Energetics of glass fragmentation: Experiments on synthetic and natural glasses

Authors


Abstract

[1] Natural silicate glasses are an essential component of many volcanic rock types including coherent and pyroclastic rocks; they span a wide range of compositions, occur in diverse environments, and form under a variety of pressure-temperature conditions. In subsurface volcanic environments (e.g., conduits and feeders), melts intersect the thermodynamically defined glass transition temperature to form glasses at elevated confining pressures and under differential stresses. We present a series of room temperature experiments designed to explore the fundamental mechanical and fragmentation behavior of natural (obsidian) and synthetic glasses (Pyrex™) under confining pressures of 0.1–100 MPa. In each experiment, glass cores are driven to brittle failure under compressive triaxial stress. Analysis of the load-displacement response curves is used to quantify the storage of energy in samples prior to failure, the (brittle) release of elastic energy at failure, and the residual energy stored in the post-failure material. We then establish a relationship between the energy density within the sample at failure and the grain-size distributions (D-values) of the experimental products. The relationship between D-values and energy density for compressive fragmentation is significantly different from relationships established by previous workers for decompressive fragmentation. Compressive fragmentation is found to have lower fragmentation efficiency than fragmentation through decompression (i.e., a smaller change in D-value with increasing energy density). We further show that the stress storage capacity of natural glasses can be enhanced (approaching synthetic glasses) through heat treatment.

1. Introduction

[2] Natural glasses form in diverse environments and under a wide range of physical conditions including under elevated confining pressures and differential stresses. Examples of glass formation at elevated confining pressures include pseudotachylites, or frictional melts, resulting from high strain rate faulting of rocks [Kendrick et al., 2012; Lavallée et al., 2012; Riller et al., 2010; Sibson, 1977], volcanic dykes [Noguchi et al., 2008], and volcanic conduits [Soriano et al., 2009]. Glass formation in the above scenarios is linked to cooling of the involved melts at rates faster than crystallization and the confining pressure is elevated as a result of the overburden of the host rock (pseudotachylites and volcanic dykes) or the weight of the deposit itself (volcanic conduits).

[3] The thermal and stress history of a melt, as it crosses over the glass transition, is recorded in the internal structure of the glass [Wilding et al., 1995] which, in turn, ultimately controls the glasses mechanical behavior. Understanding the elastic, mechanical properties of glasses under geologically relevant confining pressures may allow for more sophisticated modeling of conduit processes (e.g., dome emplacement) and the brittle disintegration of melts at and near their viscous relaxation limit.

[4] The elastic properties and fragmentation behavior of glasses at atmospheric pressures have been explored extensively [Hyodo and Kimura, 1973; Tandon and Cook, 1993; Wiederhorn, 1969; Wilantewicz and Varner, 2007]. Glasses have also been studied as materials used in military armoring. The testing done in this field involves high strain rates and very high confining pressures; up to 1 GPa [e.g., Dannemann et al., 2008 and references therein]. Studies investigating the deformation behavior of glasses under confining pressures similar to those found in volcanic conduits and the shallow crust [e.g. Ougier-Simonin et al., 2011] are surprisingly rare.

[5] Here, we present results from a series of room temperature deformation experiments designed to explore the mechanical behavior of both natural (obsidian) and synthetic (Pyrex) glass under confining pressures of 0.1–100 MPa. In each experiment, glass cylinders are driven to brittle failure under compressive triaxial stress and a constant displacement and stress rate. We use the elastic and brittle response curves to quantify the storage and release of elastic energy. We conclude with a smaller set of parallel experiments on cores of heat-treated natural obsidian to investigate the effects of heat treatment on the mechanical properties of natural silicic glasses. These are aimed at gaining insight into the potential changes in the deformation behavior of glasses formed under highly dynamic geologic processes such as volcanic eruptions. The experimental products are analyzed for their grain size distributions to explore the degree of fragmentation attending elastic failure. Our results suggest that the particle size distributions of fragmented sample cores are directly linked to the energy density within the sample core at the point of failure.

2. Experimental Materials

[6] Our experimental campaign investigates the brittle deformation of glassy materials based on samples of synthetic borosilicate (Pyrex) and natural obsidian glasses (Figure 1). We also ran experiments on the same natural obsidian glass following heat treatment. The physical and chemical properties of all sample materials are reported in Tables 1 and 2, respectively.

Figure 1.

Experimental starting materials. Cores (40.6 × 20.3 mm) of: (A) Pyrex glass and (B) Newberry obsidian; horizontal bands are flow banding. Electron backscatter images of Newberry obsidian (C) natural and (D) 660°C heat treated; the heat-treatment process does not affect core geometry and does not induce crystallization or vesiculation. Dark and light spots are vesicles and iron oxides, respectively.

Table 1. Properties of Starting Materials, Including Seismic Wave Velocitiesa
Sample NamePyrexNewberryNewberry 660
  1. a

    Density and porosity were measured using He-pycnometry.

Diameter (mm)19.90 ± 0.03120.14 ± 0.02320.1 ± 0.028
Length (mm)40.59 ± 0.03540.69 ± 0.02140.69 ± 0.054
Density (g/cm3)2.232.372.37
Porosity (%)0<2<2
Vp (km/s)5.65.525.54
Vs (km/s)3.293.333.39
Table 2. Chemical Composition of Newberry Obsidian Measured by Electron Microprobe and Pyrex as Reported by Manufacturer
Newberry ObsidianWt%PyrexWt%
SiO273.66SiO280.60
TiO20.24B2O313.00
Al2O313.77Na2O4.00
FeO(T)2.10Al2O32.30
MnO0.07Trace0.10
MgO0.12  
CaO0.80  
Na2O4.95  
K2O4.22  
P2O50.06  

2.1. Pyrex

[7] The synthetic glass deformed in this study is Pyrex® 7740, purchased from Specialty Glass Products (SGP) (Figure 1). It is a crystal free, homogenous borosilicate glass with no porosity. The glass rods were fire polished and annealed by the manufacturer. Any internally stored stresses were reduced to a minimum through the annealing process and the surface of the glass rods is assumed to be fracture free through fire polishing. The glass is reported to have a Young's modulus of 64 GPa and a Poisson's ratio of 0.2 according to the material datasheet provided by the manufacturer. Cores of Pyrex (20.3 mm diameter and 40.6 mm length) were cut and the sample ends were ground plane parallel using a high-precision (0.025 mm) grinder to avoid uneven stress concentrations at the sample ends.

2.2. Newberry Obsidian

[8] Natural glass samples (i.e., obsidian) derive from the big obsidian flow [Castro et al., 2002; Fink, 1983; Laidley and McKay, 1971] in Newberry, Oregon, USA (Figure 1B). The obsidian is flow banded on a millimeter scale and contains <1 vol % of iron oxide microlites and <2% vesicles, with no connected porosity (Figures 1C and 1D). Sample cores (20.3 mm in diameter and 40.6 mm in length) were drilled perpendicular to flow banding, in order to minimize the potential influence of flow banding on the deformation behavior of the core. The sample ends of all cores were ground plane parallel as described above. A suite of Newberry obsidian cores was reserved for a heat-treatment process (see below) designed to release elastic stresses potentially stored within the obsidian prior to the deformation experiments.

3. Experimental Methods

3.1. Heat Treatment of the Obsidian

[9] Samples of natural obsidian were placed in a Nabertherm box furnace at 660°C for 24 h and then cooled to room temperature at a rate of 5° per minute. No noticeable crystallization occurred during the heat-treatment procedure (Figures 1C and 1D). The temperature of 660°C was chosen to be at the strain point (η = 1013.5 Pa s) of this material where most stresses stored in the glass are expected to be released on a several hours time scale [Vogel, 1994]. The strain point was estimated using a viscosity calculator (http://www.eos.ubc.ca/∼krussell/VISCOSITY/grdViscosity.html) based on the predictive model developed by Giordano et al. [2008]. Slow cooling rates were chosen in order to minimize the storage of thermally induced stresses in the heat-treated glasses. The bulk chemical composition of the Newberry obsidian (Table 2) was measured using a Cameca SX-50 electron microprobe at the University of British Columbia. A defocused, low energy (20 µ, 15 nA) beam was chosen in order to avoid alkali loss and the alkali composition was determined before all other elements in order to minimize any potential influence of alkali loss in the analytical result. The recovered composition is in good agreement with previous X-ray spectroscopy measurement of the Newberry obsidian composition [Gardner et al., 1998].

3.2. Deformation Experiments

[10] The deformation experiments were carried out in a triaxial rock deformation rig (“Large Sample Rig”; LSR) located at the University of British Columbia (Figure 2). The LSR is a modified version of the triaxial rock press used at Texas A&M University [Handin et al., 1972] and is described in detail in Austin et al. [2005]. The samples used in the deformation experiments are ∼20 × 40 mm, right circular cylinders or cores (Figure 1). Their ends have been ground plane parallel using a high-precision grinder (0.025 mm) to avoid uneven stress concentrations along the sample ends. Hardened steel spacers are placed between the sample and each of the upper and lower pistons and all surfaces are coated with Molykote P37 antiseize paste to reduce friction and edge effects during deformation.

Figure 2.

Schematic diagram of the experimental apparatus used in the deformation experiments. (A) Schematic of Large Sample Rig apparatus (LSR) [after Austin et al., 2005] showing locations of the external load cell (LC) and displacement transducer (DCDT). (B) Detailed view of the confining pressure vessel housing the sample assembly (Argon gas is used as the confining medium). (C) Expanded view of the sample assembly comprising hardened H13 steel pistons bracketing sample core (grey).

[11] Experiments on synthetic and natural glasses were carried out at a constant displacement rate of 40 µ s−1 corresponding to strain rates of ∼1.5 × 10−4 s−1. The stress rates determined from the recorded displacement (or strain rates) prior to the generation of micro fractures (pre-failure stress drops) are constant and homogenous for all experiments (see linear and consistent slopes of all deformation curves in Figure 3). Experiments were run at confining pressures of 0.1, 15, 25, 50, 75, and 100 MPa. Confining pressure is applied using compressed Argon gas. A summary of all experiments and experimental conditions is given in Table 3.

Figure 3.

Summary of results from rock deformation experiments plotted as differential stress (MPa) versus % strain: (A) Pyrex cores show two deformation responses: (1) elastic loading with little to no stress drops prior to failure (solid lines) or (2) Multiple stress drops (<190 MPa, dashed lines) as precursors to final failure. Catastrophic failure (i.e., complete release of over 60% of the induced stress) occurs at the end of all experiments and deformation was stopped immediately. All Pyrex samples display similar stiffness prior to the first fracture events. (B) Natural obsidian shows the same two deformation responses as seen in Pyrex. (C) Heat-treated (660°C) cores of obsidian show a uniform elastic response and lack precursor stress drops prior to failure. Failure occurs at higher peak stresses than observed in natural Newberry obsidian. (D) Summary plot of peak stress versus confining pressure (Pc) for all experiments; there is little correlation between Pc and peak strength at confining pressures above 15 MPa. Experiments labeled with a star represent experiments that showed pre-failure stress drops.

Table 3. Summary of the Experimental Data Including Smallest Particle Size Created and Surface Area of the Fragmental Products
Sample NameConfining Pressure (Mpa)Peak Stress (MPa)Stress Drop (Mpa)Postfailure Strength (MPa)Total Energy (J)Energy Density (MJ/m3)D ValueMin. Particle Size (μm)Surface Area (m2)Pre-failure Stressdrops
Pyrex 1251277125126148.511.71−2.460.83127.52No
Pyrex 2501046842204145.711.55−2.420.8388.16Yes
Pyrex 3751051756295161.312.76−2.390.8359.32Yes
Pyrex 41001293761532179.914.13−2.430.8354.88Yes
Pyrex 6151389137218146.311.59−2.430.83184.22No
Pyrex 758358251075.55.99−2.380.8397.66Yes
Pyrex 80.1648645352.34.13−2.320.9567.90Yes
Newberry a25112910636694.47.29−2.400.8387.16No
Newberry c159579203766.34.73−2.380.8373.96No
Newberry d53813592214.71.14−2.240.9513.23Yes
Newberry 660 a251137100713175.45.84−2.400.8378.73No
Newberry 660 b501223104118291.27.82−2.390.8366.39No
Newberry 660 c58308151544.43.42−2.320.8375.23No

3.3. SEM Analysis

[12] Scanning electron microscope (SEM) analysis used a Philips XL-30 scanning electron microscope at the University of British Columbia. SEM analysis was aimed at examining variations in microstructures resulting from different styles of elastic loading prior to failure or found in different starting materials (e.g., natural versus synthetic glass). The SEM analysis also included experimental products of the heat-treated obsidian in order to assess any impact of the heat-treatment process (vesiculation, crystallization) on the experimental material. The post-experiment sample cores were impregnated in a low-viscosity epoxy resin and carefully sectioned using a Buehler Ltd. Isomet 11–1180 low-speed high-precision rock saw at a speed of 120 rpm. The recovered tabs were then polished, carbon coated, and analyzed.

3.4. Particle Size Distribution (PSD) Analysis

[13] The analysis of particle size distributions is a widely used tool for characterization of fragmental deposits and the analysis of the fragmentation mechanisms producing these deposits [Chester et al., 2005; Kennedy et al., 2009; Kueppers et al., 2006; Wilson et al., 2005]. During failure of the sample cores, a pervasive fracture network was developed. All particles associated with the development of this fracture network were consistently smaller than 8 mm in diameter. Post-experiment products were carefully removed from their polyolefin jacket and sieved to fractions of >5 mm >2 mm >1 mm >500 μm >250 μm, and <250 μm. We consistently recovered more than 99% of the mass of the initial sample weight from each experiment. Each sieve fraction >250 μm was weighed, and these weights were converted to number of particles per size fraction using the material density and an assumed average particle shape. All size fractions were kept in a desiccator for a minimum of 24 h prior to weighing.

[14] SEM analysis reveals that the particles created during deformation are generally equant and angular in shape (i.e., cube shaped; Figure 4). An aliquot of the sieve fraction <250μm was analyzed in an optical laser particle size analyzer with a measurement range from 0.01 to 10.000 µm (Mastersizer 2000; Faculty of Pharmacy at the University of British Columbia). The sample was mixed with a dispersal agent (tween) and stirred vigorously for 5 min using the Mastersizer sample dispersal unit in order to ensure break up of any agglutinated particles and complete dispersion of the sample. The data were recorded as particle volume percent for each size fraction and then converted to number of particles per size fraction. The individual PSD data sets were plotted as particle size versus number percent particles to which a power law (i.e., log linear) model was fitted to recover the slope (D-value) and intercept (λ-value) of the individual particle size distributions.

Figure 4.

Representative experimental products for Pyrex and obsidian. (A) Pyrex core from deformation path that had precursor stress drops before failure (cf. Figure 3A) dashed lines delineate the outline of the shear zone developed during failure. (B) Pyrex core result for which there were no stressdrops prior to failure. (C) Obsidian core from deformation paths that had precursor stress drops versus (D) obsidian core that experienced elastic failure without precursor stress drops (cf. Figure 3B). (E) Shear fracture developed during deformation of heat-treated obsidian core. (F) SEM image of the fracture network developed within shear fractures. Note the vertically oriented Mode I fractures outside the shear zone and the rotation of blocks containing Mode I fractures within the shear zone, indicating the formation of Mode I fractures prior to the development of the shear zone. (G and H) SEM images of Mode I fractures (σ1 vertical) prior to failure within a deformed Pyrex core, and within a heat-treated core of obsidian, respectively; both samples show little grain production attending development of the Mode 1 fractures.

4. Results

4.1. Deformation Experiments

[15] The mechanical data for all deformation experiments (Table 3) are illustrated as stress (MPa) versus strain (%) graphs for each material (Figure 3). Representative images of experimental products are shown in Figure 4. All peak stresses are reported as differential stress (σ1σ3) (Table 3). All experiments show a near-linear relationship between stress and strain before failure: that is, samples did not exhibit ductility before wholesale failure by shear faulting (Figure 3).

[16] The constant displacement rate experiments display two distinct loading responses of the glasses (Figure 3). Some samples display constant loading with little to no pre-failure stress drops (solid lines in Figure 3). Other samples are characterized by elastic loading curves that feature a multitude of small (10 to 190 MPa) stress drops prior to failure (dashed lines in Figure 3). The deformation experiments on samples of Pyrex glass and natural obsidian display both failure types whereas the experiments performed on heat-treated obsidian show no precursor stress drops (Figure 3C).

[17] Samples that show continuous loading with no precursor stress drops consistently reached significantly higher peak stresses (σPeak ∼ 1000–1300 MPa) and, at failure, released most of their stored elastic energy. Samples without pre-failure stress drops show very low post-fragmentation strengths (e.g., 18–66 MPa; Figure 7 and Table 3). Conversely, experiments characterized by multiple small stress drops along the load-displacement curve reach higher post-failure strengths compared to experiments with no precursor stress drops. The magnitude and frequency of the pre-failure stress drops increased with increasing load and wholesale sample failure occurred at lower peak stresses relative to samples that did not show precursor failure.

4.2. Laser Particle Size Analysis (LPSA)

[18] Results derived from particle size distribution (PSD) analysis are summarized as the number (%) of particles versus particle size (Figure 5A). The modeling procedure used to recover the fit parameters (i.e., D-value and λ) is shown as a subplot (Figure 5B). The D-value is the slope of the linear part of the particle size distribution data in log–log space (i.e., assumed power law relationship). The λ value is the y-intercept to the linear model.

Figure 5.

Results of laser particle size analysis of experimental products. (A) Particle size distributions (PSD) are plotted as number percent of grains (N%) versus grain size. (B) Inset showing an example of a power law fit applied to the linear portion of PSD data set (see text). (C) Inset showing the range of D-value (slope) and λ-values (intercept); 1σ confidence ellipses illustrate the strong correlation between linear model values of D and λ (see text). Neither the coarse nor the fine end of the data population are included in the fit since they are misrepresented as a result of the laser PSD analysis limits.

[19] The D-value can be used as a measure of the ratio between coarse and fine particles. The smaller the D-value, the more fine particles were created relative to coarse particles. It can, therefore, be used as a measure of fragmentation efficiency. The recovered D-values range from −2.24 to −2.46. Selected model values are plotted with their 1σ confidence ellipses in Figure 5C. The results of all experiments are remarkably similar. However, the confidence envelopes show a systematic variation in parameters that is greater than the experimental uncertainty indicating that the measured variations in D-value can be related to process. The size of the smallest particles created during fragmentation is almost identical in all experiments ranging from 0.832 to 0.955 µ (Table 3). These values are about 1 order of magnitude higher than the detection limit of the Mastersizer.

[20] The stress drop values reported in Table 3 represent the magnitude of the final stress drop that accompanied sample failure. Figure 6 shows the evolution of the D-value and the surface area created with increasing magnitude of the stress drop during failure. The D-value decreases systematically (Figure 6A) indicating that at higher stress drops more fine-grained material is produced. This is further reflected in the exponential increase in the surface area created during failure (Figure 6B). Experiments departing from this trend are ones that showed numerous precursor stress drops prior to failure. These experiments produce more fine-grained material than the trend suggests for the remaining samples. This is likely a result of the production of fine particles during pre-failure stress drops, which may produce excess particles relative to samples failing without pre-failure stress drops.

Figure 6.

Relationship between particle size distribution and the magnitude of the stress drop at failure. (A) Summary plot of D-value versus stress drop for each of experiment showing the decrease in D-value with increasing stress drop reflecting increased production of fine-grained material. The heat-treated obsidian has lower values of D than natural obsidian because it supports higher stress drops. A departure from this trend is observed for some experiments on Pyrex and is only present for experiments displaying pre-failure stress drops. We postulate that the production of particles during pre-failure stress drops results in an excess of particles relative to samples failing without pre-failure stress drops. (B) The relationship between observed stress drop and post-failure surface area calculated from PSD data shows an exponential increase in surface area with increasing magnitude of stress drops. Particles are approximated as cube shapes based on the clast shapes observed in SEM analysis.

Figure 7.

Plot of peak stress versus stress drop for all experiments. Numbers denote confining pressure (MPa) of each experiment. Samples plotting along a 1:1 line have released all elastic energy previously stored within the sample. All experiments showed some departure from this trend line; the magnitude of the excursion indicates the capacity of the post-failure material to store elastic energy. Shallow slopes indicate a high energy-storage potential whereas steeper slopes indicate low energy storage post-fragmentation. Natural Newberry obsidian experiments define a line of slope 1 (dotted), suggesting little capability to store elastic energy after failure. When heat treated at 660°C the Newberry obsidian shows an increased ability to store elastic energy (slope decreases to 0.56; dash dotted) and generally elevated peak stresses. Pyrex displays twofold behavior; some samples released all elastic energy and other samples define a trend (slope of 0.14; dashed) consistent with a very high potential for energy storage. Note that at pressures above 15 MPa, the confining pressure has little influence on the peak stress of the materials tested (see Figure 3D).

4.3. Meso and Microstructure

[21] Mesoscopic failure is expressed either by the formation of shear faults oriented 30°–40° to the direction of σ1 or as large extensional fractures, that are typically crosscut by conjugate shear fractures (Figures 4A–4E).

[22] Based on crosscutting relationships we find that core axis parallel extensional (Mode I) fractures form prior to the shear fractures (e.g., Figure 4E). Conjugate shear fractures formed in samples that did not undergo precursor stress drops (e.g., Figures 4B and 4D). Conversely, samples that underwent precursor stress drops failed by one or more shear faults but not as conjugates (Figures 4A, 4C, and 4E). We did not observe a transition from brittle to ductile deformation behavior. This is underlined by the presence of large stress drops accompanying failure, rather than the decrease in stress build up or plateauing of stress with continuous deformation as expected for ductile behavior.

[23] Experiments that showed linear loading prior to failure and failed at high peak stresses commonly featured widely distributed damage zones with many longitudinal (Mode I) fractures and conjugate shear fractures (Figures 4B and 4D). Conversely, experiments that had multiple stress drops prior to failure and failed at lower overall values of peak stress showed more localized damage zones (see shear fractures in Figures 4A and 4C). Also, samples featuring pre-failure stress drops show increasing peak stresses with increasing Pc, as anticipated for samples that contain cracks and therefore porosity [e.g., Paterson and Wong, 2005].

[24] In order to investigate the nature of the pre-failure stress drops described above we loaded a sample of Pyrex until one of the pre-failure stress drops occurred (35 MPa), continued loading until the core had just regained strength past the pre-failure stress drop and then unloaded the sample and recovered the sample for textural analysis. The loading after the stress drop was performed to verify that the stress drop did not occur at peak stress and was truly representative of a pre-failure stress drop. The recovered sample displayed three distinct Mode I fractures (Figure 4G) along the longitudinal axis of the sample core and some randomly distributed fractures near the sample ends which are likely the result of edge effects.

[25] SEM analysis of samples of both failure types, as well as the sample recovered after one precursor stress drop, shows a similar evolution of the fracture network with loading. Extensional fractures, (Mode I cracks, Figure 4) that form parallel to σ1 (highest compressive stress), along the long axis of the sample core are the first fractures to develop. Mode I fractures form during pre-failure stress drops, as they are the only fracture type present in the sample that was not allowed to reach catastrophic failure. Fracture tips of Mode I cracks are curved at their termination. Joining of these, long Mode I fractures with smaller, oblique oriented, microcracks lead to the formation of elongate rectangular fragments (Figures 4F–4H). Upon the development of a large shear fracture, these fragments are rotated and comminuted to form angular, blocky fragments of variable size (Figure 4F). With higher stresses these fractures then start to communicate and, ultimately, a shear fracture (Figure 4F) develops within the sample and failure occurs. This development is commonly observed for materials under compression and has been observed in a wide range of different materials [e.g., Bell et al., 2011; Dannemann et al., 2008; Ougier-Simonin et al., 2011]. During displacement along the shear plane, blocks of material containing the previously developed Mode I fabric are rotated and comminuted (see Figure 4F). These textures are present in samples of both deformation styles, when taken to ultimate failure, and are independent of the sample stiffness or material.

5. Discussion

5.1. Influence of Confining Pressure and Heat on Glass Deformation

[26] In fracture theory, the concentration and length of initial flaws within rocks are the principal parameters controlling their deformational behavior. These flaws can be, for example, grain boundaries or pre-existing microcracks [Ashby and Sammis, 1990; Griffith, 1921; Horii and Nemat-Nasser, 1985]. Most natural materials (e.g., rocks) have grain boundaries, void space (porosity) and microcracks. It is these natural flaws that greatly affect brittle rock deformation as, in most rocks, cracks will originate at and propagate from any inhomogeneity available (pores, inclusions, microcracks, surface scratches [Ashby and Sammis, 1990]).

[27] Under tension, a single crack will grow unstably and transect the sample leading to failure [Gurney, 1948]. In contrast, under compression, small cracks grow stably, and grow longer with increasing applied stress until they interact and coalesce to give way to final failure [cf. Paterson and Wong, 2005]. Extensional fractures propagate parallel to the maximum stress direction (σ1); these fractures open parallel to the direction of minimum stress (σ3). The overall effect of confining pressure in such scenarios is to hinder crack growth [Ohnaka, 1973; Wong, 1982]. Increasing load stress is then required for new smaller cracks to nucleate and propagate. These cracks will eventually interact and coalesce creating new microcracks that will lead to macroscopic shear failure [e.g., Lockner and Byerlee, 1977; Nemat-Nasser and Horii, 1982].

[28] The propagation of fractures is further dependent on the material's stiffness and surface energy as well as the length and aperture of the initial fracture, because these control the stress regime at the crack tip [Bieniawski, 1967]. It is easier for extensional fractures to propagate under low confining stresses than at elevated confining stresses; higher confining pressures operate to reduce stresses at the crack tip [Brace and Bombolakis, 1963]. At low confining pressures, samples having microcracks (e.g., initial flaws) generally fail by developing large Mode I fractures parallel to σ1. Under elevated confinement the propagation of these fractures is hindered and samples fail through coalescence of small Mode I micro fractures, forming a shear plane. This effect of confining pressure results in the classic coulomb failure envelope that, for most materials, shows increasing failure strength with increasing confining pressure [Lockner and Byerlee, 1977; Ohnaka, 1973]. This behavior reflects the effects of confining pressure on the stresses at the crack tip of microcracks as outlined above.

[29] Our glass samples contained no crystals (Pyrex) or negligible crystals (obsidian), have no grain boundaries, and lack porosity. We did not observe fractures emanating from microlites or porosity and therefore we assume that the effect of microlites and porosity in the natural samples is small. Furthermore, the Pyrex samples were annealed and fire polished and are, thus, virtually free of any initial fractures (flaws) except, perhaps on the sample ends. We suggest that the absence of grain boundaries or other pre-existing “flaws” such as microcracks within the sample may be an explanation for the two failure types observed during the experiments. Experiments featuring continuous linear elastic loading curves without pre-failure stress drops appear unaffected by Pc; even though Pc is increased, the peak stress does not increase. Conversely, samples featuring pre-failure stress drops show increasing peak stress with increasing Pc, as anticipated for samples containing cracks and, thus, pore space [e.g. Ohnaka, 1973; Paterson and Wong, 2005; Wong, 1982]. We interpret this to be due to the absence of micro fractures upon which the Pc could act. Samples failing after linear loading develop very long, axial Mode I fractures prior to the development of conjugate shear zones and ultimate failure [Gurney, 1948]. Upon failure, they release almost all energy introduced to the sample during loading. Samples that experienced precursor stress drops develop micro fractures during these stress drops, which ultimately coalesce, to form shear fractures (30°–40° from the direction of σ1). These samples do not release all energy introduced through loading.

[30] We propose that the fractures developed during the pre-failure stress drops create crack/pore space within the sample and thereby allow for Pc to affect the fracture propagation within the samples following the Mohr-Coulomb failure criterion, showing increased compressive strengths with increasing confining pressures [Handin, 1969; Labuz and Zang, 2012; Ohnaka, 1973]. Such deformation behavior is well known from published literature [e.g. Brace and Bombolakis, 1963; Labuz and Zang, 2012; Paterson and Wong, 2005]. If, nonetheless, a sample displays linear loading there are no precursor fractures created that could be influenced by confining pressure, allowing the samples to behave as though there were no confinement present (developing large axial Mode I fractures rather than small coalescing micro fractures prior to ultimate failure).

[31] The heat treatment of obsidian cores resulted in increased peak stresses (e.g., increase from ∼400 to ∼800 MPa at 5 MPa confinement) and post-failure strengths (e.g., ∼60 to ∼130 MPa at 25 MPa confinement). The deformation behavior changed such that no pre-failure stress drops were observed in experiments on heat-treated obsidian.

[32] Figure 7 summarizes the energetics of fragmentation by plotting the peak differential stress (σ1 − σ3) versus the stress drop. Samples plotting along a line with unit slope (M = 1) have released all elastic strain energy stored within the sample prior to failure. All materials tested in this study plot below the model curve for complete energy release and define several trends (Figure 7). The slopes of the trends defined by the experimental data set are a measure of the material's capacity to store elastic strain energy after failure. Shallow slopes indicate a high energy-storage potential whereas steeper slopes indicate low energy storage, post-failure. The energy release during failure appears to be independent of the applied confining pressure at confining pressures higher than 15 MPa (see inset Figure 7). This suggests that the trends shown in Figure 7 are a material-dependent property rather than a response to the experimental conditions imposed upon the sample. The relationship between peak stress and the magnitude of the stress drop is an indirect measure for the capability of a material to retain elastic strain energy after fragmentation.

[33] Figure 7 shows the evolution of energy retention for natural obsidian with heat treatment and the energy retention capabilities of Pyrex. Natural Newberry obsidian plots close to the complete energy release with a slope of 1, indicating that it has little capacity for storing elastic energy after failure. When heat treated at 660°C, the Newberry obsidian fails at elevated peak stress compared to nonheat-treated obsidian and shows an increased ability to store elastic energy (slope decreases to 0.56). Heat-treated obsidian consistently displays considerably higher post-failure strengths than natural obsidian (see Figure 7).

[34] Pyrex displays two different types of energy retention behavior. Some samples released virtually all elastic energy stored in the sample prior to failure. Conversely, other samples, plotting along a trend with a slope of 0.14 (Figure 7), show the highest potential for post-failure storage of elastic energy. The twofold nature of the energy budgets for Pyrex is linked to: (1) deformation without precursor stress drops for the two samples showing the highest peak stresses (1389 and 1277 MPa) and energy release and (2) low confining pressures (0.1 and 5 MPa) for the two Pyrex samples of 648 and 835 MPa peak stress, respectively.

[35] We argue that the higher post-failure energy retention capacity of heat-treated glasses is a result of the release of stresses previously stored within the glass structure. The release of such stresses during the heat-treatment process allows the glass to now sustain higher post-fragmentation stresses. We suggest that these stress differences may reflect the stresses stored within the natural glass. In natural environments such stresses could arise by quenching of unrelaxed melts undergoing shear, or relatively rapid bubble nucleation and growth. If, for example, a melt is quenched and/or is experiencing deformation within the viscoelastic regime, the elastic component of that deformation can potentially be stored within the glass structure. The investigation of the purely elastic and brittle deformation of glasses, as performed in this study, provides crucial information regarding the behavior of melts that experience strain rates within and exceeding the viscoelastic deformation regime. The storage and dissipation of deformational energy in natural, viscoelastic systems have been modeled by Ichihara and Rubin [2010] who emphasize the importance of the brittle behavior (i.e., fracture) in flowing magma and describe the viscoelastic regime in which magmas can store elastic energy before fracture [Webb and Dingwell, 1990]. This energy retention capability of glassy materials may allow for the storage of shear stresses in glassy lava domes and, therefore, delay volcanic eruptions, as this elastic energy is no longer available for fragmenting the glass. During the next stress event, if the material had no opportunity to relax the stored stresses, it can now behave weaker and failure may occur earlier than expected.

5.2. Elastic Strain Energy and the Creation of Surface Area

[36] The particle sizes and size distributions of experimental and natural products have been adopted as indicators of the energy involved in fragmentation processes in both volcanic [Alatorre-Ibarguengoitia et al., 2010; Kaminski and Jaupart, 1998; Kueppers et al., 2006; Newhall and Self, 1982] and tectonic [Chester et al., 2005; Pittarello et al., 2008] systems, as well as, processes operating in magmatic systems [Bindeman, 2005]. In the following section, we will discuss the relationship between the elastic energy stored in the sample prior to fragmentation and the fragmented experimental product.

[37] Deformation of the sample core is recorded as a function of the load applied during constant displacement rate experiments. By treating the sample as an idealized spring, the energy stored in the sample can be calculated as a function of these two parameters (load and displacement) through integration of the area under the deformation curve [He et al., 2011] (Figure 8). This may be done using the following equation:

display math(1)

where f is the applied force and x is the shortening of the sample.

Figure 8.

Schematic representation of deformation paths for elastic loading, failure and three different unloading paths (dashed lines) as used to recover the energy budgets of compressive fragmentation. Unloading paths include experimental products having the same (a), less (b), or greater (c) stiffness than the starting material. Photographs are of cores: (1) immediately prior to, and (2) after failure. Following unloading path (a) the gray, solid area shows the elastic energy stored in the sample after failure, recovered through integrating the area under the unloading curve. The gray, stippled area, therefore, represents the energy used in the fragmentation process.

[38] After failure, samples commonly continue to have the capacity to sustain loads and to store elastic energy. This indicates that the energy introduced to the sample during loading is not released completely during the fragmentation process. The amount of energy stored in the sample post-failure can be determined by completely unloading the sample after fragmentation and, then, integrating the area under the unloading path (Figure 8). The difference between the total energy introduced into the sample and the energy stored in the sample after failure is the energy consumed during fragmentation. A similar procedure has been established in material science where the energy density of ductile bulk materials is recovered [He et al., 2011].

[39] Figure 8 shows a simplified deformation curve including loading path, stress drop and an unloading path (of the same stiffness as the sample along the loading path) in solid. The dashed lines marked a and b represent a less stiff and a more stiff response of the newly created material, respectively. Inserts 1 and 2 show the state of the material immediately prior to and after failure.

[40] The amount of elastic energy consumed by fragmentation in rock deformation experiments is a combination of the energies consumed by the creation of surfaces on new grains created during fracturing, frictional sliding of clasts, and frictional heating during sliding [Kanamori and Rivera, 2006; Pittarello et al., 2008]. The total surface area created during fragmentation was calculated by multiplying the number of particles per size bin by the surface area of a cube-shaped particle with a side length of the respective size bin. Cubic particles were chosen rather than spherical particles after evaluating the fragmentation pattern using SEM analysis. Figures 4F–4H shows the majority of particles being square and only small amounts of particles within the shear zone (Figure 4F) have undergone rounding.

[41] Loading of the sample during the experiment supplies the energy available for fragmentation. The amount of new particles created during fragmentation is a function of the energy released at failure. Figure 6A shows decreasing D-values with increasing stress drops (i.e., higher energy release at failure), meaning that there are more fine particles created with higher energy release at failure. Since, the D-value is an expression of the amount of fragmentation taking place during failure, the relationship between the stress drop (i.e., energy release) and D-value can give insights to the fragmentation efficiency of the sample. Interestingly, even though the D-values change considerably with energy release, the smallest particle created during fragmentation varies little (between 0.83 and 0.95 µ for all experiments; see Table 3). The relationship between the stress drop (i.e., energy release) and the surface area created during fragmentation is shown in Figure 6B. The surface area created is directly linked to the amount of energy that was released during failure; higher energy release results in larger surface areas.

5.3. Fragmentation Efficiency: Compression Versus Decompression

[42] On the basis of the results described above we discuss the differences in fragmentation energy budgets between compression induced fragmentation (this study and Kennedy and Russell [2012]), and fragmentation resulting from decompression [Alatorre-Ibarguengoitia et al., 2011; Kueppers et al., 2006].

[43] The decompression fragmentation experiments were carried out on porous, glass bearing volcanic rock samples from Unzen (Japan) and Popocatepetl (Mexico) at high temperatures and data are summarized in [Alatorre-Ibarguengoitia et al., 2010]. During these experiments the samples are subjected to gas pressure whilst being housed in an autoclave. The pressure is allowed to equilibrate between the autoclave and the open pore space within the sample. The pressure is then released instantaneously (through opening of diaphragms) and the energy stored in the compressed gas within the sample pore space is made available for fragmentation of the sample. Alatorre-Ibarguengoitia et al. [2010] report the evolution of the fragmentation threshold to increase quasi logarithmically with decreasing porosity up to a maximum of about 30 MPa at 5% open porosity. The energy density available for the decompressive fragmentation experiments performed by Alatorre-Ibarguengoitia et al. [2010] is recovered by calculating the energy stored in the compressed gas within the pores of the samples (using an approach developed by Alidibirov [1994]; see references in Alatorre-Ibarguengoitia et al. [2010]) and relating this energy to the sample volume.

[44] The samples deformed in this study have little (natural glasses) to no (synthetic glasses) porosity (see Table 1), in contrast to the samples described in Alatorre-Ibarguengoitia et al. [2010]. The fragmentation efficiency of both processes can be evaluated by plotting the D-value versus the energy density for samples fragmented via decompression (Figure 9B) versus compression (Figure 9A). Figure 9C shows a compilation plot of both compressive and decompressive fragmentation experiments; we have also included data on the compressive fragmentation of lavas from the 2004–2008 eruption of Mt. St. Helens [Kennedy and Russell, 2012].

Figure 9.

Comparison of fragmentation efficiency of compressive and decompressive fragmentation. (A) Plot of D-value versus energy density for the glasses deformed in this study. Both fragmentation types produce lower D-values with increasing energy density (energy stored in the sample at the time of failure). (B) Compilation of PSD analysis on samples for a range of volcanic rock types fragmented through rapid decompression experiments [Alatorre-Ibarguengoitia et al., 2010]. Decompressive fragmentation shows a dependence of D-value on both, energy density and porosity (Alatorre-Ibarguengoitia, 2012. pers. comm.). (C) Summary plot of experimental results from both compressive experiments on glasses (this study), natural lavas from Mt. St. Helens [Kennedy and Russell, 2012] and the decompression experiments on natural lavas (Alatorre, personal communication). Decompression fragmentation is less efficient in creating fine particles than compressive fragmentation at low energy densities, whereas at high energy densities decompressive fragmentation is more efficient. The porosity dependence of the samples deformed through decompressive fragmentation is negligible (Alatorre-Ibarguengoitia, 2012. pers. comm.) for the glasses deformed in this study due to their low porosities.

[45] At low energy densities compressive fragmentation creates fine particles more efficiently, resulting in lower D-values (Figure 9C). At high energy densities the materials fragmented under compression lag in the creation of fine particles, resulting in larger D-values (Figure 9C). At intermediate energy densities both compression and decompression are equally efficient. From this data comparison we suggest that, at high energy densities, compressive fragmentation is less efficient at creating surfaces than fragmentation introduced by rapid decompression of pressurized pores. This demonstrates that the calculation of energy budgets for compressive (i.e., shear)-fragmentation, and decompression-fragmentation are inherently different. Therefore, the evaluation of the energetics of faulting (e.g., earthquakes) and volcanic eruptions needs to be treated differently. One implication is that the fractal dimension of fragmented material can only be related to fragmentation energies [e.g. Chester et al., 2005; Kueppers et al., 2006] when the failure mechanisms are understood.

[46] Both types of experiments have inherently different stress rates resulting from the differences in experimental methodology for decompression fragmentation and compressive fragmentation experiments. We therefore think that each approach produces unique data sets applicable to different stages of an eruption. The decompression fragmentation experiments pertain to most magmatic gas-driven eruptions where the rapid expansion of gas-filled bubbles exceed the relaxation time scale of the melt causing brittle fragmentation of the erupting magma. The compressive fragmentation experiments are pertinent to the fracture of magmas experiencing strain rates exceeding their viscous relaxation time scale in subsurface conduits and during extrusion of lava domes. Circular faults developing along the conduit margins during the extrusion of lava domes create particles through fragmentation in a shear regime and the resulting particle size distributions will therefore follow the energy efficiency trend of compressive fragmentation. It is therefore crucial, when estimating the energy involved in the generation of volcanic ash, that the fragmentation process involved in the generation of these ash particles is identified.

6. Conclusions

[47] 1. Heat treatment of obsidian affects the mechanical properties of natural glass; it reduces the potential for pre-failure fracture growth and allows for higher post-fragmentation strengths than natural obsidian. We therefore postulate that the cooling history locked in a glass when crossing the glass transition plays a significant role in shaping the mechanical properties of the resulting rock unit.

[48] 2. The grain size distribution of fragmented glassy materials is a direct reflection of the energy release during failure. Fragmented glassy materials can display different post-fragmentation strengths. The mechanical properties of the materials post-fragmentation are the result of a combination of factors such as confining pressure and particle size distribution.

[49] 3. Materials fragmented through compression display a different evolution of fragmentation efficiency (changes in D-value with increasing energy density) than materials fragmented through decompression. Compressive fragmentation is found to be more efficient at grain size reduction when the process is operating at a low energy density whereas decompressive fragmentation is more efficient at higher energy densities. We suggest that this is a result of the materials being fragmented in a purely tensile (decompression) versus an extensile (compression) stress regime. Both processes are present in volcanic eruptions; we therefore argue that any estimate of fragmentation energy from volcanic ash size distributions needs to be combined with an identification of the fragmentation mechanism.

Acknowledgments

[50] We thank Mark Davis for general discussions on the mechanical and elastic properties of glass and Lucy Porritt and Michelle Campbell for discussions related to data presentation. We thank two anonymous reviewers and G-cubed editor Cin-Ty Lee for their constructive comments and suggestions, leading to improvement of this manuscript. We further thank Doug Polson, Joern Unger, and David Jones from the EOAS staff that helped with maintaining the experimental equipment and programming. This research was funded through an NSERC Discovery grant held by J. K. Russell.

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