Introduction

The apparent mechanical and physical properties of snow depend not only on the density but also on the fabric (microstructural arrangement of the snow crystals). Reference Shapiro, Johnson, Sturm and BlaisdellShapiro and others (1997) reviewed snow mechanics and found that no current system of snow classification is satisfactory, because no unique dependence between mechanical property, density and snow type is possible. Reference Sturm, Holmgren, König and MorrisSturm and others (1997) showed the importance of microstructure for the thermal conductivity of snow. However, they found no clear correlation between snow type and thermal conductivity and concluded that current snow classification techniques cannot be used to determine physical properties. A similar problem occurs in the analysis ofmechanical properties of trabecular bones (Reference Ford and KeavenyFord and Keaveny, 1996). Stereological methods based on vertical sections are insufficient to achieve good correlations between fabric and mechanical properties. Direct finite-element simulation of elastic properties using three-dimensional reconstruction techniques by serial sectioning or X-raymicrotomography circumvents the correlation between stereological andmechanical parameters (Reference Van Rietbergen, Weinans, Huiskes and OdgaardVan Rietbergen and others, 1995). The three-dimensional reconstruction of casted and undisturbed snow samples is now possible using serial sections (Reference SchneebeliSchneebeli, 2001) or X-ray microtomography (Reference Coléou, Lesaffre, Brzoska, Ludwig and BollerColéou and others, 2001; Reference SchneebeliSchneebeli, 2002). On the other hand, direct measurement of mechanical and physical properties of snow samples is difficult due to the inherent fragility of snow and sometimes to very thin layers. In this study, a finite-element (FE) simulation technique was used to investigate the microstructural stress distribution in different snow types. The elastic stress of snow was simulated in order to derive the elastic modulus with polycrystalline ice as material. The stress concentrations in heterogeneous fabrics, such as occur in weak layers, were simulated and a possible fracture mechanism was derived.

Material and Methods

Material

Two samples of snow were used: one was natural snow sieved into a sample container, subjected to a temperature gradient and imaged at different stages of temperature-gradient metamorphism at the same spatial location without disturbing the sample, called TG; the other was from a natural weak layer of 29 January 1999, called WK. The TG sample was used to investigate the effect of microstructure on the elastic modulus without changing the density; the WK sample was used to investigate the effect of a very layered microstructure on stress distributions.

The TG sample was prepared from the snow which was stored for several months in a cold room, and was sieved into a small cylinder 48 mm in diameter. The volume fraction was 0.266 and the density 243 kgm^–3. A temperature gradient of 100 Km^–1 was applied over 6 days. The sample was scanned non-destructively with an X-ray computer tomograph (Scanco μ-CT80) with a resolution of 36 μm at four stages of metamorphism. The scanning was performed at the start (TG1), after 1day (TG 2), after 3.5 days (TG 3) and after 6 days (TG 4). The segmentation of the ice matrix from the images was done such that the porosity was constant. Total mass and mass distribution was checked at the start and at the end of the measurements. Unconnected parts of the ice matrix, which was 52% of the total ice, were removed for the FE simulation.The four reconstructed cubes had a side-length of 3.6 mm and consisted of 100³ elements (Fig. 1).

Fig. 1. Three-dimensional reconstruction of the TG sample by X-ray microtomography. The sample was artificially metamorphosed and scanned non-destructively at four stages.

The WK sample was collected in the field at the site of a snow-profile measurement. The air was replaced with diethylphtalate, and the sample was frozen with dry ice. Afterwards 500 serial sections were cut; the thickness of the slices was 30 μm. The horizontal resolution of the images was 15 μm and was resampled to 30 μm. The segmentation of the images is based on the histogram. The segmentation threshold could not be checked by an independent density measurement due to the very different density of the layers. However, because of the clear bimodal histogram and good contrast of the digital images, a 10% variation of the threshold resulted in a 55% change in the average density and was therefore not considered critical. The three-dimensional reconstruction was applied to the central part of the sample consisting of rounded snow of small crystals, a melt crust forming a single grain layer and underlain by faceted crystals. The reconstructed cube had a side-length of 7.56 mmconsisting of 252³ elements. The average volume fraction was 0.336 and varied along the vertical axis (Fig. 2).

Fig. 2. Vertical planar section through sample WK. The subsection used for the simulation is in black, and the surrounding in grey. The horizontally averaged density is shown by the black line.

Results

The simulated elastic moduli for theTG samples are shown in Table 1. Von Mises equivalent stresses are shown in the center along the xz-plane in Figure 3. Peak stresses in the ice matrix of the TG1 sample were about four times higher than in the other samples. Stress concentrations occur along structures consisting of several grains. The locations of stress concentrations show a complex pattern, and are in many cases not at constrictions.

Fig. 3. Slices in the central xz-plane of the TG samples. VonMises equivalent stresses are shown as greyscale. The pictured slice thickness is 0.36 mm (ten slices) to show the connections between the grains. Stress concentrations are indicated with arrows.

Table 1. Apparent stresses σ, apparent strains ϵ and elastic moduli E for the TG sample (density 243 kgm^–3)

For the WK sample, as shown in Figure 4, higher von Mises equivalent stresses are found in the faceted snow below the melt crust. A few connections below the melt crust showed high stress concentrations. Several connections are loaded such that bending moments must occur.

Fig. 4. Slices in the central xz (right) and yz (left) plane of theWK sample. VonMises equivalent stresses are shown in color. The pictured slice thickness is 0.3 mm.

Discussion

The elastic moduli of the TG samples were about 10–100 times higher than have been measured from tests with strain rates of 10^–3 s^–1 (Reference MellorMellor, 1975), but only 3–10 times larger than the values introduced in Figure 9 by Reference MellorMellor (1980); in the latter case no strain rates are indicated. A probable explanation is that the effective elastic modulus is strongly dependent on strain rate. This behaviour is well known for ice (Reference SinhaSinha, 1978). Reference KryKry (1975) also observed a very strong dependence of the elastic modulus on strain rate. In this simulation, strain-rate dependency was not taken into account.

The calculated stresses nearly amounted to the strength of ice (around 10 MPa (Reference SandersonSanderson, 1988)) at some locations, although the bulk strainwas small. Fracture processes occur even at very small strains, as was observed from acoustic emissions (Reference St. LawrenceSt. Lawrence, 1980). The effect of crystal boundaries cannot be considered with three-dimensional reconstructions since neither with serial sections nor with X-ray tomography are they visible. However, it is not known if this is relevant, considering the complex geometry observed in these samples. The elastic moduli, much larger than those that have been measured, are also relevant to fracture mechanics. In this study, theYoung’s modulus was taken for very high strain rates, as in a propagating fracture. This caused larger elastic moduli than measured and therefore a larger fracture toughness. Future experiments are needed to verify this numerical observation.