Determination of failure envelope for faceted snow through numerical simulations

Cold Regions Science and Technology 116 (2015) 56–64

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Cold Regions Science and Technology journal homepage: www.elsevier.com/locate/coldregions

Determination of failure envelope for faceted snow through numerical simulations Chaman Chandel a,b, Praveen K. Srivastava a,b, P. Mahajan b,⁎ a b

Snow & Avalanche Study Establishment, Plot No 1, Sec-37, Chandigarh, India Applied Mechanics Department, IIT Delhi, Hauz Khas, New Delhi, India

a r t i c l e

i n f o

Article history: Received 18 December 2014 Received in revised form 13 April 2015 Accepted 17 April 2015 Available online 22 April 2015 Keywords: Failure envelope X-ray micro-CT Representative volume element Damage

a b s t r a c t Failure of weak layer in a snowpack, lying on a slope, due to combined compression and shear loading is a major factor in release of slab avalanche. Failure envelopes for FCso and FCsf snow (Fierz et al., 2009) were determined using finite element (FE) modeling. Snow samples of FCso and FCsf from the field were taken to the laboratory and X-ray tomography was performed on these to reconstruct 3D microstructure of each snow type. From the images representative volume elements (RVEs) of each type was constructed. The RVEs were subjected to combined loading to predict the failure envelopes. These failure envelopes were compared to the data published earlier and showed similar trends. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Slab avalanches usually occur due to failure of a weak layer or an interface underlying a strong slab (McClung and Schaerer, 1993). The weak layer can be a depth hoar layer, a near surface faceted particle layer, a buried surface hoar layer or a buried graupel layer in a snowpack. The layers in a snowpack lying on a slope are subjected to simultaneous compressive and shear stresses due to self-weight and additional loads resulting from skier's weight, explosives, earthquakes or due to new snow. If the resultant of these stresses exceeds a limiting value failure can initiate in the pack. To get a qualitative idea about the weakness present in the snowpack, different stability tests such as Rutschblock test (Föhn, 1987), stuffblock test (Birkeland et al., 1996), compression test (Jamieson and Johnson, 1997) and shovel shear test (Tremper, 1994) have been practiced by skiers and avalanche researchers. The initiation of slab avalanche release is caused by the combined compressive and shear loading and due to this reason researchers started to explore the effect of combined loading on snow and avalanches. Perla and Beck (1983) and Zeidler and Jamieson (2006) conducted combined compressive and shear loading experiments and observed an increase in the shear strength with normal compressive load for different snow types. Nakamura et al. (2010) investigated the behavior of rounded polycrystals snow and measured the shear strength for a varying compressive load and reported a linear increase in shear strength with compressive pressure. Reiweger et al. (2010) developed a load controlled apparatus to study the failure of ⁎ Corresponding author at: Applied Mechanics Department, IIT Delhi, New Delhi, India. E-mail address: [email protected] (P. Mahajan).

http://dx.doi.org/10.1016/j.coldregions.2015.04.009 0165-232X/© 2015 Elsevier B.V. All rights reserved.

snow under combined loading conditions. Reiweger and Schweizer (2010) studied the failure of sandwiched surface hoar samples as well as faceted and depth hoar snow samples (Reiweger and Schweizer, 2013) using the same instrument developed by Reiweger et al. (2010). In another recent experimental study, Chandel et al. (2014b) reported failure envelopes for round grain snow (RGsr), faceted snow (FCso) and near surface faceted particles snow (FCsf) using combined compression and shear loading experiments. Podolskiy et al. (2014) developed a portable shear apparatus to assess the strength of snow interfaces under different normal and shear pressures. The focus of the above studies was to estimate the snowpack stability or strength and relating the strength trends to the microstructure or micro-mechanics was not attempted. Lately, direct 3D reconstruction of snow microstructure (Brzoska et al., 1999; Schneebeli and Sokratov, 2004) at resolutions down to a few microns has become possible with X-ray micro-computed tomography (μ-CT). Schneebeli (2004), Srivastava et al. (2010), Yuan et al. (2010) and Theile et al. (2011) applied different numerical techniques on the 3D microstructure of snow to obtain its mechanical properties/behavior. The constitutive behavior of ice determines the mechanical behavior of snow, which in turn helps to understand the slab avalanche release mechanism and hence snowpack stability. Schleef and Löwe (2013) conducted creep experiment on new snow and studied the densification and change in specific surface area (SSA) with deformation. Wang and Baker (2013) conducted compression experiments on different snow types and by using X-ray tomography described the evolution of microstructure. Köchle and Schneebeli (2014) utilized X-ray tomography and FE analysis to demonstrate the contrast in microstructural and elastic properties to identify weak layer present in the snowpack. Chandel et al. (2014a)

C. Chandel et al. / Cold Regions Science and Technology 116 (2015) 56–64

reconstructed the microstructure of RGsr snow using X-ray tomography and numerically simulated the mechanical response of RGsr snow by assigning damage based elasto-plastic constitutive law to the ice matrix of snow. Hagenmuller et al. (2014a) reported that the mechanical response of snow is directly dependent on minimum cut density (MCD) and determined its tensile strength through numerical simulations. Recently failure envelopes for weak snow layers were developed using discrete and finite element modeling (DEM and FEM respectively) from idealized 2D model (Gaume et al., 2014; Podolskiy et al., 2015) under the combined effect of compressive and shear loading. Reiweger et al. (2015) represented the failure criterion for weak snow layers in the normal stress–shear stress by Mohr–Coulomb with Cap (MCC) envelope. For small compressive stresses (corresponding to high slope) the failure envelope was modeled by conventional Mohr– Coulomb criterion. For high compressive stresses (corresponding to small slope) cap region was modeled separately and included in the failure envelope. In the present study, numerical simulations and actual 3D microstructure of FCso and FCsf snow samples are used for determination of failure envelopes under the combined effect of compressive and shear loading. The methodology described by Chandel et al. (2014a) was used to extract the deformation behavior of FCso and FCsf snow samples, and determine their strength under different loading conditions. 2. Micro-CT imaging and Image reconstruction In the experimental study, Chandel et al. (2014b) determined the failure envelopes for two weak layers, FCso and FCsf snow layers. In the present study we decided to carry out the numerical simulations on these two types of weak layers and reproduce failure envelopes. Snow and Avalanche Study Establishment (SASE) has a research station at Patseo in Great Himalayan range at an altitude of 3800 m amsl where a weekly pit study is carried out on horizontal ground to keep an eye on the evolution of snowpack. During pit observations FCso and FCsf snow layers were identified. These layers are very weak as well as very fragile at low-density and could not be extracted from the snowpack easily. Due to this problem of handling the weak layer snow samples, only higher density snow samples were extracted and transported in insulated boxes to environmental chamber SASE (Manali, India) via helicopters, for X-ray scanning. Due to dominance of grain growth direction along gravity, microstructural fabric also shows anisotropic behavior therefore a special precaution was taken during extraction of these samples such that during X-ray scanning, the vertical axis of sample and direction of grain growth coincide. The snow samples of FCso and FCsf were weighed and found to have densities of 412 kg m−3 and 251 kg m−3 respectively. These were next imaged at a temperature of −10 °C using Skyscan 1172 high resolution μ-CT system. During the scanning the specimen rotates with a fixed rotation step up to 180° and at each angular position, shadow or projection images are captured at the detector. The details of the scanning parameters are given in Table 1. From the projection images, cubic volumes were reconstructed using modified Feldkamp cone beam software (NRecon, SKYSCAN). The reconstructed 3D images of snow were gray scale images and have an isotropic resolution of 7.96 μm, resulting in 1185 images of 1185 × 1185 pixels for FCsf snow and isotropic resolution of 10.79 μm, resulting in 774 images of 774 × 774 pixels for FCso snow, which is a

57

very large volumetric data for processing. To ease the problem of processing and increase the size of sample that can be handled, the data was coarsened such that one voxel corresponds to 23.88 μm and 32.37 μm for FCsf and FCso snow respectively. These faceted snow types can have very small structural details (e.g., bonds) which may be poorly represented at coarse resolutions. Therefore to ensure that there is not much variation between the scanned microstructure and coarse resolution microstructure, structural thickness distributions (STD) for both FCso and FCsf snow were determined and plotted (Fig. 1). It was observed from the distributions that for lower structural thickness (which represents small details of microstructure) percentage at fine resolution was higher but difference was not significant. Hence microstructures with coarse resolution were used for numerical simulations. 3. Numerical simulations Snow is a porous material with a solid skeleton or matrix of ice. The mechanical properties/response of snow strongly depends on the morphology of the ice phase distribution in space. Ice volume fraction (φi), specific surface area (SSA), connectivity density (β1V) etc. are the microstructural parameters which are used to estimate the morphology of porous materials (Arns et al., 2002). These microstructural parameters for FCsf and FCso snow samples scanned in the present study are given in Table 2. The values of these microstructural parameters indicate that the matrix of ice for FCsf snow compared to FCso snow is very weak and hence more damage is expected for even very small loading. Analyzing stresses in a snowpack using complete microstructure is numerically prohibitive. Analysis methods, therefore, approximate porous structures by an equivalent homogeneous material (EHM) and the properties of which are derived using a representative volume element (RVE). 3.1. Representative volume element Maugin (1992) suggested that a quantitative relation can be determined by using homogenization methods, where the heterogeneous material is replaced by an EHM. The equivalent continuum is defined in such a way that, in a certain sense, it has the same average mechanical response as the actual heterogeneous material (Nemat-Nasser and Hori, 1993). This equivalent continuum is called a representative volume element (RVE) and for snow Chandel et al. (2014a) have used a cubical RVE. In snow studies, Srivastava et al. (2010) and Chandel et al. (2014a) used consistency analysis to determine the size of RVE with respect to ice volume fraction (V RVE φi ). Kanit et al. (2006) suggested that for heterogeneous porous material any sub-volume size can be used as the RVE such that a sufficient number of realizations are considered to obtain desired precision. Chandel et al. (2014a) used statistical RVE analysis and found that if the size of RVE (i.e., V) = 8 V RVE φi , the standard deviation for ultimate strength reduced by 50% but the error involved remained ≈ 13% for RVE sizes of 1.7913 mm3 and 3.5823 mm3 with 64 and 8 number of realizations respectively. Köchle and Schneebeli (2014) determined the size of the RVE by calculating Young's modulus, for three different snow types, with an assumption that the snow in given volume was isotropic. The RVE calculations were carried out from four different corners and the cube side length was increased

Table 1 CT parameters for scanning snow. X-ray tube

Radiograph acquisition

Volume reconstruction

Snow sample

Voltage (kV)

Current (μA)

Angular displacement step (°)

Exposure time (ms)

Pixel size (μm)

Reconstruction volume (voxels)

FCso FCsf

80 80

100 100

0.3 0.3

1178 1178

10.79 7.96

7743 11853

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C. Chandel et al. / Cold Regions Science and Technology 116 (2015) 56–64

a)

b) 20

20

FCso 7.96 m resolution FCso 23.88 m resolution

16

Frequency (%)

Frequency (%)

16

FCsf 10.79 m resolution FCsf 32.37 m resolution

12

8

4

12

8

4

0

0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Structure thickness (mm)

Structure thickness (mm)

Fig. 1. Structural thickness distribution for a) FCso snow at scanning resolution of 7.96 μm and coarse resolution of 23.88 μm and for b) FCsf snow at scanning resolution of 10.79 μm and coarse resolution of 32.37 μm.

stepwise in all three directions until the entire volume was filled. In present study the size of RVE to determine strength of snow, was selected in a similar way. It was observed that, as the sub-volume size increased, the strength showed convergence behavior similar to the ice volume fraction (Fig. 2). The coefficient of variation (CV) with respect to ice volume fraction for FCso and FCsf snow were less than 20% at cube side lengths of 4.176 mm and 4.72 mm respectively. For these sizes the CV of strength was less than 40%. This relation between the two variances can be explained by the power law relation between strength and density of faceted snow given by Jamieson and Johnston (2001)
B ρ σ ¼A s where B ¼ 2:11 ρi

ð1Þ

By differentiating and rearranging above equation, dσ dρ ¼B s σ ρs Where

ð2Þ

dσ σ

and

dρs ρs

are analogous to coefficient of variation (CV) of

strength and density respectively and indicates that the variation in the strength will be twice the variation in the density. Cubes of length 4.176 mm and 4.72 mm corresponding to strength variance less than 40% were selected as sizes of RVE (V) for FCso and FCsf respectively for numerical simulation. The selected RVE sizes were also in agreement with Hagenmuller et al. (2014b) who reported that the V with respect to MCD for faceted and other snow types lies in the range 33 to 63 mm3.

3.2. Numerical simulations of FCso and FCsf snow samples The total volume that could be reconstructed from the scanned snow samples of FCso and FCsf snow were 8.3533 mm3 and 9.433 mm3 respectively. Both of these volumes were further subdivided into eight non overlapping sub-volumes leading to 8 realizations for each snow sample. The 3D reconstructed microstructure of these sub-volumes thus obtained were further processed to remove the small unconnected regions (islands) and the remaining largest portion of microstructure, which contributes to the stress response of the snow, was selected for further analysis and meshed using eight noded hexahedral and four noded tetrahedral elements. Carney et al. (2006) and Sain and Narasimhan (2011) modeled the brittle failure of ice using a elastoviscoplastic constitutive law combined with damage and used strain rates exceeding 10−3 s− 1. The localized strain rates in the narrow constricted regions in the ice matrix of snow are expected to be at least one order higher than the macroscopic applied strain rate (Mahajan and Brown, 1993). The strain rates assigned to snow to obtain its compressive strength by Chandel et al. (2014a) was 2 × 10−4 s−1 and for tensile strength by Hagenmuller et al. (2014a) was 2.5 × 10−4 s−1. From experimental stress time curve in Fig. 3 of Chandel et al. (2014b), the shear stress rate is approximately 1.5 kPa s−1 which corresponds to the strain rate of 1.5 × 10−4 s−1 considering the shear modulus of snow layer is 10 MPa. Therefore the macroscopic strain rate and damage based elasto-plastic constitutive law (Fig. 3) for ice as described in Chandel et al. (2014a) was used for FE analysis. Here the behavior of ice is considered to be pressure independent and follow the von-Mises J2 plasticity with isotropic hardening. Due to strain hardening the yield stress keeps increasing until maximum yield stress σ yo is reached. Beyond this point, isotropic damage initiates and strain softening occurs.

Table 2 Microstructural parameters for both FCso and FCsf snow samples. S. no.

Snow type

Ice volume fraction φi %

Std Dev φi %

Specific surface area (SSA) (1/mm)

Std Dev SSA (1/mm)

Connectivity density (β1V) (1/mm3)

Std Dev β1V (1/mm3)

1 2

FCso FCsf

45.34 27.22

4.477 3.104

4.82 2.98

0.388 0.327

4.78 1.19

0.68 0.32

Ultimate Strength (kPa)

a)

30

FCso

100

Ultimate Strength Ice Volume Fraction

25

80

20

60

15 40

10

20

5 0

0 0

1

2

3

4

5

6

7

8

9

Ice Volume Fraction (%)

C. Chandel et al. / Cold Regions Science and Technology 116 (2015) 56–64

100

FCsf Ultimate Strength Ice Volume Fraction

25

80

20

60

15 40

10

20

5 0

0

0

1

2

3

4 5 6 Cube Side (mm)

7

8

9

Ice Volume Fraction (%)

Ultimate Strength (kPa)

30

Table 3 Mechanical properties of ice considered for numerical simulation. S. no.

Property

Values

Units

1 2 3 4 5 6

Elastic modulus (E) Poisson's ration (ν) Plasticity hardening modulus (hp) Yield stress (σo) pl Plastic strain at which damage initiates (ε 0 ) Fracture energy release rate (Gf)

950 0.3 95 2.25 0.01 1.05

MPa MPa MPa % J m−2

10

Cube Side (mm)

b)

59

10

Fig. 2. RVE calculations for ultimate strength (left y-axis) and ice volume fraction (ϕi) (right y-axis) vs cube side length. RVE results for one sample each of FCso and FCsf are shown. Each cube was calculated four times, starting from different corners to account for structural variability.

The damage (ϖ) evolves with plastic strain and the material fails when the damage reaches a limiting value of 1. The energy dissipated during the damage evolution process is equal to Gf. The sensitivity analysis for strain softening behavior with respect to mesh density was performed as mentioned in our previous study (Chandel et al., 2014a) and same mesh density is used. The mechanical properties of ice (Chandel et al., 2014a; Gow et al., 1988; Kermani et al., 2008; Podolskiy et al., 2013; Schulson and Duval, 2009; Timco and Frederking, 1982) used in the damage based elasto-plastic model are summarized in Table 3. The ultimate strength of snow determined from numerical simulations depended on the microstructure of snow and the value of yield stress of ice (σo), and the effect of hardening modulus and plastic strain was

negligible due to their negligible contribution (Chandel et al., 2014a). The elastic modulus plays an important role to decide at what rate the stress will increase in relation to the applied strain, but it doesn't affect the value of ultimate strength. Therefore in the present study the value of elastic modulus is selected in agreement with Chandel et al. (2014a). 3.3. Boundary conditions The faces of sub-volumes were subjected to combined compression and shear loading to obtain failure envelope for weak layers. The faces of sub-volume are marked as shown in the Fig. 4 and the faces on opposite side to A, B and C are D, E and F respectively. The simulations were carried out in two steps using explicit analysis in ABAQUS, in the first step the face ‘D’ was kept fixed (Fig. 4) and uniform compressive stress was applied on face ‘A’ in a very short time. Subsequently, in the second analysis step a shear deformation of 2 × 10− 4 s− 1 was applied on the same face in the direction ‘y’ such that the compressive load was simultaneously active. Since ice grains are elasto-plastic the order in which loads are applied on snow can have an effect on strength of snow. A few simulations on FCsf snow were therefore performed with compression and shear loads applied simultaneously. 4. Results and discussions The representative mechanical responses of one of the FCso snow sub-volumes under pure compression loading during first step is shown in Fig. 5a and during second step under combined compressive and shear loading is shown in Fig. 5b. It is observed that in both the steps, the modulus of snow changes. This change in the modulus is explained at two scales, at microstructural scale it is explained in terms of

Shear Loading

Compressive Pressure A

E C

F B z

D

y

Ux=Uy=Uz=0

x Fig. 3. Damage based elasto-plastic model for ice under uniaxial loading, where σ0 is initial yield stress, σyo is the yield stress beyond which damage initiates, ϖ is the damage varipl pl able, ε 0 is the plastic strain at which damage initiates, E is elastic modulus and ε f is the plastic strain at failure (after Chandel et al., 2014a).

Fig. 4. Cauchy's boundary conditions such that one face is assigned zero deformation and on the opposite face, compressive pressure followed by deformation rate causing shear loading is applied.

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C. Chandel et al. / Cold Regions Science and Technology 116 (2015) 56–64

a

b

6

6

5

5

4

4

Shear Stress (kPa)

Compressive Pressure (kPa)

Applied Compression Pressure (5 kPa)

3

2

Applied Compression Pressure (0 kPa) Applied Compression Pressure (1.25 kPa) Applied Compression Pressure (2.5 kPa) Applied Compression Pressure (5 kPa)

3

2

1

1

0 0.0000 0.0005 0.0010 0.0015 0.0020

0 0.000 0.005 0.010 0.015 0.020 0.025 0.030

Strain

Shear Strain

Fig. 5. a) Response of FCso snow due to application of compressive loading b) shear response of snow under the combined effect of compressive pressure and shear loading.

% of elements undergoing damage in ice matrix and at macroscopic scale it is explained in terms of isotropic macroscopic damage (D) (Lemaitre, 1992) in snow. D ¼ 1−

E E0

ð3Þ

where E0 is the Young's modulus of undamaged snow and E is the secant Young's modulus of snow in the deformed state obtained from the compression stress–strain curve in Fig. 5a. If Dc is the damage at end of compression the shear modulus, Gi is given by Gi ¼ ð1−Dc ÞGo

ð4Þ

Where Go is the shear modulus of undamaged snow and subsequent damage is measured in terms of shear modulus (G) degradation from Fig. 5b and obtained using the relation D ¼ 1−ð1−Dc Þð1−Ds Þ

where

Ds ¼ 1−

G Gi

ð5Þ

In the subsequent explanation of observed behavior the magnitude of damage is given in terms of % of elements undergoing damage and in parenthesis macroscopic damage is given. In Fig. 5a, the stress strain graph appears to be linear till 2.5 kPa as the damage is initiated in only 5.9 × 10−2% (D = 0.22) of elements. As the compressive load is ramped up beyond 2.5 kPa to 5 kPa, the percentage of elements in which damage has initiated becomes higher than double leading to nonlinearity in the curve. The shear deformation response of the same sub-volume under the effect of different compressive pressure is shown in Fig. 5b. Damage or failure of small load bearing links in the ice matrix causes drop in the stress and is reflected in the form of kinks in all the curves. Continuous drop after the ultimate strength in each graph indicates failure of the 3D structure of snow. These simulations were repeated for all the eight sub-volumes of each snow type (i.e., FCso and FCsf) and at least five simulations on each subvolume by varying compressive pressure, were carried out such that the effect of compressive pressure could be observed. It was found that the damage in microstructure was induced even with the application of very small compressive pressure but the extent of damage was negligible (Tables 4 and 5).

Table 4 Percentage of elements undergoing damage (in parenthesis, macroscopic damage is mentioned) under combined compression and shear loading conditions for a sub-volume of the FCso snow. Compressive pressure 0 kPa

1 kPa

Shear strain

% of elements undergoing damage (macroscopic damage)

0

0 (0) 0.029 (0.21) 0.119 (0.52) 0.170 (0.7) 0.270 (0.74) 0.401 (0.83)

1.3 × 10

−3

2.7 × 10−3 3.9 × 10−3 5.3 × 10−3 7.8 × 10−3

0.01 (0.14) 0.047 (0.58) 0.153 (0.75) 0.263 (0.86) 0.368 (0.89) 0.528 (0.94)

Remarks 1.25 kPa

2 kPa

2.5 kPa

5 kPa

0.014 (0.16) 0.055 (0.62) 0.145 (0.72) 0.262 (0.85) 0.365 (0.89) 0.522 (0.92)

0.050 (0.2) 0.064 (0.64) 0.132 (0.69) 0.258 (0.77) 0.357 (0.83) 0.504 (0.87)

0.059 (0.22) 0.069 (0.66) 0.123 (0.67) 0.255 (0.74) 0.351 (0.78) 0.492 (0.83)

0.14 (0.53) 0.141 (0.73) 0.247 (0.78) 0.346 (0.8) 0.442 (0.82) 0.592 (0.88)

Just after compressive loading

Bold enteries in the table highlight a decreasing pattern in % of elements undergoing damage (macroscopic damage) with applied compressive pressure.

C. Chandel et al. / Cold Regions Science and Technology 116 (2015) 56–64

61

Table 5 Percentage of elements undergoing damage (in parenthesis, macroscopic damage is mentioned) under combined compression and shear loading conditions for one of the FCsf snow sub-volumes. Compressive load 0 kPa

0.25 kPa

Shear strain

% of elements undergoing damage (macroscopic damage)

0

0.000

1.3 × 10−3

0.009 (0.280) 0.049 (0.368) 0.162 (0.373) 0.351 (0.412) 0.636 (0.472)

2.7 × 10−3 3.9 × 10−3 5.3 × 10−3 7.8 × 103

2.8 × 10−4 (0.001) 0.01 (0.374) 0.051 (0.505) 0.179 (0.596) 0.363 (0.652) 0.682 (0.795)

As the compressive pressure was increased, the damage during compression also increased. However, it was seen that with application of shear loading, the damage growth rate was lower at higher compressive pressures. In Table 4, at a shear strain of 2.7 × 10−3, the percentage of elements that have undergone damage decreased from 0.153% (0.75) to 0.123% (0.69) as the compressive pressure increased from 1 kPa to 2.5 kPa. The initial damage (at zero shear strain) was much higher at 5.9 × 10− 2% (0.22) for 2.5 kPa as compared to 1 × 10− 2% (0.14) for 1 kPa. This trend continued for higher shear strains and this decrease in the extent of damage was observed due to reduction of stresses induced during compression loading in some critical regions on application of shear deformation. This reduction in the net damage leads to increase in the shear strength of snow with applied compressive pressure. At the same time when the compressive pressure of 5 kPa was applied the percentage of elements that underwent damage during compressive loading, increased significantly and hence the shear strength reduced (Table 4, Figs. 5 and 6). It is interesting to observe that although the percentage of elements of ice undergoing damage is very small the damage variable D for snow is close to 1 indicating that damage is concentrated in few ice links. Fig. 6 shows failure envelope for FCso snow obtained from all the eight sub-volumes, and the error bars shown reflect ±2 standard deviations and represent the statistical uncertainties between the analyzed sub-volumes which correspond to relative error (εrel) b 7%. The failure envelope for FCso snow indicates

1 kPa

2 kPa

3.5 kPa

4 kPa

0.001 (0.298) 0.017 (0.652) 0.061 (0.728) 0.195 (0.770) 0.391 (0.821) 0.709 (0.892)

0.018 (0.490) 0.058 (0.651) 0.126 (0.730) 0.282 (0.774) 0.479 (0.831) 0.774 (0.898)

0.229 (0.515) 0.327 (0.780) 0.431 (0.846) 0.610 (0.882) 0.804 (0.929) 1.093 (0.955)

0.379 (0.561) 0.502 (0.836) 0.614 (0.870) 0.779 (0.919) 0.960 (0.945) 1.165 (1.00)

that the shear strength increased linearly up to applied compressive pressure σ τmax ≈2.5 kPa and exhibits Mohr–Coulomb failure criteria in the form Eq. 6. τ ¼ c þ σ tanϕ

From Fig. 6 the values of cohesion, c = 7373.34 Pa and friction angle, ϕ ≈ 22° are obtained. When the compressive pressure was increased beyond 2.5 kPa a drop in the shear strength was observed (Fig. 6) till compressive stress of 7.5 kPa when failure occurred in pure compression. This behavior indicates that Mohr–Coulomb with Cap model (MCC) describes the failure of FCso snow. MCC model for the snow can be summarized as

τ¼

8 c þ σ tan ϕ > > < sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 σ −σ τmax 2 > > : τmax 1− σ c −σ τmax

a. b. c. d. e. f. g.

6000

Numerical Simulation data MCC model max

Shear Strength (Pa)

9000

c

max

tan

Shear Strength (Pa)

5000

12000

ð6Þ

4000

if σ b σ τmax if σ N σ τmax

ð7Þ

b

Round Grain (Chandel et al. 2014 ) Faceted Snow (Zeidler and Jamieson, 2006) b FCsf (Chandel et al. 2014 , 09 Feb 2014) b FCsf (Chandel et al. 2014 , 10 Feb 2010) b FCsf (Chandel et al. 2014 , 19 Feb 2010) Cohesive Snow (Perla and Beck, 1983) Simulated response of Weak Layer (Gaume et al. 2014)

3000

2000

6000

1000 c

3000

0 0

1000

2000

3000

4000

Compressive Pressure (Pa) 0 0

2000

max

4000

6000

8000

10000

Compressive Pressure (Pa) Fig. 6. Failure envelope of FCso snow obtained from numerical simulations.

Fig. 7. Failure envelopes developed by different researchers for different snow types that are reported in literature a) RGsr snow, density 230 kg m−3 b) Faceted snow c) FCsf snow, density 100 kg m−3 d) FCsf snow, density 150 kg m−3 e) FCsf snow, density 200 kg m−3 f) Cohesive snow, density 200 kg m−3 and g) failure envelope from DEM simulations.

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C. Chandel et al. / Cold Regions Science and Technology 116 (2015) 56–64

4.0 3.5

Shear Stress (kPa)

3.0 2.5 2.0 1.5

Compressive Pressure 0 kPa Compressive Pressure 1.0 kPa Compressive Pressure 2.0 kPa Compressive Pressure 3.5 kPa Compressive Pressure 4.0 kPa

1.0 0.5 0.0 0.00

0.01

0.02

0.03

0.04

0.05

Shear Strain Fig. 8. Shear response of FCsf snow under the combined effect of compressive pressure and shear loading.

where σc is the compressive strength obtained from pure compression test and τmax is the shear strength corresponding to the common point of MC and the cap model. The observation of increase in the shear strength with applied compressive pressure is in agreement with different studies (Gaume et al., 2014; Perla and Beck, 1983; Zeidler and Jamieson, 2006). In all these studies the shear strength of snow is found to be increasing up to a certain value of applied compressive pressure. This closed cap failure envelope derived from FE analysis in σ–τ stress space is qualitatively in agreement with Gaume et al. (2014) who reported that for a weak layer shear strength increases with the applied compressive pressure up to a certain point thereafter a continuous drop in the shear strength with compressive pressure was observed (g. curve in Fig. 7). In addition to FCso snow, FCsf snow sample was also analyzed numerically and the representative shear responses of one of the FCsf snow sub-volumes under combined compression and shear loading is shown in Fig. 8. It can be deduced that the shear strength reduces with the increase in the compressive pressure and same observation can be drawn from Table 5, as the applied compressive pressure is higher, the percentage of elements undergoing damage and macroscopic damage increases and spread out of damage is more for FCsf snow as compared to FCso snow (Fig. 9). This difference of excessive damage is

supported by the microstructural parameters (Table 2) that indicates weak microstructure of FCsf snow as compared to FCso snow. The percentage of elements, in which damage was initiated due to compressive loading, increased from approx. 2.8 × 10−4% (0.001) to 0.379% (0.561) as applied compressive pressure increased from 0.25 kPa to 4 kPa. More importantly, the rate of growth of damage increases with applied compressive pressure during shear loading. In Table 5, at a shear strain of 2.7 × 10−3, the percentage of elements that have undergone damage, increased from 0.051% (0.505) to 0.431% (0.846) as the compressive pressure increased from 0.25 kPa to 3.5 kPa. The shear strength vs. compressive pressure plot in Fig. 10 indicates that the response of FCsf snow obtained from the numerical simulations is qualitatively similar to that in in-situ combined compression and shear loading tests (c., d. and e. curves of Fig. 7). For generating the results above the compression load was applied first and allowed to reach its maximum value. It was then maintained constant and a constant shear load was applied. To see if simultaneous application of compressive and shear load had any effect on strength envelope few simulations were run on one of the sub-volumes with these loads applied together. It is observed that there is no difference in the failure envelopes (Fig. 11) which are obtained by applying either sequential loading or simultaneous loading in numerical simulations. A continuous reduction in the shear strength with compressive pressure for FCsf snow is a very different behavior as compared to FCso snow. The following equation can be used to model this behavior of FCsf snow, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 σ τ ¼ τ o 1− σo

ð8Þ

where τo and σo are shear strength and compressive strength for FCsf snow. Experiments on FCsf (Chandel et al., 2014b) showed similar elliptical failure envelopes although the compression and shear strengths there were lower, as were the densities. The obtained results reveal that there is a nontrivial dependence of the shear strength on the applied compressive pressure. Podolskiy et al. (2015) used Mohr–Coulomb failure criteria for representing weak layers as interfaces and suggested a necessity of further testing for non-linear shapes of Mohr–Coulomb. This was confirmed by Reiweger et al. (2015) who modeled the failure envelope by MCC. The present work complements the above studies by producing similar results using actual 3D micro-structure of FCso snow. In FCso snow,

Damage Regions Fig. 9. SDEG is scalar damage in ABAQUS and damage contours are shown a) in the microstructure of FCso snow and b) the microstructure of FCsf snow at end of compressive loading step for compressive pressure of 2.0 kPa.

C. Chandel et al. / Cold Regions Science and Technology 116 (2015) 56–64

8000 Avg shear strength corresponding to compressive pressure Modeled failure curve

7000

Shear Strength (Pa)

6000 5000 4000

o

3000 2000 1000 o

0 0

1000

2000

3000

4000

5000

6000

7000

Compressive Pressure (Pa) Fig. 10. Failure envelope of FCsf snow obtained from numerical simulations.

the shear strength increases with the applied compressive pressure up to a certain point before dropping while in FCsf snow the shear strength tends to decrease from the beginning. For FCsf the microstructure is so weak that the compressive load produces enough damage in the microstructure which negatively affects the resistance of the sample to the applied shear load.

5. Conclusions Two snow samples, one of each FCso and FCsf were considered to obtain failure envelopes. These snow samples were investigated through numerical simulations on 3D FE models of weak snow layers prepared using X-ray scanned images. The simulations showed that the damage initiates in the microstructure even with the application of very small compressive pressure for all snow types but the extent of damage varied with snow type. For no shear load, the extent of damage in FCso snow increased with compression. During subsequent shear loading the damage growth rate reduced for higher compressive pressure leading to less damage at higher compression which in turn leads to increase in shear strength of snow with applied compressive pressure. This increase in the shear strength is observed up to a certain compressive pressure, σ τmax . For FCso snow sample under consideration it is 2.5 kPa. When 6000 Sequential Loading Simultaneous loading Modeled

Shear Strength (Pa)

5000

4000

3000

2000

1000

0 0

1000

2000

3000

4000

5000

6000

Compressive Pressure (Pa) Fig. 11. Failure envelope from one of the sub-volumes of FCsf snow, obtained through numerical simulations by application of sequential and simultaneous loads.

63

compressive pressure was further increased drop in the shear strength till compression failure was observed. This cumulative behavior of FCso snow sample is modeled using MCC model. For no shear load, the extent of damage in FCsf snow also increased with compression and same trend was maintained during subsequent shear loading. This increased damage is reflected in the lower shear strengths with increased pressure in FCsf snow and hence the response of FCsf snow is different from FCso snow. The numerical simulations gave failure envelopes qualitatively similar to those obtained from in-situ testing by Chandel et al. (2014b). The continuous drop in shear strength is analogous to the cap of MCC model and hence modeled with equation used to model shear strength in the cap. The results are qualitatively similar to the observed behavior in compression and shear. The study can be extended to tensile behavior of snow by modeling ice as anisotropic material with different damage behaviors in tension and compression. While undergoing deformation ice matrix experiences sintering and contributes toward stress relaxation, growth of structure, resistance to deformation etc. The present study has the limitation of not including these features in the modeling. Yet, FE modeling could reproduce response similar to the reported responses and gives useful insights into shape of failure envelope of snow. The obtained envelops can be scaled-up at macro-scale and even slope scale for purposes of avalanche dynamics to determine avalanche release mass. Acknowledgments Authors are thankful to Sh. Ashwagosha Ganju, the director of SASE for constant encouragement to publish this work. We thankfully acknowledge administrative members of Patseo Research Station SASE, for making our stay comfortable during winter under inhospitable conditions. Authors are also grateful to the reviewers for their sincere valuable suggestions to improve the quality of the paper. References Arns, C.H., Knackstedt, M.A., Pinczewski, W.V., Garboczi, E.J., 2002. Computation of linear elastic properties from microtomographic images: methodology and agreement between theory and experiment. Geophysics 67 (5), 1396–1405. Birkeland, K.W., Johnson, R., Herzberg, D., 1996. The Stuffblock snow stability test. US Forest Service Tech. Rep., Missoula Technology and Development Center, 96232836-MTDC 20 pp. Brzoska, J.B., Coléou, C., Lesaffre, B., Borel, S., Brissaud, O., Ludwig, W., Boller, E., Baruchel, J., 1999. 3D visualization of snow samples by microtomography at low temperature. ESRF Newsl. 32, 22–23. Carney, K., Benson, D., Dubois, P., Lee, R., 2006. A phenomenological high strain rate model with failure of ice. Int. J. Solids Struct. 43 (25-26), 7820–7839. Chandel, C., Srivastava, P.K., Mahajan, P., 2014a. Micromechanical analysis of deformation of snow using X-ray tomography. Cold Reg. Sci. Technol. 101, 14–23. Chandel, C., Mahajan, P., Srivastava, P.K., Kumar, Vinod, 2014b. The behaviour of snow under the effect of combined compressive and shear loading. Curr. Sci. 107 (5), 888–894. Fierz, C., Armstrong, R.L., Durand, Y., Etchevers, P., Greene, E., McClung, D.M., Nishimura, K., Satyawali, P.K., Sokratov, S.A., 2009. The International Classification for Seasonal Snow on the Ground. Tech. Doc. Hydrol 83. UNESCO-IHP, Paris. Föhn, P.M.B., 1987. The Rutschblock as a practical tool for slope stability evaluation. In: Salm, B., Gubler, H. (Eds.), Avalanche Formation, Movement and Effects. IAHS Publ. 162, pp. 223–228. Gaume, J., Chambon, G., Reiweger, I., van Herwijnen, A., Schweizer, J., 2014. On the failure criterion of weak-snow layers using the discrete element method. In: Haegeli, P. (Ed.), International Snow Science Workshop. Banff, Alberta, pp. 681–688. Gow, A.J., Ueda, H.T., Govoni, J.W., Kalafut, J., 1988. Temperature and structure dependence of flexural strength and modulus of freshwater model ice. CRREL Report 88-6. Hagenmuller, P., Theile, T., Schneebeli, M., 2014a. Numerical simulation of microstructural damage and tensile strength of snow. Geophys. Res. Lett. 41, 86–89. Hagenmuller, P., Calonne, N., Chambon, G., Flin, F., Geindreau, C., Naaim, M., 2014b. Charaterzation of snow microstructural bonding system through the minimum cut density. Cold Reg. Sci. Technol. 108, 72–79. Jamieson, J.B., Johnson, C.D., 1997. The compression test for snow stability. Proceedings of the 1998 International Snow Science Workshop in Banff. Canadian Avalanche Association, Revelstoke, BC, Canada, pp. 118–125. Jamieson, B., Johnson, C.D., 2001. Evaluation of shear frame test for weak snowpack layers. Ann. Glaciol. 32, 59–69. Kanit, T., N'Guyen, F., Forest, S., Jeulin, D., Reed, M., Singleton, S., 2006. Apparent and effective physical properties of heterogeneousmaterials: representativity of samples of two materials from food industry. Comput. Methods Appl. Mech. Eng. 195, 3960–3982.

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