Introduction

Snow is a non-linear compressible viscoelastic material. Modelling of the continuously changing stress regime in creeping snowpack on an avalanche slope requires a multiaxial constitutive law. Prior to 1970, the equation for a four-parameter viscoelastic fluid with linear elements was used as the most general constitutive relationship (Reference MellorMellor, 1964). Reference SalmSalm (1967) investigated multiaxial behaviour of snow using a constitutive law similar to Hooke’s law, with strain rate in place of strain. Probably the first comprehensive non-linear constitutive relation was proposed by Reference SalmSalm (1971). An approach similar to the stress–strain relationship for a Green–Rivlin material was used by Reference Brown, Lang, Lawrence and BradleyBrown and others (1973) and Reference BrownBrown (1976) to describe the non-linear deformation behaviour of snow. Reference SalmSalm (1977) developed a constitutive equation for creeping snow in a quasi-stationary state by the principle of maximum entropy production.

In this paper, we extend the phenomenological approach of Reference Szyszkowski and GlocknerSzyszkowski and Glockner (1986) for incompressible ice to compressible snow. This is achieved by defining an effective viscoelastic stress, which takes into account the effect of compressibility. The deformation of snow is divided into elastic, primary creep and secondary creep. A non-linear hereditary integral is used to describe the primary creep. Secondary creep strain is approximated by a generalized Norton’s power law. In the phenomenological approach used here, the constants appearing in the constitutive law are determined from macroscopic observations without considering microstructure or its evolution. Instead, these constants are related to the relative density. Finally, the proposed constitutive law is applied to triaxial test data to determine how well it approximates this behaviour. The snow here is assumed to be isotropic and non-age-hardening. Both these assumptions seem reasonable for the duration and environmental conditions of the laboratory experiment. Snow samples prepared and tested under laboratory conditions show comparatively less scatter in deformation behaviour and properties, thereby facilitating analytical modelling. Hence, results of cold-laboratory experiments on sieved snow samples were used both to determine the constants and to validate the law.

Multiaxial Constitutive Law

For snow we propose the existence of a complementary power potential W such that

(1)

where D_ij is the rate-of-deformation tensor which is divided into elastic and creep parts D_eij and D_cij. The complementary power potential is expressed as a function of the first and second invariant of stress tensor σ_ij and consists of three terms:

(2)

The first term represents linear elastic strain energy, and the second is recoverable viscoelastic potential, while the third term signifies permanent deformation. In Equation (2), K, G and σ_m denote the elastic bulk modulus, shear modulus and hydrostatic stress respectively. The second invariant of deviatoric-stress tensor s_pq is depicted as

(3)

The function F[P(σ)] is defined as

(4)

(5)

where [P(σ)] is the effective stress (Reference Chenot, Bay and FourmentChenot and others, 1990). Using Equations (1–4), we have

(6)

where μ is the Poisson’s ratio.

Defining

(7)

and replacing A₁B and A₂B by 1/ν₁ and 1/ν₂ respectively, the creep deformation rate can be written as

(8)

and are the primary and secondary creep deformation rates. The function j(t) is normalized such that jð0) = 1 and j(t) → 0, as t →∞; this implies that area (J) under the j vs time curve is finite, i.e.

(9)

A non-linear spring–dashpot model (Fig. 1) is used to represent the hereditary integral j(t) (Reference Szyszkowski and GlocknerSzyszkowski and others, 1986), so that

Fig. 1. A non-linear spring–dashpot model for primary creep.

(10a)

(10b)

(10c)

(α is introduced so that E₁ ≠ 0 at σ̃ = 0).

The postulated constitutive law expressed as Equation (8) implies certain invariant features for creep curves for various constant stress levels. It can be shown that

(11)

We thus obtain a stress-independent parameter t₀ (see Fig. 2):

Fig. 2. Concept of t₀ shown graphically on a typical creep curve.

(12)

From Equation (8), we obtain

(13)

A second stress-independent parameter, t₁, can be written as (see Fig. 3)

Fig. 3. Concept of t₁ shown on a strain-rate υs time plot.

(14)

When a constant stress is removed at time t = t′, the reversible portion, amounting to should be recovered while the irreversible component, ; remains as permanent viscous strain. If the time, t′, is chosen such that the creep process is in its “steady creep” stage, measurements of the “permanent” and “recoverable” viscous strains may also be used in determining the parameter, t₀, from the relation

(15)

The memory function j(t) in the proposed equation is approximated with that of the model’s j_m(t) shown in Figure 1 and expressed by Equations (10a–c). j_m(t) is a characteristic function of the model describing the hardening process during primary creep stage. For the model, the creep deformation-rate tensor D_cij can be written as

(16)

Comparing the first term with Equation (10a),

(17)

which, when used with Equations (10b) and (10c) (for spring element), gives

(18)

and

(19)

From Equation (19), by differentiation, we obtain

(20)

Substituting Equations (14) and (20) into Equation (10a), one arrives at

(21)

In the model, it may be appreciated that as t !1, σ̃′′(t) → 0, while ultimately the total stress applied to the model is taken by the spring. This condition, together with Equations (16) and (10b), leads to

(22)

from which we obtain

(23)

Using Equations (21), (23) and (12), two additional constants, α and E₁, for the model can be related to the material parameters, t₀ and t₁ (forn > 1), by

(24a)

(24b)

where

(24c)

From the above it is evident that the determination of strain in snow requires knowledge of all the constants ν₁, ν₂, t₀, t₁, E₁, α, n, c and f appearing in the constitutive law.

The authors realize that the spring–dashpot model for snow may not be very exact. In snow, primary creep occurs largely from deformation of bonds/grains, whereas secondary creep occurs both from deformation of bonds/grains and from sliding of grains after bonds break. If grain sliding is the predominant deformation mechanism, only a fraction of the primary creep strain will be recovered. On unloading, a spring–dashpot model will show full recovery over a period of time, which may not happen in the case of snow.

Experiments

Three sets of experiments were conducted for the present study: (1) Over 60 constant-load unconfined compression creep tests were conducted, results of which were utilized for the determination of all the constants appearing in the developed constitutive law. (2) Results from about 15 constant-load unconfined compression creep experiments were used to validate the determined constants. (3) About 60 constant displacement-rate tests under different confining pressures were conducted. Since the constants are determined from constant-load experiments, the form of the proposed constitutive law will be confirmed if it works for triaxial state of stress under altogether different boundary condition, i.e. constant displacement rate.

Determination of Parameters from Uniaxialtests

During uniaxial tests, from the instantaneous axial and lateral deformation at t = 0, the Young’s modulus E and Poisson’s ratio μ were determined. Based on these, three values for E, namely 4, 10 and 17 MPa, were used for snow-density ranges 230–300 kgm^-3, 300–370 kgm^-3 and 370 kgm^-3 and above, respectively.

For the uniaxial unconfined creep test, the primary creep rate reduces continuously and acquires a negligible value in 1–2 hours in a test conducted for 8 hours. With this assumption, for time t > 2 hours

(25)

where is applied axial stress.

The following form of c and f was chosen so that the constitutive law reduces to that for incompressible matrix material for relative density, i.e. _snow/_ice = = 1:

(26)

By substituting Equations (26a) and (26b) into Equation (25),

(27)

By performing creep tests on two samples of the same density, the approximate value of n can be determined from Equation (25).

Writing an equation for D^s ₂₂ similar to Equation (27), and dividing that by Equation (27),

(28)

A regression fit, using software developed by D. Hyams (http://www.ebicon.com/~dhyams/cvxpt.html), performed on Equations (28) and (27) gives c₁, c₂, q, r and ν₂. We next determine the constants ν₁, E₁, α required in primary creep equations. ν₁ is calculated from the deformation rate at t = 0 from the following equation:

(29)

The determination of E₁ and α requires evaluation of t₀ and t₁ as shown in Equations (24a–c). Table 1 lists the values of various constants appearing in the constitutive equation. The proposed law is now used to predict the deformation behaviour of snow for uniaxial confined compression tests not used in determining the constants. Fig. 4 and 5 show the comparison of actual and simulated creep response for low- and high-density snow, respectively. The constitutive law predicts the creep response accurately, though it appears that in a few cases the difference is appreciable (e.g. Fig. 6). Since, the values of c and f have been considered to be initial-density-dependent only, a different microstructure arrangement may significantly alter the snow response.

Fig. 4. Comparison of actual and simulated creep deformation for high-density snow.

Fig. 5. Comparison of actual and simulated creep deformation for high-density snow.

Fig. 6. Comparison of actual and simulated deformation for low-density snow.

Table 1. Values and units of the constants determined

Multiaxial Tests

Validation of the proposed constitutive equation was attempted along similar lines to Reference Desrues, Darve, Flavigny, Navarre and TailleferDesrues and others (1980). To determine the longitudinal stress and volume change, corresponding to the above strain rates and confining pressure, from the model, the following equations, combined with Equation (8), were solved in an iterative manner:

(30a)

(30b)

For the given strain rates, the axial stress developed was in the range 0.02–0.14MPa. Figure 7 shows the comparison of actual and simulated axial stress developed in the sample due to the subjection of the sample to constant strain rate under lateral confining pressure. It may be noted that stress develops more during the first 500 s than has been predicted. One may expect fast adjustment of grains/bonds during the initial period of strain-rate application for such moderate-density snow. Figure 8 shows a comparison of actual and simulated volume change. The theory simulates the experimental results moderately well.

Fig. 7. Simulated developed axial stress, compared with triaxial test result.

Fig. 8. Simulated volume change is slightly less than the actual one.

Conclusion

A constitutive law has been proposed, and various constants required for it have been determined from creep tests on snow. A comparison of the model predictions with uniaxial tests other than those used to determine constants showed a good match. Also, the generally overlooked primary-stage strain contribution in creep response is predicted satisfactorily. In certain cases, however (e.g. high-density sample subjected to long-duration curing and low-density samples of less age (<7 days)), the creep deformation predictions were not very accurate. In the model, parameters c and f were introduced as functions of initial density. In future we aim to relate these to a non-dimensional parameter (Reference Sethi, Srivastava and MahajanSethi and others, 2002) which characterizes snow. Experiments (with microstructure index measured) are in progress to see if these parameters can be made dependent upon the index. In nature, creep occurs over a long time and the snow density shows significant variation. To account for this it is possible to have different forms of P in primary and secondary creep. For example, “P” in primary creep (which becomes insignificant in 1–2 hours) may be taken as a function of initial density whereas it may be taken as a function of current density for secondary creep. Since density changes are not very high in the laboratory, in the proposed constitutive law P is assumed to depend on initial density only.

The triaxial tests were also performed and the constitutive model was used to predict the behaviour. Generally, the model accurately predicts the stresses developed and the volumetric change for bilaterally confined snow samples subjected to axial strain rates. The same theory can be extended to tensile tests by employing the damage factor.

The highlight of this proposed law is its ability to determine volumetric change, while a moderate degree of accuracy is obtained in the prediction of volume change.