## Introduction

There has been considerable interest in the fracture characteristics of snow for some time, since some basic understanding of this topic is needed in order to evaluate properly the conditions necessary for avalanche initiation. The earlier studies were of an observational nature in the field, where detailed studies were made of the fracture line in the crown area and the portions of the flank which were not destroyed by the avalanche.

Determination of the triggering mechanism invariably becomes a difficult task, since in a snow slope there exists a wide distribution of snow density and snow types, with complicated boundary conditions and a complex state of stress within the pack. In addition, it is difficult to ascertain just where the pack failure was initiated, i.e. in the crown, the base or the toe region.

It therefore seems quite reasonable that a respectable number of theories of failure have been put forward to explain just how an avalanche starts under natural or artificial triggering conditions, and it also appears reasonable to assume that there will continue to be considerable disagreement among avalanche researchers concerning the plausibility of the various theories.

One approach to the problem has been to look at snow simply as a material reacting to a stress or deformation situation and then to attempt to formulate a fracture criterion for the material. If successful, this criterion can then be of some use in studying the avalanche problem, since the fracture criterion can indicate the stress conditions under which snow can most readily fracture. However, this is not in itself an easy task. As a material snow is extremely difficult to characterize. In addition to the fact that it is non-linear and highly compressible, the material properties of snow also depend on temperature and strain history. The actual strength of snow (fracture stress) is dependent upon density, strain-rate, strain history, and snow type. Consequently, a general fracture criterion is an exceedingly difficult result to achieve, and quite probably the complexity of the criterion might preclude its use in engineering analysis.

Previous studies by Salm (1971), Brown and Lang (1973, [1975]) and Brown and others (1973) have considered the characterization of the fracture properties of snow. Salm (1971) utilized a non-linear mechanical model to formulate a constitutive theory for snow in order to study the stress power and conserved energy levels at fracture under uniaxial compression. Brown and others (1973) attempted to formulate a fracture criterion which would express the fracture condition in terms of the histories of volumetric and deviatoric stress power. While this criterion did prove to yield a good correlation with experiment, it was quite involved and not readily applicable to any engineering analysis. Consequently, Brown and Lang [1975]) took a different approach and studied the levels of free energy and energy dissipation associated with fracture for a variety of deformation paths. It was observed in that paper that a criterion based on the free energy ψ and dissipation σ could yield a criterion which was physically meaningful and yet would be mathematically somewhat simpler than earlier criteria.

This paper reports the results of an effort to formulate a fracture criterion which is based upon energy concepts and hence may be put in forms which are usable in engineering analysis. The intent here is the formulation of a criterion which is general enough to encompass threedimensional states of stress and variable deformation histories. Such generality necessarily makes a criterion extremely complicated, but simplification to special states of deformation will be considered.

First, the thermodynamic theory on which the fracture criterion is based will briefly be discussed. This has already been developed by Brown and Lang [(1975]) and will not be dealt with in detail here. Then the form of the criterion based on experimental evidence will be formulated, compared to test results, and discussed.

## Thermodynamic formulation

The constitutive equation for snow is based upon the assumption that there exists a functional ψ, the Helmholtz free energy, which measures the recoverable strain energy in the body under isothermal conditions. By making use of the first and second laws of thermodynamics, one can show that the Piola stress, **T**, and specific internal energy dissipation rate σ are directly expressible as Frechet (′) derivatives of ψ. This result therefore makes ψ a central state functional for this type of material. The kinematic variables describing deformation and strain are

where **F** is the deformation gradient, *x* is the deformed coordinates, *X* is the original coordinates, **E** is the Lagrangian strain tensor, **I** is the unity tensor, and **G** is defined as the Green deformation tensor. Here ∇_{
x
} is the gradient operator with the Cartesian components ∂/∂*X*_{i}
. The form for the Helmholtz free energy finally arrived at by Brown (in press) IS expressible as a functional on the strain history and is expanded into a series as follows:

where *ψ*_{I}
are *i*th order convolutions on the history of the strain tensor. The thermodynamic state of snow is deformation-history dependent, and a convenient way of expressing this is by means of convolutions. The terms *ψ*_{I}
have the forms:

The term tr (·) is the trace of the tensor quantity in the parentheses, i. e. the sum of the diagonal terms in the matrix of tensor components. In an earlier paper, Brown (1976) truncated the series to a fourth-order expansion, but later the fifth-order fit given here was found to give better results and consequently is used in this study. The above expansion involves a total of eight relaxation-memory functions, Ψ_{1} to ψ_{8}.

The Piola stress **T** consistent with this form for *ψ* is

where the functions Ψ*
*_{i}
and Φ*
*_{i}
are related on a one-to-one basis. If Ψ_{6} is assumed to be completely symmetric in its arguments, then Φ_{6} = Φ_{6}′ and there are then only eight memory functions in the constitutive equation. To evaluate the memory functions experimentally, complete symmetry in the arguments of all the memory functions Ψ_{
i
} was actually assumed. The forms finally chosen were

and the relationship between the function Φ*
*_{i}
and Ψ*
*_{i}

where the coefficients *C*_{i}
are

For a more detailed discussion of the above development, the reader is referred to Brown (1976).

A Prony series expansion is used to approximate the memory functions

and the coefficients evaluated experimentally are given in Table I for fine-grained granular snow with a density of 330±20 kg/m^{3}. Since this snow was well sintered and exhibited good strength, the stress response of this material as indicated in Figure 1 was significantly greater than the gradient metamorphosed snow studied by Brown (1976).

Table I. Material Coefficients for Fine-Grained Granular Snow

Fig 1. Comparison of fracture criterion with experimental results for unconjined compression

For a given deformation process, Equations (4) and (6) can be used to calculate the stress response and the variation of the Helmholtz free energy. The energy dissipation under isothermal conditions is given by the relation

and may readily be calculated once **T** and ψ are found.

As was mentioned earlier, the intent of this paper was to formulate a fracture criterion in terms of the volumetric and deviatoric free energies. ψ gives the recoverable energy for the total deformation, which involves both volumetric and deviatoric contributions. To find the deviatoric portion ψ_{D} of the free energy, consider the term

where , the Jacobian of the deformation, determines the relative volume change by the relation

Now consider the deformation *αX* which has the deformation gradient

where **F** is the true deformation gradient. One can readily show that this deformation is isochoric. If we define **E**′ by

one can show that

where

and **E**′ is the portion of the strain **E** which is isochoric. The functional for the free energy then can be put in the modified form

so that the free energy is expressible as the sum of a purely deviatoric part

and a part which contains the purely volumetric part and a remainder term containing coupling between the volumetric and deviatoric contributions of deformation. The difference ψ–ψ_{D} we will refer to as the volumetric free energy ψ_{v}, and the relationship between ψ_{D}, ψ_{v} and σ at fracture will then be analyzed in the following section.

## Fracture criterion

The fracture strength of snow is dependent upon the strain history, as has been demonstrated by Salm (1971) and Brown and Lang ([1975]). It was shown by Brown and Lang that for only uniaxial compression the history dependence of the fracture condition could probably be expressed in terms of ψ and σ, which respectively provide a measure of the elastic state and viscous state of the material, both history-dependent quantities. However, under multiaxial states of stress, it would appear more feasible to consider a criterion which is expressible in terms of ψ_{D}, ψ_{v}, and σ, since the mechanical properties are strongly dependent upon the nature of the volumetric deformation and the deviatoric deformation. Since the condition depends on the history of the deformation, a fun ctional representation would be appropriate, such as

i.e. the critical value of ψ_{D} depends on the instantaneous value of ψ_{v} and the history of the internal energy dissipation. Here σ^{
t
}(*s*) = σ(*t—s*), where *s* represents the length of time since a point in the past. In this study we are interested in developing simple forms of the above general criterion applicable to special types of deformation histories. The first case is a situation where the material is initially in an unstressed rest configuration and is suddenly subjected to a high strain-rate which initiates fracture almost immediately. The second case involves a high rate perturbation of a material which is already in a stressed state. These forms of strain histories are relevant, since they are characteristic of many situations leading to avalanche initiation. Such examples would include the effect of explosives, skiers, sonic boom, earthquakes, wind gusts, or stress waves, among others.

In the first case, since the material is unstressed, the value of the dissipation at the onset of the deformation is zero, and a form of the fracture criterion for this special case could be:

In the second case, if σ_{0} is the value of the dissipation at the onset of a high strain-rate, theri a first-order approximation of the criterion could be

where the initial dissipation σ_{0} is assumed to be the predominant quantity defining the dissipative state within a small time period after the high strain-rate was applied to the material. Mathematically, the above approximation amounts to assuming a Taylor series expansion of the history σ*
*^{t}(s) about the instant when the perturbation is applied to the material and then retaining only the first term in the expansion. Since ψ and σ are smooth functions of time, this approximation is valid under the restrictions indicated.

In order to determine specific forms for the fracture criterion empirically, a complete set of fracture tests were run under conditions given in Table II. In all, about 23 deformation paths were followed in tension, compression, and shear. Data from some of the paths, such as paths No. 12 and 20, were not usable, since no specimens of the correct snow type or density were tested for those experiments.

Table I. Deformation Paths used in Experimental Program

The first nine deformation paths correspond to the first case described above (called brittle fracture). Figures 1, 2, 3 and 4 show the comparison of the fracture criterion for brittle fracture. For each test, the values of ψ_{v} and ψ_{D} were calculated at the instant fracture occurred, and these results are represented graphically in Figure 4 which shows both average results and the scatter incurred. In Figures 1–3 a comparison is shown between the experimental results for stress and the value predicted by the fracture criterion.

Fig 2. Comparison of fracture criterion with experimental results for tension.

Fig 3. Comparison of fracture criterion with experimental results for shear.

Fig 4. Comparison offracture criterion with experimental resultsfor brittlefracture conditions.

The results in Figure 4 suggest that a linear relationship such as

would be a very accurate approximation to the average fracture conditions for brittle fracture.

Figure 5 and Figures 1–3 illustrate the results for the second case, ductile fracture. In Figure 5, the deviation from Equation (21) is plotted against the energy dissipation σ_{0}. Again we see a linear variation, so that the fracture condition may be approximated by

Fig 5 Comparison of fracture criterion with experimental results for ductile fracture cOI/djtions.

where γ = 1.60. The data points in this figure represent the average experimental results for each deformation path. The discrepancy between theory and experiment is here larger than for brittle fracture, although not intolerable except for deformation path No. 13. In this particular path, the material was in a very dissipative state, and the approximations used for the functional in Equation (19) probably lead to a significant error for cases involving such large values of dissipation. Figures 1–3 illustrate how well the criterion approximates the experimental values of fracture stress for compression, tension, and shear.

A word should be said about the experimental scatter encountered in this project. As indicated in the figures, the scatter is quite low when compared to the scatter usually encountered in fracture tests of snow. A great amount of care was exercised in cutting and handling the specimens, and they were also given several days curing time after cutting to allow any damage to be partially mended. Also, only snow within a narrow density range (330 kg/m^{3}) were used, since density strongly influences strength, and care was taken to use snow which was fine-grained and granular.

## Discussion and conclusions

The results of this paper do show that a fracture criterion can be developed to characterize the strength of snow under multiaxial states of stress. For the two special forms considered here, the criterion takes on the simple forms given in Equations (21) and (22). These then determine the critical value of ψ_{D} required for fracture. Therefore, when a strain perturbation is applied to snow, by calculating the variation of ψ_{v} and ψ_{D}, Equation (21) or Equation (22) will determine when the critical value of ψ_{D} is reached.

The form of this fracture criterion is similar to the Coulomb-Mohr criterion since it relates volumetric and deviatoric effects, but in snow the dissipation rate plays a very central role in determining strength and consequently must be taken into account.

The specimens here were quite small (about 1 × 10^{-3} m^{3}) and consequently size effects should be considered. Sommerfeld (1974) has shown that the fracture strength of snow varies with the sample size, and by utilizing Weibull's method, size effects may systematically be accounted for.

It should also be noted that these results are for fine-grained granular snow with a d ensity of 330 kg/m^{3}, which makes it a fairly high-strength snow. Other results for lower-density snow gave significantly lower values of fracture stress. For instance, at a density of about 200 kg/m^{3}, typical fracture stresses were only about a quarter of the values indicated in Figures 1–3.

## Acknowledgement

This work was done under the joint sponsorship of the Office of Army Research- Durham (grant No. DA-ARO-D-31-124-71-GS9) and the National Science Foundation (grant No. NSF GA-3943). The author also wishes to thank Dr T. E. Lang, W. F. St. Lawrence and Dr C. C. Bradley for their support and constructive criticism.

## References

Brown, R. L.
1976. A thermodynamic study of materials representable by integral expansions. Intemational Joumal of Engineering Science, Vol.14, No. 11, p. 1033–46.

Brown, R. L., and Lang, T. E.
1973. On the mechanical properties of snow and its relation to the snow avalanche. Proceedings of the eleventh Annual Engineering Geology and Soils Engineering Symposium, p. 19–36.

Brown, R. L., and Lang, T. E. [1975.] On the fracture properties of snow. [Union Geodesique et Geophysique Intemationale. Association Internationale des Sciences Hydrologiques. Commission des Neiges et Glaces.] Symposium. Mecanique de la neige. Actes du coUoque de Grindelwald, avril I974, p. 196-207. (IAHS-AISH Publication No. 114·)

Brown, R. L., and others. 1973. A failure criterion for snow, [by] R. L., Brown and T. E., Lang, W. F. St., Lawrence and C. C., Bradley. Joumal of Geophysical Research, Vol.78, No. 23, p. 4950-58.

Salm, B.
1971. On the rheological behavior of snow under high stresses. Contributions from the Institute rof Low Temperature Science, Hokkaido University, Ser. A, No. 23.

Sommerfeld, R. A.
1974. A Weibull prediction of the tensile strength-volume relationship of snow. Joumal of Geophysical Research, Vol.79, No. 23, p. 3353–56.

## Discussion

H. Gubler: Salm's investigation on the rheological behaviour of snow (1971) show that the dissipative terms in a fracture criterion are negligible for strain-rates above about 1 s^{-1}. This agrees well with your experiments.

R. L. Brown: That is encouraging to hear. I have not read Bruno Salm's pioneering work for quite some time, so I must admit I am not well acquainted with the numerical values in his paper.

J. W. Glen: The quantity σ_{0} in your theory is the energy dissipation taking place in the material at the instant the final strain-rate is applied. You have used the formula for stressrelaxation conditions, when stress is presumably decreasing at constant strain. If strain is constant, surely strain-rate is zero, so how do you define energy dissipation as non-zero, and how is it determined?

Brown: When the strain-rate is terminated a fter a period of deformation, the material begins to relax and the stress d ecreases in approxima tely an exponential manner, so that after an extended p eriod of relaxa tion, the stress may be only, say, 10% of the stress level attained before relaxation began. Likewise the internal energy dissipation decreases to very small values, but the dissipation does not disappear the instant the deformation stops, since the material is still under a load, and internal dissipation mecha nisms are still a ctive. As the stress relaxes, the dissipation also d ecreases in a uniform manner.

R. A. Sommerfeld: How do you get a negative hydrosta tic pressure in a solid medium?

Brown: A negative pressure here results when the volumetric stress, which is just the trace of the stress tensor, becomes positive. In the tension test, the volumetric stress is possitive, and therefore for this case we have a negative hydrostatic pressure.

A. Dyunin: Have you attempted to derive a dimensionless criterion?

Brown: No, I have not.

M. Mellor: Did you make load-cycling tests and hysteresis measurements to obtain estimates of dissipation? If so, did you use the results to differentiate between brittle and ductile fracture modes?

Brown: No, we did not, and I think we really should have. These tests could have yielded some valuable information. To perform these experiments properly, an electrohydraulic testing machine such as th e one at the U.S. Army Cold Regions Research and Engineering Laboratory would be very desirable, since a unit such as this has the capability and flexibility accurately to control the deformation cycle in almost any manner desired.

Gubler: The strain-rate resulting from skiers, explosives, etc., are very different. For example the strain-rates associated with explosives (outside the crater range) are 10^{-3} s^{-1} whereas for skiers they are ≈1 s^{-1}. From this it follows that shear waves originating from explosives will not initiate brittle fracture whereas skiers may initiate brittle fracture. Stability of a slope to brittle fracture and to ductile fracture differ widely.

Brown: I was not aware of this, and admittedly the statement in my paper which implied that explosives cause brittle fracture was based on intuition and not fact. Consequently I am surprised by this new information, which goes a long way towards explaining some of the puzzling questions about the effectiveness of explosives in snow.