AN EXPLICIT NONLINEAR NUMERICAL MODEL OF EXPLOSIVE AIR BLAST ON SNOW AND IMPLICATIONS FOR AVALANCHE CONTROL

D.A. Miller*, R.G. Tichota and E.E. Adams

Department of Civil Engineering, Montana state University, Bozeman, MT

Abstract: Recent post explosive control avalanches and the lack of analytical methods for evaluating explosive control have motivated this study to better quantify the effects explosives on snow. Currently, there are insufficient tools for predicting the effects that explosives will have on snow avalanche release. While there is significant practitioner experience, that experience has (on occasion) not adequately anticipated control response. This research seeks to develop analytic tools describing phenomenology and that can examine the “internal” snow response during explosive events. A state-of-the-art 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 over the very short blast/propagation time. The axisymmetric model predicts internal structural response during explosive events, including important parameters such as stress, strain, density changes velocity and acceleration. Various snowpack configurations and explosive charges with variable locations can be examined. The results point to a rolling shear stress wave in the snow that lags behind the explosive shock wave traveling over the surface. The intensity, effective depth and lateral extent of the stress wave may determine the control effectiveness. The analytic approach provides a tool for future detailed examination of critical avalanche control parameters.

1.0 Introduction

Since the 1950’s, explosives have been used to initiate avalanches on snow slopes. Many ski resorts and several state departments of transportation use this method (and others) in an effort 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 placement and detonation 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). During the 2009/10 winter, 2 additional people were killed by avalanches within US resort boundaries. Most of these incidents were classified “post control release” where the avalanche unexpectedly initiated after explosive

control efforts failed to clear the slope.

Two field studies targeted explosive avalanche control in particular. Gubler (1977) conducted field experiments where charge mass, snowpack stratigraphy, explosive type, charge placement relative to the surface, explosive type and ground type were considered. One significant conclusion of Gubler’s work is that charges detonated from 1 -2 m above the surface have enhanced results for releasing dry slab avalanches. This important insight has led to a common industry practice in use today. Ueland (1992) used mining seismographs to measure the vertical response of snowpacks during explosive events. His work confirmed the air blast advantage over surface or buried charges, but also examined shock attenuation through the snowpack depth. He found snow hardness, more than density, was a significant factor in shock attenuation. Hard snowpacks allowed for deeper penetration of stress waves when compared to softer snow.

While these experimental studies have provided critical insights, modern analytical tools integrating the explosive, atmosphere, terrain and snowpack have been elusive. There has been limited experimental and virtually no significant numerical or analytic research into the snow’s dynamic to explosive detonations, particularly at a avalanche slope scale. A comprehensive approach capable of examining the internal snow response is needed. Such an approach or analytical technique

*Corresponding author address: Daniel A. Miller, Department of Civil Engineering, Montana State University, 205 Cobleigh Hall, Bozeman, MT 59717; tel (406)994-6118; email dan.miller@coe.montana.edu

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would open the door for studying the critical parameters and sensitivities leading to enhanced effectiveness of explosive charges on avalanche prone slopes.

This study is a first step in such an approach applying modern analytic and numerical techniques to the avalanche control situation. The goal of this study is to develop an appropriate analytical tool capable of predicting snow response to an explosive event. The approach should allow for multiple snow and explosive types and configurations. Internal snow dynamic response must be predicted and available for analysis.

2.0 Approach

2.1 Explosive Air Blast

Following an explosion in air, a compressive shock wave is generated and radiates outward through the air at supersonic speeds. The strength of a normal shock front is commonly characterized by a pressure increase across the front, the wave propagation velocity and an increase in particle velocity in the medium. As a normal shock propagates, the air experiences an increase in pressure and velocity from the front passage. For avalanche control with air blast, this air pressure jump is the primary loading mechanism into the snowpack as the air behind the shock transversally loads the surface. For explosive shock waves, the shock strength diminishes as it propagates radially from the blast center. Eventually, the shock wave degenerates into a sound wave as the overpressure approaches zero and the wave speed becomes sonic.

Explosive blast wave characteristics depend upon the energy release and the medium through which the wave moves. For a spherical blast expanding radially into air, there is a dramatic decrease of blast pressure with distance. To create a particular blast at double the distance requires eight times the explosive energy (Kinney and Graham (1985)). While doubling the charge does enhance over pressure, it is not a linear increase. Increasing explosive weight alone may not always be the most viable or efficient way to increase effectiveness for avalanche release. Factors such as charge placement, terrain and snowpack conditions each play a role in how well the control measures work. A numeric approach for examining air blast and snow interaction is now presented.

2.2 Explicit Nonlinear Numerical Modeling

A state of the art technique utilizes an explicit nonlinear method for evaluating materials during explosive events is presented. ANSYS AUTODYN has particular strengths in modeling high energy, short duration explosions and predicting the nonlinear behavior of solids, gases and liquids along with their interactions. The powerful constitutive models and complex solid/gas interactions available in AUTODYN were encouraging for snow application. One distinct advantage AUTODYN has is the ability to handle nonlinear solids and gases simultaneously. Stress and strain distributions, snow compaction, position, velocity and acceleration fields through the snow slope based on multiply configured detonations with actual terrain features can be predicted. In the current study, a modeling approach was developed with suspended and surface based explosives. The model uses a coupled air/snow interaction and simulates the complex nonlinear interactions and responses. With this model, several variables can be studied and varied to determine an optimized solution; all without actually detonating a charge. The effects of thin snow cover, rebound of shock waves from rock walls, height of the charge above the snowpack, size of the charge, multiple charges, snow type/profile and shock wave coupling to snow are some of the topics in need of investigation. For this initial study, we demonstrate feasibility of the approach and limit the examples to a few select parameters such as charge placement and charge size on a single snowpack configuration.

2.3 AUTODYN Model

To demonstrate the approach and make preliminary investigations into explosive responses, an example configuration of a high density slab resting on a thin weak layer was developed. A snowpack comprised of a 1 m base, thin weak layer and 0.5 m top layer slab was developed. Snow densities (ref Figure 1) were selected due to the availability of explosive response data and they are representative of potential field conditions. A 2D axisymmetric element was utilized with the axis of symmetry about the vertical axis where the explosive is placed. This axis is referred to as the blast axis. The resulting simulation is a disk of snow and air with the explosive centered above the snow in the air. For this initial example, 1.8 kg of pentolite is suspended 1 m above the snow surface. A wedge

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of the model is shown in Figure 1 with air, snow and explosive identified.

2.4 Constitutive Relationships

A shock wave can be thought of as a discontinuous “jump” in states across an abrupt front. Prior to shock arrival, the original material state can be described by: the particle velocity (uo), material density (ρo), specific internal energy (eo) and pressure (Po). As the shock wave passes through the material, each of these change to new values u1, ρ1, e1 and P1 behind the front. The speed of the shock is given by U. Cooper (1996) derives relationships for mass conservation, momentum balance and energy conservation across the shock front.

For a solid, a volumetric equation of state must be found. Shock Hugoniots are experimentally derived empirical relationships for particular materials and are normally expressed as P=f(ρ), P=f(u) or U=f(u). In AUTODYN, the shock equation of state describes the volumetric material response in U=f(u). For most materials, this is a linear relationship (Cooper,1996) given by

𝑈 = 𝐶𝑜 + 𝑠𝑢1 (1)

where Co is the intercept and s is the slope of the experimental shock-particle velocity data. Limited experimental data exists for snow and suitable

substitutes are not known. Fortunately, Johnson et al’s (1993) snow gas gun experiments and analysis yielded P=f(ρ) data on a wide range of initial snow densities. Their data suggests two modes of snow loading; large deformation compaction followed by rapid load increase with little change in density. At low pressures, large deformations occur as the snow is compacted. As the density increases, there is a sudden increase in deformation resistance requiring higher pressures for further strain. The critical density where significant strain hardening begins was dependent upon the initial snow density. They found a power law relationship between pressure and density with derived coefficients for pressures 2–20 MPa. To translate for use in the model, Johnson et al’s (1993) P-ρ relationships were calculated and then converted to U=f(u) using relationships in Cooper (1996). The shock equation of state parameters for eqn (1) are given in Table 1, completing the volumetric constitutive relationships.

Table 1: Shock equation of state parameters derived from Johnson et al (1993)

Initial snow density Co s 0.4 g/cm3 53.87 m/s 1.86 0.111 g/cm3 7.63 m/s 1.43

For deviatoric deformations, the snow was modeled as linear elastic followed by brittle failure after exceeding the ultimate shear strength. Elastic moduli, shear strengths, maximum compressive normal stresses and maximum octahedral stresses were taken from snow material summaries found in Gary and Hale (1981). AUTODYN uses superposition for the volumetric (from the equation of state) and the deviatoric contributions to get the overall material state.

3.0 Results

3.1 Air Blast Pressure

In the model described in Figure 1, the 1.8 kg pentolite charge was detonated and the shock wave was allowed to propagate through the air, impinge on and travel across the snow surface. Figure 2 shows snap shots of the pressure wave in the air and across the snow surface at three different times. As the shock wave propagates, the pressure deceases due to geometric expansion. The shock wave velocity also decreases during expansion. It is noted that the

Figure 1: AUTODYN axisymmetric model with 90⁰ slice shown. Blue area is air, green is high 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 m thick and the weak layer is 1 cm thick. The explosive is 1.8 kg of pentolite and is suspended 1m above the snow surface. A 1 cm square mesh is used throughout the model.

Air

ρ=0.4 g/cm3

Explosive

ρ=0.111 g/cm3

Snow

AUTODYN Model Configuration

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pressure is intensified where the shock front impinges on the snow surface due to oblique reflection.

3.2 Snow Stresses

As the blast transits the snow surface, AUTODYN allows the snow to be loaded and for the calculation of several engineering parameters. In particular, the stresses that develop in the snow are of interest for examining the effectiveness of

avalanche control measures. As the shock wave moves across the snow, a stress wave develops at the surface and moves into the snow. The analysis points to a “rolling” shear stress wave that moves through the snow lagging behind the shock wave. Figure 3 shows the shear stresses in the snow cross section at two different times, coincident with Figure 2 (B) and (C). The view is a 3.7m wide cross section of the snowpack with the shock wave originating on the blast axis (right) and moving left.

A shear stress wave moves through the snow and concentrates above the weak layer; very little shear load is transmitted to the snow below the weak layer. In this case, the shear stress was sufficient to cause weak layer brittle failure in shear, eliminating the layer’s ability to transmit shear load. In avalanche control, this is the desired outcome by creating a layer failure and releasing the slab above. While not presented in

Figure 3: Snow cross sectional shear stress contours at times after detonation coincident with Figure 3 (B) and (C) respectively. The planar view is of the snowpack. The current position of the shock wave is noted. The detonation originated along the right hand side of the plots and the shock is traveling to the left. The dashed line is the weak layer.

Figure 2: Air pressure contours at times after detonation of 0.25 ms (A), 2.4 ms (B) and 4.55 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 snow surface.

A

B

C

B

C

Shock wave

3.7m

Shock wave

1.0m

0.5m

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detail here, normal stress (in the vertical direction) developed as expected. The density increased according to the prescribed equation of state. In the area directly beneath the explosive, the snow density increased from 0.400 g/cm3 to ~0.480 g/cm3. Unlike shear loads, compressive normal loads were transmitted through the failed weak layer. It is believed that the depth, intensity and extent of this shear stress wave may be a primary factor in determining the effectiveness of explosive avalanche control.

3.2 Surface Blast

In the next analysis, the problem described in Figure 1 is repeated, but with the 1.8 kg charge placed on the snow surface. As expected, a crater developed along with high snow stresses in the vicinity of the crater. The compaction of the snow can be monitored through material density. Figure 4 shows the snow density contours at two different times after the surface detonation.

Recall the slab and base snow layers had an initial density of 0.4 g/cm3 and the weak layer density was 0.111 g/cm3. In Figure 4(A), a compaction zone has moved through the slab below the crater resulting in density increases. The density behind the stress wave has rebounded somewhat after the stress wave passage. It is also noted that the peak density decreased from (A) to (B) as the stress wave attenuated and the snow rebounded somewhat. The weak layer is evident and has been compressed, but to a lesser extent based on its lower initial density.

The crater in Figure 4 formed by analytical material failure and by releasing material when critical high strain values were obtained. In this case, the crater did not extend below the weak layer, but the compaction zone did extend to a depth ~0.8 m below the surface. The stresses in close proximity to the crater zone were very large when compared to stresses in the same snowpack location from the suspended charge. As the shock wave propagated away from the surface detonation, the shear stresses deceased quickly. The air blast shear stresses were generally higher than for the surface blast at distances beyond ~1 m from the blast axis. For example, when the shock wave is located at 3.65 m from the blast axis (ref Figure 2 (C) and Figure 3 (C) for air blast), the maximum shear stresses in the upper slab layer were ~50% greater for the air blast when compared to surface detonation, supporting

Gubler’s (1977) experimental observations for suspended charges.

Shear stresses are one good measure of explosive effectiveness in snow layers, but there are other dynamic parameters that can be examined. Particle velocity is the velocity of a particle as the medium transmits waves. In this case, particle velocity is found as the stress wave moves through snow and is a measure of snow disturbance. Additionally, particle acceleration (the time derivative of particle velocity) can be used to evaluate forces in the snow from Newton’s Second Law. Table 2 summarizes the peak vertical (downward) velocity and acceleration at the middle of the top layer slab at various radial locations for the two scenarios.

Figure 4: Snow density for a surface blast at t=3ms (A) and t=7ms (B) after detonation. In (A), the shock wave location is shown, but in (B), the shock wave has exited the view through the left boundary.

A

B

Shock wave

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Distance from blast

axis (m)

Velocity (m/s)

Acceleration (m/s2)

1m Surf 1m Surf 0.25 18.7 46.6 139.0 453.1 0.5 16.2 12.2 115.6 53.8 1 9.8 4.8 69.9 34.1 3 4.6 3.7 67.1 26.6

At very close range to the blast, the peak velocity and acceleration in the middle level of the snow slab are much greater for the surface blast. In particular, the velocity and acceleration at 0.25 m are significantly higher for the surface blast. Between 0.25 m and 0.5 m, the suspended charge becomes the dominant method for inducing snow velocity and acceleration. As distance from the blast axis increases, the advantage for the suspended explosive is evident in higher peak velocity and accelerations. The analysis again supports the well known avalanche control advantage of suspended explosives.

3.3 Charge Size

Next, the problem described in Figure 1 is repeated but with a 0.9 kg pentolite charge in the 1 m suspended position. This scenario has half of the net explosive weight of the previous suspended explosive example. To compare two different charge sizes, the shear stresses in the middle of the top slab layer (25 cm below the snow surface) are plotted when the shock wave is in the position represented in Figure 2 (B). Since the net explosive weights are different, the shock velocities and shock arrival times are also different, so the comparison is based on equivalent location of the shock wave. A profile plot of shear stress across the middle of the top slab layer is presented in Figure 5.

It is noted while the peak shear stresses from the larger explosive are higher, they are not so in proportion to the net explosive weights. When the shock wave later reached the location shown in Figure 2 (C), the shear stress wave for the smaller charge was beginning to dissipate and lose definition while the shear wave from the 1.8 kg

charge maintained a distinct concentration above the weak layer (ref Figure 3 (C)). While the stresses may not be proportionally higher when the net explosive is doubled, the effective range is likely increased, but to what extent is currently not known.

4.0 Discussion

Using AUTODYN as a primary evaluation tool for snow’s response to explosive events has the advantage of handing nonlinear interactions of gases and solids with complex material responses. The application to air and layered snowpacks provides an avenue for explosive avalanche control study. The model allows for internal examination of snowpack parameters during all phases of explosive events. The examples selected for this study demonstrate the utility and power of the approach, but are far from comprehensive from an avalanche control perspective. Snow stress, density, velocity, and acceleration were selected for examination and comparison, demonstrating the modeling versatility and strengths. There are many other potential parameters that could be examined in future analyses, again, a testament to the approach.

For explosive avalanche control, the snow response for a particular air blast scenario revealed a traveling shear stress wave created by the shock front as it propagated over the snow surface. It is believed that the intensity, depth and lateral extent of this stress wave may be a

0

5

10

15

20

25

30

35

0 0.5 1 1.5 2 2.5 3 3.5

Shea

r str

ess

(kPa

)

Distance from blast axis (m)

0.9 kg pentolite

1.8 kg pentolite

Figure 5: Shear stresses vs distance from the blast axis for suspended 0.9 and 1.8 kg pentolite charges. The shock wave is located 2.45 m from the blast axis in each case.

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

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fundamental measure of explosive control effectiveness. In the specific case presented in section 3.2, the shear stress was intense enough to fail the weak layer throughout the model which prevented significant shear forces to transmit to the base layer. Other snowpack conditions can be envisioned where this isn’t the case, and an explosive would not sufficiently fail a layer with extent to release an avalanche. The development of a crater and snow compaction was also shown on a surface blast example. The stresses, velocity and acceleration in the vicinity of the blast were much greater than for a suspended equivalent charge. As the shock wave propagated away from the blast center, the suspended charge produced enhanced dynamic response in the top slab layer. These results suggest that there may be an optimal configuration above the snowpack to maximize dynamic snow response. If the charge is suspended too high, the decay in air pressure due to spherical expansion will eventually outweigh the height of burst advantage. Conversely, if the charge is too close to the surface, excessive explosive energy may be apportioned near the blast thereby limiting widespread effect. The increased effectiveness for explosives suspended 1 -2 m above the snowpack has been experimentally developed and routinely employed, but the current approach provides the opportunity to study cause and effect for a variety of parameters in search of that optimal placement for particular snowpack conditions. If a particular avalanche control situation presents a well defined localized trigger point with a high probability of propagation, a surface placement may be sufficient or even preferred. Reducing the explosive size in air blast revealed a slight reduction in stresses in the vicinity of the blast as well as a reduction in effective range of the rolling shear stress wave.

While the methods developed here are sound and appropriate for the problem, this study is just scratching the surface for evaluating avalanche control effectiveness. The approach is fundamentally a tool that can be used to focus future studies. The numerical examination of optimal charge size and placement, influence of various snowpack conditions, use of available terrain features, evaluation of buried charges or artillery are examples of future studies that can now be analytically examined. One important aspect is to align the approach with measured field data. In 2010, we developed an explosive research site with instrumentation to measure the blast pressure and snowpack dynamic response to

explosives (Tichota et al, 2010). We plan to closely couple the measured responses and the analysis to further establish the techniques and to refine critical parameters for increasing avalanche control effectiveness. The analytical model merged with the experimental campaign comprises our new explosive avalanche control research program. Both projects are in their infancy, but have established new insights and techniques for research.

5.0 Conclusions

The application of ANSYS/AUTODYN in explosive avalanche control allows for examination of variable charge and snowpack configurations. For the first time, detailed internal snowpack responses can be predicted and analyzed throughout explosive events. The future application of this technique to specific control situations may reveal insights that increase the effectiveness of explosive avalanche control. Initial studies confirm the established advantage of suspended explosives and the dependence of charge size in avalanche control.

Acknowledgments: The authors would like to thank the National Avalanche Center and snow safety personnel from Bridger Bowl, the Yellowstone Club and Big Sky ski areas for their insights and support of the project.

6.0 References

Abromeit, D., 2010, “Inbounds incidents & fatalities 2008/9”, Avalanche Review, vol 28, no 23, pp 26

Cooper, P.W.,1996, Explosives Engineering, Wiley-VCH, New York Chichester Weinheim Brisbane Singapore Toronto, ISBN 0-471-18636-8

Furnish, M.D. and M.B. Boslough, 1996, “Measuring Hugoniot, Reshock and Release Properties of Natural Snow and Stimulants”, Sandia National Laboratory Report, SAND92-0985, UC-703

Gray, D.M. and D.H. Hale, 1981, Handbook of Snow: Principles, Processes, Management & Use Pergamon Press, Toronto Oxford New York Sydney Paris Frankfurt, ISBN 0-08-025375-X

Gubler, H., 1977, “Artificial Release of Avalanches by Explosives,” Journal of Glaciology, v 19, n 81, pp 419-429

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Johnson, J.B. and D.J. Solie, 1993, “Shock Response of Snow”, Journal of Applied Physics, vol 73, no 10, pp 4852-4861

Johnson, J.B., D.J. Solie and S.A. Barrett, 1994, “The response of a seasonal snow cover to explosive loading”, Annals of Glaciology, vol 19

Kinney, G.F. and K.J. Graham, 1985, Explosive Shocks in Air, Springer-Verlag, Berlin Heidelberg New York Tokyo, ISBN 3-540-15147-8

Tichota, R.G., D.A. Miller, R. Larson and D. Richmond, 2010, “An experimental investigation of explosives and snowpack dynamic response”, 2010 International Snow Science Workshop, Lake Tahoe, CA

Ueland, J., 1992, “Effects of Explosives on the Mountain Snowpack”, 1992 International Snow Science Workshop, Breckenridge, CO

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