The tensile fracture of heterogeneous earth materials such as snow, ice, and rocks can be characterized by two fracture parameters—the fracture toughness and the fracture process zone length. The latter length scale characterizes the zone of microcracking surrounding a crack tip in a heterogeneous material. For alpine snow, these two fracture parameters influence the release dimensions and thus destructive potential of slab avalanches. In general, it is difficult to determine these parameters concurrently, and most experimental methods are based on first-order scaling laws that have considerable errors unless very large test specimens are used. Here we introduce a simple experimental method based on a higher-order quasi-brittle scaling law that has never been applied to snow nor any other geophysical material. We conducted hundreds of beam bending experiments using natural cohesive snow samples to produce the most comprehensive measurements to date of the tensile fracture toughness and effective process zone length of snow. We also adopt a new penetration resistance gauge to index the fracture toughness data, addressing a longstanding need for better proxy measurements to characterize snow structure. The peak penetration resistance met by a thin blade proved better than the bulk snow density for predicting fracture toughness, a finding that will improve field predictions and facilitate comparisons of results across studies. The tensile fracture process zone, previously a highly uncertain length scale related to avalanche fractures, is shown to be about 5–10 times the snow grain size, implying nonlinear fracture scaling for the majority of avalanches.
 Slab avalanches initiate with the failure of a weak layer or weak interface beneath a cohesive snow slab [McClung and Schweizer, 2006; Schweizer et al., 2003]. As this fracture propagates beneath the slab, tensile stresses are built up in the slab until it eventually fractures in tension [McClung, 1981]. Once the slab has fully fractured through its thickness in tension, the now isolated slab is free to accelerate downslope. The tensile fracture properties of the slab relative to the shear (or mixed-mode) fracture properties of the weak layer determine the release dimensions and thus destructive potential of the avalanche [McClung, 2009a]; therefore, measuring these properties is of fundamental importance to avalanche forecasting efforts.
 The fracture properties of the weak layer are commonly indexed using field tests which measure primarily the susceptibility of the layer to fracture or the propensity to continue propagating once a fracture is initiated [e.g., McClung, 2009b; Gauthier and Jamieson, 2008; Sigrist and Schweizer, 2007]. Such tests have rapidly gained acceptance in avalanche forecasting operations, though it is generally not possible to determine the tensile fracture properties of the overlying slab using these field tests. Instead, the tensile properties of the slab are commonly calculated from regression relations fit to independent experimental data using the mean density of the snow as the primary in situ index property [e.g., McClung, 2007; Sigrist et al., 2005; Shapiro et al., 1997]. Indeed, most mechanical properties of snow continue to be expressed solely as a function of density [e.g., Nakamura et al., 2010; Marshall and Johnson, 2009; Camponovo and Schweizer, 2001], even though it has long been recognized that density is a poor indicator of the mechanical integrity of a snow pack [e.g., Borstad and McClung, 2011b; Shapiro et al., 1997; Fukue, 1977; Mellor and Smith, 1996]. This often confounds the comparison of results from different studies due to (often unreported) differences in other important properties such as snow hardness, temperature, grain size, and strain rate.
 The uncertainty in calculated values of slab properties using only the density to index the mechanical state of the snow is compounded when considering the many experimental methods and fracture theories that have been applied to calculate tensile fracture properties from experimental data. Linear Elastic Fracture Mechanics (LEFM) was applied in early studies of the fracture toughness of snow [e.g., Kirchner et al., 2000; Schweizer et al., 2004], though it was later recognized that a large fracture process zone necessitates a nonlinear theory such as Quasi-Brittle Fracture Mechanics (QBFM) over most operative length scales for slab avalanches [Sigrist et al., 2005; Sigrist, 2006; McClung and Schweizer, 2006; McClung, 2009b; Borstad and McClung, 2011a]. This is consistent with the literature on other earth materials such as rocks [Bažant and Kazemi, 1990], sea ice [Dempsey et al., 1999], and glacier ice [Pralong and Funk, 2005] that also follow the nonlinear scaling relations of QBFMowing to their inherent heterogeneity. Nonetheless, for snow a large portion of the published tensile fracture data continue to be from studies that applied LEFM whether appropriate or not.
 There is thus considerable uncertainty and variability in the available tensile fracture data for snow slabs relevant to avalanches, which translates into high uncertainty in predicting how large and destructive an avalanche may be for a given slab depth. This uncertainty has implications for avalanche forecasting and hazard mapping in avalanche terrain. Improvements in both field-based methods and tools for indexing the mechanical state of snow and linking these index properties with greater confidence to laboratory fracture data—using more than just the snow density—would benefit the fields of snow mechanics, avalanche research, and operational avalanche forecasting.
 In this study, we determine the tensile fracture toughness and process zone length of dry cohesive snow using an experimental method which has never been applied to snow nor any other heterogeneous earth material. This method—the “zero-brittleness” method of Bažant and Li , originally developed for concrete—uses a higher-order scaling law than any previously applied to calculate fracture parameters for snow and has several experimental advantages over previous engineering approaches adapted to snow. We calculate the fracture toughness and process zone length with greater confidence than obtained in previous studies. We find that the process zone is slightly smaller than estimated previously, though not small enough to permit the direct application of LEFM. We demonstrate that a newly introduced measure of thin-blade penetration resistance is a better index measurement than density for fracture toughness, allowing for more consistent field indexing of the laboratory data and more objective comparison of results across studies.
 We begin with a review of literature on measurements and calculations of tensile fracture properties of snow, followed by the derivation of the quasi-brittle scaling law. The experimental methods are then described in detail, followed by a presentation of the results for each of the 11 test series conducted. An analysis of the resulting fracture parameters is then presented. We conclude with a discussion of the limitations and advantages of the zero-brittleness method and possible applications to other heterogeneous earth materials.
2 Tensile Fracture Properties of Snow Relevant to Avalanches
2.1 Experimental Determination
 The tensile strength of snow has been studied widely since as far back as the 1930s, when investigators in Switzerland began to measure the strength properties of snow, borrowing techniques from the soil sciences (Haefeli (1939), translated in Bader et al. ). However, it was not until recently that tensile fracture(rather than strength) properties were measured, using a variety of engineering test methods. The first calculations of tensile fracture toughness came from notched cantilever beam tests, in which beam-shaped samples of cohesive snow were cantilevered and a notch progressively cut into the top of the beam until the cantilever failed under gravitational load. Kirchner et al.  and Schweizer et al.  performed such tests and applied LEFM to calculate the fracture toughness as a function of the notch depth at failure. These early studies found fracture toughness (KIc) on the order of about 0.1–1 kPam1/2depending on the density of the snow, remarkably low values for a natural material.
 A method known as the “size effect” method, in which fracture specimens of the same geometry but different size are tested to failure and fit to a quasi-brittle scaling law [Bažant and Planas, 1998], was later adopted for snow. This method does not require an assumption about the applicability of LEFM, since LEFM is contained as an asymptotic limit of the quasi-brittle scaling law [Bažant, 2005]. Studies employing this method cast doubt on whether the cantilever beam samples (or any samples practical for laboratory testing) were large enough for direct applicability of LEFM [Sigrist et al., 2005; Sigrist, 2006; Borstad, 2011]. The fracture process zone in these studies was inferred to be on the order of 1–10 cm, a length scale which is not negligible for the tensile fractures in slab avalanches since the majority of avalanches have a slab thickness less than 1 m [Perla, 1977; McClung and Schaerer, 2006]. The fracture toughness calculated in these studies was on the order of 0.1–10 kPam1/2, somewhat higher than the earlier estimates. The higher KIcvalues from these studies are consistent with the physical interpretation of a large and diffuse fracture process zone as having a blunting influence on fracture [e.g., Cotterell and Mai, 1996].
 The size effect method has drawbacks when working with a fragile material such as snow, however. Extracting, transporting, and handling of snow specimens is difficult, especially for the largest sizes necessary to fit the data to the scaling law. Due to the natural spatial variability of a material such as snow, it is also difficult to ensure that natural samples of different sizes have the same bulk properties. Furthermore, beam-shaped samples of different size fail at different nominal strain rates when loaded at a constant rate of displacement, an effect that may be negligible for some materials but is important for a rate-sensitive material near its melting temperature such as snow. The fracture parameters calculated using the size effect method are sensitive to scatter in the data, a feature that partially explains the large uncertainties in the size effect results of Sigrist  and Borstad.
 An alternative method of calibrating a quasi-brittle scaling law was developed by Tang et al.  and involved varying the notch depth of specimens of the same shape and size in order to sufficiently vary the brittleness number to fit a size effect law. The brittleness number is a nondimensional measure of the amount of energy absorbed or dissipated by a material prior to fracture, with low numbers representing more ductile failure and higher numbers more brittle failure [e.g., Bažant and Kazemi, 1990]. The “variable notch” method works well for some test geometries, but in general varying the notch length alone does not sufficiently vary the brittleness number to allow accurate calculation of fracture parameters [Tang et al., 1999].
 To further expand the range in brittleness numbers achievable using a single specimen size, Bažant and Li  refined the variable notch technique to also include specimens without notches. This method was termed the “zero-brittleness” method because unnotched specimens are always characterized by brittleness numbers of zero, according to Bažant's definition of the brittleness number [Bažant and Kazemi, 1990]. The zero-brittleness method utilizes a quasi-brittle size effect law that satisfies appropriate asymptotic properties for failures at crack initiation (unnotched tests) and failures after stable crack growth (notched tests) [Bažant and Li, 1996; Bažant, 2005]. The fracture parameters calibrated for a particular material using the zero-brittleness method are the same at any scale; the transition between asymptotic solutions is entirely determined by the size of the fracture process zone relative to the specimen size. This study is the first to demonstrate the applicability and promise of this method for heterogeneous earth materials.
2.2 In Situ Index Properties
 The majority of the literature on mechanical properties of snow use the bulk density as the primary index variable to characterize the snow [e.g., Shapiro et al., 1997; Sigrist et al., 2005; McClung, 2007; Marshall and Johnson, 2009]. This is in spite of the longstanding recognition that density does not adequately characterize grain boundary strength or the structural integrity of the snowpack [e.g., Mellor and Smith, 1996; Fukue, 1977; Shapiro et al., 1997].
 To date, the small number of studies on tensile fracture properties of snow have continued this tradition of focusing solely on density as an index variable [Kirchner et al., 2000; Schweizer et al., 2004; Sigrist et al., 2005], though Schweizer et al.  separated fracture toughness data into bins according to the hand hardness of the snow as a secondary classification. It was not until recently that any alternative quantitative measure was used as an index for fracture toughness. Borstad  demonstrated that the peak penetration resistance of a thin, wide blade was a better predictor than density for fracture toughness determined using the size effect method. This is consistent with earlier results that indicated thin-blade penetration resistance was superior to density for predicting tensile strength [Borstad and McClung, 2011b]. In this study, we determine fracture toughness using a new experimental technique and pair these experiments with thin-blade penetration resistance measurements to confirm that penetration resistance is better than density for predicting fracture toughness.
3 Quasi-Brittle Theory
 Quasi-brittle fracture mechanics is based on the equivalent crack concept, in which the actual configuration of the crack in a heterogeneous material is modeled using a slightly longer brittle crack embedded in a homogeneous specimen of the same dimensions. This is a simple framework to account for the relatively large and nonlinear fracture process zone that many heterogeneous materials possess. Determining the appropriate equivalent crack length is the crux of the technique; the crack length is sought which achieves some sort of far-field equivalencebetween the actual and equivalent specimens. The zero-brittleness method of Bažant and Li  was developed by applying principles of equivalent elastic fracture mechanics to derive a scaling law which bridged cases where fractures emanate from a stress concentration (notched tests) versus initiating from a smooth boundary (unnotched or modulus of rupture tests). This method seeks equivalence between the peak loads in the actual and equivalent specimens [Bažant and Li, 1996], though equivalence can also be sought between measures other than or in addition to the peak load [Bažant and Planas, 1998]. The term Quasi-Brittle Fracture Mechanics (QBFM) in the context of this study represents equivalent elastic fracture mechanics with peak load equivalence [Bažant and Planas, 1998; Bažant, 2005].
 The equivalent crack length aeis related to the length of the actual crack (or notch) a∘ via
where cf is the effective process zone length or critical equivalent crack extension, “critical” in the sense of being defined here at peak load. This length scale (cf) is considered to be a material property, independent of specimen size or shape, provided that it is calibrated using specimens that are not too small compared to cf [Bažant and Planas, 1998]. It should be noted that cfis not the actual physical length of the fracture process zone [Mindess, 1991] but rather a length scale that satisfies the desired equivalence scheme, hence the reference to this length scale as the “effective” process zone length. However, cfcan be empirically related to the actual fracture process zone length given independent measurements of its extent [e.g., Cotterell and Mai, 1996]. For concrete, it is commonly held that the actual process zone length is about 2cf[Bažant and Planas, 1998]. However, no direct or indirect measurements of the process zone length for snow have ever been reported.
 Given the newly defined equivalent crack length from equation (1) in place of the actual crack or notch length, the framework of LEFM is applied using the same geometry and loading as the actual specimen. However, the introduction of the intrinsic length scale cfrenders the problem nonlinear. This nonlinearity appears after expanding the LEFM solution in series about cf[Bažant and Planas, 1998], as follows. First, the general LEFM stress intensity factor solution for a single crack loaded in tension is written and then expanded in a Taylor series. By truncating the expansion beyond terms of second order in cf/D, Bažant and Li  derived the following form for the nominal strength σNu as a function of the fracture toughness KIc and the effective process zone length cf,
where in the context of the present study D is the beam depth and the terms D∘ and Db are defined as
where g∘ is a dimensionless LEFM geometric function found in handbooks [e.g., Tada et al., 2000] and the prime indicates differentiation with respect to α=a∘/D. The Macaulay brackets <X>=max(X,0) are used to exclude from the scaling analysis any specimen geometries for which the maximum stress is not at the surface but rather increases with distance from the surface (i.e., exists in the interior of the specimen) [Bažant and Li, 1996]. For bending beams, the maximum tensile stress from beam theory is always at the surface, so g∘′′≤0 and thus −g∘′′ is nonnegative. See Appendix A for additional detail on the derivation of equation (2).
 The length scale D∘in equation (2) is known as a “transitional size” that defines the center of the size range in which the strength of a cracked or notched specimen transitions from following strength of materials theory for D<<D∘ to LEFM for D>>D∘ [Bažant and Planas, 1998]. In the vicinity of D∼D∘, the strength scaling is nonlinear and can be expressed using an asymptotic matching size effect formula such as that of Bažant . Note that for unnotched specimens the geometric function g∘=0 and thus D∘=∞.
 The parameter Dbis defined as a boundary layer length scale which represents the length over which the average elastic tensile stress is equal to the tensile strength [Bažant, 2005]. This is the length scale over which the tensile crack first coalesces in an unnotched fracture specimen, a scale that depends on the heterogeneity of the material. Note that for notched beams the term <−g∘′′>=0 and thus Db=0.
 Following a series expansion of equation (2), the introduction of a horizontal asymptotic value of the nominal strength for D→0 (governed by ηDb), and some rearrangement, equation (2) can be written in the form
which has been termed a “universal” size effect law because it applies to both notched and unnotched fracture specimens [Bažant and Li, 1996; Bažant, 2005]. This equation is thus the foundation for the zero-brittleness method which combines notched and unnotched fracture data. The relation can be further generalized to account for statistical effects which may play a role in some cases [e.g., McClung and Borstad, 2012], though such effects are not considered here. A value of η=0.5 is commonly used in equation (5), a value that corresponds to the size limit (D=0.5Db) below which a rectangular beam fails by quasi-plastic collapse rather than propagating fracture [Bažant et al., 2007].
 Equations (2) and (5) are nearly identical to the expression derived by Bažant and Li , except that in the present study the value of cfwas taken to be the same in notched and unnotched tests. Bažant and Li  assumed that cf is 40% larger in unnotched tests. However, Borstad  found no evidence on which to base a similar assumption for snow. In numerical simulations of fracture experiments, Borstad and McClung [2011a] found that using a constant value for the nonlocal interaction radius, a length scale related to the process zone length, led to good agreement with measured peak loads and stable portions of load-displacement curves for both notched and unnotched experiments.
4.1 Numerical Method
 Equation (5) was rearranged into a form y=Ax+Cthat could be solved using an iterative linear least squares algorithm [Bažant and Li, 1996] with
 In equation (6), the nominal strength σNuis calculated from the peak load and geometry of the test specimen. The remaining parameters of y and x are functions of geometry alone. The resulting regression constants A (slope) and C (intercept) were used to calculate the fracture parameters KIc and cf,
 These relations differ from the regressions outlined in Bažant and Li  in that they are written here as a function of the fracture toughness instead of the fracture energy. Solving for the fracture energy requires knowledge of the appropriate value for Young's modulus or an effective elastic modulus using an elastic-viscoelastic correspondence principle. The modulus is a highly uncertain term in snow mechanics; thus, solving for the fracture toughness here avoids this uncertainty. However, for other materials for which the modulus is well known, calculating the fracture energy follows the same procedure but with regression equations as reported by Bažant and Li .
 The unknown parameter cf enters equation (6) through the definitions of D∘and Db(equations (3) and (4), respectively) which appear in equation (7). This makes the equation system nonlinear. This system was solved by writing an iterative solution procedure, in which χ=1 (the limit for large sizes) was prescribed for the first iteration. Following the first solution of the linear equation y=Ax+C, a first estimate of cf was obtained and used to update the value of χand thus y for a subsequent iteration. This procedure was repeated until the L2-norm of χ was less than 10−4. The algorithm typically converged in less than 10 iterations.
 The value of η in equation (5) was varied in the limit 0<η<1 and found to have little effect on the regression results. The value of η=0.5 was therefore adopted for consistency with other studies.
4.2 Experimental Method
 The beam experiments were carried out in 2008 and 2009 at Rogers Pass in the Columbia Mountains of British Columbia. The snow was first characterized in situ using standard stratigraphic profiling techniques [Canadian Avalanche Association (CAA), 2007; Fierz et al., 2009], including measurements of the snow density, hand hardness, temperature, grain size, and grain type for each distinct stratigraphic layer in the snowpack. Once a particular layer within the snowpack had been identified for testing, beam-shaped snow samples were extracted from the layer using stainless steel rectangular cutter boxes. The dimensions of the samples were 50 cm long by 10 cm deep by 10 cm wide. These dimensions were chosen based on prior experience which indicated that this size was the most consistent from the perspective of extraction, handling, mounting, testing, and uniformity of size. This size also struck a balance between being large enough to ensure homogeneity while not being so large as to limit the number of samples that could be extracted in a reasonable amount of time during a short winter day. Additionally, this size enabled the use of a small enough sampling area to avoid significant changes in snow properties due to natural spatial variability.
 Following extraction, the samples were transported to a nearby cold laboratory and fractured on the same day. To minimize the metamorphic change to the snow crystals upon entering a new thermodynamic environment, the samples were stored for no more than about 6 h in the cold lab prior to testing. The temperature of the lab was set to mimic the in situ temperature of the layer from which the samples originated. Prior to testing, the sample dimensions were measured, and the sample was weighed to determine the bulk density. Following a fracture test, the temperature was measured, and the type and size of the snow crystals were determined under magnification. For all but one test series, measurements of thin-blade penetration resistance were conducted on an undisturbed portion of each sample immediately after being fractured [e.g., Borstad and McClung, 2011b, Figure 7]. Details of the design and method of using the thin-blade gauge are in Borstad and McClung [2011b], but a brief explanation is given here for convenience. A 10 cm-wide by 0.6 mm-thick blade with a blunt leading edge was attached to a digital force gauge. The blade was manually pushed into the snow samples to a depth of approximately 3–5 cm at a penetration speed of about 10 cm/s. The peak force of penetration measured by the force gauge was recorded as the “blade hardness index” and represented by the symbol B.
 A benchtop universal testing machine was used for the bending experiments, which were conducted under displacement control. Four-point bending (4PB) was used for one test series, whereas the remainder of the tests were three-point bending (3PB). The support span for the 3PB tests was selected as half the beam length such that there was no gravitationally induced bending moment in the central cross section of the beam. This was done to prevent viscoelastic sagging of the beam after mounting on the supports but prior to testing, since the gravitationally induced bending moment can contribute as much as half of the bending moment needed to fracture a snow sample if it is supported near the ends of the beam [Sigrist, 2006]. This reduced support span, necessary for proper weight compensation, limited the amount of space left in the middle of the beam for multiple load points, especially since flat loading plates (rather than standard round pins) were necessary to prevent significant crushing at the contact points. The single 4PB test series was an effort to address these issues—and adopt the commonly preferred test method—by orienting the testing machine horizontally rather than using the common vertical benchtop orientation. The samples were then supported from beneath by smooth polycarbonate tables while the bending test was conducted. In this manner, weight compensation was achieved while increasing the support span to allow two sufficiently separated loading points. The friction between snow and the polycarbonate tables was measured independently and used to correct the measured peak loads for these experiments.
 The crosshead of the testing machine was mounted with an HBM RSC S-beam load cell with a capacity of 1000 N and a resolution of 0.5 N, calibrated with dead weights. Rocker supports made of 2.5 cm-wide pieces of polycarbonate were mounted to the machine to support the snow samples and hinge during the test while the beam flexed. For the 3PB tests (central loading), a polycarbonate loading plate of the same size and shape as the rocker support plates was bolted to the load cell along with an aluminum stiffener. The 4PB tests used a large aluminum plate mounted to the load cell, onto which two additional rocker supports were mounted for load application.
 Two linear variable differential transformers (LVDTs) were mounted to measure the deflection of the beams in addition to the measurement of load-point displacement of the actuator. One LVDT was mounted below the center of the beam and one on top of the beam above one of the supports. The load-time signals from these LVDTs confirmed a dynamic instability in the test setup at peak load, a result of the relatively compliant testing machine and of using open loop displacement control rather than closed loop servo control. This instability was observed at all loading rates and may have contributed some additional scatter in the peak load measurements above that due to the inherent variability for a heterogeneous natural material such as snow. Borstad and McClung [2011a] discuss in greater detail the test setup for a similar set of experiments and show evidence of the instability at peak load in load-displacement and load-time curves.
 Eleven beam bending test series were conducted and are summarized in Tables 1 and 2. In each series, the specimen dimensions and loading conditions were held constant, but in alternating samples a notch was cut into the bottom of the beam using a thin blade identical to that used for the penetration resistance measurements. According to the underlying quasi-brittle theory, it should not matter what notch depth is chosen, provided that the overall specimen dimensions are sufficiently large compared to cf[Bažant and Planas, 1998]. For the majority of the test series, the notch was cut to a depth of 3 cm. This relative notch depth (0.3) was chosen because the geometric stress intensity factor relations of LEFM are less sensitive to notch-depth variability for this relative notch depth compared to shallower or deeper notches. In two test series (Z2 and Z3), multiple notch depths were used, but in the regressions the combined weight of all the notched tests was set to be the same as the unnotched tests since the variation in the brittleness number achieved by varying only the notch length was small [Bažant and Li, 1996]. Inspection of the fracture surfaces following notched experiments indicated that the notch tips were relatively sharp and free of crushed or deformed grains.
Table 1. Notched/Unnotched (Zero-Brittleness) Test Dataa
Date is in yymmdd format, other column labels include the number of tests (n), mean snow density (ρ̄), hand hardness index (R), mean blade hardness index (B̄), mean snow temperature (T̄), crosshead speed (V) with number of tests in parentheses for series Z5 and Z6, and beam span-to-depth ratio (S/D). All samples had a beam depth D=10 cm. All uncertainties are standard deviations from the mean.
Grain forms and grain size according to the International Classification for Seasonal Snow on the Ground [Fierz et al., 2009]. Key: RG = rounded grains; DF = decomposing and fragmented grains; FCxr = mixed rounded and faceted crystals; FC = faceted crystals.
Peak loads listed as mean plus or minus standard deviation. Relative notch depth listed in parentheses for series Z2 and Z3, 0.3 for all other series.
33 ± 5
12.0 ± 1.0
120 ± 6
59 ± 2.2 (0.1)
36 ± 2.5 (0.3)
21 ± 1.7 (0.5)
20 ± 1.4
13.8 ± 0.8 (0.1)
8.7 ± 0.4 (0.3)
6.1 ± 0.4 (0.5)
11.5 ± 1.3
5.2 ± 0.4
82 ± 4
27 ± 2.3
87 ± 5
28 ± 1.0
79 ± 4
27 ± 2.0
90 ± 7
29 ± 1.6
100 ± 17
32 ± 2.2
30 ± 2.8
10.6 ± 0.6
87 ± 8
27 ± 1.7
 For two test series (Z5 and Z6), subsets of notched-unnotched tests were conducted at different rates to investigate rate effects on the resulting fracture parameters. The crosshead speed for the remainder of the experiments was 1.25 cm/s. The nominal tensile strain rate in the outer fiber of a bending beam is proportional to DV/S2, where D is the beam depth, V is the loading rate, and S is the beam span [Timoshenko, 1940]. At the typical loading rate of 1.25 cm/s, the nominal strain rate is therefore on the order of 10−1 s−1. This rate is well above the creep-to-fracture or ductile-to-brittle transition rate for snow [e.g., Mellor and Smith, 1996; Narita, 1980], and thus it was assumed that the majority of the deformation at this rate was elastic.
 The nominal strength for each test was calculated according to beam theory using the measured peak load and beam dimensions via the equation
where P is the peak load, b the beam width, and
for 3PB and
for 4PB. The correction term in parentheses in equation (12) accounts for the differing stress field caused by the concentrated central load compared to the pure bending beam solution [Timoshenko, 1940].
 Data from each of the 11 test series were fit to the size effect law of equation (5) using the iterative linear regression procedure outlined above. Figure 1 shows the experimental results expressed in the form of yfrom equation (6) versus x from equation (8). In general, the regression fits were quite good. The two variable notch-depth series (Z2 and Z3) showed more scatter and worse regression fits than nearly every other series. In most cases, the unnotched data showed less scatter than the notched data within a series, a feature that may have been due to variability in the manually cut notches.
 Table 3contains the calculated values of fracture toughness KIc, equivalent elastic crack extension cf, the geometric parameters D∘ and Db, and the brittleness number β=D/D∘. The fracture toughness values were mostly in the range 1–4 kPam1/2, a narrower range than produced in previous studies using first-order methods [Kirchner et al., 2000; Schweizer et al., 2004; Sigrist et al., 2005; McClung and Schweizer, 2006; Sigrist, 2006]. The effective process zone length cfwas around 0.5 cm or less in most cases, lower than estimated by McClung and Schweizer  by comparing fracture toughness and tensile strength data from different studies, though this discrepancy would be reduced with the use of more recent tensile strength data [e.g., Sigrist, 2006; Borstad, 2011]. Note that cf has not been reported in most studies that utilized quasi-brittle fracture theory; rather, parameters such as D∘ [McClung and Schweizer, 2006; Sigrist, 2006] and Db[Borstad and McClung, 2009] are more frequently given, both of which depend on geometry and are therefore not generally comparable across studies.
Table 3. Fracture Parameters Determined by Fitting Data to Equation (5)a
KIc (kPa m1/2)
Subsets of series Z5 and Z6 are for fast (f), medium (m), and slow (s) loading rates as defined in Table 1. The fracture toughness KIcand equivalent elastic crack extension cf are applicable for the full data sets. The transitional size D∘listed is only applicable for the notched tests, as D∘=∞for α=0. The brittleness number β=D/D∘is for the notched tests only. The boundary layer length scale Dblisted is only applicable for the unnotched tests, as Db=0 for α>0.
Mean value for test series which had different notch depths, as each notch depth led to a different value of D∘and Db.
 The coefficients of variation (CoV) of fracture toughness for the data in Table 3were between 3% and 10% for all but series Z3, which had a poor regression fit and a resulting CoV of 21%. In contrast, the CoV of the fracture toughness data of Sigrist  from size effect tests were all in the 15–25% range, partly a consequence of the lower-order size effect scaling law and perhaps also a result of experimental errors associated with notch sensitivity and the lack of weight compensation in the bending tests. In general, the length scales cf, D∘, and Db are more sensitive to scatter in the data than KIc. The two series with the poorest linear regression fits (Z3 and Z4) had the largest calculated values of all three length scales. This is the same trend observed in the data of Sigrist , in which the linear regression fits were poor and large values of D∘ were reported. This sensitivity of calculated length scales to scatter in the data is a general feature of both the size effect and zero-brittleness scaling laws and is not specific to the snow data here [Bažant and Planas, 1998]. We assume that most of the scatter observed in the experiments is the result of material heterogeneity, though the relatively compliant and open loop test setup may have contributed to the observed scatter, a factor that would have also affected the results of Sigrist.
6.1 Predictors for Fracture Toughness
 The fracture toughness correlated better with the blade hardness index than with the density (Figure 2). The relation between fracture toughness and penetration resistance was linear, and there was much less scatter around the regression model as a function of the blade hardness index (Figure 2b) than density (Figure 2a). The only notable outlier in the fracture toughness versus hardness plot (Figure 2b) was series Z8, which was also an outlier in the density plot (Figure 2a).
 A linear least squares regression through the fracture toughness KIc versus blade hardness index B (solid line in Figure 2b) took the form
with KIc in kPam1/2and B in N. This model was obtained by weighted linear least squares regression using the inverse of the variance of the toughness values in each data series as the weights (thus giving less weight to the data with greater scatter). Both the slope and intercept terms were statistically significant at the α=0.05 level. The overall goodness of fit, characterized by the coefficient of determination, was very high (adjusted r2=0.97). The residual structure of the model was good except for the presence of the data point for series Z8, a strong outlier that influenced the statistical tests for normality and independence.
 The best nonlinear regression fit through the fracture toughness data as a function of density (Figure 2a), using the common power law density formulation, was
where ρ is the density of the snow and ρice the density of solid ice, with KIcagain in kPam1/2. This model was obtained by weighted nonlinear regression using the inverse of the variance of the toughness values as the weights, as for the blade hardness model. Both regression coefficients were statistically significant, and model residuals were normal and independently distributed at the α=0.05 level. The fit had a nonlinear R2=0.79, where the nonlinear R2is defined as 1 minus the ratio of residual sum of squares to total sum of squares (which is not necessarily equivalent to the common coefficient of determination in linear least squares regression but is in this case since a constant model can be embedded in the regression model).
 The regression model of equation (15) is compared to published relations in Figure 3. At low densities, equation (15) predicts higher fracture toughness than any other relation. With increasing density, however, the curves diverge. This is primarily due to the difference in scaling law exponents, which range from a minimum of 1.5 in the present study to a maximum of 2.4 in the calculations of McClung and Schweizer . A similar scaling exponent was found by Sigrist et al. ; this relation is not shown in Figure 3 as the associated data partially overlap with those of Sigrist , which are shown. The wide scatter in published results expressed as a function of density is not surprising given that density is a poor predictor for mechanical properties of snow [Mellor and Smith, 1996; Fukue, 1977; Shapiro et al., 1997]. No secondary variables describing the snow were consistently reported in the studies represented in Figure 3 to enable a statistical analysis of variance at a given density.
 The fracture toughness calculated by Kirchner et al.  and Schweizer et al.  were obtained by assuming the direct applicability of LEFM to experimental data. These results all predict lower fracture toughness than the quasi-brittle scaling relations of subsequent studies [Sigrist et al., 2005; McClung and Schweizer, 2006; Sigrist, 2006, and present study]. This trend is consistent with the interpretation that a diffuse fracture process zone acts as a toughening mechanism in quasi-brittle materials [e.g., Cotterell and Mai, 1996; Bažant and Planas, 1998], which should therefore lead to higher predictions of fracture toughness than LEFM when applied to the same experimental data.
6.2 Rate Effects
 The fracture toughness decreased with increasing strain rate, though the effect was weak (Figure 4). However, the trend is consistent with the observed decrease in the strength of snow with increasing loading rate above the creep-to-fracture (or ductile-to-brittle) transition as the relative importance of creep diminishes [Mellor and Smith, 1996; Narita, 1980]. The peak loads in Table 2display the same trend, with decreasing peak load for increasing loading rate for series Z5 and Z6. Other quasi-brittle materials display similar trends. For example, the fracture toughness of concrete decreases with increasing strain rate due to the diminishing influence of creep-related crack blunting within and surrounding the fracture process zone [Bažant and Gettu, 1992]. We note the very large difference in homologous temperature between concrete (∼0.15Tm, where Tm is the melting temperature in Kelvin) and snow (>0.95Tm) which may lead to completely different microstructural sources of the observed rate effects, even if the trends are the same.
 No dependence of cfon loading rate was apparent in either of the series for which rate effects were tested (Figure 5). The value of cf was not significantly different for either series Z5 or Z6 when the nominal strain rate was varied by 1–2 orders of magnitude. The lowest strain rates in these tests were above the creep-to-fracture transition of about 10−4 s−1for snow in tension [Narita, 1980]. The relative influence of creep versus distributed microcracking may indeed vary with rate over these scales without translating into an appreciable change in the size of the effective fracture process zone.
6.3 Size Limit for LEFM Applicability
 The LEFM size limit for tensile fracture can be quantified using the brittleness numbers in Table 3. The weighted mean brittleness number βfor all the data, giving more weight to the data with less scatter, was around 5. This is somewhat higher than reported in the studies of Sigrist et al.  and Sigrist , both of which found β∼(1), though from data with much greater scatter. LEFM is first applicable for brittleness numbers of at least 10–25 [Bažant and Kazemi, 1990; Bažant and Planas, 1998], which indicates that beam depths of at least 2–5 times the laboratory scale, or 20–50 cm as a lower bound, are necessary before LEFM can be applied. Since the grain size enters into many scaling relations, an alternative and perhaps more general approach would be to scale the LEFMlimit by the snow grain size. Given the typical grain size of the zero-brittleness data here (0.5–1 mm), the lower-bound LEFM limit can be expressed as around 200–1000 times the grain size.
 An important distinction must be drawn between a tensile fracture in a slab avalanche and that in a beam bending test. The quasi-brittle scaling law for tensile crack initiation can be expressed as a function of the strain gradient over which the crack first initiates [Bažant, 2005]. The form of the tensile strain gradient in a slab avalanche is generally unknown; therefore, the beam depth from the laboratory data may not directly compare to the slab depth in an avalanche [Borstad, 2011]. Therefore, it is only strictly appropriate to claim that beam depths of at least 20–50 cm are necessary for the direct applicability of LEFM to beam bending experiments on snow. The LEFM limit as a function of slab depth in avalanches will depend on the layered nature of the snow slab and the failure mechanism in the weak layer, both of which will be primary influences on the strain gradient through the thickness of the slab. Given these sources of uncertainty, we suggest that a conservative lower-bound slab depth of 1 m is necessary for the direct applicability of LEFM. However, since the quasi-brittle scaling law contains LEFM as a large-size asymptotic limit, there is no need to make an assumption about the applicability of LEFM a priori. This makes the quasi-brittle theory a more general and appropriate foundation for analyzing the tensile fracture of heterogeneous materials.
 The zero-brittleness method applied in the present study is based on a higher-order scaling law (second order in terms of cf/D) than either the quasi-brittle scaling law for failures at crack initiation [Bažant and Li, 1995] or failures at crack propagation [Bažant, 1984], both of which are truncated after terms first order in cf/D. All previous tensile fracture mechanical studies of snow have been based either on LEFM or on one of these first-order scaling laws, which partially explains the greater degree of scatter in the results of previous studies. For example, in the present study the error term (cf/D)3 (the first truncated term in the scaling law derivation) was on the order of 0.1% with a maximum of about 0.5%. For comparison, the error terms for the notched size effect law of Bažant  applied to the notched experimental data of Sigrist  and Borstad  were on the order of 10% or more.
 Another explanation for the higher confidence in the results of the present study lies in the experimental advantages of the one-size method. Using a single specimen size was convenient for repeated, consistent sampling of natural snow, and the same could probably be said if this technique were used for other natural materials. Rate effects that can arise when testing differently sized specimens are avoided. The influence of spatial variability when sampling a natural heterogeneous material is more manageable from the perspective of sampling strategies when a single specimen size is used.
 The zero-brittleness method demonstrated in the present study can be applied to other heterogeneous earth materials provided that the effective fracture process zone, a function of the scale of heterogeneity of the material, is sufficiently small compared to the allowable laboratory specimen size. In concrete, the maximum aggregate size is often considered the limiting factor in determining the minimum allowable specimen size [Bažant and Planas, 1998]. For crystalline or granular earth materials, a specimen size sufficiently large compared to the maximum grain size would likely be appropriate for applying the zero-brittleness method.
 The difficulty in consistently cutting notches in the snow samples, and associated notch sensitivity, was the reason that a single notch size was used in most test series. The relative notch depth of 0.3 used for the majority of tests was selected because the dimensionless LEFM geometry functions are relatively insensitive to variability in notch depth around this value, at least for the chosen beam geometry. The scatter in the variable-notch data may in part be explained by notch-depth variability for the shallow and deep notches. Sigrist  used a shallow relative notch depth of 0.1 for which the geometric functions are very sensitive to variability, which likely contributed to the large amount of observed scatter. An additional source of the observed scatter may have been the fact that these two test series were also the only ones that had faceted snow grains, a crystal form that leads to more variable results than rounded grains according to thin-blade penetration resistance measurements [Borstad and McClung, 2011b].
 The relative roles of the various microstructural processes and scales that contribute to failure, from intergranular cracking to intercrystalline or transcrystalline cracking within grains to creep within and between grains, may all have different rate sensitivities, and explaining the microstructural source of the observed (or not observed) rate effects is not possible from the methods of the present study. Whatever the microstructural explanation, the interplay between creep and fracture is likely the source of the observed decrease in fracture toughness for increasing strain rate, consistent with studies of the rate dependence of fracture toughness of ice [Xu et al., 2004] as well as other mechanical properties of snow [Salm, 1971; Brown and Lang, 1975; Fukue, 1977; Narita, 1980].
 The effective elastic analysis of the fracture data was considered appropriate because the bulk creep strain at failure was sufficiently small compared to the instantaneous elastic strain [e.g., Bažant and Gettu, 1992]. The mean time to failure for the experiments carried out at the fastest crosshead speed was on the order of 0.1s. The creep strain for this failure time is on the order of 0.05% of the total strain, calculated using the viscoelastic tensile creep parameters determined by Shinojima . Even for the tests at the slowest loading rate, which had failure times of 1–10s (and corresponding strain rates on the order of 10−3 s−1), the creep strain at failure comprised no more than 5% of the total failure strain.
 The likelihood of triggering a slab avalanche and its resulting release dimensions can be related to the ratio between the tensile fracture toughness in the snow slab to the shear fracture toughness in the underlying weak layer [McClung and Schweizer, 2006]. Given the strong correlation between the tensile fracture toughness and thin-blade penetration resistance (Figure 2) and the possibility of indexing the fracture toughness of weak layers using the same measure, the thin-blade penetration gauge could be an effective field-based tool for avalanche forecasting rooted in fracture mechanics. The demonstration of the strong connection between thin-blade resistance and tensile strength [Borstad and McClung, 2011b] and now fracture toughness addresses a longstanding need in the field of snow mechanics for a simple, repeatable index measure for characterizing the mechanical properties of snow.
 Size-independent fracture parameters were calculated for snow using combined notched and unnotched test data from specimens of a single size. The tensile fracture toughness of snow applicable to slab avalanche release varied in the range 1–4 kPam1/2, decreasing slightly with increasing strain rate. For the first time, a quantitative measure other than bulk density was used to index tensile fracture results. Thin-blade penetration resistance was found to be better than density for correlating with fracture toughness, indicating the promise of this simple hardness measure for in situ applications and for comparing results from different studies. The effective fracture process zone length was determined to be about 5–10 times the grain size and did not vary with strain rate. This length is smaller than estimated in earlier studies but not sufficiently small over most length scales of interest in tensile fractures related to avalanches to permit the direct application of Linear Elastic Fracture Mechanics. The higher-order scaling law underlying the zero-brittleness method and experimental advantages of using a single specimen size led to reduced scatter and higher confidence in the results than previous studies of tensile fracture parameters of snow. The success of the zero-brittleness method applied to a fragile material such as snow indicates its promise for application to other heterogeneous earth materials.
Appendix A: Derivation of Universal Size Effect Law
 The general expression for the mode I stress intensity factor KI for a crack loaded by a remote opening (tensile) stress is
where σis the applied stress, a is the crack length (or half length, depending on the geometry), and f (a/D) is a dimensionless function of the specimen geometry that takes as an argument the ratio of the crack length to a representative length D [e.g., Bažant and Planas, 1998; Cotterell and Mai, 1996]. For a beam in bending, D is the beam depth. By defining the ratio α=a/Dand, for convenience in analyzing the experimental data, a nominal stress measure σN as a function of an applied moment or load, equation (A1) can be rewritten as
which shows the explicit dependence of the fracture on the specimen size D[Bažant and Planas, 1998]. For further convenience, the last two terms on the right-hand side of equation (A2) can be lumped into a single dimensionless function k(α)=παf (α). Expressions of the function k(α), or, more commonly, f (α), are tabulated in handbooks for various standard test geometries [e.g., Tada et al., 2000].
 Identifying the critical stress intensity factor (the fracture toughness) KIc as the value of KI at the ultimate value of nominal stress σNu(defined at peak load or ultimate bending moment), we have
 A polynomial expression for the geometry function k(α) as a function of the beam span-to-depth ratio given by Bažant and Planas  was adopted in the present study.
 A generalization of equation (A3) which accounts for the possible presence of a large fracture process zone can be achieved by expressing the relative crack length a as a combination of the true initial crack or notch length a∘ plus an equivalent elastic crack extension cf. Expressing equation (A3) in terms of the nominal stress and writing the crack terms nondimensionally, we have
where g(α)=k2(α) has been defined for convenience in the following expansion. Expanding g(α) in a Taylor series about α∘ and introducing the notation g(α∘)=g∘, we can write a general form of the nominal strength of the equivalent elastic specimen as
where primes indicate differentiation with respect to α∘ [Bažant and Li, 1996]. For specimens which are neither notched nor precracked prior to testing, the geometric function g∘=g(α∘)=0. Thus, when considering both notched and unnotched experimental data together, all three terms (at minimum) should be retained. Keeping only the terms shown, equation (A5) can be rearranged as [Bažant, 1995]
 This relation is equivalent to equation (2), with D∘and Dbdefined as in equations (3) and (4), respectively.
 We are grateful for the financial support of the Natural Sciences and Engineering Research Council of Canada, Canadian Mountain Holidays, and the University of British Columbia. In-kind support was graciously provided by the Avalanche Control Section of Parks Canada at Rogers Pass.