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Cold Regions Science and Technology
journal homepage:www.elsevier.com/locate/coldregions
Temporal changes in the mechanical properties of snow related to crack propagation after loading
Karl W. Birkeland
a,⁎, Alec van Herwijnen
b, Benjamin Reuter
c, Bastian Bergfeld
baUSDA Forest Service National Avalanche Center, Bozeman, MT, USA
bWSL-SLF Swiss Institute for Snow and Avalanche Research, Davos, Switzerland
cDepartment of Civil Engineering, Montana State University, Bozeman, MT, USA
A R T I C L E I N F O Keywords:
Avalanche Fracture Loading
SlabParticle tracking velocimetry Propagation Saw Test
A B S T R A C T
Most dry slab avalanches occur during or immediately following loading by snowfall or wind deposition. In the absence of further loading avalanche activity decreases over time. This suggests that loading favorably changes snow cover properties for avalanche release over short time scales (e.g., minutes and hours), and that changes over longer time scales (days or longer) help to stabilize the snowpack. This study quantifies both: 1) the effect of increasing load on the interaction of the slab and weak layer over short time scales, and 2) the longer term stabilizing changes following loading. We developed a field method to rapidly increase the load on existing weak layers, and conducted two different sets of experiments. For the first set of experiments we used a cardboard frame the dimensions of a standard Propagation Saw Test (PST) and added 5, 10, 15, or 20 cm of disaggregated snow on top of PST columns on 11 sampling days. We allowed the added snow to sinter for approximately 30 to 60 min before completely isolating the block and performing a PST. In the second set of experiments we used the same technique to add 10 cm of disaggregated snow on over 30 isolated columns. We then conducted PSTs in the minutes, hours and days following isolation, with tests ranging from 15 min to 4 days. For both experiments we filmed each test at 120 fps for particle tracking velocimetry analysis. We also utilized a model simulating the experiments to better interpret our results. In the first set of experiments, critical crack lengths dramatically decreased with increasing load while crack propagation speed increased, a finding consistent with previous work. In the second set of experiments, we found that critical crack lengths increased rapidly at first and then more slowly over time. Simulations of the experiments suggest that changes in critical crack length over time are caused by an increase in slab elastic modulus in the first hours following loading, and then caused by both increasing slab elastic modulus and weak layer specific fracture energy in days following loading. Overall, our results help to illustrate changes in critical crack lengths immediately after and in the days following loading.
Our results are consistent with field observations of increasing avalanche activity during and immediately fol- lowing loading events and decreasing avalanche activity afterwards.
1. Introduction
The first avalanche observers undoubtedly realized that increases in load, i.e. the mass of snow supported by a buried weak layer, typically increases avalanche likelihood over short time scales. Written sources as early asSeligman (1936)andBader et al. (1939)recognized loading as critically important for avalanche formation, and half ofAtwater and Koziol's (1952)original ten contributory factors for avalanche forma- tion were related to loading. Since avalanche release is facilitated by ongoing or recent loading, it follows that loading must favorably change snow cover properties for failure initiation and/or crack pro- pagation, key processes for avalanche release (e.g., Schweizer et al.,
2003;van Herwijnen and Jamieson, 2007c). However, over longer time scales after loading the snowpack tends to stabilize, so temporal changes in the absence of loading must change snowpack properties to make failure initiation and/or crack propagation less likely.
Increasing load simultaneously affects many snowpack properties which, in turn, affect failure initiation and crack propagation processes necessary for avalanche release (van Herwijnen and Jamieson, 2007a;
Reuter and Schweizer, 2018). For example, a large body of research demonstrates that increasing load leads to increases in weak layer shear strength over periods ranging from days to months (Jamieson, 1995;
Schweizer et al., 1998; Jamieson and Johnston, 1999;Chalmers and Jamieson, 2001;Chalmers and Jamieson, 2003;Zeidler and Jamieson,
https://doi.org/10.1016/j.coldregions.2018.11.007
Received 23 July 2018; Received in revised form 7 November 2018; Accepted 12 November 2018
⁎Corresponding author at: USDA Forest Service National Avalanche Center, P.O. Box 130, Bozeman, MT 59771, USA.
E-mail address:kbirkeland@fs.fed.us(K.W. Birkeland).
Cold Regions Science and Technology 159 (2019) 142–152
Available online 13 November 2018
0165-232X/ Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
T
2006b; Zeidler and Jamieson, 2006a; Jamieson et al., 2007; Logan et al., 2007;Horton et al., 2014;Conlan and Jamieson, 2016). While many time series suggest shear strength increases linearly over time, some field data show more of a power-law type relationship (Logan et al., 2007).Conlan and Jamieson (2016)used a controlled environ- ment and fit temporal shear strength increases after loading to a power law; they hypothesize that observations of linear changes result from a mismatch between the process scale and the observation scale of the work (Blöschl and Sivapalan, 1995).
In addition to affecting weak layer shear strength, the deeper slabs associated with increasing load may lower the frequency of skier trig- gering (van Herwijnen and Jamieson, 2007c) because the additional stress applied by a skier strongly decreases with depth (Schweizer and Camponovo, 2001;Thumlert and Jamieson, 2014). These findings are reinforced by observed increases in Extended Column Test (ECT) scores with deeper weak layers (Hoyer et al., 2016) and similar increases for Compression Test (CT) scores (van Herwijnen and Jamieson, 2007b).
However, these studies do not explicitly investigate the role of loading on crack initiation and propagation, but rather suggest that triggering tends to be more difficult for deeper slabs.
The deeper slabs associated with loading will also affect crack propagation propensity. Certainly the increased slab depth, hardness, and tensile strength associated with denser slabs tend to favor crack propagation (van Herwijnen and Jamieson, 2007c; Schweizer et al., 2014; Gaume et al., 2015;Reuter and Schweizer, 2018). In summar- izing a large amount of work on the Propagation Saw Test (PST), Gauthier and Jamieson (2012) conclude that thicker slabs are asso- ciated with longer critical crack lengths, suggesting that although a crack may propagate further under denser slabs, a longer critical crack length is required for the onset of crack propagation. However, similar to the studies on the ECT and CT, researchers collected these data over large spatial and long temporal scales, and they thus represent the average conditions of deeper weak layers generally being associated with longer crack lengths. Conversely,Schweizer et al. (2016)made approximately weekly field measurements and documented a case where crack length decreased shortly after a large snowfall increased the load and slab depth. To better study how crack length is affected directly after loading we need to better match our observation scale with the process scale (Blöschl and Sivapalan, 1995) of increased likelihood of avalanching in the minutes, hours and days immediately following loading.
Snow is a rate dependent material and its fracture is best described as quasi-brittle (McClung, 1979;McClung, 1981;Bazant et al., 2003).
Thus, linear elastic fracture mechanics do not fully account for the complex material behavior of snow. Still, snow is often treated as linear elastic as a first approximation, and the associated mechanical theories provide potential insights into the role of increasing load on crack propagation. FromAnderson (1995)the energy release rateG, defined as the rate of change in total energy per unit crack area, for homo- genous, linear elastic materials under tension is given by:
G r
~ E2 (1)
whereσis the stress,ris the crack length, andEis the elastic modulus of the material. While this concept has been developed for brittle failure, it can be extended to include non-recoverable deformation, yielding higher values of the critical energy release rate (Anderson, 1995). At the onset of rapid crack propagation, r=rcand the critical value of energy release rateGcequals the material resistance to crack growth. Of course, snow is neither strictly linear elastic nor homogeneous. Instead, it is a laminate material consisting of a slab (often with multiple layers) and a weak layer that fails under mixed mode loading prior to the re- lease of an avalanche. For this more complicated situation,Gaume et al.
(2017)showed that the critical crack length decreases if stress increases due to increasing load, or if either the weak layer specific fracture en- ergy or the slab elastic modulus decreases.
Schweizer et al. (2016) conducted a numerical parametric study based on the work ofHeierli et al. (2008)on the interplay of weak layer specific fracture energy, slab elastic modulus, and load on the critical crack length in PSTs, concluding that “the most influential parameter seems to be the load” (p. 2650). Analyzing numerous PSTs collected over large spatial and long temporal scales, van Herwijnen et al.
(2016a)andvan Herwijnen et al. (2016c)found that weak layer spe- cific fracture energy increases with increasing load. These studies sug- gest thatGauthier and Jamieson's (2012) observations of increasing crack length with increasing load over long temporal and large spatial scales may be due to either increasing slab stiffness, increasing weak layer specific fracture energy, or, more likely, both.
Thus far no field work has focused on the short-term effects of loading on the snow's mechanical properties related to crack propaga- tion, nor has any field work explicitly investigated those properties during the stabilization process in the hours and days following loading.
The purpose of this research is to use field experiments to investigate changes in critical crack length following loading, and to also use a modelling approach to study the driving processes behind our field observations. We developed a simple field technique to change slab properties by rapidly loading the snowpack, and then conducted PSTs to investigate changes in the mechanical properties associated with propagation. Our research addresses two related questions:
1. How does increasing load affect critical crack length, weak layer specific fracture energy, and slab elastic modulus immediately fol- lowing loading?
2. Once a snowpack is loaded, how do the above properties change in the minutes, hours, and days following loading?
In essence, our goal is to conduct a field-based parametric study to investigate the influence of recent loading on crack propagation in snow to provide insights into the processes involved in the rapid de- stabilization of the snowpack immediately following loading and the slow stabilization afterwards.
2. Field area and methods 2.1. Field sites
We conducted our fieldwork at two sites in southwestern Montana, U.S.A. (Fig. 1). The first is near Bacon Rind Creek approximately 80 km south of Bozeman (44°58′13”N, 111°5′50”W), a site also utilized by Birkeland et al. (2014). This is an open, wind-protected, easterly-facing meadow at an elevation of 2700 m. We conducted experiments on layers of buried surface hoar in this area during the 2014/15, 2015/16, and 2016/17 winters. The second field site is near the summit of Mount Ellis, about 12 km southeast of Bozeman (45°34′43.06″N, 110°57′18.60″W). This site is a relatively uniform, mostly wind-pro- tected northeasterly-facing meadow at an elevation of 2500 m, and we conducted experiments on a layer of depth hoar here during the 2015/
16 winter. Both sites contain abundant terrain with slope angles from about 15° to 25°, enabling safe access during return visits and changing conditions.
2.2. Field data collection
Our field work involved two separate sets of experiments. The first focused on how changes in the amount of loading affect the mechanical properties associated with crack propagation at the temporal scale of minutes. For the second set we kept the added load constant and in- vestigated the temporal effects by sampling in the first minutes, hours and days after loading to examine how various mechanical properties change.
2.2.1. The short-term effect of loading on mechanical properties
Based on the work ofHeierli et al. (2008)andGaume et al. (2017), we hypothesize that increasing load – which increases stress – should result in decreasing critical crack length (rc) over short time scales since both the critical value for the energy release rate (and thus weak layer specific fracture energy (wf)) and slab elastic modulus (Eslab)may not change substantially over short time periods.
We collected data on 11 days over two winters at the two field sites
to investigate how changes in loading affect mechanical properties. In order to add load to the snowpack in a semi-controlled manner, we built a cardboard frame 100 cm long, 30 cm wide and 25 cm tall and we marked the inside walls of this frame at 5 cm increments. We placed this frame on the snowpack and gently pressed it 5 cm into the snow by pushing it down to the first line marked on the inside of the frame. We then added 5, 10, 15, or 20 cm of disaggregated snow into the frame (Fig. 2a).
An effective technique to add snow involved cutting blocks out of the lower snowpack, holding them in one hand over the frame, and rubbing them with our other hand to disaggregate the grains which then fell on the snow inside the frame (Fig. 3). Initially we attempted to sieve the snow, but this proved to be too time consuming for the vo- lumes of snow required. These blocks of snow from the lower snowpack typically consisted of predominantly rounding depth hoar grains, with a Fig. 1.The Bacon Rind and Mount Ellis field sites are located south of Bozeman,
Montana, USA.
Fig. 2.(a) Fieldwork for our first set of measurements, focused on the effect of adding varying amounts of load on mechanical properties, involved adding additional load to the snowpack with a cardboard frame. Here we have added slabs of 10, 15, and 20 cm to the snowpack (from right-to-left) and we are preparing the 10 cm slab for a PST. (b) Fieldwork for our second set of measurements, focused on investigating temporal changes mechanical properties after loading, involved adding 10 cm of disaggregated snow to the existing snowpack on over 30 columns and then isolating the columns with a snow saw. As shown, we kept about 10 to 15 cm of snow around the columns to help protect the layers from changes in temperature and incoming radiation.
Fig. 3.The most efficient technique for adding load to the snowpack proved to be adding disaggregated grains by hand to a cardboard frame, and then al- lowing those grains to sinter prior to testing.
K.W. Birkeland et al. Cold Regions Science and Technology 159 (2019) 142–152
small proportion of rounded grain snow. In some cases we stamped with our feet on depth hoar to thoroughly disaggregate the grains, and then added the disaggregated grains with a shovel. In this manner we could quickly add 5, 10, 15 or 20 cm of relatively uniform snow into the frame. We then let this snow sit inside the frame and sinter for more than 5 minutes before gently cutting around the added snow with a snow saw to remove the frame. We allowed the added snow to sinter for at least 30 minutes and sometimes more than an hour before completely isolating the column and conducting the PST. Since disaggregated snow rapidly sinters within the first hour (van Herwijnen and Miller, 2013), there were likely some differences in hardness of the added snow due to the inconsistent sintering times.van Herwijnen and Miller (2013)found similar sintering rates for disaggregated depth hoar and rounded grains, and these were the grains that we typically disaggregated when adding load.
After allowing the added snow to sinter, we performed Propagation Saw Tests (PSTs) (Greene et al., 2016) on the columns with 5, 10, 15, and 20 cm added as well as a column with no added load. Thus we conducted five tests on each of our 11 sampling days for a total of 55 PSTs for this part of our study. We conducted our tests within < 5 min of the time we completely isolated the columns. We also placed plastic markers in the snowpack and filmed the PSTs with a high speed video camera (120 frames per second at 640 × 480 pixel resolution) for later particle tracking velocimetry (PTV) analysis (van Herwijnen et al., 2016b). In all cases, we cut the ends of the PST normal to the slope to facilitate PTV analysis. The critical crack length,rc, was obtained from the videos. See section 2.3 for more detailed information on the PTV analysis.
Following each PST, we measured the load of the snow above the weak layer by inserting a Snowmetrics density tube (5.5 cm in diameter and 30 cm long with an estimated accuracy of 1–3% (Snowmetrics, personal communication, 2018)) slope normal to the snow surface down to the weak layer in three different places along the fractured block and averaged the three measurements. For slabs deeper than our 30 cm sampling tube, we took two measurements and summed them to obtain a sample from the entire slab. The stiff added slab overlying softer snow below created some challenges to our density measurement technique since occasionally the stiffer snow would get stuck in the tube.
On some field days soft snow from the original snow surface col- lapsed while adding snow in the frame. In these cases, it was difficult to tell if the buried weak layer was affected by the collapse above and we did not use those measurements. On two days we mitigated this by placing the frame and then gently compressing the existing snow sur- face layers with a shovel prior to adding snow. We then pushed the frame down gently and began adding the additional disaggregated snow. This process clearly increased overall slab density, slab hardness, and slab elastic modulus on those sampling days, but our technique was appropriate because we treated each test on a given day the same.
On each sampling day we also collected a manual snow profile as outlined inGreene et al. (2016).
2.2.2. Temporal changes in mechanical properties following loading To investigate how mechanical properties change following a loading event, we conducted tests on four days over a five day period in January 2017 at our Bacon Rind study site (Fig. 1). The upper layers of the snowpack consisted of 14 cm of soft (Fist hardness), low density (averaging 135 kg/m3), new snow (consisting of stellars (PPsd inFierz et al. (2009)), decomposing stellars (DFdc), and some small near-sur- face facets (FCsf)) overlying a layer of 5 mm surface hoar. On top of this natural slab we added 10 cm of disaggregated snow using the cardboard frame and methods discussed above (Fig. 2b). We allowed the snow to sinter for about 5 min before carefully cutting around the cardboard with a snow saw and then gently removing the frame. We then isolated the column for our PST completely by cutting around all four sides to well below the surface hoar layer with a snow saw, noting the block
location and time of isolation in our field book.
We conducted a total of 33 PSTs with the time between the isolation of the column and the cutting of the weak layer ranging from 15 min to four days. All tests propagated to END except for the 15 min test which slab fractured (SF) due to insufficient sintering of the added slab. We analyzed this test with the rest of our data sincevan Herwijnen et al.
(2016c)showed there is no significant difference between SF and END when measuring critical crack length. We removed one test that our video showed was incorrectly cut outside the weak layer, resulting in an anomalously long critical crack length. Thus, for our analyses we had a total ofn= 32 tests. We also conducted tests without adding load at the beginning and end of our sampling, both of which resulted in slab fractures (SF).
One challenge we faced was ensuring the blocks were isolated, while still minimizing any external influences on the snow in the iso- lated columns we planned to test. After isolating the blocks with a saw, we removed most of the snow around the columns on three sides but we retained about 10 to 15 cm of snow around all the edges (Fig. 2b). This technique helped to protect the sides of our columns from air tem- perature changes, incoming radiation, and other external factors. Each morning when we returned to the study site we would run a snow saw around each column that we had not yet tested to ensure the snow in our columns was not sintering to the surrounding snow. Since we could easily run our saw around the columns, we are confident that the col- umns stayed effectively isolated for the duration of our field work. A high pressure system led to stable weather during our sampling period, resulting in daytime temperatures at our field site staying below −2 °C, partly sunny to mostly overcast skies, light winds, and < 1 cm of total new snowfall over the five day period.
In the minutes, hours, and days following the loading of the snowpack, we conducted PSTs as described above, including the PTV analysis (see Section 2.3). After completing a PST we used a Wasatch Touring 100 sharpened metal tube (measuring approximately 3.8 cm in diameter and 9.2 cm long) to measure the density of the added slab (in three places along the column) and the density of the snow between the surface hoar weak layer and the added slab (in one place). The metal tube was effective for density measurements of the well-sintered added slab, which exceeded Pencil hardness by the end of the sampling period.
Conger and McClung (2009)found this density cutter under sampled by about 1%.
We collected a manual profile of the upper snowpack layers on the first and last days of our sampling following the methods outlined in Greene et al. (2016).
2.3. Particle tracking velocimetry
We utilized a particle tracking velocimetry (PTV) algorithm (Crocker and Grier, 1996) to determine the displacement of the markers during the experiments and derive several fracture mechanical para- meters (Fig. 4).van Herwijnen and Jamieson (2005)pioneered the use of PTV to investigate weak layer fracture in snow. For a recent review, seevan Herwijnen et al. (2016b), who summarized and built on nu- merous studies using PTV. For this work, we utilized the video re- cordings of the experiments to determine the critical crack length (rc) from the videos by locating the position of the saw in the last image prior to rapid crack propagation (Fig. 4a). Following van Herwijnen et al. (2016c), we also computed the mechanical energy of the system from the measured displacement field and fitted that to an adjusted analytical expression ofHeierli et al. (2008)to determine the elastic modulus of the slab and the weak layer specific fracture energy (Fig. 4e). Finally, during crack propagation (Fig. 4d), there is a delay between the slope normal displacement of subsequent markers. The time delay between the onset of movement between markers is pro- portional to the distance between the markers and was used to calculate the crack propagation speedc(Fig. 4f), as outlined in van Herwijnen and Jamieson (2005).
2.4. Modelling of the propagation saw tests 2.4.1. Model description
We used a modelling approach to investigate the mechanisms leading to observed changes in critical cut length. To assess the pro- pensity of the slab/weak layer configuration to support spontaneous crack propagation, we calculated the critical crack lengthrcfollowing techniques developed byReuter et al. (2015)andReuter and Schweizer (2018). Based on the work of Heierli et al. (2008), Schweizer et al.
(2011) derived an expression for the specific fracture energy, which was solved forrc:
= + + + +
w E r D
E w w r
D w r
D w r
D w r
( , ) D
2 ,
slab
f c
slab 0 1 c
2 c 2
3 c 3
4 c 4
(2)
with = = + + +
= + + = =
( )
w w w
w w
, 3 ,
3 , 3 , 3 ,
0 3
4 2
1 3
2 2 2 2
2
2 9
2 2 2
3 2
4 2
2
whereD is the slab thickness,τ= −ρ g Dsin(α) is the shear stress, σ= −ρ g Dcos(α) is the normal stress, = 4(1+ )/5 ,γ= 1, and ν= 0.25. We derived an elastic modulus of the slabEslab– taking into account the slab layering – for the situation of a layered snow slab bending over the edge of a rigid substrate. To obtainEslabwe fit an expression for the mechanical energy to finite element derived values as described byReuter et al. (2015). We used the expression for the me- chanical energy presented byvan Herwijnen et al. (2016c), including the correction factor for the slope angle and the ratio of crack length to slab thickness.
2.4.2. Model input and settings for added load dataset
Snow layer density and thickness comprised our primary inputs for deriving the variables needed to run the model. We used the measured
average density and observed stratigraphy according toGeldsetzer and Jamieson (2001)to derive layer densities. We drove simulations for the first set of field experiments – where we looked at how added load changed mechanical properties – with an elastic modulus for the original slab derived from snow density for each snow layer according to the re- lationship ofScapozza (2004). For our artificially added layer, we adjusted the elastic modulus, E (Pa), by extendingScapozza's (2004)relationship with a time-dependent sintering law based onGerling et al. (2017):
=
( )
E e t
1.8 105 67 t
0 0.243
(3) with densityρ(kg m−3), time since adding the top slab layert(s), and, sinceScapozza (2004)sampled 1-week old snow,t0= 604,800 s. We used a timet= 3600 s between the slab building and the testing because this was approximately the average time for our experiments.
The fracture energy was assumed constant for each day, and was computed from the first – and unloaded – measurement for that day using Eq.(2).
2.4.3. Model input and settings for temporal change dataset
To drive the simulations for the second set of field experiments – where we investigated temporal changes in mechanical properties fol- lowing loading – we essentially had a two layer slab above the weak layer. The original, natural slab overlaid the weak layer, with the added slab as the topmost layer. We measured the density of both the original slab and the added slab for each test. We usedScapozza (2004)for the modulus of our original slab, and we used Eq.(3)to account for time- dependent sintering for the modulus of the added slab.
Similar to the slab elastic modulus, we also assumed the specific fracture energy of the weak layer,wf, increases over time according to a power law. We used the empirical relationship for shear strength found byConlan and Jamieson (2016)and approximated the fracture energy Fig. 4.PTV analysis to derive mechanical quantities from PST experiments. (a) Displacement field (arrows) at the last frame before crack propagation. Colors correspond to the displacements curves shown in c and d. The position of the saw was determined to obtain a value of the critical crack length rc. (b) Displacement field after crack propagation. (c) Slope parallel displacement with time. The red and green dashed lines indicate the frames shown in a and b. The red dots represent the key frames at which the saw cut length r was estimated from the images to derive the mechanical energy shown in e. (d) Slope normal displacement with time around crack propagation. The dark blue dashed lines show the displacement threshold range used to derive the crack propagation speed shown in f. (e) Derived mechanical energy with crack length r. The black line represents the best theoretical fit to the experimental data to derive slab elastic modulus and weak layer specific fracture energy, and the black dashed-dotted lines the 95% confidence interval. (f) Crack propagation speed c with slope normal displacement threshold. The dark blue dots show the values of c used to obtain a mean value.
K.W. Birkeland et al. Cold Regions Science and Technology 159 (2019) 142–152
according toGaume et al. (2014)(Eq. B7):
= +
w r
D E
t t
f c t
slab2 f a
0 0.26 2
(4) wherercis the measured critical crack length,Dthe slab depth,Eslabthe elastic modulus of the slab,σfthe initial weak layer failure strength,tis the time between adding the top slab layer and the time of the test (d), tathe age since weak layer burial (d) (in our case 7 days), andt0is 1d.
We calculated a value forEslabthe same way as for the added load dataset. Since we did not have values for the initial strength or density of the weak layer, we chose an initial weak layer strength of 580 Pa, which was reasonable given our estimated density (150 kg-m−3) and the data ofJamieson and Johnston (2001).
3. Results
3.1. The effect of increasing load on mechanical properties 3.1.1. Data overview
We collected data to assess how changes in the amount of loading affect mechanical properties on 11 sampling days over two winters at our two field sites (Table 1;Fig. 1). We conducted all tests on slopes ranging from 20° to 27°, and every test had a weak layer of either surface hoar or depth hoar. Our depth hoar measurements were all on the same depth hoar layer. The initial slope normal slab depth ranged from 12 to 40 cm with average slab densities ranging from 157 to 246 kg-m−3. Adding disaggregated snow increased average slab den- sities; after loading 20 cm of disaggregated snow our average slab densities ranged from 253 to 340 kg-m−3(Table 1). On average, adding 5, 10, 15, and 20 cm to the snowpack was equivalent to adding a load of 0.21, 0.43, 0.65, and 0.79 kPa.
For our PTV analyses, we found that occasionally the added slab fractured on the original surface snow immediately following weak layer fracture. This caused snow to fall in front of the markers, making assessing all of the mechanical properties difficult or impossible, similar to challenges faced byvan Herwijnen and Birkeland (2014). Further, in a few cases the observer was in front of some of the markers during cutting, preventing calculations of the slab elastic modulus or the weak layer specific fracture energy. In total, we did five tests on each of 11 days for a total of 55 tests. While in all cases we could measure values for critical crack length and fracture speed (n= 55), we could only derive values for slab elastic modulus or weak layer specific fracture energy with our PTV analyses in about half of the tests (n= 25). Consequently, our modelling approach became important for estimating slab elastic modulus and weak layer specific fracture energy because the model filled in these missing values.
3.1.2. The effect of increasing load on critical crack length, weak layer specific fracture energy, slab elastic modulus, and fracture speed
Measured critical crack length decreased dramatically when load was added to the existing snowpack (Fig. 5a). In addition, the model's simulated critical crack lengths (rc) matched our field observations well, with an R2= 0.62 and p < 0.01 (Fig. 6). These results give us con- fidence that the model may provide useful insights about changes in weak layer specific fracture energy (wf) and slab elastic modulus (Eslab) during our tests.
Both measured and modelled critical crack lengths decreased dra- matically when load was added to the existing snowpack (Fig. 5a and b). We illustrate data trends by graphing the data for each day and fitting a linear least squares line. The lines are general approximations because they are based on a small sample size (a single day with n = 5).
Despite this limitation, both the measured and modelled data clearly show critical crack lengths generally decreasing as load increases.
Other variables that might affect critical crack length as load increases are changes in the weak layer specific fracture energy and the slab elastic modulus. Our field data for specific fracture energy and slab elastic modulus were limited and did not show any significant increasing or de- creasing trends (p > 0.05 for our all data grouped together) (not shown).
For the model, weak layer specific fracture energy was constant as we did not anticipate substantial changes over the short time scales of these ex- periments. Our results show no obvious trends in the modulus during our experiments, with most least squares fit lines nearly flat (Fig. 5c). Though one would expect a clear relationship between the average density (Table 1) and the slab modulus, the slab modulus depends on the modulus of the added part of the slab and the original part of the slab. Since the added part of the slab had insufficient time to sinter, the slab modulus is still largely controlled by the modulus of the original slab.
Adding load resulted in higher average slab densities, and these higher slab densities also strongly correlated with increases in fracture speed (c) (R2= 0.29, p < 0.01) (Fig. 7). When taken on its own, we did not find a significant correlation between the modulus and the fracture speed (R2< 0.01, p = 0.64) nor did we find a significant re- lationship with weak layer type (Spearmanr= 0.17, p = 0.21).
3.2. Temporal changes in mechanical properties following loading 3.2.1. Data overview
We collected data to investigate temporal changes in mechanical properties following loading on a gentle (14° slope angle), southeast- facing section of our Bacon Rind Field Site (Fig. 1). We collected data on four days during a five day period in January 2017. We loaded the existing snowpack (described in Section 2.2.2) with 10 cm of dis- aggregated snow, isolated the 32 columns, and conducted.
PSTs; the elapsed time between isolating the column and testing it ranged from 15 min to more than 5600 min (approximately four days) (Table 2). Critical crack lengths ranged from 4.5 to 25.8 cm in our Table 1
Overview of the field data for investigating the effect of adding varying amounts of load on mechanical properties, including the date, field site, slope angle, type of weak layer (SH = surface hoar, DH = depth hoar), height of the original slab measured normal to the snow surface (D), average density of the original slab (ρ), and the average density of the slab when 5, 10, 15, and 20 cm of disaggregated snow was added to the original slab (ρ+5cm, ρ+10cm, ρ+15cm, and ρ+20cm).
Day Date Field site Slope (deg) WL D (cm) ρ (kg-m−3) ρ+5cm ρ+10cm ρ+15cm ρ+20cm
A 27 Jan 2015 Bacon Rind 22 SH 12 157 215 250 284 314
B 30 Jan 2015 Bacon Rind 23 SH 18 170 182 214 238 338
C 12 Jan 2016 Mount Ellis 20 DH 34 214 259 263 288 287
D 15 Jan 2016 Mount Ellis 21 DH 40 215 258 274 285 281
E 27 Jan 2016 Mount Ellis 24 DH 27 195 212 230 249 253
F 5 Feb 2016 Mount Ellis 23 DH 37 213 254 264 279 295
G 9 Feb 2016 Mount Ellis 24 DH 35 246 269 265 276 N/A
H 19 Feb 2016 Mount Ellis 26 DH 32 245 267 302 313 306
I 22 Feb 2016 Bacon Rind 27 SH 34 185 219 244 263 283
J 26 Feb 2016 Bacon Rind 23 SH 23 197 245 278 299 293
K 9 Mar 2016 Bacon Rind 23 SH 33 213 278 296 318 340
different tests.
The measured density of our added snow ranged from 313 to 397 kg-m−3, with an average of 354 kg-m−3(Table 2). Consequently, the average load added by our artificial slab was 0.35 kPa. The added slab quickly hardened, with a hand hardness of P- after only 30 min, P after about 90 min, and P+ or harder after 24 h. Thus, the snow we ultimately tested was 10 cm of hard (generally P hardness), dense (generally 350 kg-m−3) snow over about 14 cm of soft (F hardness), low density (around 150 kg-m−3) snow on top of a layer of 5 mm surface hoar. This is an optimal stratigraphy for crack propagation (Schweizer et al., 2011).
3.2.2. Temporal changes in critical crack length, weak layer specific fracture energy and slab elastic modulus following loading
Measured critical crack lengths increased following loading, with a
rapid increase in the first few hours after loading followed by a more gradual increase in the following days (Fig. 8). The model showed a similar trend, and reasonably reproduced some of the spread seen in our measurements during the initial hours following loading. For example, considerable spread exists in the two measurements made around 0.5 h, but the model reproduces the observed difference well (Fig. 8b). In this case the average density of the added slab for the shorter critical crack length was higher (tests #2 and #3 inTable 2), so the greater added load drove the shorter critical crack length.
Simulated critical crack lengths matched up well with our measured values (R2= 0.98, p < 0.01) (Fig. 9). Note that this match does not represent the ability of the model to reproduce critical crack lengths from other field measurements. Indeed, we used ideal parameteriza- tions and model settings for our data. Interestingly, the model could only adequately simulate changes in critical crack length if we took into Fig. 6.Comparison of modelled and measured critical crack length for our
added load experiments (red symbols = two surface hoar weak layers and black symbols = a single depth hoar weak layer that was sampled on several days).
(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7.Fracture speed increases with increasing density (R2= 0.29 and p < 0.01). The color of the symbols represent weak layer crystal type (SH = surface hoar and DH = depth hoar) and the size of the symbols re- presents slab elastic modulus (ranging from a minimum of 1.0 MPa to a max- imum of 6.4 MPa).
Fig. 5.As load increased, both (a) measured and (b) modelled critical crack length decreased and (c) modelled slab elastic modulus remained largely unchanged. The colors represent individual sampling days, the dots are the data, and the lines are a least squares fit for the data for each day.
K.W. Birkeland et al. Cold Regions Science and Technology 159 (2019) 142–152
account changes in both weak layer specific fracture energy and slab elastic modulus. If we only considered increases in slab modulus (i.e., assuming a constant fracture energy), the model underestimated critical cut lengths with an increasing disparity between our field data and the modelled values at higher cut lengths (results marked as “S” inFig. 9).
Further, if we only considered weak layer strengthening (i.e., dropping the time dependency term in Eq.(3)) the modelled critical cut length values were systematically overestimated (results marked as “W” in Fig. 9).
Since the temporal change of both weak layer specific fracture en- ergy and slab elastic modulus in our model followed power laws (Eqs.
(3) and (4)), our modelled values for these parameters increased rapidly in the first hours of loading and then more slowly over the following days (Fig. 10).
These model result differed from trends observed in our PTV mea- surements. Our PTV results indicated a great deal of variability in weak layer specific fracture energy in the first hours after loading, including both our lowest and our highest values, and after that it is unclear if there is any increasing or decreasing trend (Fig. 10a). The slab elastic modulus rose rapidly in the first few hours; after that rapid rise it is difficult to discern any trends given the uncertainty in our measure- ments (Fig. 10b).
4. Discussion
4.1. The effect of increasing load
Our added load results quantitatively demonstrate the short term effects of increasing load on crack propagation. Increased loads result in decreased measured and modelled critical crack lengths (Fig. 5a and b),
so shorter initial cracks in the weak layer are required for the onset of rapid crack propagation. In addition, we observed increasing fracture propagation speed with added load. This was strongly correlated with the increase in the overall slab density (Fig. 7), a finding consistent with previous work (van Herwijnen and Birkeland, 2014; Gaume et al., 2015;van Herwijnen et al., 2016b). We suspected that the slab elastic modulus could influence fracture speed by providing the deformation energy during crack propagation, but we did not find a statistically significant relationship between the two (Fig. 7). Theory proposed by Heierli (2005)suggests that the relationship between fracture speed, slab density and slab elastic modulus is:
=
c E
H (5)
wherecis the fracture speed,Eis the slab elastic modulus,ρis the slab density, and H is the slab thickness. Indeed, when we statistically compare fracture speed with the square root of the ratio of the elastic modulus to slab density in our dataset, we get highly statistically sig- nificant results (R2= 0.40, p < 0.001).
We also expected that fracture speed might be related to weak layer properties, with weaker structures supporting higher fracture speeds.
For our data this was not the case. The surface hoar was weaker than the depth hoar layers, yet the range of fracture speeds between the two weak layers was similar (Fig. 7) (p = 0.21).
The decreasing critical crack lengths we observed could also be affected by other factors besides the load, including changes in the elastic modulus of the slab or in the weak layer specific fracture energy (Gaume et al., 2017). However, since the time delay between loading, column isolation, and testing was short, we assumed a constant weak layer specific fracture energy during each day's field experiments.
Table 2
Overview of the field data for investigating temporal changes in mechanical properties after loading. All of the data are from the Bacon Rind Field site, and the weak layer was surface hoar buried 14 cm deep with a 10 cm added slab for a slope normal Hslabof 24 cm. We collected the data on four days over a five day period. The added average slab density was calculated using three density measurements, and values for critical crack length (rc), slab elastic modulus (Eslab), weak layer specific fracture energy (wf), and crack propagation speed (c) are from our PTV analyses. Cases with “N/A" are where we were unable to extract values from the PTV. All tests except for Test #1 propagated to END.
Test Date of test Time since column isolation (min) rc(cm) Eslab(MPa) wf(J-m−2) c (m/s) Average added slab ρ (kg-m−3) Total average slab ρ (kg-m−3)
1 28 Jan 2017 15 13.3 0.53 0.30 3.8 313 218
2 28 Jan 2017 29 5.3 N/A N/A 31.8 370 233
3 30 Jan 2017 32 12.3 0.73 0.21 12.3 337 219
4 28 Jan 2017 55 6.6 N/A N/A 7.5 353 226
5 26 Jan 2017 59 4.7 0.10 0.55 4.7 357 227
6 27 Jan 2017 59 5.0 0.88 0.07 5.0 377 236
7 30 Jan 2017 59 8.9 0.24 0.40 8.9 338 220
8 26 Jan 2017 74 8.1 0.19 0.55 8.1 333 218
9 28 Jan 2017 87 14.0 0.72 0.31 14.0 397 244
10 26 Jan 2017 116 5.9 0.32 0.18 5.9 360 229
11 30 Jan 2017 124 11.5 0.47 0.30 11.5 350 225
12 26 Jan 2017 128 4.5 0.14 0.38 4.5 350 225
13 30 Jan 2017 182 9.5 0.52 0.27 9.5 353 226
14 26 Jan 2017 200 8.7 0.34 0.28 8.7 333 218
15 26 Jan 2017 212 6.8 N/A N/A 6.8 350 225
16 30 Jan 2017 344 14.5 1.04 0.21 14.5 375 235
17 28 Jan 2017 1376 13.5 0.83 0.24 13.5 363 230
18 27 Jan 2017 1387 8.8 0.15 0.63 8.8 357 227
19 27 Jan 2017 1394 18.8 0.86 0.43 18.8 360 229
20 27 Jan 2017 1403 17.4 N/A N/A 17.4 337 219
21 28 Jan 2017 2858 18.3 1.13 0.34 18.3 363 230
22 28 Jan 2017 2872 12.9 1.18 0.13 12.9 333 218
23 30 Jan 2017 2875 20.2 1.03 0.48 20.2 363 230
24 28 Jan 2017 2893 6.8 N/A N/A 6.8 348 224
25 28 Jan 2017 2969 15.6 0.27 0.90 15.6 353 226
26 30 Jan 2017 4398 24.0 2.47 0.28 24.0 362 229
27 30 Jan 2017 4437 21.3 1.17 0.45 21.3 368 232
28 30 Jan 2017 4443 24.1 1.68 0.42 24.1 373 234
29 30 Jan 2017 5612 24.0 1.99 0.32 24.0 357 227
30 30 Jan 2017 5630 25.8 2.08 0.37 25.7 347 223
31 30 Jan 2017 5643 21.1 2.50 0.18 21.1 363 230
32 30 Jan 2017 5652 21.1 0.86 0.50 21.0 343 222
Modelling the crack lengths based on that assumption, we do not find a change in slab elastic modulus with increasing load (Fig. 5c). Since a relatively consistent elastic modulus does not change critical crack lengths, it appears that in our experiments it is primarily the load that drives the observed decreases in critical crack length.
When comparing our model results to our measured values of cri- tical crack length, there are three primary outliers where the modelled value of critical crack length greatly exceeded the measured values. All of these outliers are from the same sampling day, 9 March 2016. On this day the initial crack length (prior to loading) was 0.7 m, resulting in a modelled weak layer specific fracture energy for that day more than two times greater than any other day. Since the critical crack length is modelled based on the fracture energy, all modelled values for this day are high. It is not surprising that these results are inconsistent with our other measurements because the model cannot adequately characterize cases where critical cracks are longer than about one third of the PST column length. In fact,Bair et al. (2014)found that in order to elim- inate the far edge effect in PSTs that the crack to beam length ratio should be < 0.20 andvan Herwijnen et al. (2016c)showed that col- umns should be longer than three times the critical crack length since the slab deforms up to a distance of at least two times the critical crack length. For our data collection we elected to use the “standard” PST beam length of 1 m (Gauthier and Jamieson) for two reasons: 1) Longer beam lengths would have taken considerably more of our already Fig. 8.Changes in measured (gray squares) and modelled (black dots) critical
crack lengths over time following loading. In (a) we plot time linearly to show the rapid increase in crack length in the initial hours after loading, followed by much slower increases between days one and four. In (b) time is plotted loga- rithmically to better visualize the data in the early hours of data collection.
Fig. 9.Comparison of measured and modelled critical crack length for our temporal change experiments. Our model took into account both slab stiffening and weak layer strengthening (dots). Model results only taking slab stiffening into account are represented by “S” and model results only taking into account weak layer strengthening are represented by “W”.
Fig. 10.Changes in modelled and measured values of (a) weak layer specific fracture energy and (b) slab elastic modulus over time. Dots represent the model results for individual tests, lines are fit to the model data, and crosses represent the PTV measurements. Data are normalized by the first measurement in our time series. The error bars are based onvan Herwijnen et al. (2016c)and are equal to +/− 16% for weak layer fracture energy and +/− 25% for slab elastic modulus.
K.W. Birkeland et al. Cold Regions Science and Technology 159 (2019) 142–152
limited field time to complete due to the sifting of the snow in our artificial slabs, and 2) We wanted to use a consistent column length with all of our tests, and for both of our experiments we expected short critical crack lengths.
4.2. Temporal changes following loading
While we designed the added load part of our study to look at loading over short time frames, our temporal change results aimed to quantify changes following loading. Observations of avalanches fol- lowing loading clearly show a pattern of decreasing avalanche activity through time if no further loading occurs (e.g., Seligman (1936);
Atwater and Koziol (1952)).
In our experiment we added snow quickly, and the new load sin- tered into a stiff slab sitting above the existing slab and the underlying weak layer of surface hoar. Over the first several hours critical crack lengths rapidly increased, followed by more gradual changes in the following days (Fig. 8). Our PTV measurements suggest little change in weak layer specific fracture energy over the days of our experiment, though considerable uncertainty exists (error bars inFig. 10a). A con- sistent weak layer specific fracture energy following loading would be contrary to previous work demonstrating increasing weak layer shear strength following loading (e.g., Conlan and Jamieson (2016)). Our PTV measurements also showed a temporal trend in slab elastic mod- ulus, with a rapid initial increase caused by the sintering of the added slab (Fig. 10b). Following this initial increase the temporal trend is not clear given the uncertainties involved in our measurements, but our hand hardness measurements showed relatively rapid hardening of the slab as the hand hardness increased to P+.
We used our model to investigate how we expect the different parameters to change. The slab elastic modulus and the weak layer specific fracture energy increase in our model according to the re- lationships described in Section 2.4. An increase in the modulus makes sense because disaggregated grains – such as the ones we used to add load to our columns – rapidly sinter and strengthen into harder, stiffer layers (Szabo and Schneebeli, 2007;van Herwijnen and Miller, 2013;
Gerling et al., 2017). Likewise, we expect weak layer specific fracture energy will increase with time following load since a great deal of re- search shows that shear strength increases as load increases over time periods from days to months (Jamieson, 1995;Schweizer et al., 1998;
Jamieson and Johnston, 1999;Chalmers and Jamieson, 2001;Chalmers and Jamieson, 2003; Zeidler and Jamieson, 2006b; Zeidler and Jamieson, 2006a; Jamieson et al., 2007; Logan et al., 2007; Horton et al., 2014;Conlan and Jamieson, 2016). Finally, in order to effectively simulate the decreasing critical crack length over time with the model, we needed to take into account both weak layer strengthening and slab stiffening since leaving out either of these two processes prevented us from making realistic simulations (Fig. 9). Thus, our best evidence from the model is that both weak layer specific fracture energy and slab elastic modulus are driving our observed increases in critical crack length after loading.
There are inconsistencies between our PTV measurements and our model results (Fig. 10). In general, the PTV results suggest that stif- fening is more important than we have modelled and that the strengthening of the weak layer is less important than we have mod- elled. However, for the model we used the best available literature parameterizations for the two processes. Further, to increase the effect of slab stiffening would require increasing the modulus of the added top layer of the slab to unrealistically high values. Nevertheless, the loading may have affected the modulus of the original slab. We did consider this in the model because we measured the changing density of this layer in the field, but it is possible that other microstructural changes in the layer are affecting the modulus. In the end, our best evidence suggests that both processes are contributing to the changes in critical crack length over time. More clearly understanding how much each of the processes contributes to the observed decreases in critical crack length
will require the collection of additional temporal change datasets from the field, possibly with more detailed measurements of mechanical properties. In addition, lab experiments would also be useful for better understanding how microstructural changes affect fracture mechanical properties.
5. Conclusions
We developed a simple field technique to assess both the short term effect of loading and the temporal changes following loading on crack propagation as measured by changes in the critical crack length in PSTs.
This technique involved adding disaggregated snow on top of an ex- isting snowpack with a previously buried weak layer. In addition to our field data, we utilized a model to study the driving processes behind our field observations. In short, our results demonstrate:
- Over short time frames (< 1 h), increasing load leads to a decrease in critical cut length. Our field data and model results suggest that the decreasing critical cut length is largely due to the increase in load rather than changes in slab elastic modulus or weak layer specific fracture energy.
- Over longer time frames (15 min to four days), critical cut length increases in a power law following loading, with a rapid initial in- crease followed by a more gradual increase. The initial rapid in- crease appears to be primarily related to the sintering of the added slab, while the more gradual increase over longer time scales is a combination of slab and weak layer strengthening.
- With additional load we also observed an increase in fracture speed.
Consistent with recent theory, fracture speed was highly correlated with the square root of the ratio of the elastic modulus to the slab density.
To the best of our knowledge, this work represents the first field- based parametric study of crack propagation in snow, which is affected by the complex interplay of loading, slab stiffness, weak layer specific fracture energy, and time. Our results confirm previous research (Schweizer et al., 2016;Gaume et al., 2017), and may be useful for calibrating future efforts to model crack propagation. Indeed, our model reasonably simulated our field results. For practitioners, our field methods provide a possible technique for safely testing near-surface weaknesses that may not otherwise be fracturing in stability tests. Re- sults might then be roughly extrapolated to areas where the weakness is more deeply buried, such as at higher elevations or in wind loaded areas. In addition, the method may be useful for providing approximate guidance for how an existing snowpack might respond to an anticipated new snow load. Finally, practitioners may be able to couple the model in our study with a snowpack model to help predict stability.
Acknowledgements
Gabrielle Antonioli, Chris Bilbrey, Doug Chabot, Nina Hance, Eric Knoff, Alex Marienthal, and Drew Seessel assisted with fieldwork, and animated field discussions with all of them greatly improved the focus and direction of this work. Kami Crootof's assistance onFig. 1 is ap- preciated. Our paper benefited from the thorough and constructive reviews provided by Johan Gaume and Ned Bair. The Swiss National Science Foundation supported both B. Reuter (Grant P2EZP2_168896) and B. Bergfeld (Grant 200021_169424).
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