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An explicit numerical model for the study of snow's response to explosive air blast D.A. Miller , R.G. Tichota, E.E. Adams Department of Civil Engineering, Montana State University, Bozeman, MT, United States abstract article info Article history: Received 19 November 2010 Accepted 6 August 2011 Keywords: Avalanche control Snow explosives Explicit model In this paper, an analytic tool is used to examine the internal dynamic snow response during explosive events. An explicit nonlinear dynamic model (using ANSYS/AUTODYN) is presented where the explosion, shock propagation through air and snow response is simulated in a single analysis. This versatile approach handles the complex interactions from explosive events and solids, gases and liquids. Nonlinear interactions and responses are modeled during the detonation and subsequent propagation. The model predicts internal structural response during explosive events, including important parameters such as stress, strain, density changes, velocity and acceleration. Snow shock Hugoniots are used for volumetric constitutive relationships with the deviatoric relationships modeled as linear elastic. The analysis shows stress waves in the snow resulting from the explosive shock wave traveling over the surface. While the normal load transits the weak layer, a shear stress wave concentrated above the weak layer develops. The intensity at depth and lateral extent of the stress wave may be an important consideration for initiating avalanches with explosives. Examples with charges on and above the snow surface support the well known air burst advantage, but also show that the dynamic enhancement is not due to peak air pressure alone. Results for two explosive congurations support enhanced dynamic response with increased air pressure impulse, providing further insight into the suspended charge advantage. Charge size is briey examined with larger explosives providing an advantage in stress wave intensity and range. Various snowpack congurations and explosive charges with variable locations can be examined with this approach. The analytic approach provides a tool for future detailed examination of critical avalanche control parameters. © 2011 Elsevier B.V. All rights reserved. 1. Introduction Explosives are routinely used to initiate avalanches to stabilize snow on slopes. Many ski resorts and highway departments use this method as part of efforts to maintain public safety by reducing avalanche hazards. Understanding the dynamic response of a snowpack during shock wave interaction is important for effective use of avalanche control explosives. Currently, the probability of success in inducing an avalanche is based largely on the personnel experience in charge type, size, placement and timing coupled with historical performance of a particular slope. Unintended avalanche release, after explosive control efforts, has recently had a devastating toll. During the 2008/09 winter, 4 people were killed and another 18 buried within US ski resort boundaries by avalanches (Abromeit, 2010). Since 2009, avalanches within US resort boundaries have killed three additional people. Most of these incidents were classied post control releasewhere the avalanche unexpectedly initiated after explosive control efforts failed to stabilize the slope. While we cannot yet identify all the causal mechanisms of these events, they have motivated new efforts looking at responses of snowpacks to explosive events. Historically, there has been signicant experimental investigation of explosives detonated on or above the snow surface (eg: Gubler, 1977; Ingram, 1962; Joachim, 1967; Wisotski and Snyer, 1966). These studies revealed the enhanced effectiveness of suspended explosives in inducing avalanches. As a result, suspended techniques are commonly practiced in the avalanche control industry. Buried explosives dedicate signicant energy to crater formation with local dissipation yielding little momentum transfer away from the crater (Johnson et al., 1994). Detonations on and above the surface are examined here, but the numerical techniques could accommodate analysis of buried explosives in snow covers. Detonations in the air produce a spherically expanding shock wave that decreases in intensity with distance from the blast due to geometric wave expansion and medium attenuation. As the wave travels over the snow surface, energy from the air shock is transmitted to the snowpack inducing shock in the pores and stress waves within the ice network. Within the snow, this produces a spherical shock that dissipates not only due to geometric expansion but also due to snow compaction. As detonation height above the snow is increased, the air pressure immediately beneath the blast decreases, but the air pressure at distances away from the blast will increase to a maximum value and then decrease as the explosive is raised further. Johnson et al. (1994) analyzed experimental air blast data from Ingram (1962) and Wisotski Cold Regions Science and Technology 69 (2011) 156164 Corresponding author at: Department of Civil Engineering, Montana State University, PO Box, 173900, 205 Cobleigh Hall, Bozeman, MT 59717, United States. Tel.: +1 406 994 6118. E-mail address: [email protected] (D.A. Miller). 0165-232X/$ see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2011.08.004 Contents lists available at SciVerse ScienceDirect Cold Regions Science and Technology journal homepage: www.elsevier.com/locate/coldregions
Transcript
Page 1: AUTODYN_2010_Miller_An Explicit Numerical Model for the Study of Snow's Response to Explosive Air Blast

Cold Regions Science and Technology 69 (2011) 156–164

Contents lists available at SciVerse ScienceDirect

Cold Regions Science and Technology

j ourna l homepage: www.e lsev ie r.com/ locate /co ld reg ions

An explicit numerical model for the study of snow's response to explosive air blast

D.A. Miller ⁎, R.G. Tichota, E.E. AdamsDepartment of Civil Engineering, Montana State University, Bozeman, MT, United States

⁎ Corresponding author at: Department of CivilUniversity, PO Box, 173900, 205 Cobleigh Hall, BozemTel.: +1 406 994 6118.

E-mail address: [email protected] (D.A. M

0165-232X/$ – see front matter © 2011 Elsevier B.V. Adoi:10.1016/j.coldregions.2011.08.004

a b s t r a c t

a r t i c l e i n f o

Article history:Received 19 November 2010Accepted 6 August 2011

Keywords:Avalanche controlSnow explosivesExplicit model

In this paper, an analytic tool is used to examine the internal dynamic snow response during explosive events.An explicit nonlinear dynamic model (using ANSYS/AUTODYN) is presented where the explosion, shockpropagation through air and snow response is simulated in a single analysis. This versatile approach handlesthe complex interactions from explosive events and solids, gases and liquids. Nonlinear interactions andresponses are modeled during the detonation and subsequent propagation. The model predicts internalstructural response during explosive events, including important parameters such as stress, strain, densitychanges, velocity and acceleration. Snow shock Hugoniots are used for volumetric constitutive relationshipswith the deviatoric relationships modeled as linear elastic. The analysis shows stress waves in the snowresulting from the explosive shock wave traveling over the surface. While the normal load transits the weaklayer, a shear stress wave concentrated above the weak layer develops. The intensity at depth and lateralextent of the stress wave may be an important consideration for initiating avalanches with explosives.Examples with charges on and above the snow surface support the well known air burst advantage, but alsoshow that the dynamic enhancement is not due to peak air pressure alone. Results for two explosiveconfigurations support enhanced dynamic response with increased air pressure impulse, providing furtherinsight into the suspended charge advantage. Charge size is briefly examined with larger explosives providingan advantage in stress wave intensity and range. Various snowpack configurations and explosive charges withvariable locations can be examined with this approach. The analytic approach provides a tool for futuredetailed examination of critical avalanche control parameters.

Engineering, Montana Statean, MT 59717, United States.

iller).

ll rights reserved.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Explosives are routinely used to initiate avalanches to stabilizesnow on slopes. Many ski resorts and highway departments use thismethod as part of efforts to maintain public safety by reducingavalanche hazards. Understanding the dynamic response of asnowpack during shock wave interaction is important for effectiveuse of avalanche control explosives. Currently, the probability ofsuccess in inducing an avalanche is based largely on the personnelexperience in charge type, size, placement and timing coupled withhistorical performance of a particular slope. Unintended avalancherelease, after explosive control efforts, has recently had a devastatingtoll. During the 2008/09 winter, 4 people were killed and another 18buried within US ski resort boundaries by avalanches (Abromeit,2010). Since 2009, avalanches within US resort boundaries have killedthree additional people. Most of these incidents were classified “postcontrol release” where the avalanche unexpectedly initiated afterexplosive control efforts failed to stabilize the slope. While we cannotyet identify all the causal mechanisms of these events, they have

motivated new efforts looking at responses of snowpacks to explosiveevents.

Historically, there has been significant experimental investigation ofexplosives detonated on or above the snow surface (eg: Gubler, 1977;Ingram, 1962; Joachim, 1967; Wisotski and Snyer, 1966). These studiesrevealed the enhanced effectiveness of suspended explosives in inducingavalanches. As a result, suspended techniques are commonly practicedin the avalanche control industry. Buried explosives dedicate significantenergy to crater formation with local dissipation yielding littlemomentum transfer away from the crater (Johnson et al., 1994).Detonations on and above the surface are examined here, but thenumerical techniques could accommodate analysis of buried explosivesin snow covers. Detonations in the air produce a spherically expandingshock wave that decreases in intensity with distance from the blast dueto geometric wave expansion and medium attenuation. As the wavetravels over the snow surface, energy from the air shock is transmitted tothe snowpack inducing shock in the pores and stress waves within theice network. Within the snow, this produces a spherical shock thatdissipates not only due to geometric expansion but also due to snowcompaction. As detonation height above the snow is increased, the airpressure immediately beneath the blast decreases, but the air pressure atdistances away from the blast will increase to a maximum value andthen decrease as the explosive is raised further. Johnson et al. (1994)analyzed experimental air blast data from Ingram (1962) and Wisotski

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157D.A. Miller et al. / Cold Regions Science and Technology 69 (2011) 156–164

and Snyer (1966) (summarized in O'Keeffe (1965) and Mellor (1985))and presented a scaled curve of maximum air pressure as a function ofburst height and radius from the blast vertical axis. This curve identifiesthe scaled burst height for maximum pressure at particular scaleddistances. The data supports ~40% increase in overpressure as thedetonation height is increased from the surface to the optimal height(for a particular range and net explosive weight). The maximumoverpressure was relatively insensitive to scaled detonation heightsabove ~2 m/kg1/3. Gubler (1977) conducted field experiments wherecharge mass, snowpack stratigraphy, explosive type, charge placementrelative to the surface and ground type were considered. One significantconclusion of Gubler's work is that 1 kg charges detonated from 1 to 2 mabove the surface have enhanced results for releasing dry slabavalanches. Ueland (1992) used mining seismographs to measure thevertical response of snowpacks during explosive events. His workconfirmed the air blast advantage over surface or buried charges, butalso examined shock attenuation through the snowpack depth. He foundsnow hardness, more than density, was a significant factor in shockattenuation. With the relationships of range, explosive weight and blastheight established; what peak pressure at what range is required foravalanche release? Few data exist to answer this question, but Mellor(1973) suggests loading the anticipated avalanche release zone with atleast 3.5 kPa, but detailed analysis or empirical data justifying this limit islacking. A numerical approach that can predict internal snowpackresponses would be valuable for investigating load distributions throughthe depth of the snowpack at various ranges from the blast center.

Brown (1981) developed jump equations to describe the change insnow physical parameters across a shock wave. Of particular interest,he predicted the change in snow density across the shock. He found itdifficult to validate his approach due to a shortage of good shock wavemeasurements in snow. Johnson (1991) developed a momentummodel to predict shock wave attenuation in snow. He showed thatsnow attenuation was largely dependent upon the snow pressure–density relationship. To predict the snow response to explosives,volumetric constitutive relationships must be used to describesnowpack compression during loading. Haehnel and Shoop (2004)present a capped Drucker–Prager model simulating the load-deformation characteristics of snow. Their model provides upperand lower “bounds” for the response of low density snow loaded athigh strain rates, but was applied to tire movement through snow, notto explosive loading. Some experimental research has focused onsnow response to explosives to better develop constitutive relation-ships. Furnish and Boslough (1996) conducted impact tests on snowsamples and material simulators to experimentally derive shockHugoniot states, reshock characteristics and release properties. Theyreport reliable snowHugoniot states, with snow density of 500 kg/m3,at stresses up to ~4 GPa. Johnson and Solie (1993) conducted gas gunimpact tests (with analysis) to determine the pressure–densityrelationships for several initial snow densities with stresses up to40 MPa. In all of the tests, snow had significant shock attenuation.Johnson and Solie (1993, 1994) discuss snow's large load hysteresisdue to its small volume recovery during unloading. This compactioncharacteristic is very important for understanding snow's response toexplosives. In close proximity to the blast, the snow will compact to acritical density value before accepting significant stress. The pressurerequired to compact snow to a final density value increased withdecreasing initial density.

To initiate slab avalanche release, Heierli et al. (2008) present atwo stage process for fracture and subsequent avalanche. In the firststage, normal and shear stress combine in loading a preexisting flaw.When the mechanical energy reaches a critical value for a particularcrack nucleus, fracture progresses with a mixed mode anticrack. Thisfailure mode is driven by a volumetric collapse of a layer containingthe crack nucleus. After fracture, the nature of contact forces betweenthe crack faces dictate if the slab will slide down the hill. This theorypredicts the frequently observed phenomenon of crack propagation

with no accompanying avalanche (i.e. whumpfs). While Heierli et al.(2008) focuses on gravitational snow loading for the shear andnormal comments of mechanical energy, they do briefly mentionother “triggers” such as explosives.

While these studies have provided critical data and insights,modern analytical tools integrating the explosive, atmosphere, terrainand snowpack have not seen application. Historically there has beenexperimental investigation of explosives on and over snow, but thereexists virtually no numerical or analytic research into the snow'sdynamic response to explosive detonations, particularly at theavalanche slope scale. A comprehensive approach capable ofexamining the internal snow response during explosive loading isneeded. Such an approach or analytical technique would open thedoor for studying the critical parameters and sensitivities eventuallyleading to enhanced effectiveness of explosive charges on avalancheprone slopes.

This study is a first step in such an approach applying modernanalytic and numerical techniques to the avalanche control situation.The goal of this study is to develop an appropriate analytical toolcapable of predicting snow response to an explosive event. Theapproach should allow for multiple snow and explosive types andconfigurations. Internal snow dynamic response must be predictedand available for analysis.

2. Approach

2.1. Explosive air blast

Following an explosion in air, a compressive shock wave isgenerated and propagates radially through the air at supersonicspeeds. Across a shock wave, the air experiences an increase intemperature, pressure and velocity from the front passage. Thestrength of a normal shock front is commonly characterized by thesize of the pressure increase across the front, the wave propagationvelocity and the increase in particle velocity of the medium. Otherthermodynamic parameters such as stagnation pressure and stagna-tion temperature are also used. For avalanche control with air blast,this pressure jump is the primary loading mechanism into thesnowpack as the high pressure air behind the shock loads the surface.For explosive shock waves in air, the shock strength diminishes as itpropagates radially from the blast center. Eventually, the shock wavedegenerates into a sound wave as the wave speed becomes sonic.

Explosive blast wave characteristics depend upon the energyrelease and the medium through which the wave moves. Theperformance of a common explosive, TNT, has been studiedextensively, consequently, its performance is well understood anddocumented. Scaling law can be used to estimate the blast wavecharacteristics of other explosives. Kinney and Graham (1985)present a comprehensive TNT scaling law approach. Their method ispresented here to introduce basic air blast characteristics of acommon avalanche control explosive. The method utilizes geometricsimilitude (spherical for this case) and conservation of momentum. Ascaled distance allows other explosives at actual distances to be scaledup or down based on equivalent TNT distance. The scaled TNTequivalent distance z (m/kg1/3) is given by

z =fdzoW

13

; ð1Þ

where zo (m) is the actual distance of interest, W (kg) is the TNTequivalent explosive mass and fd is an atmospheric transmissionfactor. Considering air as an ideal gas, fd is given by

fd =PPo

� �13 To

T

� �13; ð2Þ

Page 3: AUTODYN_2010_Miller_An Explicit Numerical Model for the Study of Snow's Response to Explosive Air Blast

158 D.A. Miller et al. / Cold Regions Science and Technology 69 (2011) 156–164

where Po (Pa) and To (K) are reference pressure and temperature andP (Pa) and T (K) are the actual atmospheric pressure and temperature.Once the TNT equivalent distance is found, the explosive parametersin question may be scaled to the well known TNT equivalent. InEq. (1), the scaled distance is inversely proportional to the cube root ofthe TNT equivalent yield (due to spherical expansion), implying thatto create a particular blast at double the distance requires eight timesthe explosive energy. As an example, Fig. 1 shows the air blastoverpressure as a function of distance from the blast for 0.9 and 1.8 kgpentolite charges.

The dramatic decrease of blast pressure with distance is shown asis the influence of doubling the explosive weight for pentolite castboosters, a common avalanche control charge. While doubling thecharge does enhance over pressure, it is not a linear increase.Increasing explosive weight alone may not always be the most viableor efficient way to increase effectiveness for avalanche release. Factorssuch as terrain and snowpack conditions play a role in control efficacy.While scaling law helps quantify the first order influence of chargesize and distances, it does not adequately address the morecomplicated scenario of detonating explosives on or over snowpacksand is not operationally utilized. A numeric approach for examiningair blast and snow interaction is now presented.

2.2. Explicit nonlinear numerical modeling

The mining community has made significant research investmentin the use of explosives for the fracture and movement of rock. Thecurrent state of the art approach utilizes an explicit nonlinear methodfor evaluating rock movement during explosive events. AUTODYN is acommercial explicit modeling program produced by ANSYS, Inc.AUTODYN has particular strengths in modeling high energy, shortduration explosions and high velocity impacts by predicting thenonlinear behavior of solids, gases and liquids along with theirinteractions. One distinct advantage AUTODYN has for the currentapplication is the ability to handle nonlinear solids and gasessimultaneously. Preece and Lownds (2008) present a 3D method foranalyzing rock blasting using explicit computational models. Theirwork motivated the current study in applying AUTODYN tosnowpacks and explosives. While the response of snow is verydifferent from rock, the powerful constitutive models and complexsolid/gas interactions available in AUTODYN are encouraging. Stressand strain distributions, snow compaction, position, velocity andacceleration fields through the snow slope based on multiply

0.1

1

10

100

1000

10000

100000

0.1 1 10 100

Pre

ssu

re (

kPa)

Distance from blast (m)

0.9 kg Pentolite

1.8 kg Pentolite

Fig. 1. Pressure behind the shock wave vs distance from the explosive for 0.9 and 1.8 kgof pentolite. Atmospheric pressure was 76 kPa and temperature was −7 °C.

configured detonations with actual terrain features can be predicted.In the current study, a modeling approach was developed withsuspended and surface based explosives. The model uses a coupledair/snow interaction and simulates the complex nonlinear interac-tions and responses. With this model, several variables can be studiedand varied to determine parameter sensitivities and optimizedsolutions. The effects of thin snow cover, rebound of shock wavesfrom rock walls, height of the charge above the snowpack, size of thecharge, multiple charges, snow type/profile and shock wave couplingto snow are some of the topics in need of investigation. For this initialstudy, we demonstrate feasibility of the approach and limit theexamples to a few select parameters such as charge placement andcharge size on a single snowpack configuration.

2.3. AUTODYN model

To demonstrate the approach andmake preliminary investigationsinto explosive responses, an example configuration of a high densityhard slab resting on a thin low density layer supported by a highdensity base was developed. A snowpack comprised of a 1 m base,1 cm weak layer and 0.5 m top slab was developed. Snow densities(shown in Fig. 2) were selected due to the availability of explosiveresponse data (presented later) and they are representative of a hardslab supported by a weak layer. A 2D axisymmetric square mesh(10 mm) was utilized for both the air and snow with the axis ofsymmetry about the vertical axis where the explosive is placed. Thisaxis is referred to as the blast axis. The resulting simulation is a disk ofsnow and air with the explosive centered on or above the snow in theair. For this initial example, 1.8 kg of cast pentolite is suspended 1 mabove the snow surface. A wedge visualization of the model is shownin Fig. 2 with air, snow and explosive identified.

In Fig. 2, the air is modeled with Eulerian elements where theexplosive gases and shock wave travel though the mesh. The snow istreated as a Lagrangian solid where the mesh deforms with thematerial. At the snow and air interface, Lagrange-Euler interactionsare defined allowing the air to interact with the snow. The interactionof the solid and gas phases with the Lagrange–Euler interface sets thisapproach apart and allows for direct loading of the snowpack fromshockwaves traveling through the air. While each approach takenindividually is common, their simultaneous implementation makesthis a particularly powerful approach.

2.4. Constitutive relationships

To adequately describe the snow response to loading, volumetricand deviatoric constitutive relationships must be defined. A shockwave is a discontinuous jump in states across an abrupt front. Prior to

Air

ρ=400 kg/m3

Explosive

ρ=111 kg/m3

Snow

AUTODYN Model Configuration

Fig. 2. AUTODYN axisymmetric model with 90° slice shown. Blue area is air, green ishigh density snow, black is a low density weak layer and red is the explosive. The snow/air disk is 10 m in diameter, the upper snow layer is 0.5 m thick, the bottom layer is 1 mthick and the weak layer is 1 cm thick. The explosive is 1.8 kg of pentolite and issuspended 1 m above the snow surface. A 1 cm square mesh is used throughout themodel.

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Table 1Shock equation of state parameters derived from Johnson and Solie (1993).

Initial snow density Co s

400 kg/m3 53.9 m/s 1.86111 kg/m3 7.63 m/s 1.43

159D.A. Miller et al. / Cold Regions Science and Technology 69 (2011) 156–164

shock arrival, the original material state can be described by: theparticle velocity (uo), material density (ρo), specific internal energy(eo) and pressure (Po). As the shock wave passes through thematerial,each of these change to new values u1, ρ1, e1 and P1 behind the front.Cooper (1996) derives relationships for mass conservation, momen-tum balance and energy conservation across the shock front, yielding(in order) the following relationships (with uo=0):

ρ1

ρo=

UU−u1

; ð3Þ

P1−Po = ρou1U; ð4Þ

e1−eo =12

P1 + Poð Þ 1ρo

− 1ρ1

� �; ð5Þ

where U is the shock wave velocity. In Eqs. (3), (4) and (5) with theinitial state known, there are five unknown variables. To solve theserelationships, more information is needed. For air, the ideal gasassumption is used. For a solid, a volumetric equation of statedescribing equilibrium states in terms of energy, pressure and densitywould provide the necessary relationships. If an equation of stateexisted such that e=f1(P,ρ), then it could be used in the energyequation (Eq. (5)) eliminating e leaving P=f2(ρ). Unfortunately, suchan equation of state is unknown for most solids and in particular forsnow. An expression relating any two of the four unknowns in themass (Eq. (3)) and momentum (Eq. (4)) equations, that includedequilibrium states, would provide an alternative to the equation ofstate. With an equation of state and using the shock velocity, U, as aboundary condition, the equations can be solved. Shock Hugoniots areexperimentally derived empirical relationships for particular mate-rials and are normally expressed as P=f3(ρ), P=f4(u) or U=f5(u). InAUTODYN, the shock equation of state describes the volumetricmaterial response as U=f5(u). For most materials, this is a linearrelationship (Cooper, 1996) given by

U = Co + su1; ð6Þ

where Co is the intercept and s is the slope of the experimental shock-particle velocity data. Limited experimental data exists for snow andsuitable substitutes are not known. Fortunately, Johnson and Solie(1993) snow gas gun experiments and analysis yielded P=f3(ρ) dataon a wide range of initial snow densities. Their data suggests twomodes of snow loading; large deformation compaction followed byrapid load increase with little change in density. Large deformationsoccur as the snow is initially compacted. As the density increases,there is a sudden increase in deformation resistance requiring higherpressures for further strain. The critical density where significantstrain hardening begins was dependent upon the initial snow density.They found a power law relationship between pressure and densitywith derived coefficients (a and b) for pressures 2–20 MPa as

P = a ρ−ρoð Þ b: ð7Þ

For pressures up to 40 MPa, additional exponential terms wereadded to Eq. (7). All of the snow densities considered in this studywere included in Johnson and Solie (1993), but are of the form ofEq. (7). The AUTODYN shock equation of state requires the data in theform of Eq. (6). To translate for use in the numerical model, Johnsonand Solie (1993)P–ρ relationships were calculated and then convertedto U–u. By rearranging Eqs. (3) and (4) and eliminating u1, the shockvelocity is given by

U2 = P1−Poð Þ�

ρo−ρ2o = ρ1

� �: ð8Þ

And by eliminating U using Eqs. (3) and (4), the particle velocityafter the shock is given by:

u21 = P1−Poð Þ 1

ρo− 1

ρ1

� �: ð9Þ

Using Eqs. (8) and (9) with Johnson and Solie (1993) data, linearU=f(u1) curves were developed. A linear regression yielded thecoefficients in Eq. (6) for implementation in AUTODYN. Thistranslationwas completed for each initial snow density with excellentfit (R2N0.99). The shock equation of state parameters for Eq. (6) aregiven in Table 1, completing the volumetric constitutive relationships.

For deviatoric deformations, the snowwas modeled as linear elasticfollowed by brittle failure using the maximum principal stress failurecriterion for each snow density. While the equation of state describesthe element volumetric response, the deviatoric deformations result inchanges to element shape. In this case, the deviatoric stress/strainrelationship is linear up to a critical maximumwhere failure is assumedto be instantaneous and complete. The assumption of deviatoric brittlefailure is common for snow and implicit in fracture analysis(macroscopically). While deviatoric plastic deformations are notincluded, the equation of state does consider significant volumetriccompaction. Elastic moduli, shear strengths, maximum compressivenormal stresses and maximum octahedral stresses were taken frommaterial summarized in Mellor (1975) and are based on snow density.For the current example, the slab and base layers (ρ=400 kg/m3):Young's modulus=60MPa, maximum compressive normal stress=2 MPa, maximum shear stress=0.5 MPa and maximum octahedralstress=0.2 MPa. For the weak layer (ρ=111 kg/m3): Young's mod-ulus=0.2 MPa, maximum compressive normal stress=0.03 MPa,maximum shear stress=0.04 MPa and maximum octahedralstress=0.015 MPa. AUTODYN uses superposition for the volumetric(from the equation of state) and the deviatoric contributions to get theoverall material state. While the example represented here is limited todistances 5 m from the blast axis, it is believed that this approachwouldbe appropriate for greater distances and is being considered in futureefforts.

The explosive is modeled with a Jones–Wilkins–Lee (JWL)equation of state using pentolite data based on manufacturerprovided JWL empirical parameters, detonation velocity and chemicalenergy. This model is used in conjunction with the modeled castbooster physical configuration to predict the high explosive detona-tion and rapid expansion of gas products. The resulting high pressuregases are allowed to operate in the Eulerian air elements.

3. Results

3.1. Air blast pressure

The shock wave in the air, for a suspended 1.8 kg pentolite charge,moves through the air, impinges on and travels across the snowsurface. Fig. 3 shows snap shots of the pressure wave in the air andacross the snow surface at three different times after detonation. Theshock front produces enhanced pressure at the snow surface as thewave reflects obliquely and acts on air that has already been impactedby the primary shock. As the shock wave propagates, the pressure andwave velocity decease due to geometric expansion.

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Fig. 3. Air pressure contours at times after detonation of 0.25 ms (A), 2.4 ms (B) and4.55 ms (C) respectively. Snow is shown in green (no data) at the bottom for reference.In A, the shock wave is entirely contained in the air and has not yet reached the snowsurface.

X

Y

B

C

Shock wave

Shock wave

3.7 m

1.0 m

0.5 m

Fig. 4. Snow cross sectional vertical normal stress contours at times after detonationcoincident with Fig. 3 (B) and (C) respectively. The planar view is of the snowpack. Thecurrent position of the shock wave is noted. The detonation originated along the righthand side of the plots and the shock is traveling to the left. The dashed line is the weaklayer.

160 D.A. Miller et al. / Cold Regions Science and Technology 69 (2011) 156–164

3.2. Snow stresses

As the blast transits the snow surface, the Euler–Lagrangeinteraction feature in AUTODYN allows the air to load the snow andfor the calculation of several physical parameters. In particular, thestresses that develop in the snow are of interest for examining theeffects the explosive has internally on the snowpack. As the shockwave moves across the snow, a stress wave, consisting of normal andshear stresses, develops at the surface andmoves into and through thesnowpack. The vertical normal stresses along a radial slice of themodel are shown in Fig. 4 at two different times, coincident with Fig. 3(B) and (C). The view is a 3.7 m wide cross section of the snowpackwith the shock wave originating on the blast axis (right) and movingleft (+Y direction). A normal compressive stress wavemoves throughthe snow as the shock wave transits the surface. The compressivestress wave decreases in intensity through the snow depth andradially with shock movement. In the area directly beneath the

explosive, the snow density increased from 400 kg/m3 to ~480 kg/m3

(according to the equation of state described in Section 2.4).When thewave reached the weak layer, the compressive normal loadstransmitted through the weak layer from the slab to the underlyingbase layer. While the weak layer did not significantly impede thecompressive normal stress, it did have an interesting impact on theshear stress.

The analysis points to a “rolling” shear stress wave that movesthrough the snow concentrated in the slab near the weak layer. Fig. 5shows the shear stresses in the snow cross section at two differenttimes, also coincident with Fig. 3(B) and (C). In Fig. 5, a shear stresswave moves through the snow and concentrates above the weaklayer; very little shear load is transmitted to the snow below the weaklayer. In this case, the stress was sufficient to cause weak layer failure,eliminating the layer's ability to transmit shear load. It is believed thatthe depth, intensity and extent of these stress wavesmay be a primaryfactor in determining the effectiveness of explosive avalanche control.

3.3. Surface blast

In the next analysis, the scenario described in Fig. 2 is repeated, butwith the 1.8 kg charge placed on the snow surface. As expected, acrater developed along with high snow stresses in the vicinity of thecrater. The compaction of the snow (according to the constitutiverelationships described in Section 2.4) was monitored here throughmaterial density. Fig. 6 shows the snow density contours at twodifferent times after the surface detonation.

In Fig. 2, the hard slab and base layers had an initial density of400 kg/m3 and the weak layer density was 111 kg/m3. In Fig. 6(A), acompaction zone has moved through the slab below the crater, due tocompressive normal stresses, resulting in density increases. The

Page 6: AUTODYN_2010_Miller_An Explicit Numerical Model for the Study of Snow's Response to Explosive Air Blast

X

Y

Shock wave

Shock wave

B

C

Fig. 5. Snow cross sectional shear stress contours at times after detonation coincidentwith Fig. 3 (B) and (C) respectively. The planar view is of the snowpack. The currentposition of the shock wave is noted and the geometry is identical to Fig. 4. Thedetonation originated along the right hand side of the plots and the shock is traveling tothe left. The dashed line is the weak layer.

A

B

Shock wave

Fig. 6. Snow density for a surface blast at t=3 ms (A) and t=7 ms (B) after detonation.In (A), the shock wave location is shown, but in (B), the shock wave has exited the viewthrough the left boundary.

Table 2Peak vertical velocity and acceleration in the middle of the snow top layer slab atvarious ranges from the blast axis for a 1.8 kg pentolite charge detonated from 1 mabove the surface (1 m) and from the surface (Surf).

Distance fromblast axis (m)

Velocity (m/s) Acceleration (m/s2)

1 m Surf 1 m Surf

0.25 18.7 46.6 139 4530.5 16.2 12.2 116 53.81 9.80 4.80 69.9 34.13 4.60 3.70 67.1 26.6

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density behind the stress wave rebounded somewhat after the stresswave passage. Johnson and Solie (1993) use analysis and post testdensity measurements to estimate release moduli and report a largerange of values that vary widely with pressure. It is also noted that thepeak density decreased from (A) to (B) as the stress wave attenuatedand the snow rebounded somewhat, consistent with the hysteresisobserved in the experiments of Johnson and Solie (1993). The weaklayer is evident and has been compressed, but to a lesser extent basedon its lower initial density.

The crater in Fig. 6 was formed by releasing material afteranalytical material failure when the snow experienced very highstrain. In this case, the crater did not extend below the weak layer, butthe compaction zone did extend to a depth ~0.8 m below the originalsurface. The stresses in close proximity to the crater zone were verylarge when compared to stresses in the same snowpack location fromthe suspended charge. As the shock wave propagated away from thesurface blast location, the stresses deceased quickly. The air blastshear and normal stresses were generally higher than for the surfaceblast at distances beyond ~1 m from the blast axis. For example, whenthe shock wave is located at 3.65 m from the blast axis (ref Figs. 3 and5 (C) for air blast), the peak shear stress experienced by the upper slablayer was ~50% greater for the air blast when compared to surfacedetonation, supporting Gubler's (1977) experimental snowpackobservations for suspended charges. While snow stresses were notgenerally reported, Johnson et al's (1994) summaries suggest ~40%increase in peak air pressure from raising the explosive above thesurface to the optimal height. This increase is based onmaximizing airpressure for a particular range while the current simulation is basedon a typical avalanche control placement, making direct comparisondifficult. As discussed later, peak air pressure alone does not explainthe increases in shear stress due to explosive elevation.

Stresses are one good measure of explosive effectiveness in snowlayers, but there are other dynamic parameters that can be examined.Particle velocity (u1) is the velocity of a particle as the wave transmitsthrough the medium. In this case, particle velocity is found as thestress wave moves through snow and is a measure of snowdisturbance. Additionally, particle acceleration (the time derivativeof particle velocity) can be used to estimate forces in the snow fromNewton's Second Law. Table 2 summarizes the peak vertical(downward) velocity and acceleration at the middle of the top layerslab at various radial locations for the two scenarios.

At very close range to the blast, the peak vertical velocity andacceleration in the middle level of the snow slab are much greater forthe surface blast. In particular, the velocity and acceleration at 0.25 mare significantly higher for the surface blast. Between 0.25 m and0.5 m, the suspended charge becomes the dominant method forinducing vertical velocity and acceleration. As distance from the blastaxis increases, the advantage for the suspended explosive is evident inhigher peak velocity and accelerations.

While this analysis supports the well known avalanche controladvantage of suspended explosives, it has not yet provided insight intowhy this is the case. The only difference between the two scenarios is

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the location of the explosive. This location difference results in a variedload input to the snowpack. To investigate the differences, air pressureis now examined. Fig. 7 shows the peak air pressure and percentdifferences (from surface blast) at various distances from the blast axislocated 3 cm above the snow surface. Since the evaluation points areclose to the snow surface, the surface detonation has significantlyhigher peak air pressure in the vicinity of the blast due to closeproximitywith slant range affecting the suspended results. As the rangefrom the blast increases, the suspended blast has higher peak pressureswith the decay of the two pressure peaks appear to reasonably trackeach other. The suspended blast has slightly larger peak pressures atranges beyond ~2.2 m.While there are differences in the two scenarios,the suspended charge advantage (at large ranges from the blast axis)doesn't appear to be from enhanced peak pressure alone.

The air pressures in Fig. 7 are higher than the open air blastpredicted by scaling law alone (Fig. 1), again pointing to the limitedapplication of scaling approaches. The air pressure contours in Fig. 3(B) and (C) show pressure intensification near the snow surface,when compared to the open air scaled estimates. In part, this pressureenhancement is from previously discussed (experimentally verifiedsummary in Mellor (1985)) shock reflection off the snow surface, butmodeling differences cannot be ruled out as a factor.

The shock wave is an N-shaped pressure wave with a nearlyinstantaneous jump in pressure followed by decay as the wave movespast a fixed position. The pressure impulse, the area under theoverpressure–time curve, is found by integrating the overpressureover the time it acts at particular points. The pressure impulseaccounts not only for the peak pressure, but is also influenced bypressure wave action time and shape which can vary with shockengagement geometry, detonation velocity and gas volume.

Fig. 7. Peak air pressure at 3 cm above the snow surface for suspended and surfacedetonations at various ranges from the blast axis. The peak air pressure percentdifference, as compared to the surface charge, is shown.

As shown in Fig. 8, the pressure impulse from the surface blast ismuch greater than the suspended pressure impulse in close proximityto the blast axis, but the suspended impulse quickly exceeds thesurface values as range increases. At distances N3 m from the blastaxis, the impulse is ~50% greater for the suspended charge. Much ofthe surface charge energy is dedicated to crater generation, leavingless momentum spreading to the far field (Johnson et al. (1994)). Thepressure impulse represents the integrated pressure over time atpoints above the snow surface and is the primary loading input to thesnow. It is likely that this increased impulse (at far field) created bysuspending the charge is a contributor in increasing snowpackdynamic responses. In addition, the shape of the N wave may alsobe important since manywave shapes can have identical impulses butmay produce different results.

3.4. Charge size

Next, the scenario described in Fig. 2 is repeated but with a 0.9 kgpentolite charge in the 1 m suspended position. This scenario has halfof the net explosive weight than the previous suspended explosiveexample. To compare the two different charge sizes, the shear stressesin the middle of the top slab layer (0.25 m below the snow surface)are captured when the shock wave is in the position represented inFig. 3 (B). Since the net explosive weights are different, the shockvelocities and shock arrival times are also different, so the comparisonis based on equivalent geometric location of the shock wave. A profileplot of shear stress across the middle of the top slab layer is presentedin Fig. 9.

In Fig. 9, it is noted while the peak shear stresses from the largerexplosive are higher, they are not so in proportion to the net explosive

Fig. 8. Pressure impulse at 3 cm above the snow surface for suspended and surfacedetonations at various ranges from the blast axis. The pressure impulse percentdifference, as compared to the surface charge, is shown.

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weights. In this particular location, the doubling of the net explosiveweight resulted in only a 21% percent increase in peak shear stress.When the shock wave later reached the location shown in Fig. 3 (C),the shear stress wave for the smaller charge was beginning todissipate and lose definition while the shear wave from the 1.8 kgcharge maintained a distinct concentration above the weak layer (refFig. 5 (C)). While the stresses may not be proportionally higher whenthe net explosive is doubled, the effective range is increased, but towhat extent is currently not known and is a topic for further research.

4. Summary

Using AUTODYN as an evaluation tool for snow's response toexplosive events has the advantage of handing nonlinear interactionsof gases and solids with complex material responses. The applicationto air and layered snowpacks provides an avenue for snow responsesto explosives. The model allows for internal examination of vitalsnowpack parameters during all phases of explosive events. Theexamples selected for this study demonstrate the utility and power ofthe approach, but are far from comprehensive from an avalanchecontrol perspective. Various parameters such as snow stress, density,velocity, acceleration, air pressure and pressure impulsewere selectedfor examination and comparison, demonstrating the modelingversatility and strengths. There are many other potential parametersthat could be examined in future analyses, again, a testament to theapproach.

The snow response for a particular air blast scenario revealed atraveling stress wave created by the shock front as it propagated overthe snow surface. Both normal and shear stress were presentsupporting current thought in layer fracture and avalanche release.It is believed that the intensity, depth and lateral extent of this stresswavemay be a fundamental measure of explosive effectiveness. In thespecific case presented, the shear stress was intense enough to fail theweak layer throughout the model, which prevented transmission ofsignificant shear forces to the base layer. Other snowpack conditionscan be envisioned where an explosive would not sufficiently fail alayer or provide enough force to initiate an avalanche after fracture.The development of a crater and significant snow compaction wasalso demonstrated on a surface blast example. In our example, thestresses, velocity and acceleration in the vicinity of the blast weremuch greater than for a suspended equivalent charge. As the shockwave propagated away from the blast center, the suspended chargeproduced enhanced dynamic response in the top slab layer. Theseresults suggest that there may be an optimal configuration above thesnowpack tomaximize dynamic snow response thatmay not be basedon peak air pressure alone. If the charge is suspended too high, thedecay in air pressure due to spherical expansion will eventuallyoutweigh the height of burst advantage. Conversely, if the charge is

Fig. 9. Shear stresses vs distance from the blast axis for suspended 0.9 and 1.8 kgpentolite charges. The shock wave is located 2.45 m from the blast axis in each case.

too close to the surface, excessive explosive energy may beapportioned near the blast and reduce shock geometric advantagethereby limiting widespread effect. The increased effectiveness forexplosives suspended 1–2 m above the snowpack has been experi-mentally developed and routinely employed, but the currentapproach provides the opportunity to study cause and effect for avariety of parameters in search of that optimal placement forparticular snowpack conditions. The balance between charge sizeand placement may be used to optimize pressure impulse tomaximize the snowpack loading over a large area. If a particularavalanche control situation presents a well defined localized triggerpoint with a high probability of propagation, a surface placement maybe sufficient or even preferred. Reducing the explosive size in air blastrevealed a slight reduction in stresses in the vicinity of the blast aswell as a reduction in effective range of the rolling shear stress wave.

While the methods developed here are sound and appropriate forthe problem, this study is just scratching the surface for evaluatingavalanche control effectiveness. The approach is fundamentally a toolthat can focus future studies. The numerical examination of optimalcharge size and placement, influence of various snowpack conditions,use of available terrain features, evaluation of buried charges orartillery are examples of future studies that can now be analyticallyexamined. One important aspect is to align the approach withmeasured field data. In 2010, we developed an explosive researchsite with instrumentation to measure the blast pressure andsnowpack dynamic response to explosives (Tichota et al., 2010). Weplan to closely couple the measured responses and the analysis tofurther establish the techniques and to refine critical parameters forincreasing avalanche control effectiveness. The analytical modelmerged with the experimental campaign comprises a new explosiveavalanche control research program. Both projects are in their infancy,but have established new insights and techniques for supportingresearch.

Acknowledgments

The authors would like to thank the National Avalanche Center andsnow safety personnel from Bridger Bowl, Yellowstone Club and BigSky ski areas for their insights and support of the project. We wouldalso like to acknowledge the two reviewers and the technical editorwho provided excellent recommendations for improving the paper.

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