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Snow MechanicsReview of the State ofKnowledge and ApplicationsLewis H. Shapiro, Jerome B. Johnson, Matthew Sturm,and George L. Blaisdell August 1997

Abstract: A review of snow mechanics indicates that,with the exception of avalanche studies, it is seldomused. In this report we give our interpretation of whythis is the case, and suggest ways to help expand therange of problems to which snow mechanics can beapplied. Until the late 1960s, most experimental workin snow mechanics was devoted to finding values ofthe parameters for equations of linear elasticity, vis-cosity, and viscoelasticity. In about 1970, work onthat approach stopped and since then the emphasishas been on 1) the development of nonlinear theoriesto describe the deformation and fracture of snow, and2) attempts to develop constitutive relationships basedon the study of the microstructural aspects of snow

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Cover (clockwise from top right): Exploration—Blasting to open snow-bridged crevasses in shear zone betweenRoss and McMurdo Ice Shelves, Antarctica, in search of a safe route for heavy tractor trains (R.G. Alger, MichiganTechnological University); Buildings—Compacted snow foundation supports huge Defense Early Warning radarstation on Greenland Ice Cap (W. Tobiasson, CRREL); Utilities construction—Milling machine bores long unlinedtunnel and vacuum system removes snow chips 10 m below snow surface for Amundsen-Scott South PoleStation sewer line (M.R. Walsh, CRREL); Snow control—Snow fence array on steep slopes above village in SwissAlps retains snow and controls avalanches (E. Wengi, Swiss Federal Institute for Snow and AvalancheResearch); Surface transportation—Military vehicles in deep seasonal snowfields in Alaska rely on efficientcompaction and shearing of snow to be mobile; Logistics—LC-130 Hercules (ski-wheel) aircraft provide the onlymeans of supplying Amundsen-Scott South Pole Station with fuel, food, and other cargo; they operate from agroomed skiway (G.L. Blaisdell, CRREL).

deformation. We believe that the best hope of encour-aging more applications for snow mechanics in thenear term lies in improving and expanding the data-base on the response of snow to applied loads, andorganizing it in a manner that makes it easy for poten-tial users to determine the anticipated deformationalbehavior of snow in any particular application. To dothis, we suggest developing a classification of snowbased on physical properties and index parametersthat give information about the bonding and micro-structure. Mechanical properties, constitutive relationsunder various loading conditions, and other relevantinformation can then be associated with each class.

CRREL Report 97-3

Snow MechanicsReview of the State ofKnowledge and ApplicationsLewis H. Shapiro, Jerome B. Johnson, Matthew Sturm,and George L. Blaisdell August 1997

Prepared for

OFFICE OF THE CHIEF OF ENGINEERS

Approved for public release; distribution is unlimited.

US Army Corpsof EngineersCold Regions Research &Engineering Laboratory

®

US Army Corpsof Engineers®

Cold Regions Research &Engineering Laboratory

PREFACE

This report was prepared by Dr. Lewis H. Shapiro, Geologist and Consultant; Dr.Jerome B. Johnson, Geophysicist; Dr. Matthew Sturm, Research Physical Scientist; andGeorge L. Blaisdell, Research Civil Engineer, Applied Research Division, Research andEngineering Directorate, U.S. Army Cold Regions Research and Engineering Laboratory.

Funding was provided by DA Project 4A762784AT42, Design, Construction and Opera-tions Technology for Cold Regions, Task CS, Work Unit M09, Engineering Snow Mechanics.

The authors wish to thank K. Jones, Dr. R. M. Lang and Dr. B. Salm for their reviews ofthis report. They especially appreciate the helpful and constructive comments from Dr.Salm.

The contents of this report are not to be used for advertising or promotional purposes.Citation of brand names does not constitute an official endorsement or approval of the useof such commercial products.

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CONTENTSPage

Preface ................................................................................................................................. iiIntroduction........................................................................................................................ 1

Overview.................................................................................................................... 1Plan of the report ...................................................................................................... 2

Review of previous work ................................................................................................. 2Background ............................................................................................................... 2Constitutive equations and parameters ................................................................ 2Microstructural studies............................................................................................ 4A descriptive model for snow deformation ......................................................... 6

Some case histories illustrating the use of snow mechanics ...................................... 6Snow creep forces on avalanche structures.......................................................... 6Vehicle mobility in snow ......................................................................................... 7Snow roads and runways ........................................................................................ 9

Assessment of the current state of snow mechanics.................................................... 9An approach to snow mechanics research .................................................................... 10

Introduction ............................................................................................................... 10Establishing independent variables or index properties for snow

microstructure ..................................................................................................... 11A classification of snow for applications .............................................................. 12Testing and test data ................................................................................................ 13

Recommendations and conclusions ............................................................................... 13Literature cited ................................................................................................................... 14Appendix A: Constitutive relationships used to describe snow deformation ........ 21Appendix B: Review of the literature on mechanical properties of snow with

compilation of data ............................................................................................. 25Appendix C: Determination of microstructural variables by plane section

stereology ............................................................................................................. 31Appendix D: Possible index properties ......................................................................... 33Abstract ............................................................................................................................... 37

ILLUSTRATIONS

Figure

1. Cohesion stress vs. density and specific grain contact surface Sk for snowsamples with grain sizes as indicated ......................................................... 4

2. Unconfined compressive strength vs. deformation rate for 15-cm-longsnow samples of various ages ...................................................................... 4

3. Classification of snow from Bader et al. (1939) ............................................... 12

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INTRODUCTION

OverviewIn this document we describe the state of

knowledge of the field of snow mechanics* andsuggest studies to help expand its range of appli-cations. The work was motivated by a review ofthe literature, which showed that, aside from ava-lanche studies (i.e., release mechanisms, particlemovement and impact effects, and effects on ava-lanche defenses), relatively infrequent use is madeof snow mechanics. However, possible applica-tions in other areas include 1) design of equip-ment for snow removal, 2) calculating loads onstructures (roofs, towers, snow fences, cables,walls, etc.), 3) the use of snow as a constructionmaterial for roads, airstrips and foundations, 4)over-snow vehicle traffic including sinkage, shearstrength, resistance to motion and design of tracksand tread patterns for vehicles, and 5) militaryapplications such as those based on the capabilityof snow to absorb projectile impacts, and the prob-lems presented by snow-covered minefields.

We agree with Brown (1989), who pointed outthat “...the properties of snow are not yet wellenough known for use with a high degree of confi-dence. Snow, as a natural geological material, isfound in a wide range of densities, stages of meta-morphism, free water content, etc., and its prop-erties have been determined only for a few cases.”

Perhaps the most important reason why this is sois that there are few commercial or governmentalactivities that absolutely require knowledge ofsnow properties and processes. For example, inthe case of snow removal, heavy equipment de-signed for road construction and maintenance isavailable, although it may be significantly over-powered for the task of clearing snow (Minsk1989). Similarly, in the design and construction ofstructures that must contend with snow loading,overdesign can be substituted for knowledge ofsnow properties because the additional construc-tion costs are a small fraction of the total for anysingle project. In general, the economic incentivesfor any one project or agency are insufficient toencourage the research necessary to improve thebody of snow mechanics information. However,the economic benefits would be significant whenthe entire range of potential applications are con-sidered. Minsk (1989) noted that an average sav-ing of 10% in the cost of snow removal alonewould save about $100,000,000 per year in theU.S.A.

A further impediment to progress has beenthat the community of researchers in the field hasalways been relatively small and scattered. As aresult, there has never been broad awareness norinterest in the field within the scientific commu-nity in general. The breadth of intellectual activ-ity that could lead to expanded financial supportfor basic research on snow mechanics has fre-quently been lacking.

Thus, despite the potential for practical, eco-nomically viable applications, development ofsnow mechanics has been limited. Our purpose isto determine why this is so and what can be doneto extend the range of applications.

Snow MechanicsReview of the State of Knowledge and Applications

LEWIS H. SHAPIRO, JEROME B. JOHNSON,MATTHEW STURM, AND GEORGE L. BLAISDELL

*We define “snow mechanics” in a similar fashion to Jaegerand Cook (1976, p.1) but change the word “rock” to “snow”:“Snow mechanics is the theoretical and applied science of themechanical behavior of snow; it is that branch of mechanicsconcerned with the response of snow to the force fields of itsenvironment.”

Plan of the reportIn the following sections we review the litera-

ture on descriptive and experimental studies ofsnow mechanics and snow deformation, and giveour view of the current state of the subject. Weconclude that the field is relatively static at present,particularly in the area of applications to engi-neering problems. We then argue that there islittle hope for improvement in the near future,unless special efforts are made to make data onthe deformational behavior of snow available topotential users in an accessible format. A primesource of difficulty is that data on mechanicalproperties and deformational behavior* have usu-ally been organized and presented as functions ofthe snow density. However, we will show fromthe literature that snow density is not a reliableindicator of these properties. Instead, for a giventemperature and loading condition, the responseto load depends primarily on the bonding andmicrostructure, and the geometric characteristicsof the grains. This was recognized in early studiesof snow deformation, but developing a method ofusing microstructural properties as an indicatorof deformational response to load still remains tobe done. We propose that this can be done bybuilding a classification of snow based on a com-bination of microstructural properties and physi-cal characteristics, with the classes then corre-lated to characteristic deformational behavior. Weargue that the critical microstructural propertiescannot be established by stereologic work (App.C). Instead, we suggest that index properties (theresults of tests designed to be sensitive to thestate of the microstructure) are the best way torepresent the critical microstructure. We describeseveral possible index tests, but suggest that amodification of a blade penetration measure ofsnow hardness (Fukue 1979) may be most useful.With a classification established, tests can be runto obtain stress–strain–time–strength data to es-tablish the characteristic deformational behaviorfor each class of snow.

We have limited this report to the propertiesof dry snow in order to avoid dealing with the

problems introduced by the presence of free wa-ter. For brevity, we have not considered frictionbetween snow and other materials, acoustic prop-erties, properties of snow in motion, and shockwaves in snow.

REVIEW OF PREVIOUS WORK

BackgroundMost of the literature on snow mechanics has

been summarized in reviews by Bader (1962a),Mellor (1964, 1975, 1977) and Salm (1982). Weused these extensively. For discussion, we sepa-rate the field into two areas. The first area in-cludes the descriptive and experimental studiesthat established the basic ideas about snow defor-mation and snow as a material, and efforts toestablish constitutive relationships for snow. Inmuch of this work, the objectives were to describehow snow responds to applied loads, to measurethe strengths of various types of snow under dif-ferent loading conditions, and to find numericalvalues of the parameters required by the variousconstitutive relationships. Early experiments andconstitutive relations were based on measure-ments of macroscopic deformation. Later, recog-nition of the importance of snow microstructuralinfluences on deformational behavior led to thesecond area of research: studies of microstruc-tural scale processes that operate during defor-mation. The purpose of these studies has been todescribe and quantify the changes in grain andbond relationships that occur during deforma-tion as the grains rearrange, fracture, recrystallizeor sinter, and then to use the results as the basisfor developing constitutive relationships for snow.

Constitutive equationsand parameters

Most of the descriptive and experimental stud-ies were done between 1930 and 1980 and aredescribed in the reviews by Bader (1962a), Mellor(1964, 1975) and Salm (1982). The early studies inwestern Europe were primarily motivated by theneed to understand and predict the occurrence ofavalanches, and to mitigate their hazards. Simi-larly, the problems posed by the heavy seasonalsnow cover in parts of Japan provided the incen-tive for the systematic studies of snow propertiesby researchers at the Institute of Low Tempera-ture Science at the University of Hokkaido. In theU.S.A., the work by SIPRE and CRREL in-vestigators between the late 1940s and continuing

*We use “mechanical properties” to refer to parameters suchas the constants of the familiar stress–strain relationships ofelastic, viscous and viscoelastic materials, or the strength invarious loading modes. The expression “deformational be-havior” is intended to mean the nature of the response toload in a general sense. For example, Young’s modulus is amechanical property, while deformational behavior is thebulk deformation of the material.

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to the late 1960s was largely in response to needsarising from the expansion of U. S. military activi-ties in the polar regions. The sheer size of theSoviet Union, and its range of arctic, subarcticand alpine environments, made the study of snowmechanics important in that nation, although, un-fortunately, only a small fraction of the resultingliterature is available in translation.

The objective of most of the work through thisperiod was to determine the parameters requiredfor application of linear elasticity, viscosity andviscoelasticity to problems involving snow me-chanics. The effort followed the recognition thatsome patterns of deformational behavior in snowsamples in a laboratory or field setting could bedescribed by linear relationships. For example,Bader et al. (1939) discussed the creep of snow(they used the term “plasticity”) in connectionwith investigations of snow settlement. They didexperiments on samples in both uniaxial confinedand unconfined compression, but since they didnot attempt to formulate a constitutive relation-ship to describe the process there was no frame-work within which parameters could be defined.Thus, they made no mention of any particularmechanical property or constitutive relationship,although the patterns of deformation certainlysuggested a combination of linear elastic and vis-cous behavior. In fact, Yosida et al. (1956) wereable to use data from Bader et al. (1939) to calcu-late values for the coefficient of Newtonian vis-cosity of snow.

The most general constitutive relationship usedfor snow prior to about 1970 was the equation fora four-parameter viscoelastic fluid with linear ele-ments (App. A). According to Yosida et al. (1956),it was first used in snow mechanics by deQuervain (1946) to interpret the results of torsionexperiments.* Bucher (1948) included a sketch ofa Maxwell model (a spring and dashpot in seriesas shown in Fig. A1 in App. A) and used theconstitutive relationship for a linear viscous fluidto find the coefficient of Newtonian viscosity forcompacted snow as a function of temperature,duration of loading and a variety of types of snow,grain sizes, and ages. Interestingly, although theMaxwell model includes a spring element, Buchermade no mention of the elastic properties (or lackof them) of snow, although Yosida et al. (1948)

did measure Young’s modulus of snow in staticuniaxial compression tests. Later, Yosida et al.(1956) discussed the interpretation of the four-parameter model and found the parameters for itfrom creep tests on snow under uniaxial com-pressive stress. Bader (1962a) also suggested thatthe one-dimensional hyperbolic sine relationship:

ddt

Aoε ε σ= ( )sinh (1)

(where ε is the strain, σ is the stress and t is timeand εo and A are constants) might be used todescribe creep in snow; that is, it could replacethe linear relationship for the dashpot of the Max-well element of the four-parameter model. Mellor(1964) introduced an additional term into eq 1 bydividing the coefficient of the hyperbolic sine bya viscosity coefficient, η. He also discussed theuse of exponential and power relationships torepresent compactive viscosity (i.e., the viscositydetermined from the compaction of natural snow-packs, or from confined compression experimentsin the laboratory) in terms of the snow density asderived from data sets collected by various inves-tigators. Other determinations of the constantsfor the four-parameter model from creep test datahave been done in Russia by Kuvaeva et al. (1967)and by Shinojima (1967). Parameters for these lin-ear relationships, along with the available values,are summarized in Appendix B.

Even as efforts continued to find parametersfor linear relationships, it was apparent that theranges were too limited to solve many problemsin snow mechanics. Bader (1962a) recognized theproblem and suggested that the ranges of the lin-ear relationships might be extended if they wereapplied incrementally, as the values of the pa-rameters change with deformation. We have foundno references in which attempts to use this ap-proach were made, although Desrues et al. (1980)did devise a similar method involving simple non-linear relationships. Mellor (1975) stated that therewere still no alternatives to linear relationships,and that 1) there were no constitutive relation-ships for use in solving problems involving mul-tiaxial stress states, and 2) the data to developsuch relationships did not exist. He credited B.Salm with initiating efforts to address the needfor such relationships. In fact, Salm (1967) didconsider the extension of the hyperbolic sine rela-tionship to cases of the creep of snow in triaxialstress states. Later Salm (1971) used the relation-ships in exponential form to develop a failurecriterion based on energy storage and dissipa-

*Kuvaeva et al. (1967) reported that the viscosity of snow wasfirst determined by “the group of K. S. Zavriev in 1937.”Unfortunately, the reference they gave for this work appearsto be incorrect and we could not locate the paper.

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tion. Work by Brown et al. (1973), Brown andLang (1975) and Brown (1976, 1977) had the ob-jective of deriving constitutive relationships andfailure criteria for snow from theories of nonlin-ear viscoelastic materials. However, little addi-tional work in this direction has been publishedsince these papers appeared. Instead, the effortappears to have shifted to finding constitutiverelationships for snow from studies of its micro-structural properties and processes.

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Microstructural studiesWhen the parameters of linear constitutive re-

lationships or the strength of snow are plottedagainst density, typically the scatter is large (App.B). This situation prompted a question to M.Mellor, following his presentation at the 1974Grindelwald International Symposium on SnowMechanics (Mellor 1975) as to whether the largescatter might be reduced if the influence of thesnow “texture” could be accounted for. Melloragreed, and there are many reports in the litera-ture that indicate it is necessary to characterizethe microstructure along with determining thedensity in order to derive indicators of the me-chanical properties of snow. Yosida et al. (1956)used plate penetration experiments to demonstratethe differences in deformation between snowsamples of the same density but different degreesof bonding. Voitkovsky et al. (1975) showed thatthe cohesion of a particular suite of samples wasindependent of the density, but linearly related tothe contact area between grains (Fig. 1). Similarly,from his own experiments and a review of theavailable data, Fukue (1979) argued that snowdensity is not a reliable predictor of the uniaxialcompressive strength, and showed (Fig. 2) thatthe unconfined compressive strength of manufac-tured snow samples increased by a factor of 10 asthey sintered at constant density. Armstrong(1980), in a study of the densification of an alpine

Figure 2. Unconfined compressive strength vs. defor-mation rate for 15-cm-long snow samples of variousages (from Fukue 1979).

Figure 1. Cohesion stress vs. density andspecific grain contact surface Sk for snowsamples with grain sizes as indicated. Sk isdefined as the net area of intergrain contactsper unit volume. The cohesion shows little de-pendence on the density but is approximatelylinearly related to Sk. Figure modified from Voit-kovsky et al. (1975).

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snowpack, showed that a layer of fine-grained,sintered snow deformed at a rate 10 times greaterthan that of an adjacent layer of depth hoar, al-though the densities of both layers were the same.Finally, de Montmollin (1982) argued that break-ing and rapid redevelopment of bonds, even dur-ing the course of an experiment, is important insnow deformation. These examples show that re-gardless of the specific mechanisms involved, thebonding between snow grains (and not the den-sity) is the critical factor in determining the re-sponse of the snow to applied loads.

The importance of snow microstructure to def-ormational processes has been known for manyyears. Bader et al. (1939) made thin sections ofsnow after it was deformed in order to search forchanges in grain orientation that might have beenattributed to deformation. Kragelski and Shakhov(1949) also recognized the importance of bond-ing. Yosida et al. (1956) referred to bonding intheir interpretations of test results, Bader (1962a)discussed snow deformation in terms of bondingin a general way, and Kinosita (1967) showed thedifference in microstructural-scale process be-tween high- and low-rate tests in uniaxial com-pression. In addition, several theories based onassumptions about the processes that affectchanges in bonding during deformation or overtime have been derived to describe snow consoli-dation (Feldt and Ballard 1966, Ebinuma andMaeno 1987, Alley 1987, Wilkinson 1988) andstrength (Ballard and McGaw 1965, Ballard andFeldt 1966).

Keeler (1969a,b) was the first investigator tosystematically study the relationship between mi-crostructural changes in the fabric of snow dur-ing deformation and metamorphism. He creditedEugster (1952) with doing the first thorough fab-ric study of snow in thin section, and Kinosita(1960) with introducing the parameter of “jointorder” (the number of intersections of lines halv-ing connecting grains), which is important in theanalysis of snow fabrics. However, it was Nakaya(1961) who first tried to relate microstructure tomechanical properties when he experimentallydetermined the relationship between the dynamicYoung’s modulus and the density of processedsnow, and interpreted the results in terms of thedegree of bonding between grains.

Kry (1975a,b) tried to determine how the mi-crostructure of snow changes with deformation,and how the changes affect the mechanical prop-erties. He developed techniques and definitionsto quantitatively describe the grain and bond

structure of snow (see also Good 1975). In hisexperiments, snow samples were repeatedly de-formed in uniaxial compression by rapid loadingand unloading to determine values of the staticYoung’s modulus. Next the samples were allowedto creep at constant stress and the coefficient ofNewtonian viscosity was found from the rela-tionship between stress and the “steady-state”strain rate (Kry 1975b). Each sample was deformedin stages until the strain reached about 30% andobservations of the bond structure were madefrom samples collected at several stages. The re-sults showed that the stress is transmitted throughonly a fraction of the grains, and that these aregrouped into chains. Kry (1975b) hypothesizedthat the chains should be regarded as the basicunit of snow structure, and used that concept tointerpret the variations in viscoelastic properties.At the same time, Akitaya (1974) described the“skeleton” structure of some types of depth hoarin which grains were bonded primarily in thevertical direction providing strength in verticalloading, but virtually none for lateral loads. Thisskeleton structure is clearly similar to the “chains”identified by Kry (1975b). Subsequently, Gubler(1978a,b) extended the idea of chains, and used itto interpret data on the tensile strength of snow.

St. Lawrence (1977, 1980), St. Lawrence andBradley (1975) and St. Lawrence and Lang (1981)have used acoustic emissions as indirect evidenceof microstructural changes to develop constitu-tive equations for snow. Similarly, Brown (1979,1980) derived a constitutive relationship based ona model of collapsing pore spaces to describe thevolumetric compaction of snow. Alley (1987) useda grain boundary sliding model to describe thedensification of highly porous firn. Wilkinson(1988) used a density at which particle rear-rangement can no longer act (about 600 kg m–3)and a multi-mechanism theory of pressure sinter-ing to describe the densification of polar firn toice. Hansen and Brown (1986, 1987) and Hansen(1988) have also derived constitutive relationshipsfor snow based on theoretical considerations ofmicroscopic deformation mechanisms and mea-sured geometric parameters of the bonds. Edensand Brown (1991), and Brown and Edens (1991)have studied the deformation of the grain bondsand produced a mathematical model to describesome of the processes involved. However, thegeneral application of constitutive relationshipsbased on models of the deformation on a micro-structural scale is still considered to be some yearsin the future (Weeks and Brown 1992).

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A descriptive model forsnow deformation

Based on the studies cited above, the micro-structural deformation of a snow sample during ahypothetical experiment in uniaxial compressioncan be described. Our purpose is to highlight theproblems involved in characterizing the deform-ational properties of a material like snow that canoccur in many forms. For simplicity, temperatureand loading rate are assumed to be constant andthe grains in the sample are initially bonded intochain structures (Akitaya 1974, Kry 1975b).

There are several paths that the deformationcan follow as the load increases. The bonds be-tween grains can deform so that they thicken orthin according to their orientation with respect tothe loading. Alternatively, fracture of the bondswill permit grains to be displaced with respect toeach other and grains can break, changing thegrain size distribution (Kinosita 1967). Bond ge-ometry can also change by sintering at a rate thatdepends on the temperature and the pressure atgrain contacts. In fact, for a test in which thedeformation mechanisms operate at low rates, itis possible that the changes in bonding from sin-tering can be more important than those due tothe deformation.

The overall effect of the deformation is totighten the structure and increase the density ofthe snow. Concurrently, the bonding changes sothat the mechanical properties of the snow canvary through a wide range of values, dependingon the deformation path. For example, Salm (1977)found a 20% change in the viscosity of a snowsample due to 1% deformation in uniaxial com-pression.

As the deformation process continues, an ap-parent relationship between density and mechani-cal properties may be established. The reason thisrelationship seems to exist is that both the me-chanical properties and the density depend onthe nature of the bonding/grain contacts. Thus, itis the bonding, and not the density, that is thecritical variable, suggesting that some parameterthat represents the influence of the bonding shouldreplace the density in plots of snow strength orother properties.

The macroscopic deformation of a snow samplereflects the accumulated deformation on the scaleof the grain size. The relationship between themacroscopic deformation and the stress is used todetermine the parameters for constitutive rela-tionships. In general, if tests on natural snow areof short duration, then strains are small and

changes in bond structure limited. In such cases,the constitutive equations for linear-elastic, vis-cous or viscoelastic materials can be used to inter-pret the test results as described previously (see,for example, Yosida et al. 1956, Shinojima 1967,Kuvaeva et al. 1967, Kry 1975b). However, if thesnow has been compacted, plowed, wind-blown,or otherwise processed and is well-bonded and ofhigh density, then even relatively large stressescan be sustained without significant deformationor changes in bonding (Abele and Gow 1976). Inthese cases, linear relationships are probably ap-plicable over a relatively large range of stresses.Unfortunately, in most applications involvingnatural snow, the strain is large enough that sig-nificant changes in the bonding and deformationalproperties occur throughout the deformationalprocess and linear relationships apply only over alimited range of deformation. Thus, either gen-eral nonlinear constitutive relationships that spanthe entire range of behavior are needed or, assuggested by Bader (1962a) simple linear rela-tionships may be used incrementally as deforma-tion increases.

SOME CASE HISTORIESILLUSTRATING THE USEOF SNOW MECHANICS

Despite the impediments that exist and makesnow mechanics difficult to use, it has been ap-plied to a diverse range of problems (vehicle mo-bility, foundations, tunnels, creep loading of struc-tures, roads and runways, snow removal, impactand explosive shock loading, avalanche release,construction of snow structures, and interpreta-tion of seismic and acoustic signals). Here wegive an overview of three engineering topics thatillustrate the success, and the recurring problems,in trying to apply snow mechanics.

Snow creep forces onavalanche structures

One of the problems which prompted the startof formal study of snow mechanics was the deter-mination of snow creep forces acting on fixedstructures designed to prevent snow avalanches(Bader et al. 1939). Initial attempts to calculatesnow forces on avalanche structures were madeby assuming that a snow block, between two infi-nitely long containing walls, with dry sliding re-sistance at the base, acted on the downslope re-sisting structure. The model predicted a linear

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increase in snow pressure with slope distance be-tween the two containing walls because of theassumption that the downslope retaining struc-ture supported the entire weight of the snow blockless the basal frictional restraint (i.e., no frictionalresistance along the side walls) (Salm 1977).

Snow deforms by viscous creep under the ac-tion of gravitational forces to increase the pres-sure acting on the avalanche defense structure.The contributing factors to this pressure are thestatic load, the creep motion of snow particlesdownslope, the glide of the snow cover along theground, and the friction between the structureand the snow. Static load is produced by the trans-ference of the vertical stresses laterally (the mag-nitude of the lateral stress is determined by theviscous analog to Poisson’s ratio). The pressuredue to creep and glide is caused by the retarda-tion of downslope movement of the snow by astructure. The effect of this retardation is greatestat the structure and decreases with slope distanceaway from the structure (called the back pressurezone). Structure/snow friction retards the settle-ment of the snow producing a force parallel to theupright face of the structure.

Avalanche defense structures with a finitecross-slope length display three-dimensional flowof snow around the structure and will produceend-effect forces as well. End effects increase theforce acting on the structure compared to an infi-nitely long structure (the force acting on an iso-lated structure asymptotically approaches that ofthe infinite structure as the ratio of the structurelength to snow depth increases). When multiplefinite length avalanche defense structures are usedtheir force influences may overlap, allowing areduction in the structural forces (the magnitudeof force reduction depends on the separation be-tween structures).

Initial theoretical work was done by Bader etal. (1939), Haefeli (1948, 1951) and Bucher (1948)to obtain engineering formulae to estimate thecreep pressures and forces acting on infinite andfinite length avalanche defense structures. Thesewere modified as a result of later studies andempirically adjusted to include the possible rangeof effects due to creep, snow density, and snowdepth that might occur throughout Switzerland(Salm 1960, de Quervain and Salm 1963). The theo-retical developments and field measurements offorces on avalanche defense structures (Kummerli1958) are the main basis for establishment of theSwiss guidelines for avalanche control in the start-ing zone (Switzerland 1990). The guidelines are a

detailed engineering primer for constructing ava-lanche control structures and were first issued in1955. They have been revised when warranted bynew findings.

The problem of determining creep pressureson avalanche defense structures has attracted sig-nificant attention beyond the Swiss effort to es-tablish engineering guidelines. Ziegler (1963, 1975)applied plasticity theory to determine the pres-sures acting on avalanche defense structures andthe resulting length of the backpressure zone.Theories assuming Newtonian or non-Newtonianviscosity have been used along with field mea-surements to develop estimates of the pressureand pressure distribution on the upright face ofavalanche defense structures and the backpressurezone around such structures (McClung 1974, 1976,1982, 1984, McClung and Larson 1989, McClunget al. 1984, Brown and Evans 1975, Bader andSalm 1989, Larson et al. 1985, Olagne and McClung1990).

When the Japanese tried to apply the Swissguidelines to construct avalanche control struc-tures in their country, they found that the Swissguidelines were not always adequate to preventsnow creep damage to avalanche defense struc-tures under Japanese conditions. Katakawa et al.(1992) conducted a study to determine the appro-priate design factors for Japan and found thatglide factors and pressure distributions on struc-tures there were significantly higher (about 1.7times greater) than in Switzerland. These resultspoint to the empirical nature of avalanche struc-ture design and the fact that the results of theextensive Swiss efforts were fully applicable onlyto snow very similar to that found in Switzerland.

Vehicle mobility in snowMobility of ground vehicles is defined as the

efficiency with which a vehicle travels betweentwo points of interest. While this may include abroad range of factors, the essence of mobility isthe balance of traction and motion resistance. Trac-tion is the ability of the vehicle’s running gear toengage the terrain and the strength of the terrainto resist horizontal shear deformation. These com-bine to generate horizontal thrust from which avehicle may move forward, accelerate, tow loads,climb hills, or do other useful work.

Many sources give rise to resistance and theyconstitute a tax on the vehicle’s available power.Resistance sources internal to the vehicle are var-ied and generally well known (e.g., drive traingear losses, tire flexing or track bending resis-

7

tance), allowing vehicle manufacturers to designfor them. For travel in deformable terrain, such asmost snow, the dominate source of resistance isdeformation and displacement of the surface. Byvirtue of the terrain surface not being able to sup-port the running gear contact pressure, the ve-hicle must sink to a level where adequate supportcan be found. Thus, the vehicle is perpetuallyattempting to drive itself out of a rut.

The mechanical aspects of snow important tomobility are the ability to support vertical loadsand its resistance to horizontal shear displace-ment. These two requirements are closely related,and hinge on the bearing capacity and shearstrength of natural and compacted snow. The ad-hesion between a vehicle’s running gear and thesnow is also a factor in some situations, but isusually only a small contributor to traction. Effec-tive running gear will shear the snow within thesnowpack, since the thrust available there is nearlyalways greater than adhesion.

The bearing capacity of natural snow (densityless than 400 kg m–3) is usually very low com-pared to the needs of a vehicle for support. Thisresults in considerable snow deformation (nearlyall in the form of compaction) leaving the vehiclefounded some distance below the snow surface.Compaction and vertical sinkage proceed untilthe pressure bulb (sharply defined zone of influ-ence under the running gear) reaches a heightthat can provide enough vertical shear area tomake up the difference between the natural snow’sbearing capacity and the load placed by the run-ning gear.

Occasionally, during growth, the pressure bulbencounters a firm snow layer or the base of thesnowpack. This increases greatly the effectivebearing capacity and thus reduces the verticalshear area required. When a very firm base (ei-ther soil, pavement, or ice) is contacted by thebase of the pressure bulb, the snow depth is con-sidered shallow (note that this depth is depen-dent on the combination of vehicle and snow type).For some shallow snow conditions it is possiblefor vehicle loads to force the pressure bulb ofcompacted snow beyond the confines of the verti-cal projection of the edges of the running gear.This only occurs when vehicle load is signifi-cantly greater than snow bearing capacity andthe depth to a firm substrate is small. In allother circumstances the pressure bulb maintainsessentially vertical side walls that are alignedexactly with the lateral boundaries of the run-ning gear.

For shallow snow it has been shown the pres-sure bulb has virtually constant properties for avery wide range of vehicles and snow types(Blaisdell et al. 1990). Sinkage z is predicted by

z = h [1 – (ρ0/ρf)] (2)

where h = snow depthρ0 = undisturbed snow densityρf = the pressure bulb density.

Pressure bulb density ρf for shallow snow is es-sentially a constant (critical density) at 500 kg m–3

(Young and Fukue 1977). A balance of forces re-quires that

L = Sbc Ah + Ss Av (3)

where L = vertical loadSbc = the natural snow’s bearing capacitySs = the shear strength of the natural

snow/pressure bulb interfaceAh and Av = the horizontal and vertical areas of

the pressure bulb.

This describes vertical equilibrium in the snow-pack. The height of the pressure bulb H can beincorporated in the Av term; sinkage z and H arealso related. However, the usefulness of eq 3 islimited by the unknowns Sbc and Ss. Further limi-tations are the unknown properties of the pres-sure bulb and the fact that Ss is a shear forcedeveloped between two dissimilar snow masses.

Once sinkage equilibrium is reached, the vehiclerunning gear can engage to produce horizontalshear to generate forward thrust. In shallow snowthe available horizontal shear strength was de-rived empirically in Blaisdell et al. (1990). This waspossible because of the “constant” pressure bulbproperties found in the shallow snow condition.

Results of field mobility tests in deep snow arevery limited. The deep snow case is considerablymore difficult since the pressure bulb has no firmbase to assist in supporting the normal and shearforces. Additionally, during horizontal shearingit is common for some portion of the top of thepressure bulb to be removed by shear displace-ment. This upsets vertical equilibrium (eq 3) andthe vehicle suffers greater sinkage. This process iscalled slip sinkage and explains why tracked ve-hicles operating at even a small degree of slipalways assume a “bow up–tail down” attitude indeep snow.

While most mobility researchers agree that mo-tion resistance in snow is related principally to

8

means used to routinely generate strong snowpavements. Using additives has always beenpopular; however, these rarely provide a long-term benefit (Lee et al. 1989) Studies of snowpavement technology at the time of those reports,and continuing to now, were largely empirical.

Since Abele’s review, there have been a fewadvances. A successful experimental effort wascompleted to build a snow runway on deep snowat the Australian Antarctic base Casey (Russell-Head and Budd 1989). Compaction in layers wasused to build up a pavement of snow that with-stood proof rolling by a cart that simulated aloaded C-130 Hercules aircraft. A prototype snowrunway was also produced in the Ross Sea areaof Antarctica using sequential compaction effortsgoverned by seasonal ambient temperaturechanges (Blaisdell et al. 1992). This group tookadvantage of warming temperatures to place in-creasing loads on thin (10-cm) snow layers via aheavy pneumatic tire roller (glacial ice provideda rigid reaction base for the roller). Rest periodsof at least 24-hours were interspersed betweencompaction rolling to allow new interparticlebonds to form. Densities of about 600 kg m–3 werethe maximum attained and strengths adequate tosupport a test landing by C-130. Lack of near-melting temperatures and the ever-present strongtemperature gradient limited bond developmentand thus the ultimate strength of the snow.

Lang et al. (in press) performed a series of testsusing a variety of snow processing tools. Theyused snow tillers, of the type used by the skiindustry for reconstituting ski slopes, and a snowblower. Minimal compaction was done, in con-trast to the emphasis in all prior studies. Thisstudy appears to be the first to attempt to identifythe intergranular processes occurring as a resultof processing and subsequent aging. Using stere-ology and mechanical index tests (penetrometer),Lang et al. (1996) tried to correlate intergranularbond and grain size changes with strengthchanges. They also correlated these changes withambient temperature changes.

The study by Lang et al. (in press) was success-ful in producing some of the strongest snow pave-ments ever recorded. However, snow strengthwas difficult to quantify, owing to the difficultyin using a penetrometer in hard, dense snow.Larger scatter in the data were apparent, and oc-casionally, the penetrometer could not be forcedinto the snow. In addition, it was found that thecurrent state of stereological software is inad-equate for making determinations of snow’s me-

the sinkage (volume of compacted snow) therehas yet to be a reliable mathematical description.Richmond et al. (1995), Richmond et al. (1990),and Blaisdell et al. (1990) have tried many empiri-cal and analytical possibilities but acknowledgeno better than 25% accuracy on average. Somevehicle types show much larger divergence be-tween measured and predicted resistance. Mobil-ity measurements can even differ widely for snowsthat have the same density and similar physicalcharacteristics. Closer inspection usually showsthat these differences are the result of differencesin the internal strength of the snows brought onby variable compaction or sintering histories. Pen-etrometer and direct shear tests have occasionallybeen used in an attempt to document snowstrength. However, these are isolated attemptsand none have been shown by themselves to ac-curately determine expected snow/vehicle behav-ior. Thus, numerical models have begun to ap-pear (Mohamed et al. 1993 and Xu et al. 1993).These models hold promise for greater accuracyand insight in describing mechanical interactionbetween the snow and a vehicle’s running gear.However, these models are currently limited bythe need for complex snow load response datathat in general does not exist. For use, sophisti-cated and case-specific tests are performed to ob-tain these data. No systematic library of these testdata is maintained.

Snow roads and runwaysThe most practical, and perhaps widespread

application of snow mechanics is for the creationof snow roads and runways. Animal herds pro-duced the first snow “roads,” having recognizedthe reduced energy expenditure associated withtraveling along narrow compacted paths. Humanstraveling over snow-covered terrain followed thisapproach and, using snow shoes, skis, or boots,packed trails to increase travel efficiency. Me-chanical techniques were sought by humans toproduce robust snow roads beginning when beastswere harnessed to conveyances and continuingwhen mechanical locomotive devices evolved.Upon the refinement of motor vehicles and theadvent of aircraft, the focus moved from modify-ing the snow to removing the snow. Today, onlypersons interested in off-road travel and polaroperators still required snow roads and runways.

Abele (1990) produced a thorough review ofthe topic of snow roads and runways. His reviewhighlights the fact that compaction and snow mill-ing (with snowblowers) were the only successful

9

chanical properties. Some positive correlation wasfound among the stereology results and mechani-cal tests, but stereology factors were misleading.

ASSESSMENT OF THE CURRENTSTATE OF SNOW MECHANICS

Our assessment of the current state of snowmechanics is pragmatic: Is snow mechanics beingused for practical engineering, and if not, whynot? We compare snow with other materials, bothnatural and man-made. For those materials wheremechanics is being used, we find that there areextensive compilations of data and tabulated pa-rameters for constitutive relationships that de-scribe the deformational behavior under manyloading conditions. No comparable compilationsexist for snow, and existing parameters for con-stitutive relationships are either limited in rangeof applicability, or untested. Further, we do notsee the research activity necessary for rapid im-provement. In short, our view is that the field islittle used and relatively stagnant at present.

The most comprehensive source of both dataand parameters for linear constitutive relation-ships for snow are the reviews by Mellor (1975,1977). There have been few new determinationsof values since his were published (see App. B).Mellor (1975, 1977) recognized the importance ofthe microstructure in controlling snow’s mechani-cal properties, but no data relating them to themicrostructural features existed. Therefore, Mellorhad to present the results plotted against snowdensity. But, as we discussed above, density is apoor predictor of the mechanical properties. Notsurprisingly, the values of constitutive param-eters show large scatter: commonly 100% to 300%(Fig. B1–B7; App. B). As a result of the large scat-ter, an engineer seeking to use linear constitutiverelationships to solve problems cannot expect sat-isfactory solutions.

The usefulness of the nonlinear constitutiverelationships, mostly developed between the 1960sand late 1970s, is also limited. While it has beendemonstrated in the literature that reasonableequations can be derived to fit particular datasets, there are no examples of the resulting consti-tutive relationships having been shown to fit otherdata sets. Without such independent tests, therecan be little confidence that the nonlinear rela-tionships can be applied generally to solve prob-lems. Also, many of the nonlinear constitutiverelationships require parameters for which no

compilations of numerical values exist. An engi-neer wishing to use the relationships for a par-ticular application would be faced with the formi-dible task of having to determine the parametersfor the particular type of snow of his or her appli-cation.

Developing constitutive relationships based onsnow microstructure and micro-mechanical pro-cesses is still a relatively young field and the ques-tion of its ultimate utility is still open. Two points,however, are already clear: 1) stereological analy-sis is both difficult and tedious, and 2) there isuncertainty in how measured stereological val-ues relate to the actual microstructural state of thesnow. The latter point implies that stereology ismore suited for establishing microstructural in-dexes than for describing the true microstructuralstate of the snow (App. C). Also, microstructuraldescriptions of snow deformational behavior suf-fer from the same lack of data and independenttesting as the nonlinear constitutive relationshipsand theories. As a consequence, any possibility ofderiving constitutive relationships for general usefrom microstructural analysis is far in the future.

In summary, our general view of the state ofthe field is that snow mechanics is in a relativelystatic condition at present. We think that this re-flects the fact that the existing experimental dataare limited and constitutive relationships are notsufficiently developed to describe the behavior ofsnow over its full range of deformation and load-ing conditions. This situation is partly due to themany different types of snow that exist over awide range of environmental conditions and thebroad range of deformation behaviors. In addi-tion, the majority of existing data do not includeindependent variables that reflect the influence ofthe microstructure or include sufficient informa-tion about the characteristics of the snow to whichthe data apply. Finally, there is not, at present, aworkable method of relating the easily observedphysical features of snow (cf. Colbeck et al. 1990),to its deformational response to an applied load.Finally, funding for snow mechanics research,which has always been sparse, has further de-clined. This, in turn, has reduced the number ofworkers in the field and limited the research op-portunities of those who remain. As a result, thescope of research at present is relatively narrowand the prospects for expanding applications arelimited.

For the field of snow mechanics to find widerapplication, investigators must be able to identifyand classify the type(s) of snow involved in a

10

problem, locate information on the expected de-formational behavior for the conditions of theproblem, and have access to numerical values ofthe parameters for constitutive relationships thatare applicable. Our suggested approach to fillingthis need is given in the next section.

AN APPROACH TO SNOWMECHANICS RESEARCH

IntroductionWe believe that for the present, the goal of

engineering snow mechanics research should beto develop a comprehensive source of data on themechanical properties of interest and analytic toolsthat can be used to solve engineering problems.*This would make it possible for investigators to1) identify the types of snow involved in a par-ticular problem, 2) anticipate the response of thatsnow to applied loads under the conditions of theproblem using various measures or indices of themechanical property of interest (see Abele 1990),3) guide the selection of an appropriate constitu-tive relationship and test its usefulness, 4) findnumerical values of the parameters of that rela-tionship, and 5) determine the strength of snowin different loading modes if that is relevant tothe problem. This clearly requires new data onsnow in a format that currently does not exist inthe literature. To provide it requires that a classi-fication of snow be developed relating the physi-cal characteristics of snow (e.g., grain size, grainsize distribution, grain shape, density and othermeasures) to specific deformational behavior (e.g.,compressive strength and deformation underload) that operate over known ranges of en-vironmental conditions. The data on the defor-mational behavior will be needed to select appro-priate constitutive relationships and theirparameters for various snow types under the con-ditions of specific problems.

The classification must be based on featuresthat can be determined objectively and repeatablyby direct observation or by simple measurements.The physical characterization of snow should be

familiar (e.g., International Classification of SeasonalSnow on the Ground [Colbeck et al. 1990]), andmeasures of mean grain size, grain size distribu-tion, snow crystal morphology, bulk snow struc-ture, and density are appropriate. To categorizesnow types by their deformational behavior, theclassification should also include information onmicrostructure and bonding that most influencedeformational processes. Unfortunately, there areno suitable variables that provide unambiguousmicrostructural information (App. C) so it will benecessary to use index properties that depend onmicrostructure instead. These are not true prop-erties of the material, but are the numerical re-sults of simple tests that are correlated with thedeformational behavior of interest.

Once a classification is established, thedeformational behavior of each class of snow canbe characterized. This would involve collectingrepresentative stress–strain–time-strength data forsamples of snow from each class in different load-ing modes and rates. When available over a suffi-cient range of conditions, the data would be use-ful for selecting constitutive relationships anddetermining their parameters. Initially, the test-ing might be restricted to a representative rangeof conditions to demonstrate the styles of defor-mation for each class of snow.

Establishing independentvariables or index propertiesfor snow microstructure

Index properties are the results of simple teststhat are correlated with the deformational behav-ior for snow. According to Salm (as cited inOakberg 1982) in order for the results of some testto be useful as an index property it is necessary toestablish the following:

1. The results of the test depend on the micro-structure of the snow, although it is notnecessary to know exactly how that depen-dence arises.

2. The results are repeatable and can be donein a field setting, either in-situ or with por-table equipment that minimizes the need tohandle the snow.

3. The numerical range of the test results islarge enough to discriminate across thescope of possible seasonal snow types asthey appear in various environmental con-ditions.

4. The test results can be shown to vary sys-tematically with the mechanical propertiesby demonstrating, for example, that they

*A mechanical property of interest is defined as that propertymost relevant to a particular snow mechanics application (e.g.,high rate uniaxial compaction for impact and explosive prob-lems, compaction and shear deformation for mobility and ava-lanche release studies, creep deformation behavior to deter-mine loads on snow fences and structures, and other data usefulin dealing with a particular engineering problem).

11

are correlated with a parameter such as theuniaxial compressive strength at selectedrates of loading.

Several types of measurements that might serveas index properties are described in Appendix D.They include electrical properties, disaggregationenergy, sonic wave propagation velocity, and vari-ous methods for measuring the penetration hard-ness. Based on previous experimental results, allbut the disaggregation energy have some prom-ise as index measures of microstructure, but webelieve that an adaptation of the blade penetra-tion force suggested by Fukue (1979) is the best ofthese. Fukue (1979) used a relatively thick, shortblade to demonstrate that the penetration forcewas linearly related to the uniaxial compressivestrength (App. D, Fig. D3). Other penetrating de-vices (the most common of which is the Ramm-sonde penetrometer) require that relatively largevolumes of snow be compacted or displaced aheadof the advancing penetrometer (Huang et al. 1993).Thus, they are sensitive to the shape and rate ofadvance of the penetrometer, the properties ofthe snow, and the manner of interaction betweenthe penetrometer and snow (which can vary dur-ing a test). Rammsonde penetration is compli-cated and does not appear to be consistently re-lated to any particular mechanical property (seediscussion in App. D). However, we believe thatthe reason that Fukue (1979) obtained good re-sults was that his blade penetrometer interactedwith the snow on a scale that was not much largerthan the microstructural elements. Even betterresults may be possible by using a thinner, longerblade that brings the scale of the interaction closerto that of the microstructure and increases thenumber of bonds and grains that the blade con-tacts. We have done preliminary experimentswhich indicate that a thin-walled cylinderpenetrometer may work as well as a blade, yetprovide sufficient strength to penetrate hard snow.

The snow microstructure involves the proper-ties of the bonds between grains and the mannerin which they are coupled into larger structures(such as chains), the shapes, sizes and size distri-bution of the grains, and other variables. Thus,since no single physical property uniquely de-fines the microstructure, it is reasonable to expectthat more than one index property will correlatewith various modes of deformational behaviorthat are controlled by the microstructure. For ex-ample, in addition to the penetration force ofblades or thin-walled tubes described above, in-

dex properties based on sonic wave propagationspeed, electrical conductivity, and some stereo-logically derived variables may also correlate wellwith the deformational behavior of snow in someregimes. Because of this we anticipate that valuesof good index properties may correlate with eachother and the microstructral factors that affectdeformational behavior. This also allows for thepossibility that test results related to index mea-sures may eventually be related directly tomicrostructurally important variables, when ac-curate methods to determine them are more fullydeveloped.

A classification ofsnow for applications

A classification of snow for engineering appli-cations consists of a physical classification (i.e.,snow crystal size, shape, type, structure, free wa-ter content, density, and other relevant features)combined with a deformational classification. Thedeformational classification would be obtainedfrom index property measurements as describedin the last section and would give informationabout the microstructure and bonding of the snow.In practice the classification would provide ameans to develop classes of snow in whichdeformational behavior and physical characteris-tics are related. Experiments to acquire stress–strain–time–strength data for the classes in theclassification would then give the range ofdeformational behavior for each snow class.

A possible model for such a classificationwas given by Bader et al. (1939). They sepa-rated snow into 10 classes using qualitative mea-sures of grain size and bond strength as dis-criminators (Fig. 3). In effect, this is a classificationbased on a physical property of snow and aparameter that may be an index property of themicrostructure. Bader et al. (1939) intended theclassification for use in identifying snow typesin the field, and were not attempting to classifyby deformational behavior. However, Kuvaevaet al. (1967) and Fukue (1979) have suggestedclassifications of seasonal snow types accord-ing to anticipated deformational response to ap-plied loads. Both authors require only fourclasses of snow which are similar in both classi-fications and are comparable to some of theclasses in the classification of Bader et al. (1939).Neither author reported having done any sys-tematic work leading to establishing the classi-fication, but the fact that they are similar andwere derived independently may indicate that

12

the number of classes in a classification such aswe suggest might not be excessive.

Testing and test dataWe anticipate that initially the experiments to

demonstrate the range of deformational behaviorof the various snow classes will be unconfinedand confined uniaxial compression at variousloading rates. The confined compression testswould provide useful information on compactionof different types of snow, and the data from theunconfined tests (including the compressivestrength) could be used for comparison with ear-lier work, and for determination of the param-eters of some of the linear stress-strain relation-ships.

Regardless of which parameters are selectedas discriminators in the classification, there willbe gradients rather than sharp boundaries be-tween snow classes. In addition, it is likely thatstress–strain–time–strength data for some classeswill overlap under some conditions, indicatingthat the deformational behavior of those types ofsnow are similar for those cases. This might re-duce the number of classes needed to sort snowtypes according to deformational behavior, al-though it would still be necessary to establish therange of conditions over which the overlap oc-curs. The effort required to do this might ul-timately be equivalent to constructing a “defor-mation map” for each snow class. These would besimilar to the maps used to show how the defor-

mation mechanisms for specific materials varywith test conditions (as an example, Wilkinson[1988] includes a recent deformation map for pureice). Information needed to evaluate the qualityof testing includes details about the test appara-tus, the sample, test procedure, measurementmethods and errors, and data analysis methods.A preliminary discussion of the information thatshould accompany an experiment to measuremechanical properties is given in Smith et al. (1982).

RECOMMENDATIONSAND CONCLUSIONS

We believe that the most important step thatcan be taken to encourage the expanded use ofsnow mechanics in applied problems is to de-velop a readily accessible source of data on theresponse of snow to applied loads. This wouldprovide the information to allow engineers to treatproblems using available numerical data in theabsence of reliable constitutive relationships andwould make information available to assist in thedevelopment of constitutive relationships. We rec-ognize that developing such a comprehensive dataset is a daunting task, but we do not think thesituation will improve unless such an effort ismade. The approach that we recommend involvesthe following steps:

1. Develop methods to measure index proper-ties that are sensitive to variables of the snow

3bt

Coarse-grainedold snow,

medium hard

3at

Depth Hoar

3ct

Coarse-grainedold snow,

hard

3dt

Sun Crust

2bt

Medium-grainedold snow,

soft

2at

Powder Snow

1bt

Wind slab,soft

1at

New snow,loose

1ct

Wind slab,hard

1dt

Wind Crust

2ct

Medium-grainedold snow,

hard

d

Ice Horizon

Loose Medium Hard Hard Very Hard

Fin

e G

rain

Med

ium

Gra

inC

oars

e G

rain

Incr

easi

ng G

rain

Siz

e

3

2

1

dcba

Increasing Bonding Strength

Figure 3. Classification of snow from Bader et al. (1939)

13

microstructure that influence deformationalbehavior. The methods should be suitablefor field use, and their relevance as indi-cators of response to load needs to be dem-onstrated by comparison with the results ofexperimental measurements of properties,such as the strength under some standardset of conditions.

2. Develop a classification system for snowin terms of familiar, descriptive physicalproperties and values of the index proper-ties described above. Suitable physicalproperties might include mean grain size,grain size distribution, snow crystal mor-phology, bulk snow structure, density,and/or other properties that can be deter-mined in the field.

3. Conduct tests to gather stress–strain–time–strength data on representative samples fromthe various classes in the classification.

LITERATURE CITED

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20

The motive for making many of the measure-ments of the mechanical properties of snow hasbeen to find parameters for applying the constitu-tive relationships for ideal elastic solids and vis-cous fluids to problems involving snow deforma-tion. The purpose of this appendix is to describethese “ideal” materials and show how the rela-tionships that describe their response to stress arerelated to more general constitutive relationships.Note that we use the term “ideal” to describethese materials and their constitutive relationshipsbecause they are idealizations based on data fromexperiments on many types of materials. Thus,their origin is empirical, rather than analytical.

For our purposes we only need to considerideal elastic solids and viscous fluids under iso-thermal conditions. An ideal elastic solid is de-fined by the property that the strain at any timedepends only on the instantaneous magnitude ofthe stress and is independent of the stress history.Further, if the stress is removed from a deformedsample of an ideal elastic material, then the straindisappears, the sample returns to its original stateand all the strain energy stored during deforma-tion is recovered. Note that the definition doesnot require the stress-strain relationship to be lin-ear. However, determining a value of Young’smodulus (E) for snow from a static test in uniaxialloading includes the assumption that it is linear,because the experimental data are fit to the one-dimensional form of Hooke’s law, σ = Eε where σis stress and ε is strain in the same direction.Similar arguments apply to the other elastic con-stants (the shear and bulk moduli, Poisson’s ratioand Lame’s constant, λ).

Ideal linear viscous behavior is represented bya constitutive equation in which the stress is pro-portional to the strain rate. As a result, the strainat any time depends on the complete history ofthe stress, rather than its instantaneous magni-tude. Further, when the load is removed from asample of a linear viscous fluid undergoing de-formation, the strain rate goes to zero and none ofthe strain is recovered. Thus, no strain energy isstored during deformation, and all of the workdone by external stresses is nonrecoverable.

The one-dimensional constitutive relation-ship for a homogeneous, isotropic, linear vis-

*The discussion of the section titled “Composite, ViscoelasticSubstance Disclosing Recovery Strains” in Nadai (1963, p. 166),begins “This leads us to propose a third, ideal, composite, viscoelas-tic, recovery-sensitive substance...” (italics his). Nadai then contin-ues to describe the model of the four-parameter viscoelasticfluid as given in the text above. Further, in a footnote on p. 170,Nadai noted the spring-dashpot models analyzed by Burgersand referred to one that “...demonstrates our composite sub-stance having the three types of strain...” Note that Nadai (1963)is a version of a volume which was published originally in 1931,and in revised form in 1951. There is no similar discussion ineither of the earlier editions nor is there reference to earlierpublication of the model, although it is was included withoutreference in Jaeger (1962).

APPENDIX A: CONSTITUTIVE RELATIONSHIPS USEDTO DESCRIBE SNOW DEFORMATION

cous fluid is σ = η(dε/dt) where η is the coeffi-cient of Newtonian viscosity. Unlike the casefor elasticity, the parameters for nonlinear con-stitutive equations for viscous fluids have beendetermined under some conditions for bothsnow and ice. The relationships that have beenused are the power law

d dt A nε σ/ =( ) , the ex-

ponential law

d dt Be bnε σ/ =( ) and the hyperbolicsine law d dt C cε σ/ sinh=( ) where A, B, b, C and care constants that may depend on the temperature,pressure and physical properties.

The idea of using the law for what is nowcalled the four-parameter viscoelastic fluid (calledthe Burger’s material or general linear substanceby some authors) to represent the deformation ofa material was apparently first proposed by Nadai(1963, p. 166*) based on observations of experi-mental creep curves. He defined a material thatresponds to loads by “three distinct types of strainε1, ε2, ε3 and two types of stress, σ1, σ2 ...” Thestrains are 1) an ideal elastic strain that respondsinstantly to changes in stress, 2) a component ofpermanent strain that changes as a function oftime and load, and 3) a semi-permanent, recover-able strain which represents a time-dependentelastic response to the applied load. The familiarspring-dashpot model for this material (shown inFig. A1 with linear springs and dashpots) illus-trates how the total strain results from the sum-mation of the “three distinct types of strain...”listed above. The first two are from the Maxwellmodel, while the third is the Voigt model, (calledthe Kelvin, firmoviscous, or Bingham material bysome authors). Summing the strain components

21

Strain

Time

ε + ε21

ε1

σ 0

E2

σ 0

E1

=dεdt

σ 0

η1

ε 3 ε 2 ε1

σ

σ1

σ2

η1

E1

η2

E 2

σ1

σ2

σ

Voigt Model Maxwell Model

in this manner implies the assumption that theyare independent of each other so that, for ex-ample, changes in the magnitude of the perma-nent strain component with time do not affect theparameter that determines the instantaneouselastic response. To our knowledge, this effecthas never been studied experimentally, but itseems likely that at some strain, the assumptionwill no longer be satisfied. Another point of inter-est is that, since the three strains are independentof each other, it is easy to omit one or two of themdepending upon the application. Thus, for ex-ample, for a problem in which a small stress isapplied for a long period of time, the permanentstrain would become much larger than the com-bined instantaneous and time-dependent elasticstrains. As a result, the elements that contributethese strain components can be ignored. Con-versely, if the loads are applied for only a shorttime, then the permanent strain can be neglectedwith only the elastic components being retained.

Note also that the “two types of stress” re-ferred to by Nadai (1963) are simply the stressesacross the arms of the Voigt model (Fig. A1). Equi-librium requires that they sum to the magnitudeof the stress, σ, applied across the model.

The summation and integration of the strainsfor a constant stress σo applied at time t=0 to the

Figure A1. The four-parameter spring-dashpot model ofa viscoelastic fluid showing nomenclature, stress-strainlaws and relationships for determining values of theparameters from a creep curve.

model with linear elements, leads to (Fig. A1)

ε σ

η ηt

Et

EE t( ) = + + − −

0

1 1 2

2

2

1 11 exp . (A1)

A plot of this equation approximates a creep curve,so that, as described in Yosida et al. (1956), Bader(1962a), Nadai (1963) and Mellor (1964), andshown in Figure A1, the parameters in eq A1 canbe determined from a single experimentally de-rived creep curve (although, obviously, morecurves would be required for accuracy). This wasdone by Yosida et al. (1956) and Shinojima (1967)from experimental data for snow deformed tosmall strains. A similar procedure could also beused (although more experiments would be re-quired) to determine the parameters if some or allof the model elements followed nonlinear stress-strain rate relationships.

Next, we show the relationship between thefour-parameter viscoelastic fluid model and themore general constitutive relationships for non-linear viscoelastic materials. The intent is to showthe assumptions required to make the transitionbetween the different constitutive relationships.

The constitutive relationships for the four-parameter model with linear elements can be de-rived from conditions of equilibrium and the sum-

22

mation of the strains. As shown in Mellor (1975),in differential form it is

∂ σ∂ η η η

∂σ∂ η η

σ2

2t

E E E

t

E E+ + +

+

M

M

M

K

K

K

M K

M K

∂ ε∂ η

∂ε∂

2

2Et

E E

t= +

M

M K

K. (A2)

in which σ and ε are understood to be functionsof time. Equation A2 is a special case of the gen-eral relationship,

Pσ = Qε (A3)

where P and Q are the differential operators

P a a

ta

ta

t

n

n= + + + +0 1 2

2

2∂∂

∂∂

∂∂

K n

Q b b

tb

tb

t

m

m= + + + +0 1 2

2

2∂∂

∂∂

∂∂

K m .

The forms of the constants a0, a1,..., b0, etc., can bedetermined for the four-parameter model by com-parison with eq A2.

Equation A3 can be derived from the integralform of the one-dimensional stress-strain relation-ships for linear viscoelastic fluids,

σ τ ∂ε∂τ

τ= −( )− ∞∫ G t dt

(A4)

ε τ ∂σ∂τ

τ= −( )− ∞∫ J t dt

where G and J are relaxation and creep functionsrespectively. However, there is some loss of gen-erality in making this step (Christensen 1971) sincethe complete spectrum of possible creep or relax-ation times in the integrals in eq A4 is replaced bya discrete number in the differential form (eq A3).

Equations A4 are the one-dimensional formsof the integrals

σ τ ∂ε

∂ττij ijkl

kl= −( ) =− ∞∫ G t d i jt

, , , .1 2 3 (A5)

and its equivalent for strain in terms of stress,which are the three-dimensional forms of the con-stitutive relationships for linear viscoelastic ma-terials.

The problem of defining constitutive relation-ships for nonlinear viscoelastic materials has been

approached through both empirical and analyticmeans (see Lockett 1972). The former involvesfitting relatively simple mathematical expressionsto experimental data, which gives results withlimited ranges of application. In contrast, the ana-lytical approach is more rigorous, originating infundamental axioms of physics. An example ofan empirical relationship follows from the obser-vation in Lockett (1972) that the equation

ε ε σ

σε σ

σt ee

dd

=

+

sinh sinhtn , (A6)

where σ is stress, εt is the total strain, εe, εd, σe, andσd are constants, is a good representation of thecreep of many plastics at the constant stress, σ.Note that if σe is large, then

sinh ,

σσ

σσe e

≈

so that, with the substitution (εe/σe) = 1/E, eq A6describes a Maxwell material with a linear elasticspring and a dashpot which, for n=1, follows thehyperbolic sine relationship for the relationshipbetween stress and strain rate. Equation A6 or itsversion with the linear spring can be extended tothe four-parameter model by simply adding thecontribution of the Voigt model shown in FigureA1, with either linear or nonlinear elements de-pending upon the data which is to be fitted to theequation. Thus, there is some flexibility to theempirical relationships with respect to fitting data,but the resulting equations could become incon-venient for solving boundary value problems.

As an illustration of the analytical approachwe use the example of the stress–strain relation-ship for a Green-Rivlin material, because it is simi-lar to a relationship used by Brown et al. (1973)and Brown (1976) to describe the nonlineardeformational behavior of snow. After assumingisotropy, homogeneity and deformation under iso-thermal conditions, the relationship can be writ-ten in matrix form as (Lockett 1972)

σ ψ ψ τt T dt( ) = +{ }−∞∫ I M1 1 1 1 1

+ +{−∞−∞ ∫∫ I Iψ ψ3 1 2 4 12T T Ttt

+ + }M M Mψ ψ τ τ5 1 2 6 1 2 1 2T d d

+ +{−∞−∞−∞ ∫∫∫ I Iψ ψ7 123 8 1 23T T Tttt

23

+ + +M M M Mψ ψ ψ9 1 2 3 10 12 3 11 1 2 3T T T T

+ } +ψ τ τ τ12 1 2 3 1 2 3M M M d d d K (A7)

where

ij

ij

ijij

ijij ij

ijij ij ij

ij

I

M

=

=

=( )

=( )

=( ) ( )

=( ) ( ) ( )

== ≠= −

1 0 00 1 00 0 1

01

σ σ

∂ε τ∂τα

δ∂ε τ

∂τ

δ∂ε τ

∂τ

∂ε τ

∂τ

γ δ∂ε τ

∂τ

∂ε τ

∂τ

∂ε τ

∂τ

δ

αα

αα

α

αβα

α

β

β

αβα

α

β

β

γ

γ

T

T

T

for i j

for i j

.

The functions ψa are relaxation functions(analogous to G(t–τ) in eq A4 in which ψ1 and ψ2are functions of t–τ1, ψ3,…,ψ6 are functions of t–τ1

and t–τ2, and ψ7,…,ψ12 are functions of t–τ1, t–τ2,and t–τ3. Depending upon their form, each ψacould include several constants which, if the phys-ics were known completely, might be determinedanalytically. However, in general, they are foundby fitting the equation to experimental data. Thus,the approach is really semiempirical, rather thancompletely analytical. The equation could be writ-ten in inverted form, with strain as a function ofstress. In that case, the functions ψa would becreep functions analogous to J(t–τ) above. Notethat the equation is written in three dimensionsalthough it is often reduced to one dimension forapplication as was done in Brown et al. (1973)and Brown (1976).

Finally, to demonstrate the connection betweenthe relatively simple relationships in eq A3 andA4 as compared to A7, we note that, for isotropicmaterials, eq A5 can be simplified to (Lockett 1972)

σ δ λ τ

∂ε∂

τijij

ij T= −( )

−∞∫ t dt

µ τ∂ε∂

τij

T+ −( )

−∞∫ t dt

2 (A8)

which, for constant λ and µ, reduces to the famil-iar relationship for a linear elastic material. How-ever, introducing the substitutions ψ1 = λ(t–τ)and ψ2 = µ(t–τ) puts eq A8 into the notation ofeq A7 and shows that it is the linearized form ofthe constitutive equation for the Green-Rivlinmaterial.

24

HISTORICAL DEVELOPMENT

In this appendix we use published reviews ofthe field of snow mechanics to trace the develop-ment of data on the mechanical properties of snow.Following the discussion, the data are given in aseries of figures which have been updated fromMellor (1975, 1977).

The first attempt at a comprehensive review ofthe mechanical properties of ice and snow in En-glish was presented in SIPRE Technical Report 4(Mantis 1951). The only data given in that reportwere 1) the values of a static Young’s modulus forsnow from Yosida et al. (1948), 2) the snow vis-cosities from Bucher (1948), 3) some values forfriction, and 4) a few tensile and shear strengthmagnitudes. There was also mention of unsuc-cessful attempts by Corps of Engineers personnelto determine some of the mechanical propertiesof snow by dynamic methods, although no de-tails or references were given. SIPRE TechnicalReport 7 (Bader et al. 1951) included the samevalues of the tensile and shear strengths of snowas in Mantis (1951) but also reported the firstvalues of the cross section number “m” (the recip-rocal of Poisson’s ratio). There was an obviouslack of quantitative information on the mechani-cal properties of snow available when these re-ports were prepared. However, there was enoughinformation on the qualitative aspects of the be-havior of snow during deformation that it waspossible to outline a broad experimental program,including the design of testing equipment, to ac-quire additional data (Bader et al. 1951).

The next review and summary of the data onsnow mechanics was published by Bader (1962a).His primary objective was to describe the generalbehavior of snow as a material but he also pre-sented data on mechanical properties from statictests by Butkovich (1956) and Jellenik (1957), andfrom dynamic tests on polar snow by Nakaya(1959a). In addition, Bader (1962a) discussed theinsights to understanding the processes of snowdeformation contributed by the creep experimentsunder uniaxial and multiaxial stresses byLandauer (1955, 1957), and the plate indentationtests by Landauer and Royse (1956) and Yosida et

al. (1958). It is of interest that the numerical val-ues for properties determined by Yosida et al.(1956) and some of those from Jellenick (1957)were the only new data for mechanical propertiesof nonpolar snow included in Bader (1962a) thathad appeared in the literature since the publica-tion of Mantis (1951).

Two years later, Mellor published his first re-view of the mechanical properties of snow (Mellor1964) as part of the snow engineering section ofthe CRREL monograph series. In it, Mellor col-lected and organized data on seasonal, polar andvarious types of processed snow, but the onlywork on seasonal snow reported by Mellor (1964)and not by Bader (1962a) was from Yosida (1963).There were, however, several contributions onthe properties of polar and processed snows(Butkovich 1962; Bender 1957a; Brunke 1959; Lee1961; Nakaya 1959b, 1961; Ramseier 1963;Ramseier and Pavlak 1964; Mellor and Hendrick-son 1965).

The most comprehensive review of the litera-ture on mechanical properties of snow was doneby Mellor (1975), supplemented by additional datain Mellor (1977). Aside from the results from Abeleand Gow (1975, 1976) on the relationship betweendensity and maximum principal stress, the new-est reported data on the mechanical properties orstrength of snow in either review paper was thatby Kovacs et al. (1969). In addition, the only dataon seasonal snow that had not appeared in theearlier reviews were some values of Young’smodulus from Kojima (1954) that had been over-looked earlier but had appeared in Yosida et al.(1956), and the results of creep tests in torsion,uniaxial tension and uniaxial compression byShinojima (1967).

Figures from Mellor (1975, 1977) are still themost comprehensive sources of data on the me-chanical properties of snow. Some of these arepresented later in this appendix with supplemen-tal data added where possible.

The most recent review of the field of snowmechanics was done by Salm (1982). However, itwas devoted mainly to describing progress inunderstanding the mechanics of fracture of snowand in the development of constitutive relation-

APPENDIX B: REVIEW OF THE LITERATURE ON MECHANICALPROPERTIES OF SNOW WITH COMPILATION OF DATA

25

ships including fracture criteria. The only studiesof snow properties cited in Salm (1982) areMcClung’s (1977) work on the shear strength ofsnow, Narita (1980) and Watanabe (1980) on thetensile deformation and strength, and an earlierpaper by Tusima (1973) on the densification ofsnow under repeated loading.

An extensive body of work done in Russia onsnow properties and processes by Kuvaeva et al.(1967) appeared in English translation in 1975 butwas overlooked in the reviews described above.It is a compilation and summary of studies donebetween 1948 and 1962 in the Caucasus Moun-tains and includes material on 1) the thermal prop-erties of a snowpack, 2) snow metamorphism, 3)avalanche forecasting, effects and defense, and 4)the mechanical properties of snow. The tests ofmechanical properties were done between 1958to 1962, and involved snow types ranging fromnewly fallen snow (ρ ≈ 60–80 kg/m3) to wind slaband firn (ρ ≈ 400–500 kg/m3). Representativesamples were tested but the data from individualtests were not generally given. Instead, the re-sults were presented as parameters for variousconstitutive relationships (i.e., as an equation forYoung’s modulus with density and temperatureas parameters) that were stated to be within, forexample, 10 or 15% of the data.

Another important work that was not reviewedearlier is the thesis by Fukue (1979), some of whichwas previously published in Yong and Fukue(1977). The study was broad, but emphasized theinterplay between small-scale processes (friction,adhesion, sintering and creep of individual grains)and the macroscopic behavioral characteristics ofsnow in a variety of loading modes. The workinvolved experiments designed to illustrate thosecharacteristics, but which were not comprehen-sive enough to define the form or parameters ofconstitutive relationships. However, the resultswere synthesized into a qualitative model of snowstructure which provides a useful framework forconsidering and anticipating snow deformationunder a variety of conditions. Fukue (1979) alsoproposed a classification of snow for engineeringpurposes, and developed and tested an experi-mental procedure, which we believe will be im-portant for indexing snow properties.

The data on specific parameters are presentedin the following sections. Note that in most of thefigures the parameter is plotted against the den-sity, which is the usual procedure.

Figure B1. Young’s modulus and Poisson’s ratio vs.density for dry, coherent snow (modified from Fig. 2 inMellor 1975). Data sources cited in the original figureare (A) Pulse propagation or flexural vibration at highfrequencies, –10° to –25°C (Smith 1965; Nakaya1959a,b; Bentley et al. 1957; Crary et al. 1962; Lee1961; Ramseier 1963). (B) Uniaxial compression,strain rate approximately 3 × 10–3 to 2 × 10–2 s–1,temperature –25°C (Kovacs et al. 1969). (C1) Uniaxialcompression and tension, strain rate approximately 8× 10–6 to 4 × 10–4 s–1, –12° to –25°C. (C2) Static creeptest, –6.5° to –19°C (Kojima 1954). (D) Complexmodulus, 103 Hz, –14°C (N. Smith 1969). (S) Quasi-static measurements of Poisson’s ratio (Salm 1971).Additional data added for this report; (K) plotted fromequation for best fit curve to data for static Young’smodulus and quasi-static measurements of Poisson’sratio from Kuvaeva et al. (1967).

26

YOUNG’S MODULUS ANDPOISSON’S RATIO

The plot of the data for both static and dy-namic measurements of Young’s modulus forsnow from Mellor (1975), supplemented by addi-tional data from Kuvaeva et al. (1967), is shown inFigure B1. In addition, the only data availablefrom dynamic or quasistatic determinations ofPoisson’s ratio for snow are also included in thefigure.

VISCOSITY

The difficulty of determining Poisson’s ratio inrapid-loading compression tests, coupled with theextreme compressibility of low density snow, ledto the introduction in Bader et al. (1951) of theparameter called the “cross section number (thereciprocal of Poisson’s ratio).” Mellor (1975) iden-tified this parameter as the viscous equivalent ofPoisson’s ratio and presented the available data(Fig. B2).

Mellor (1975) separated the data on viscosityinto the categories of “axial” and “compactive”viscosity. The former refers to the viscosity de-termined from the “steady state” creep rate inexperiments under constant uniaxial compressivestress. In terms of the four-parameter model,

the axial viscosity is the viscosity of the dash-pot of the Maxwell model. The compactive vis-cosity is determined either from the results ofexperiments in confined compression (uniaxialstrain), or from measurements of compaction as afunction of time for natural snowpacks.

The data on axial viscosity are shown in FigureB3. The range of values for this parameter is large,even allowing for the differences in the physicalproperties of polar and seasonal snow. It shouldalso be noted that, in addition to determining theaxial viscosity, Shinojima (1967) also did creepexperiments in torsion and uniaxial tension andused the results to determine the parameters forthe four-parameter viscoelastic fluid model for allthree loading modes. In addition, the data wereused to define the viscosity of the lead dashpot ofthe model as a function of temperature and snowdensity over the range of temperatures from 0° to–40°C and densities from 125–300 kg/m3. Notethat these results were used by Lang andSommerfeld (1977).

Mellor (1975) determined the compactive vis-cosity of seasonal snow from data from field mea-surements by Kojima (1967), and both field andlaboratory studies by Keeler (1969a). He combinedthese results with data for polar snow from Bader(1962b), experimental work by Mellor andHendrickson (1965) and other field studies, intohis Figure 11 (p. 266). Ambach and Eisner (1985)

Figure B2. Mellor’s (1975) summary of data on the viscous analog of Poisson’s ratio.Data sources were de Quervain (1966), Roch (1948), Shinojima (1967), Yosida (1963)and Bader et al. (1951).

Calculated from Data by DeQuervain (1966)

Uniaxial Tension*

Uniaxial

*Shinojima (1967)

Yosida (1963)

Bader et al.(1951)

Data Summarized by Roch (1948)

?

5

4

3

2

1

0 200 400 600 800 1000

Density (kg m )–3

υ ,

Poi

sson

's R

atio

v

Incompressible

Compression*

0.5

0.4

0.3

0.2

0.1

Calculated from Data by de Quervain (1966)

27

Figure B3. Axial viscosity of snow vs. density from Mellor’s (1975) Figures 8 and 9. The data sourcesin (a) are (A) Ramseier and Pavlak (1964), (B) Mellor and Smith (1967), (C) Bucher (1948), (D)Shinojima (1967), (E) Mellor and Smith (1967) and Mellor and Testa (1969). Data sources in (b) asshown.

Figure B4. Compactive viscosity vs. den-sity, modified from Mellor (1975, Fig. 11)by Ambach and Eisner (1985). Data from(A) Greenland and Antarctica at –20° to–50°C (Bader 1962b); (B) Seasonal snow inJapan at 0° to –10°C (Kojima 1967); (C)Alps and Rocky Mountains (Keeler 1969a);(D) Uniaxial-strain creep tests at –6° to–8°C (Keeler 1969a); (F) Uniaxial straincreep tests at –23° to –48°C (Mellor andHendrickson 1965); (G) Dorr and Jessberger(1983) and Ambach and Eisner (1985);(H) Seasonal snow in Japan at 0° to –6°C(Endo et al. 1990).

28

updated that figure to include field data de-veloped by them, as well as the results ofearlier work by Dorr and Jessberger (1983).That figure is shown here in Figure B4 to whichwe have added data from Endo et al. (1990).

STRENGTH

In his discussion of failure, Mellor (1975)discussed the ambiguity of the term andadopted the definition that failure is “relatedto the maximum deviatoric stress that can bereached at a given strain rate...”. He used thisdefinition because it does not distinguish be-tween brittle and ductile cases, and thus ap-plies across the spectrum of possible failuremodes for snow. However, he presented dataonly for the brittle regime (i.e., at high rates ofloading) in uniaxial tension and compression,and some data for shear from various exper-iments and from estimates based on theanalysis of avalanche fractures. The datafrom Mellor (1975) are shown in Figures B5and B6 to which we have added data fromlater studies by McClung (1977) and Narita(1980). Note that a compilation of in situmeasurements of tensile strength of snowhas recently been prepared by Jamieson andJohnston (1990) and has been added to Fig-ure B5.

Figure B5. Uniaxial compressive and tensile strengths ofsnow under rapid loading rates from Mellor (1975). Orig-inal data (M-compression and M-tension) from Bucher(1948), Butkovich (1956), Haefeli (in Bader et al. 1939),Hawkes and Mellor (1972), Keeler (1969a), Keeler andWeeks (1967), Kovacs et al. (1969), Mellor and Smith(1965), Ramseier (1963), and Smith (1963, 1965). Addi-tional data are (A) for strain rates greater than 5 × 10–4 s–1

from Narita (1980) and (B) the compilation of in situ ten-sile strength data in Jamison and Johnston (1990).

Figure B6. Shear strength of snow from Mellor (1975).Original data from Ballard and McGaw (1965), Butkovich(1956), Haefeli (in Bader et al. 1939), Keeler (1969a), Keelerand Weeks (1967).

29

CONFINED COMPRESSION(UNIAXIAL DEFORMATION)AND BEARING STRENGTH TESTS

The relationship between the major principalstress and the density in uniaxial strain has beenof concern in the subject of snow mechanics be-cause it simulates, to some extent, the process ofdensification of natural snowpacks and is impor-tant in considerations of the bearing strength ofsnowpacks. Bader (1962a) included a discussionof the data from plate bearing tests by Landauerand Royse (1956), Yosida et al. (1957) and Bucherand Roch (1946) as they relate to this subject.Mellor (1964) discussed some of the material onplate indentation, and treated the combined sub-ject thoroughly in Mellor (1975). Experimentalwork has since been done on the stress vs. densityrelationship by Abele and Gow (1975, 1976) (re-viewed in Mellor 1977) and Yong and Metaxas(1985).

Mellor (1975) produced a plot of maximumprincipal stress vs. density for the range of densi-ties from less than 100 kg/m3 to 900 kg/m3 (thecomparable stress range is from about 0.1 kPa to 1MPa) by combining 1) densification curves fromnumerous unspecified locations, 2) miscellaneousdata on the compression of ice and firn, 3) thedata which had appeared in Bader (1962a), 4)shock experiments and 5) new data by Kinosita(1967). Data from static loading experiments covera significant range of the density, which was ex-panded further in the work of Abele and Gow(1975, 1976). Those authors included the effects ofinitial density, temperature, strain rate and aging.A plot from the latter paper which shows thepresent state of the data is given in Figure B7. In alater work, Yong and Metaxas (1985) studied theeffect of strain rate and aging on the deformationof snow in confined compression and direct shear.The range of strain rates they used (about 3 × 10–3

to 1 × 10–2 s–1) overlapped part of the range ofrates used by Abele and Gow (1975), so the re-sults may be comparable. The data in Yong andMetaxas (1985) showed the effect of strain rate,with the stress at a strain of 35% elevated at thelower rates; however, the differences may not besignificant for strain less than about 20–25%. In

Figure B7. Stress vs. snow density data fromvarious sources from Mellor (1975) as modi-fied by Abele and Gow (1976). Data fields are(A) natural densification of snow at –1° to –48°C,(B) Slow natural compression of dense firn andporous ice (from depth/density curves for polar icecaps), (C) Slow compression of solid ice, (E) Calcu-lated values for plane wave impact at 20–40 m s–1.(F) Hugoniot data for explosively generated shockwaves (impact velocities 1 to 12 m s–1) at –7° to –18°C; (J) Compression at approximately constantstrain rate e ≈ 10–4 s–1 at –7° to –18°C (Kinosita1967); (K) Compression in uniaxial strain withincremental loading to collapse, –2° to –3°C (Bucherand Roch 1946). Heavy lines labeled –1°C and–34°C are the boundaries of the data of Abele andGow (1975, 1976).

addition, the influence of aging of the sampleswas clearly shown by the increase in the stress atany strain as the samples aged. Abele and Gow(1975, 1976) did not find any influence of strainrate in the range from about 2 × 10–3 s–1 to 11 s–1,but their data do show the same trend of theinfluence of aging as those of Yong and Metaxas(1985).

30

APPENDIX C: DETERMINATION OF MICROSTRUCTURALVARIABLES BY PLANE SECTION STEREOLOGY

Independent variables measured for snow un-dergoing mechanical testing should represent fun-damental states or conditions of the snow, includ-ing relevant macroscopic and microscopicfeatures. Examples of such state variables mightbe temperature, density, mean grain size (and orgrain size distribution), mean number and radialcross-sectional area of grain bonds per grain, themean number and length of unsupported chainsof grains, the mean distance between grains, andother physically measurable characteristics. Whiledetermining macroscopic state variables can bedone reliably, measuring microscopic variablesand determining their significance is problematic.

Attempts to define and determine state vari-ables using plane section and thick section stere-ology have demonstrated the importance of snowmicrostructure on the response of snow to an ap-plied load (Kry 1975a,b; Gubler 1978a,b; Alley1986; Good 1987; Dozier et al. 1987; Hansen andBrown 1987; Hansen 1988; Edens and Brown 1991;Brown and Edens 1991). However, there are twodifficulties with using stereological methods thatneed to be taken into account when evaluatingtheir usefulness. First, the accuracy with whichstereological methods can be used to determinemany of the parameters they are intended to rep-resent is, in general, unknown. While bulk den-sity and parameters related to the two-dimen-sional plane section can be determined, extendingthese measures to a three-dimensional represen-tation of microstructural measures requires as-sumptions about the geometry of the materialwhich cannot be rigorously tested (Alley 1986).The accuracy of the determination of the averagenumber of grains per unit volume (Nv) stronglydepends on the grain shape. Errors of more thanan order of magnitude in the value of Nv canresult from only small variations in grain shape(Dehoff and Rhines 1961), and seasonal snow con-sists of grains of complex and variable shapes.Thus, estimates of Nv, and variables that dependupon values of Nv from plane section stereologyshould be considered inaccurate (Alley 1986). Ex-

amples of such variables are measures relating tochains of grains, the number of bonds per grainand others.

Other problems arise because the assumptionsused to convert measurements from plain sectionor thick section images into the desired microstruc-tural parameters are often overly simplified andmay contain free variables (Gubler 1978a,b; Alley1986). In addition, the lack of objective methodsfor identifying many structural features in snowusing plane section and thick section stereologyhas been well recognized (Kry 1975a, Alley 1986,Dozier et al. 1987).

Stereological methods require relatively largeinvestments in time and effort to construct planesection or thick section samples, obtain good qual-ity images, and analyze the results. Because ofthese difficulties few experimental studies relat-ing stereological parameters to the deformationalbehavior of snow have been published since thework of Voitkovsky et al. (1975) and Kry (1975a,b).Voitkovsky et al. (1975) presented a relationshipbetween cohesive force and the number ofintergrain contacts per unit volume, based onabout 50 tests in five distinct density categorieswhile Kry’s (1975b) results were based on repeatedexperiments on only five snow samples. Gubler(1978a,b) presented results for the tensile strengthin terms of microstructural variables from fewerthan twenty tests, and changes in microstructureunder large deformations have been examined intwo studies (Edens and Brown 1991, Brown andEdens 1991).

We consider it unlikely that microstructuralmeasures truly represent those features that di-rectly control deformational behavior. They mayeventually be used to derive indexes of deforma-tional behavior, in the same manner as the indexparameters discussed in the main text in Estab-lishing Independent Variables or Index Properties forSnow Microstructure, but the effort required tomeasure microstructural variables suggests thatother more easily measured properties should besought.

31

ELECTRICAL PROPERTIES

Kuroiwa (1962) described an experiment inwhich the components of the complex dielectricconstants and the AC conductivity of a snowsample were measured several times over a pe-riod of 143 hours while the snow sintered at atemperature of –3°C. Over the frequency rangefrom less than 103 to about 104 Hz the constantsvaried significantly and the conductivity increasedby a factor of 10 during the experiment. Thechanges were interpreted as due to the shorten-ing of the electrical paths because of bond growth(Fig. D1), suggesting that the AC electrical con-ductivity is sensitive to the bonding and could bea useful index property. In addition, since it is adirectional property, the AC conductivity mayalso give information about the degree of anisot-ropy of the snow. In other work, Keeler (1969a)found no relationship between the dielectric prop-erties and the structure of the snow he studied.Denoth (1985) found that the static dielectric con-stants depended strongly on porosity grain shape

and liquid water content; the effects of sinteringwere not directly observed.

The DC conductivity also shows some possi-bility of being useful as an index property, al-though it is more sensitive than the propertiesnoted above to the presence of impurities in thesnow (Mellor 1977).

ELASTIC WAVE VELOCITY ANDDYNAMIC ELASTIC MODULI

Nakaya (1961) showed that the dynamicYoung’s modulus of processed snow was relatedto the bond structure of the snow, and Voitkovskyet al. (1975) found that the wave propagation ve-locity of snow samples undergoing creep defor-mation increased with time as the snow densityand grain contact area increased (Fig. D2). These

APPENDIX D: POSSIBLE INDEX PROPERTIES

Immediately after Packing

After143 Hours

ICE BONDING

143 hr

11370

40

16

0–3° C

ρ = 450 kg m –3

Frequency (Hz)

AC

Con

duct

ivity

107

106

105

104

103

102 103 104 105 106

Figure D1. AC conductivity of an aging snow sample asa function of frequency. Measurements were made at theindicated elapsed times and the change in structure of thesample over 143 hours is indicated schematically by theaccompanying sketch (from Kuroiwa, in Bader 1962a).

a.

0.30

0.20

0.10

0

1500

1000

500

0

Str

ain

Ela

stic

Wav

e V

eloc

ity (

m s

)

–1

ρ = 360 kg m–3

380

390

400410

b.

2.4

2.0

1.6

1.2

0.8

0.4

0 2 4 6 8Time (days)

800

1200

400

0

Ave

rage

Gra

in S

ize

(mm

)

S

(m

)

–1K

Figure D2. (A) Creep strain at constant stress andelastic wave velocity (V) with changing density (ρ),(B) average grain size (d) and specific grain contactsurface (Sk) as a function of time (t) in days. Measure-ments of ρ, V, d and Sk were made periodically as a series ofsamples deformed at constant stress (from Voitkovsky 1975).

33

results indicate that seismic velocity (or the dy-namic elastic moduli calculated from the velocitydata and the density) is a potential index prop-erty.

SNOW HARDNESS

Hardness, as indicated by a penetrating cone(e.g., Swiss Rammsonde, Russian AARI penetrom-eter), flat plate penetrometer (Kragelski 1949) orby the hand-hardness scale (de Quervain 1950),has been used for many years as an index param-eter in a semi-quantitative sense. Quantitative in-terpretations have been hampered by difficultiesin interpreting the results of measurements withthese devices because the results depend on theshape of the penetrometer, the size of penetrom-eters of the same shape (Kragelski 1949) and othervariables. Abele (1963) attempted to derive a quan-titative correlation between Rammsonde hardnessand the uniaxial compressive strength of snowbut was not successful. Later, Waterhouse (1967)reanalyzed Abele’s data using alternative ap-proaches but was still not able to establish a reli-able correlation. He concluded that his inabilityto account for the structure of the snow was amajor reason for the lack of success. In fact, Gubler(1975) concluded that it is not possible to define aunit of hardness that is independent of the in-strument used to measure it. However, if it ispossible to establish a relationship between thenature of the bonding in a range of snow samples,and the hardness of those samples as measuredby any particular instrument, then the hardnessvalues as determined by that instrument could beuseful as an index parameter.

Snow hardness is relatively easy to measure inthe field and for that reason it is worth seeking ameasurement of that type that can be quanti-tatively related to the snow structure. A test thatmay be useful for the purpose is the measure-ment of the blade penetration force as suggestedby Fukue (1979). He measured the force requiredto push a thick blade into a snow sample at aconstant rate and then showed that the resultswere related to both the bonding between grainsand the uniaxial compressive strength (Fig. D3).Similarly, Kovacs (1976 and pers. comm.*) notedthat the resistance to driving hollow piles in polarsnow was closely correlated with the uniaxial com-pressive strength of the snow. This suggests, by

Figure D3. Uniaxial compressivestrength of snow samples vs. bladepenetrating force (from Fukue 1979).The linear relationship indicates thepotential value of the blade penetratingforce as an index property.

200

100

0 20 40 60Blade Penetration Force (N)

Unc

onfin

ed C

ompr

essi

ve S

tren

gth

(Pa)

Density (kg m )–3

350420480

*A. Kovacs, CRREL, personal communication 1992.

analogy, that it might be useful to experimentwith penetrometers in the form of thin-walledhollow tubes.

Kuvaeva et al. (1967) described another pen-etration technique but, unfortunately, without giv-ing the dimensions of the equipment. They pusheda small spherical penetrometer into snow samplesup to some value of a parameter they define asthe hardness, H (see Fig. D4). They then allowedthe penetrometer to settle further under its ownweight and calculated H at a series of later times,using the weight and dimensions of the penetrom-eter and assumptions about the manner in whichthe displacement was partitioned between differ-ent mechanisms. Eventually, the hardness becameasymptotic to some value, designated as H∞, whichwas presented in a table along with the tensileand shear strengths of the samples and informa-tion on grain sizes and bonding. The sampleswere of different densities and grain sizes, andthe temperatures varied through the tests. How-ever, the plot of H∞ against the values for thestrength measurements (Fig. D4) shows that therelationships may be linear, which suggests thatthis technique should be investigated further as apossible index test.

34

0.4

0.3

0.2

0.1

0 10 20 30

H

, H

ardn

ess

(MP

a)∞

Stress (kPa)

Shear StrengthTensile Strength

360-400 450

Density (kg m )– 3

Figure D4. Snow hardness, H∞ (see text for defini-tion) vs. shear strength and tensile strength atdensities indicated. Data from Kuvaeva et al. (1967).

PERMEABILITY

Permeability has long been regarded as an im-portant property of snow. Bader et al. (1939)showed that several types of snow fall into dis-tinct fields on a plot of permeability vs. porosity,and permeability has also been shown to vary

with grain size and morphology (Bender 1957b;Bader 1962a; Chacho and Johnson 1987). It is pos-sible that the contact area between grains couldaffect the permeability, since increases in contactarea can decrease the fraction of the cross sectionof a sample that a fluid can pass through, andpossibly increase the tortuosity. However, changesin permeability might be independent of the ex-tent of grain bonding rather than simply the con-tact area. Thus, two snow samples with the samedensity, grain size distribution and grain mor-phology might have the same permeability butvery different mechanical properties. For thesereasons, we do not consider permeability alone tohave potential as an index property. However, itmay be of use when interpreted in conjunctionwith other index properties.

ENERGY OF DISAGGREGATION

A device to measure the energy of disaggrega-tion (in effect, the work done in separating thegrains in a snow sample) was described in Mantis(1951) and was used to establish a relationshipbetween that parameter and the uniaxial com-pressive strength of snow (Bender 1957a, as de-scribed in Mellor 1964). Other investigators havehad less success (see discussion in Mellor 1964)and the method has received little attention sincethat time. However, it was suggested in bothMellor (1977) and Oakberg (1982) that the energyof disaggregation be studied again as a possibleindex parameter.

35

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Snow Mechanics: Review of the State of Knowledge and Applications PE: 6.27.84APR: 4A762784AT42TA: CS

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43Classification Electrical properties SnowDeformation Microstructure

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

A review of snow mechanics indicates that, with the exception of avalanche studies, it is seldom used. In thisreport we give our interpretation of why this is the case, and suggest ways to help expand the range ofproblems to which snow mechanics can be applied. Until the late 1960s, most experimental work in snowmechanics was devoted to finding values of the parameters for equations of linear elasticity, viscosity, andviscoelasticity. In about 1970, work on that approach stopped and since then the emphasis has been on 1) thedevelopment of nonlinear theories to describe the deformation and fracture of snow, and 2) attempts todevelop constitutive relationships based on the study of the microstructural aspects of snow deformation. Webelieve that the best hope of encouraging more applications for snow mechanics in the near term lies inimproving and expanding the database on the response of snow to applied loads, and organizing it in amanner that makes it easy for potential users to determine the anticipated deformational behavior of snow inany particular application. To do this, we suggest developing a classification of snow based on physicalproperties and index parameters that give information about the bonding and microstructure. Mechanicalproperties, constitutive relations under various loading conditions, and other relevant information can then beassociated with each class.

For conversion of SI units to non-SI units of measurement consult ASTM Standard E380-93, Standard Practice for Use of theInternational System of Units, published by the American Society for Testing and Materials, 1916 Race St., Philadelphia, Pa.19103.