high surface area to volume ratios, and therefore reduce clast boundary roughness and increase clast circularity with continued exposure to the surrounding magma. A mature magmatic breccia therefore shows very different shape and size characteristics than its initial morphology, and careful analysis is required to evaluate its evolution.
 The Shatter Zone of Mount Desert Island, Maine, gives us a unique opportunity to study a spatial gradient of magmatic breccia development. Field observations and clast size distribution analysis suggest that the breccia initially formed by a volcanic reservoir explosion, but was quickly intruded by superheated magma. Clast size distribution, boundary roughness, and circularity data all suggest that magmatic intrusion caused partial removal of clasts with the highest degree of modification adjacent to the magma reservoir. Partial melt rims and late-stage cracking of clasts observed in the field support this conclusion.
 Magmatic breccias exposed in other localities can be analyzed using similar methods applied in this paper, and such analysis will further advance our understanding of the mechanics of active hydromagmatic systems.
 We gratefully acknowledge support from the National Science Foundation grants EAR- 0911150 and EAR-1118786. We thank Bob Wiebe for guidance in the field, and Bob Wiebe and Chris Gerbi for helpful comments on the evolving manuscript that led to improvements. We gratefully acknowledge David Okaya for discussions about non-dimensional heat transfer analysis. We thank Tom Blenkinsop for his journal review that led to additional improvements.