CREEP INSTABILITY OF THE WEAK LAYERAND NATURAL SLAB AVALANCHE TRIGGERINGS

Francois Louchet*1LTPCM • UMR CNRS 5614 I INP de Grenoble - Universite Joseph Fourier

St Martin d'Heres (France)•

ABSTRACT: The weak layer connecting the snow slab with the older snow substrate is modelled. as an open cell ice foam made of an array of ice bonds. The comparison of bond rupture and

rewelding rates under load show that the weak layer experiences stable flow for a wide range ofloads of bond brittleness and of bond rewelding rates. However, a critical point may be reached,

corresponding to a finite shear rate of the weak layer, beyond which the shear rate increasescatastrophically, leading to a natural avalanche release. This critical situation is equivalent to aductile to brine transition, whose activation energy is onehalf of that of self diffusion in ice. A

measurement of the exponent in the shear strain vs stress relation may give a hint on theinminence of the avalanche release. A similar calculation performed at the tip of a preexistingbasal crack gives a simple criterion for crack stability and an analytical expression for snow

fracture toughness.

KEYWORDS: brittle ductile transition,creep, ice, snow, avalanche, toughness.

1. INTRODUCTION

It is well known that slab avalanchesresult from a failure of the weak layer thatconnects the slab to an older snow substrate.More precisely, human triggered slabavalanches are most often initiated by theexpansion in shearing mode of a "basal" crackin the weak layer (McClung 1981, Gubler1992). The corresponding time scales are shortenough to allow considering snow as a brittlemedium, in which fast crack expansion isgoverned by Griffith's criterion (Louchet 1999,Michot and Kirchner 1999, Louchet 2000 a). Bycontrast, time scales involved in snow coverevolution that may initiate natural avalanchesare by far much longer. The behaviour of theweak layer has to be treated in this case intenns of creep rather than in terms of brittlefracture. The simple model proposed in thepresent paper aims at giving a possiblephysical basis for analysing the weak layercreep stability and the conditions required forcatastrophic flow initiation.

I Corresponding author address: Francois Louchet,LTPCM, ENSEEG, BP 75, Domaine Universitaire,38402 St Martin d'Heres cedex (France); tel: +33(0)4 768266 09;fax: +33 (0)4 76826644;email: jrancois.louchet@ltpcm.inpg!r

273

2. CREEP MECHANISM

The system is schematised as shown infigure 1. The weak layer bonding the slab tothe substrate is described as an op~n cell icefoam, as proposed (or snow by Michot et al.(1999), Le. as a network of ice bonds, with a

porosity (} defined as the average bondseparation.

Figure 1: The weak layer is treated as an opencell foam made cif a network of ice bonds.Under shear loading, it experiences creep flowthrough a balance between bond rupture andrewelding events.

These bonds are prone to break understress, but also, since the material is close to

its melting point, broken bonds may reconstructthrough self diffusion welding if they are facingeach other for some time. Creep is thussupposed to result from a balance betweenbond rupture and welding rates. For the sake ofsimplicity, we neglect here sintering lIldercompressive loading. These startingassumptions are somewhat similar to those atthe basis of a recent numerical simulation ofdiphasic flow in semi-solid alloys by Perez et al(2000), except that here, instead of computinga viscosity at constant applied shear rate, weaim at studying possible creep instabilities atimposed shear stress, through a derivation ofbond breaking and rewelding kinetics.

3. RATE EQUATIONS

Let n be the proportion of unbrokenbonds. The coresponding proportion of brokenbonds is thus (1 - n). If n = 1 ,all bonds areintact, and each of them experiences-a shear

stress T. This stress is related to the

macroscopic shear stress TO by:

T = TO (;r (1)

where 0 is the bond separation, w the bond

thickness, and (w I 0)2 the surface fraction ofbonds. In the current case, n is comprisedbetween 0 and 1, and each bond experiencesa stress TIn larger by a factor 1 I n thanthat in the case where n = 1 . We considerthat the bond breaking rate, i.e. the number ofbreaking events per second, is proportional toboth the number of unbroken bonds n and thisstress TIn, which means that it is proportional

to T, with a proportionality factor a supposed tobe independent of temperature.

Let us now consider the bond weldingrate. Since two half bonds have to meet toachieve the welding process, the welding rate

is proportional to the square (1 - n)2 of thebroken bond fraction (1 - n). The welding rateis also taken proportional to the time duringwhich the two broken bonds are facing eachother, i.e. inversely proportional to the shear

274

strain rate. It is also proportional to a

f3(T) that accounts for ice welding k'varies exponentially with temperaturedoes the diffusion of water molecules 'proportional to exp (-Q I kT), Whereactivation energy Q is the self diftus'and k is the Boltzman's constant.

It can be shown (Louchet 2000 b), thabalance between breaking and weiervaries as shown in fig. 2. The curve isfor three different values of the loadsnow cover. Let us consider the iltecurve. In the positive parts of this Curvewelding is faster than breaking, and coin the negative parts. The exact babetween breaking and welding rates isthe two points where the curve intehorizontal axis, and correspond to stevalues of the fraction n of unbrokenpoint on the right corresponds to a ssteady state: if for some reason n decfrom its steady state value, the balancebecomes positive, i.e. welding becomesthan breaking, which restores n to its itvalue. For the same reason, the poiltleft corresponds to an unstable state,be disregarded in the following.

Each breaking event producesbonds that reweld somewhere in its vicinresulting in some local creep. The steadycreep rate dy I dt can be determined as

product of the breaking rate (proportionalby the corresponding creep distance, i.e.separation between unbroken bonds 1 Iwhere ii is the proportion of lIlbrokenthe stable steady state point. The steadycreep rate is therefore proportional to r: I

4. CREEP INSTABILITY

Let us now discuss the influence.O

snow load (through shear stress T) and W

layer properties on both creep rate andstability. It can be easily shown (Louchet

b) that an increase in stress T (due torain fall for instance) will shift the curvedownwards, bringing the steady state va

(4)

broken bond fraction ii to a smaller value.un . rt' It/­Silce the strain rate IS. propo lona 0 T n,th corresponding strall rate gradually. ereases. However, a further increase of snow:d may bring the curve to the ~wer positi.onshOwn in fig. 2, tangent to the honzontal axiS,

t which both stable and unstable steady:tates merge together. Beyond this stage, thebalance becomes negative, which means thatbreaking starts becoming faster than welding; ncontilUOusly decreases with time, which makesthe balance even more negative, rapidlyleading to catastrophic flow, that may befurther enhanced by local frictional melting. Theconditions for natural avalanche release are

thus met for a critical load i", corresponding toa fraction of unbroken bonds of 1/3, asillustrated in fig. 2, which can be shown to be:

T* = ~J;: (2)

or, in terms of the macroscopic load TO :

T'" - T,"(~)2 = .!.-(~)2 J4 P (3;0- 0 a 0 27

2 2--<E-----=--I-()( '" If!

Figure 2: Schematic variations of the balancebetween welding and rupture rates as afl.-.ction of the proportion of unbroken bonds n,at zero stress (upper curve), in the generalcase (intermediate curve), and in criticalavalanche release conditions (lower curve).

In a same way, a decrease of weaklayer temperature is expected to slow down

275

the ice welding rate f3(T), and an evolution ofthe weak layer morphology (under temperaturegradientfor instance) may increase bondbrittleness through bond thinning, bothpromoting avalanche release.

This critical situation may be consideredas a ductile to brittle transition of the weaklayer. It was recently suggested (Kirchner,Michot and Suzuki 2000) that the temperaturedependence of the activation energycharacteristic of the ductile to brittle transitionof snow in tension should be the same (0.6 eV)as that for self diffusion in ice (Ramseier1967). An interesting point is that in the case ofshear examined here the critical stress T' can

be shown to be proportional to -Jp (T) (Louchet2000 b), which means that the activationenergy of the ductile to brittle transition in

shear is expected to be ~ = 0 .3 eV, Le. one

half of the value proposed by Kirchner et al. fortensile failure, based on a quite differentmechanism.

It can also be shown that theconstitutive law accounting for the steady statecreep in the vicinity of the critical point can bewritten:

[2]1 I 2

TO / Y _ 1 = _2_ 1 _~TO '" / y'" .j3 TO *2

where TO is the macroscopic shear load

experienced by the weak layer, TO '" its value atthe critical point corresponding to theavalanche release, y the shear strain rate, and

f its value at the critical point. This law, and inparticular the value of the exponent 1/2, arevalid only in the vicinity of the critical point. Adetermination of the exponent of theconstitutive equation, obtained from fieldmeasurements of both snow load and driftvelocity, might perhaps give a hint on theimminence of unstable flow.

5. INFLUENCE OF A PREEXISTING CRACK

In the above calculation, the weak layeris considered as homogeneous at a scale

larger than the porosity. Large scale defects,as preexisting stable basal cracks, shouldmodify the above analysis through anenhancement of the creep rate due to stressconcentration at the crack tip. A simple way tosolve the problem is to multiply the shearstress in the balance equation by a

concentration factor )0: n where (0 I n) is

the actual porosity in the flow direction, and athe size of the preexisting crack. Thecalculations, whose details are not given herefor the sake of brevity, show that creepinstability at the crack tip (corresponding toy ~ 00) is reached for a macroscopic criticalshear stress:

TO = ~ ; If (5)

Beyond this stress value, the breaking rate atthe crack tip becomes larger than the weldingrate, which results in a catastrophic growth ofthe basal crack size, and in avalanche release.This scenario is similar to a type B overcriticaltriggering described in (Louchet 2000 a) inwhich the tensile rupture of the slab takesplace quasi-simultaneously along the wholecrown crack line, after a fast expansion of thebasal crack.

Since the crack size a is large as

compared to the parallel porosity 0, the criticalstress for avalanche release is significantlyreduced as compared to the homogeneouscase, as expected. Eq. (5) is equivalent to saythat unstable basal crack propagation takesplace when the stress concentration factor

TO *~ reaches a critical value:

Kc = ~ (Jf JllPO (6)

which can be considered (by definition) as thefracture toughness of the weak layer. Thisfracture toughness obviously increases withsnow "compacity" wi o. It is again proportional

to JjJ, and the associated activation energy isexpected to be one half of self diffusion in ice,as for the critical stress in the homogeneouscase.

276

6. CONCLUSIONS AND PERSPECl

This simple model shows that,reasonable assumptions, downhill sloa snow slab can be described throughstate creep of the weak layer. Thebased rate dependent constitutiveobtained for the weak layer may alsofor other types of snow under shearprovided specific values are taken forbrittleness a and rewelding rate p.quite recent measurements of ice on .friction by Kennedy, Schulson and(2000), qualitatively accounted for onsimilar ideas of making and breaking ofmay possibly be discussed in tenns ofpresent calculation.

An increase of the weight of thecover, due to snow or rain fall, may gbring the system to a critical point atnatural avalanche release takes placerapid strain rate softening. This criticalcan also be reached through either aof weak layer rewelding rate (colder wor an increase of bond brittleness (e.g.frost development resulting from terngradient metamorphism). The criticalcorrespond to a finite shear strain rate, .finite slab drift velocity, which increasestemperature.

These critical conditions for naavalanche triggering can be consideredsharp ductile to brittle transition, whosetemperature dependence can be descran Arrhenius law, with an activation enequal to one half of that of self diffusion

A measurement of the critical eof the stress vs strain rate law mightbe used to forecast the imminence of theavalanche release.

The stability of a preexisting basalcrack can be studied in a similar way,a correction of the shear stress at the cby a stress concentration factor. Avarelease takes place for a smaller streSSthe previous case, and an analytical e~can be obtained for snow toughness, with

sarne activation energy as above.

The present approach differs from

McClung's (McClung. 1979, MCCI~g ~ 981)ntially in that it IS based on kinetic

esse .olution equations and not on statIc rupture

:~iteria, even if time is introduce~ by McClung. the slab itself, through a supermposed~ergy release rate. To this respect, since wetake into account rate-dependent healing in theweak layer, we are dealing here with strain ratesoftening rather than with strain softening.

The present model may be transposedto landslip problems. Though being relevant ofgranular rather than of cellular media, themechanisms involved in wet landslips mayildeed be treated in a fairly similar way, grainssliding over each other through a series ofcapillarity controlled debonding and reweldingevents. The load is directly related to theamount of rain, whereas humidity due to eitherrail, snowmelt or springs may affect bondrupture and rewelding rates.

At this stage, the present approach ishowever fairly rough, in that it deals with anidealistic case (planar and infinite slope,without any boundary conditions, ...). Atemperature dependence of ice brittleness(GUbler and Bader 1989), that may change thecreep behaviour in the vicinity of the meltingpoint, should be introduced. Future work,requiring the use of numerical simulations,should try to reproduce the presentmechanisms, and explore more realisticsituations.

REFERENCES

McClung D. 1979. Shear fracture precipitatedby strain softening as a mechanism of dry slabavalanche release. J. Geophys. Research 84,

277

no B7, 3519-3524.

Mc Clung D. 1981. Fracture Mechanical Modelsof Dry Slab Avalanche Release. J. Geophys.Research 86, n° B11, 10783-10790.

Gubler H. 1992. Measurements and modellingto improve our understanding of avalancheformation. Universite Europeenne d'Ete 14-25sept. 1992, Cemagref Editions; Chamonix, 89­97.

Gubler H. and Bader H.P. 1989. A model ofinitial failure in slab avalanche release. Annalsof Glaciology 13,90-95

Kennedy F.E., Schulson E.M., and Jones D.E.,2000. The friction of ice on ice at low slidingvelocities. Phil. Mag. A 80, n05, 1093-1110.

Kirchner H.O.K., Michot G., and Suzuki T.,2000. Fracture toughness of snow in tension,Phil. Mag. A 80, noS, 1265-1272.

Louchet F., 1999. Deux modes possibles dedepart d'avalanches de plaque. ColloquePlasticite, March 29-31, Nancy (France).

Louchet F., 2000 a. A transition in dry snowslab avalanche triggering modes, Proc. Int.Symp. on snow, avalanches and impact on theforest cover, Innsbruck, 22-26 may 2000,Annals of Glaciology, in press.

Louchet F., 2000 b. Submitted to Cold RegionsScience and Technology.

Michot G. and Kirchner H.O.K., 1999. Plasticiteet rupture de la neige, la transition ductile­fragile. Colloque Plasticite, March 29-31, Nancy(France).

Perez M., Barbe J.C., Neda Z., Brechet Y., andSalvo L. 2000. Computer simulation of themicrostructure and rheology of semi-solidalloys under shear, Acta Mater. 48, n014, 3773­3782

Ramseier A.O. 1967. Self diffusion of Tritium innatural and synthetic ice monocrystals. J. Appl.Phys. 38, 2553. '