Introduction

The mechanical properties of snow can exhibit a strong influence on many winter activities such as vehicle travel, winter recreation, and avalanche prediction and control. One of the most commonly observed features of snow is its compressibility. It can undergo large volumetric deformations when subjected to moderate bulk stresses. These deformations are basically irreversible. Its ability to undergo large deformation is in large part due to its porosity. In its natural state, snow exhibits a wide range of grain shapes and densities. (Reference Abele, and GowAbele and Gow 1975, Reference Abele, and Gow1976) have demonstrated the effect initial snow density has on its load-bearing ability. From their experiments, it was clearly demonstrated, that for undisturbed snow with densities from 100 to 300 kg m⁻³ and subjected to high-rate deformation, lower initial densities resulted in lower final densities for a given final stress than that with a higher initial density. They attributed this to the irregular-shaped grains found in low-density snow and the more uniform rounded nature of the higher-density material. It was found that the initial density was a significant factor for final stresses <1 MPa. Beyond this, the effects of higher initial density became less obvious. Their studies also showed the rate-dependent nature of the load-bearing ability of snow. These results have also been verified with a constitutive theory (Reference Brown,Brown, 1979, Reference Brown,1980).

Modern constitutive theories are now using micro-structural parameters, which may include bond radius, neck length, grain-size and three-dimensional coordination number, among others. The bond is the minimum constriction in the necked region connecting two grains. The neck is defined as the total constricted region between the two grains. It is assumed to begin at the point where the surface curvature changes from convex outward to concave outward as viewed in a surface section. In cases where a sharp change in surface slope exists, this is used to define the juncture of the neck and grain. Figure 1 illustrates a neck connecting two grains, and Figure 2 illustrates a surface section with a number of grains. Admittedly, the definition used here to define the neck contains a degree of subjectivity and at times is difficult to implement. The three-dimensional coordination number is simply the average number of bonds per grain in the material.

Fig. 1. Schematic of neck and bond as they appear in a surface section.

Fig. 2. Example of basic counting techniques. Twenty-one grains are intercepted for a total line length of 4×L

A volumetric constitutive law based on a neck-growth model was developed by Reference Brown,Brown (1980). This model also included structural parameters for change in neck radius, neck length, grain radius and coordination number. The resulting equation described qualitatively the behavior for snow subjected to arbitrary deformation histories for a range of initial densities. In order to obtain accurate quantitative description of the strains, empirical adjustment was required. A more general constitutive law based upon non-equilibrium thermodynamics and the mechanical properties of ice has been developed by Reference Hansen, and BrownHansen and Brown (1986) for snow subjected to high-rate multi-axial deformations. The microstructural properties of snow in this theory are included in the form of internal-state variables. This theory was shown to predict behavior such as rate-dependence, stress relaxation and strain recovery. Over the past 10 years, other investigators have formulated constitutive relations with the use of microstructural variables. This includes Reference Maeno, and EbinumaMaeno and Ebinuma (1983), Reference Ebinuma, and MaenoEbinuma and Maeno (1985), Reference Alley,Alley (1987), Reference Wilkinson,Wilkinson (1988) and others.

Accuracy of any constitutive relation is dependent upon understanding the behavior of the microstructural variables included in the theory. Data are needed for the evaluation of empirical parameters. In addition, most constitutve laws based on microstructural parameters are too complicated to be of use in practical applications. Simplifications of these constitutive laws for special situations are best accomplished with knowledge of which microstructural parameters can be eliminated. At the present time,little or no data on microstructural behavior exist for snow densified by large deformation to densities in the 500-650 kg m⁻³ range. The work of (Reference Abele, and GowAbele and Gow 1975, Reference Abele, and Gow1976) provides a strong basis for understanding bulk behavior of snow subjected to high deformation rates and large strains, but little or no data were obtained relating the microstructure to the deformation. (Reference Gubler,Gubler 1978a, Reference Gubler,b) and (Reference Kry,Kry 1975a, Reference Kry,b) carried out extensive microstructural evaluation in their work, but these studies were generally confined to final densities below 500 kg m⁻¹ and to small deformations.

It is the major purpose of this investigation to evaluate the changes in microstructural variables that result from large deformations. In addition, it is intended to measure microstructural variables which until the present have not been adequately measured and mathematically described. An empirical equation relating the number of bonds per volume to the mean pore length will be given and its physical significance discussed. A new definition of band width in Reference Gubler,Gubler’s (1978b) work will also be given. The results obtained will contribute to currently available data and to the understanding of the microstructural processes responsible for material deformation in granular materials.

Stereology and Microstructural Parameters

Stereology is a statistically based method by which quantities measured directly from a two-dimensional plane intersecting a three-dimensional body are used to obtain average values for the three-dimensional structure of the body. The relationships between two and three dimensions, and the resulting values, can then be used as a means for understanding the average microstructural properties of the body. The task then is to select those parameters most important to characterizing the material behavior and making the pertinent section measurements. The variables selected to represent the micro-structure of snow under large deformations must be able to characterize the average re-arrangement of the micro-structure. These variables must be able to represent dominant mechanisms such as pore collapse, bond fracture, intergranular glide, and neck and bond growth.

Measurements from a section plane are generally based upon three counting methods: point, lineal and areal. The main assumptions necessary for validity of measurements are homogeneity and isotropy. Reference HansenHansen (1985) considered the microstructure of snow by representing it as a three-phase mixture. The phases were the air, neck and ice-grain phases. The subscripts used to represent each phase are respectively a, b and g. The required measurements of each phase in the section plane are represented by the following variables:

• N_gL Number of grains intercepted per unit length of test line.

• N_bA Number of bonds per unit area.

• N_gA Number of grains per unit area.

• A_aA Area fraction of air phase appearing in the surface section.

• A_bA Area fraction of neck phase in the surface section.

• A_gA Area fraction of ice grain phase in the surface section.

The remaining surface-section measurements required are:

• a Ratio of ice density to snow density, p_i/p_s

• D₂ Mean bond diameter.

• E Harmonic mean of bond diameter.

• f0 Fraction of grains with 0 bond appearing in the section plane.

• f1 Fraction of grains with 1 bond appearing in the section plane.

• f2 Fraction of grains with 2 bonds appearing in the section plane.

• f3 Fraction of grains with 3 bonds appearing in the section plane.

The definition used for a grain bond is that given by Reference Kry,Kry (1975a). A grain bond intercepted by the section plane will appear as a line joining two opposite edges of the ice in the plane. To identify this line as a bond, three criteria must be met. First, there must be a minimum constriction of 67% of the ice in the section plane. Secondly, the constriction must be present on both edges of the ice. Finally, the constrictions must point approximately towards each other. In practice, applying this definition will involve some subjectivity. As long as the consistency in applying the definition is observed, comparison of measurements should not be seriously affected.

The neck is considered to be that part of the ice joining two ice grains and lying between the points where the surface goes from convex to concave with respect to the outward normal. The application of this definition also contains an element of subjectivity, since identifying the point of change in curvature sign is difficult. The neck regions tend to contain less than 2% of the total section area and errors in location of the change in curvature or identification will have only small effect on other measurements. The region enclosed by the large rectangle in Figure 2 shows all phases of interest for an image-enhanced surface section. The image is enhanced in the sense that the ice grains, necks and pore space are given different consistent shades of gray. The ice grains are given a black shade, while the pore space and necks are, respectively, white and an intermediate gray shade. Any blemishes in the surface section are also removed. The following parameters can be directly determined from surface-section measurements.

Experiment and Results

Freshly fallen snow was collected and stored in a cold room at 10°C. Confined compression tests were carried out at 10°C within a period of 2-0 months after the samples were first placed in the cold room. After 2 months, the original sample volume had decreased by nearly 50%, and volume changes occurring after this were generally very small. The snow samples consisted of snow from 2 years, so that the large-grained samples (sample numbers 10k01, 5k001 and 2k501) were from the earlier year (1980) and the fine-grained samples (sample numbers 5k002 and 2k502) were from the following year (1987).The initial densities of the samples were all between 250 and 320 kg m⁻³, so there was not a wide spread in this variable among the samples tested. The notation used for identifying samples was:

• (1) Maximum load to which a sample would be subjected is indicated by the first four terms (2k50, 2500lb (11.2kN)).

• (2) p or f indicates if the sample is pre-compressed or in its final compressed state.

• (3) 1 or 2 indicates if the sample was tested in the first or second year, respectively.

The first part of the experiment involved subjecting samples to a constant deformation rate, in this case at the nominal strain rate of 2.75 10⁻³ s⁻¹, until a predetermined maximum load value was reached. The chosen load maxima were 44.8, 22.4 or 11.2 kN, corresponding, respectively, to Cauchy stresses of 1.55, 0.77 and 0.387 MPa. Once this load was attained, the test was stopped. Surface sections were then immediately prepared from the uncompressed and compressed samples. Surface preparation was based on the procedure given by Reference Perla,Perla (1982). Prints of the sections were made with the image as large as possible, while maintaining good contrast and avoiding graininess. The magnifications used ranged between 10 and 20. Using the digitizing software supplied with the Thunder Scanner on a Macintosh computer, it was then possible to run the image file through a filter to remove blemishes and maximize contrast. In order for the digitized images to be compatible with the image-analysis software, the section contrast had to be set to make the ice grains black and pore spaces white.

The image scale was first measured. The region to be analyzed was delineated by marking off a rectangular region. Next, each grain bond was measured by marking a rectangular region enclosing the neck. The bond diameter, the plane and neck area were then automatically measured. After all bonds connected to a particular grain had been measured, the number of these bonds was tallied by selecting the value of cither 1, 2, 3, >3 or 1/2 (for those bonds intersected by the region boundary). After all bonds had been measured, the number of grains not having a bond was tallied and assigned the value of 0.

Two areas of each image were analyzed. The results were then averaged to obtain mean values and relative errors. The major factors limiting the size of each area analyzed were the actual photograph size and the part of the image for which each feature’s identity could be accurately determined. The resolution of the digitized images was better than 0.0333 mm pixel⁻¹.

In the results, it is important to keep in mind that two distinct ranges of grain-size exist. The large grains, corresponding to the number 1 samples, have values of L₃ ranging from 0.34 to 0.39 mm in the uncompressed state, whereas the small-grained (number 2) samples range from 0.19 to 0.22 mm in the uncompressed state. The ratios of L ₃/L ₃₀ vs ρρ/ρ₀ are plotted in Figure 4. The "0" subscript indicates initial value. ρρ/ρ is the change in density resulting from the loading. As indicated in Table 1, neck length tends to remain relatively constant throughout the deformation process. An increase in coordination number does occur, however, and will be discussed later. The factors which collectively govern the change in neck length are fairly complicated, so that predicting neck-length change is difficult. This topic will be discussed in detail in a later paper. It suffices to say that, for the densifications considered here, no definite trends in neck length could be unambiguously determined.

Fig. 4. L₃/L₃₀ vs ρρ/ρ₀. Ratio of mean final and initial mean intercept length versus change in density. The straight line is a linear regression curve with a standard error of s₁ = 0.032. The error bands represent the total data scatter.

Changes in grain-size did show a definite trend as the material was compressed. Figure 4 illustrates the gradual decrease in grain-size with increasing density. Since the strain rates used in these tests were large, some apparent grain fracture did result. The straight line is the linear regression curve, and the vertical bars represent the entire range of the data scatter.

Table 1 also includes values of λ, the mean distance across a pore from grain surface to grain surface. Looking at the values of λ for the compressed samples, the ratio of λ/L ₃ decreases from 0.99 to 0.62 as the final stress ranged from 0.387 to 1.55 MPa. It is evident that as the grains become more immobile as the load-bearing ability increases.

Two- and three-dimensional bond diameters and radii are given in Table 2. The two-dimensional diameters show a definite decrease from the pre-compressed to compressed state. This indicates that bond growth is less significant than bond fracture and formation of new bonds. In fact, few of these bonds appear to survive the deformation process.

As the grain mobility is decreased, there should be some point at which the rate of bond breakage decreases and an increase in average bond radius begins. Figure 5 shows some indication for this for the largest density change, A complementary factor to the size of the grain bond is the number of bonds per volume N _gV Over the range of loading conditions in this study, the increase in N _bv showed a 2.5-fold increase at the 11.2 kN load (0.387 MPa) and 6.9-fold increase at the 44.8 kN load (1.55 MPa). The values of N_bV are shown in Table 4.

Fig. 5. MR₃/R₃₀ vs ρρ/ρ₀. Ratio of final and initial three-dimensional bond radius versus change in density. The curve is a cubic least-squares fit to the data, and the vertical lines represent the total range of the data scatter. The standard error of the least-squares fit is S_R = 0.025.

If the effects of grain-size are factored out, a better correlation between the mean pore length and the number of bonds per volume should exist. To check this, N _bV L ₃ ² was plotted versus λ L ₃ ² was used to normalize the effects of grain-size. The results are shown in Figure 6. A least-squares curve was fitted to the data. The resulting equation is

This equation can be rewritten in the form

Finally, introducing three constants C ₁, C ₂ and C ₃ in place of 0.347, the physical meaning of the equation becomes clearer. The resulting equation is

It can be seen that, by including the constant with the inverse of the cube of the mean intercept length, the inverse of the average grain volume will result. The inclusion of grain volume certainly makes sense as a factor influencing the number of bonds per volume, since the average grain volume determines the number of grains that can fit into a given volume. The second group of terms measures the grain mobility, which would be strongly influenced by the ratio

. As the mean pore length decreases relative to the grain-size L ₃, the probability of grains coming into contact and forming bonds should increase. The last set of terms is a weak function of the pore size, and further experimental data might result with this part being nearly constant. At this point, a conjecture as to its significance will not be made.

Fig. 6. N_bV L² us λ. Product of number of bonds per volume and the square of the mean intercept length versus the mean pore length. The standard error for the fit is S_N = 0.2. As in the other figures, the error bands represent the total data scatter.

The discussion of the terms appearing in Equation (24) indicates the validity of this equation on physical grounds. As a result, Equation (24) should be significant in determining the mean coordination number of a sample undergoing large deformation.

The values of N ₃, N _gV, V and S_v obtained for each of the five pre-compressed samples are shown in Table 3. The resulting mean coordination number N ₃ for these samples ranged from 2.28 to 2.67 bonds/grain. These values appear to be independent of the mean grain-size. In obtaining values of N _gV for the compressed samples, the following equation was used to obtain a first estimate:

where the subscripts f and i refer to initial and final respectively, and p is the density.

Figure 7 demonstrates the variation of the three-dimensional coordination number, N ₃, for the density ranges covered in the testing program reported here.

Fig. 7. Three-dimensional coordination number N₃ vs density. The curve is a least-squares fit to the data with a standard error S_N = 0.12. The vertical lines give the total data scatter.

Values of S _V in Table 3 and 4 indicate an increase in the surface area per unit volume. This is shown in Figure 8. As would be expected, S _V increases steadily for much of the deformation where bond breakage is frequent and where there is some grain fracturing. It then levels out as the density approaches 600 kg m⁻³ ρρ/ρ approaches 1.75-2.0). At densities which are higher than those studied here, this ratio should begin to decrease as bond breakage all but ceases, and bond mobility approaches zero.

Fig. 8. Variation of normalized surface area per unit volume with change in density, ρρ/ρ₀. The total data scatter is given by the vertical error bands.

Fig. 9. Axial and radial stress vs Lagrangian strain.

One should not confuse the results shown in Figure 8 with what one would expect with thermodynamic consolidation of snow. Here, we are studying the large-scale compaction of snow at large enough rates to fracture grain bonds as well as grains. As a consequence, new surface area is exposed due to the fracturing at high deformation rates. This is in contrast to the slow consolidation of snow as it attempts to reduce its free-surface energy by grain growth and densification. In this case, deformations are so slow that no fracturing occurs, and a gradual reduction of free-surface area results.

Discussion and Conclusions

At the maximum density attained, there is approximately a 3.5-fold increase in coordination number. The increase in coordination number does not increase very rapidly until densities of 500 kg m⁻³ are reached. This corresponds to strains of approximately 33%. This occurs at the point where the stress response begins increasing dramatically during the compression. The stress is about 0.1 MPa at this point. At the maximum stress, the density (639.4 kg m⁻³) represents a 25% increase over that at 0.1 MPa (500kgm⁻³). In this interval, the axial stress increased 16-fold and the coordination number was doubled. There is approximately a 60% decrease in the grain mobility as indicated by λ/L ₃ at 500 and 639.4 kg m⁻³. The largest relative decrease in bond radius (22%) occurs at this maximum stress. This is a 38% reduction in original bond area, but this is countered by the increased coordination number, with a net increase in bond area by a factor of 1.9 over the entire deformation process. The material stiffness is not a linear function of total bond area, since the structural rigidity of the ice matrix is strongly influenced by the coordination number in the sense that a structurally indeterminant structure is much stiffer than one that is structurally determinant. Therefore, the coordination number is a dominant factor determining the material stiffness and strength. Small changes in microstructural variables such as coordination number can have a measurable effect on the load-bearing ability of snow. As indicated by Reference Brown,Brown (1979), the effects of reduced grain mobility due to pore collapse may account for much of this at high densities.

The neck-length parameter (h) has not been described in a satisfactory manner in this study. The results of the testing program and image analysis gave an erratic behavior, as illustrated in Table 1. In some cases, the neck length is seen in increase in value, while in other instances the results showed the neck length decreasing. It was expected that the neck length would decrease with compressive deformation, since compressive stresses on the necks would shorten them. Formation of new bonds (and subsequently new necks) during densification would also serve to cause the mean neck length to decrease, since necks with nearly zero length would be added to the averaging process. There are a number of possible explanations for this apparent discrepancy. One could be that an image-analysis system with a better resolution is needed to provide a more accurate measurement of the neck length. Also, the definition of the neck length is difficult to implement and leads to a subjective criterion for measuring h. Changes in the sign of the surface curvature as seen in the surface sections is difficult to determine with the system used. This problem is now being studied and will be reported on in a later paper.

The neck-length parameter (h) has not been described in a satisfactory manner in this study. The results of the testing program and image analysis gave an erratic behavior, as illustrated in Table 1. In some cases, the neck length is seen in increase in value, while in other instances the results showed the neck length decreasing. It was expected that the neck length would decrease with compressive deformation, since compressive stresses on the necks would shorten them. Formation of new bonds (and subsequently new necks) during densification would also serve to cause the mean neck length to decrease, since necks with nearly zero length would be added to the averaging process. There are a number of possible explanations for this apparent discrepancy. One could be that an image-analysis system with a better resolution is needed to provide a more accurate measurement of the neck length. Also, the definition of the neck length is difficult to implement and leads to a subjective criterion for measuring h. Changes in the sign of the surface curvature as seen in the surface sections is difficult to determine with the system used. This problem is now being studied and will be reported on in a later paper.

There are several experimental results that would complement this study. Values of L ₃ for density changes of 25, 50, 150 and 200% will help verify any trends in grain-size change that are occurring (Fig. 4). It appears that the average grain-bond radius may begin to increase at a density change of 150% (Fig. 5). Measurements of R ₃ at 130, 150 and200% are needed to confirm this. The other measurement needed is for the coordination number at a final density of 450kg m⁻³. This should be sufficient to define the point of transition from low to high coordination numbers. In addition, the methods presented here can also be applied to evaluation of changes in microstructure of snow due to thermodynamic sintering and also for low strain rates where grain and bond fracturing is much less significant.