Discrete element model for high strain rate deformations of snow
Thiemo Theile, Denes Szabo, Carolin Willibald, Martin Schneebeli
DDiscrete element model for high strain rate deformations of snow
T. Theile , D. Szabo, C. Willibald, M. Schneebeli WSL Institute for Snow and Avalanche Research SLF, Flüelastrasse 11, Davos Dorf, Switzerland, [email protected]
Keywords: snow mechanics, snow modeling, discrete element modeling, angle of repose, direct shear test
Abstract
In engineering applications snow often undergoes large and fast deformations. During thesedeformations the snow transforms from a sintered porous material into a granular material. In orderto capture the fundamental mechanical behavior of this process a discrete element (DE) model is thephysically most appropriate. It explicitly includes all the relevant components: the snowmicrostructure, consisting of bonded grains, the breaking of the bonds and the followingrearrangement and interaction of the loose grains. We developed and calibrated a DE snow modelbased on the open source DE code liggghts. In the model snow grains are represented by randomlydistributed elastic spheres connected by elastic-brittle bonds. This bonded structure corresponds tosintered snow. After applying external forces, the stresses in the bonds might exceed their strength,the bonds break, and we obtain loose particles, corresponding to granular snow. Model parameterscan be divided into temperature dependent material parameters and snow type dependentmicrostructure parameters. Material parameters are elastic properties of the particles and bonds,coefficient of friction and coefficient of restitution of the particles and strength of the bonds.Microstructure parameters are density of the initial packing, rolling friction of the particles anddiameter of the bonds. The model was calibrated by angle of repose experiments and several highstrain rate mechanical tests, performed in a cold laboratory. We demonstrate the performance of theDE snow model by the simulation of a combined compression and shear deformation of differentsnow types with large strains. The model successfully reproduces the experiments. Mostcharacteristics of the mechanical snow behavior are captured by the model, like the fracturebehavior, the differences between low and high density snow, the granular shear flow or thedensification of low density snow. The model is promising to simulate arbitrary high strain rate rocesses for a wide range of snow types, and thus seems useful to be applied to different snowengineering problems.
Introduction
A model, describing the mechanical behavior of snow, is useful for many snow related engineeringproblems. From snow removal equipment such as snow plows or snow blowers, constructions madeof snow to mobility on snow with winter tires or chain drives. The prediction of reaction forces,deformation and failure of snow can help optimizing such snow related products. However, a generalmechanical snow model does not exist. Mainly three characteristics of snow make its modelling adifficult task: the wide range of different snow types, the complex rheology of ice and under highstresses the transformation from a sintered solid material to a granular material. Snow on the ground consists of irregularly shaped, bonded ice crystals which form a complex threedimensional continuous microstructure. The wide range of different snow types follows from thewide range of different microstructures. The size of the single ice crystal varies from a tenth of amillimeter to several millimeters, the shape varies from rounded to faceted and even hollow cup-likecrystals. The bonds between the single crystals grow over time and vary from unbonded to stronglysintered. And the relative density varies from 5% for fresh snow to 60% for compacted snow. Thiswide range of snow types and microstructures needs to be considered in a mechanical snow modelsomehow. Beside the microstructure also the rheology of ice determines the mechanical behavior ofsnow. Ice is an elasto-viscoplastic material. Even under small stresses ice continuously creeps.Especially this creep behavior is difficult to model. It is nonlinear, highly anisotropic (on the crystalscale) and temperature dependent. In contrast, the elastic behavior is linear (Hooke’s law) and showsonly slight anisotropy and temperature dependence. Elastic deformation dominates only for fastdeformations, for strain rates faster than 10 -3 Methods
Model
In this section the most important ingredients of the snow model are presented: the simulation toolwe used, the material model and the material parameters, the microstructure approximation and theboundary conditions. imulation tool and DE model
As a simulation tool we used the open source DE model liggghts (Kloss et al., 2012) with a bond-extension [retrieved from https://github.com/richti83/LIGGGHTS-WITH-BONDS]. The bond-extensionis based on a publication by Potyondy and Cundall (Potyondy and Cundall, 2004). We used an elastic-brittle material model. The snow grains are represented by monodisperse elastic spheres connectedby elastic-brittle bonds. Viscous deformations are not included. This limits our model to high strainrate deformations of snow, where creep behavior can be neglected. When the stresses in the bondsreach their strength, the bonds break and disappear. The unbonded spheres deform following aHertzian contact model (Hertz, 1882) and interact by Coulomb friction and rolling friction.
Particle packing and microstructure approximation
The snow microstructure is approximated by monodisperse spheres, where each sphere representsone snow grain. The initial particle packing is created by ballistic deposition and following removal ofrandom particles. Thus the density of the model snow can be adjusted to a desired value between250 kg/m and 570 kg/m . By ballistic deposition a random close packing with a volume fraction of0.62 is obtained. This volume fraction corresponds to a snow density of 570 kg/m . From this randomdense packing random particles are removed until the desired density is reached. Particles can onlybe removed if the connectivity of the packing is not violated. In a last step neighboring particles areconnected by bonds if their distance (from particle center to particle center) is smaller than a certainthreshold. This threshold is the parameter “bonding distance”, which controls the coordinationnumber of the structure. Fig. 1 shows a comparison between our model and a real snowmicrostructure imaged by micro computer tomography. It is obvious that there are large differencesbetween model and reality. Parameters
The model parameters can be divided into material parameters and microstructure parameters. Thematerial parameters are based on literature values of ice and summarized in Table 1. All materialparameters are fixed parameters for all load cases and snow types and depend only on temperature.All microstructure parameters are variable depending only on the snow type and are summarized inTable 2. We did not consider different temperatures. All experiments were performed at -5 °C. The determination of the parameters is the most crucial aspect of the model. There are someparameters which are obvious to interpret and to choose and there are other parameters which aredifficult to choose and will be fitted by experiments. The latter parameters are indicated by anasterisk (*) in table 1 and table 2. bvious parameters are the material parameters density, Young´s modulus and Poisson´s ratio of theparticles and strength of the bonds. These parameters correspond directly to literature values of ice.Also the microstructure parameter “density of the packing” is an obvious parameter. Density is aparameter which can easily be measured in real snow. Thus density will be matched between modeland real snow.All other parameters are difficult to choose. Either because they are abstract DE specific parameterswhich do not have a clear counterpart in real snow, like the rolling friction, or because the parametercan not definitely be obtained from literature values, like the coefficient of ice-ice friction. In thefollowing we discuss these parameters and how they were chosen, in order of appearance in table 1and table 2. For the coefficient of ice-ice friction a wide range of values from 0.01 to 0.7 can be found in literature(Schulson and Fortt, 2012)(Yasutome et al., 1999). Therefore we fitted this parameter with an angle ofrepose experiment. The coefficient of restitution of ice varies from 0 to 0.9 depending on particlevelocities (Higa et al., 1996). We have chosen a small value to have more damping and thus anumerically more stable system. The Young´s Modulus of the bonds is adjusted from the Young´smodulus of ice to obtain a better agreement with experimental results. The value is reduced in ourmodel by two orders of magnitude compared to values for ice found in literature (Schulson andDuval, 2009). This can be justified by the fact that snow exhibits a certain creep contribution even athigh strain rates. The Young´s modulus of snow determined with high strain rate experiments is up totwo orders of magnitude smaller than finite element simulations based on the Young´s modulus of ice(Köchle and Schneebeli, 2014), which were recently shown to correspond to the true elastic modulusof snow (Gerling et al., 2017). The diameter of the spheres is fixed to 1 mm for convenience. For realsnow the size of the grains ranges from 0.2 mm to 2 mm. The shape of the snow grains is consideredby the rolling friction parameter and will be fitted by the angle of repose experiment. The bonddiameter is an important parameter which can be adjusted to consider differently sintered snow. Thisparameter will be fitted by compression tests of differently sintered snow. The bonding distancewhich controls the coordination number of the model snow is fixed to 1.1 mm. This is a strongsimplification, since the coordination number has a strong impact on the mechanical behavior ofgranular materials (Gaume et al., 2017) and might differ for different snow types. The calibration experiments will be explained in the next section.
Boundary conditions
The boundary conditions are chosen to correspond with the experiments. Dimensions of the snowsamples, external forces and external displacements are exactly reproduced in the models. Externalorces and displacements are applied on the model snow by rigid objects of arbitrary geometry,which can be defined as a 3D geometry in the STL-file format.
Table 1 Material parameters
Model Ice property (from literature)Density of particle 917 kg/m
917 kg/m Young´s modulus of particle 9 GPa 9 GPaPoisson´s ratio of particle 0.3 0.3Coefficient of friction * 0.3 0.01 – 0.7Coefficient of restitution * 0.2 0 - 0.9Young´s modulus of bond * 90 GPa 9 GPaStrength of bond 2 MPa 1 – 4 MPa
Table 2 Microstructure parameters
Model Real SnowDensity of packing 250 – 550 kg/m
30 – 570 kg/m Particle diameter * 1 mm 0.2 – 2 mmCoefficient of rolling friction * 0.2 – 0.3Bond diameter / particle diameter * 0.05 – 0.5Bonding distance / particle diameter * 1.1
Fig. 1:
Comparion real snow (left) with model snow (right). The side length of the cubic volumes is 5 mm and the density of both volumes is 270 kg/m . The real snow sample shows rounded snow and was imaged by computer tomography. Single grains were identified by image processing (Theile and Schneebeli, 2011) and color labeled. xperiments Several experiments were conducted to calibrate and verify the model under different load cases.Emphasis was placed on experiments with large strains where both, the sintered and granular state ofsnow occurs. During compression and shear deformation large strains can be applied and threephases of deformation can be distinguished: First the elastic deformation of the sintered snow,followed by failure and finally the granular behavior, which is dominated by the interaction of loosesnow grains. The failure of the sintered snow will be referred to as fluidization in the following. Toanalyze the granular behavior separately, we conducted angle of repose experiments.
Snow types
Basically three different snow types were used for the different experiments: rounded snow, facetedsnow and crushed ice (Fig. 2). Rounded snow develops in several weeks from fresh snow underisothermal conditions by equilibrium metamorphism. Faceted snow grows in a few hours or daysunder high temperature gradients by kinetic metamorphism. The rounded and faceted snow wascollected from an alpine snowpack. The crushed ice was created by freezing tab water and crushingthe ice with a commercial ice crusher. All snow types were sieved before the experiments. Only grainswhich passed the first, large sieve and did not pass the second small sieve were kept for theexperiments. Finally the grains were sieved into the sample holder, compressed to a defined densityand left for sintering for a defined sintering time. Details about the snow types used for theexperiments are summarized in table 3.
Fig. 2:
Shape of single snow grains. Image a) shows a faceted grain. Image b) shows rounded grains. And image c) shows grains from crushed ice. The scale in the background is in millimeters. All images have the same scaling.
Angle of repose
For the angle of repose experiments snow was sieved onto a cylinder with 50 mm diameter. Theangle of the heap, which formed on the cylinder, was measured. The sieve was placed 40 mm abovehe cylinder. Two different snow types were used: rounded snow and faceted snow (Fig. 2). Thetemperature was set to -5 °C. Images of the heap from six perspectives were taken to measure theangle all around the heap. The heap which formed is not a perfect cone, it is flattened on the top dueto the impacting particles. The angle was measured up to the point where the heap has a constantslope. Fig. 3 a) indicates how the angle was measured. The goal of this experiment was to calibrate the particle-particle friction and rolling friction parameterfor different snow types.
Table 3 Snow types used for experiments
Experiment Snow type Sieve (mm) Sintering time(hours) Density (kg/m ) Specific surface area (1/mm) Angle of repose Rounded grains 0.7 – 1.4 No sintering 370 11Angle of repose Faceted grains 0.7 – 1.4 No sintering 400 11.5Unconfined compression Rounded grains 0.0 – 1.4 0.03 - 100 350 25Unconfined compression Faceted grains 0.0 – 1.4 0.03 – 100 400 20Direct shear Crushed ice 0.5 – 0.7 3 500 30Direct shear Crushed ice 0.5 – 0.7 25 300 30 (a ) (b) (c)Fig. 3:
Comparison of angle of repose experiment and model. Fig. a) and Fig. b) both show an angle ofrepose of 34°. Fig. a) shows the angle of repose of faceted snow at -5 °C. Fig. b) shows the modeled angle of repose with the model parameters particle-particle friction of 0.3 and rolling friction of 0.2. As a comparison Fig. c) shows the modeled angle of repose of 24° with the model parameters particle-particle friction 0.1 and rolling friction 0.1. The red drawings on Fig. a) indicate how the angleof repose a was determined from the images. The angle was measured up to the point where the heap has a constant slope. At this point a certain radius r of the cone is reached. Unconfined compression
At a temperature of -5 °C snow was sieved into a cylindrical sample holder of 10 mm height and40 mm diameter. Two different snow types were used: rounded snow and faceted snow. The samplesested for 1 to 2000 minutes for sintering. Afterwards the surface of the sample was cut using ascraper to obtain a flat, horizontal surface. Subsequently, the side wall of the sample holder wasremoved. The displacement controlled compression test was executed by pushing a stiff plate with avelocity of 10 mm/s onto the snow sample. Fig. 4 show a snapshot of an unconfined compressionexperiment. Forces were measured with a precision of 0.5 N and a frequency of 10000 Hz. Thehighest stress during the first 10% strain is defined as the compressive strength of the sample.The goal of this experiment is to further calibrate the microstructural parameters of the model,especially how the bond diameter has to be adjusted as a function of sintering time.
Fig. 4:
Snapshot of unconfined compression experiment (top) and model (bottom) of a well sintered rounded snow sample after about 30% of strain.
Direct shear experiment
At a temperature of –5 °C a rectangular block of snow was placed in a confined shear device. First avertical compression with a constant velocity of 50 mm/s was applied until a normal stress of 170 kPawas reached, after half a second of pure compression additionally a shear deformation with aconstant velocity of 50 mm/s was applied. Snow samples with two different densities were used, “lowdensity” snow with a density of about 300 kg/m and “high density” snow with 500 kg/m . Thedimensions of the snow block were 30 mm x 30 mm x 27 mm for the low density snow and 30 mm x10 mm x 33 mm for the high density snow. The goal of this experiment is to verify the model with a complex mechanical experiment with highstrains where both, the sintered and granular behavior is important. esults Angle of repose
At a temperature of -5 °C an angle of repose of 36° for the rounded snow and 33.4° for the facetedsnow was measured. The experiments were repeated eight times and the standard deviation is about1°. The measured angles were matched in the simulation with an ice-ice friction of 0.3 and a rollingfriction of 0.2 for faceted snow and a rolling friction of 0.3 for the rounded snow. As a comparison,with a rolling friction of 0.1 and a particle-particle friction of 0.1 we obtain a simulated angle ofrepose of only 24°. Fig. 3 shows a comparison of different heaps which formed during the angle ofrepose experiment and during the simulation.
Unconfined compression
The measured stress-strain curves for three unconfined compression experiments with differentlysintered rounded snow samples are shown in Fig. 5. The lowest curve corresponds to a sintering timeof 2 minutes and reaches a strength of 1 kPa. The highest curve corresponds to a sintering time of1000 minutes and reaches a strength of 10 kPa. For this curve three different phases of deformationcan be distinguished: 1. elastic deformation of the sintered snow; 2. failure; 3. granular behavior. Theexperimental curves can be matched by adjusting the relative bond size in the model to 0.1, 0.2 and0.3. From this matching we obtain a relation between sintering time and model bond size. Fig. 6shows this relation for rounded and faceted snow. This relation follows a power law with an exponentof 0.15 for the rounded snow and 0.2 for the faceted snow. Fig. 7 shows the strength over sinteringtime for all experiments with rounded snow (red circles) and faceted snow (grey diamonds), as wellas the matched simulation results for the two snow types. Since there are no side walls in a confined compression experiment, we are able to visually observehow the snow fails under pressure. This can also be visualized with the model. Fig. 4 shows aqualitative comparison of the failure behavior in the experiment and in the model. Clearly the cracksin the real and modeled snow sample can be seen. ig. 5:
Comparison between simulated and measured stress-strain curves of the unconfined compression. The measured curves differ by sintering time. The simulated curves differ by bond size.
Fig. 6:
Relation between relative bond size in the model and sintering time. In order to model faceted snow which sintered for 100 hours a relative bond size of 0.3 should be chosen in the model. ig. 7:
Experimental results of the compressive strength for different sintering time for rounded and faceted snow. The dashed lines show the fitted simulation results.
Direct shear experiment
The shear experiment can be divided into two phases: compression and shear deformation (Fig. 8 and9). First the normal pressure is applied on the snow sample, after half a second the sheardeformation starts. The high density snow (500 kg/m ) shows almost no compression at pressures upto 300 kPa. In contrast the low density snow (300 kg/m ) is densified to about 600 kg/m at thesepressures (Fig. 8). The following shear behavior is also fundamentally different for the low and highdensity snow. The low density snow shows ideal plastic behavior with pronounced stick-slip duringshearing (Fig. 9). In contrast the high density snow shows a brittle fracture behavior with a clear peakstress (shear strength) followed by a drop in shear stress (Fig. 10). The comparisons of the measuredand modeled stresses show good agreements. Almost all the characteristics of the compression andshear deformation are captured by the model (Fig. 8-10). Only the stick-slip behavior of the soft snowis not reproduced. The simulations were repeated with a rolling friction of 0.3 (solid black line) and0.2 (dashed black line). ig. 8: Compression of soft snow. Comparison simulation with experiments. The experiment was repeated 14 times (blue curves). The simulation results with a rolling friction of 0.2 (dashed black line) and 0.3 (solid black line) are shown.
Fig. 9:
Shearing of soft snow. Comparison simulation with experiments. The experiment was repeated14 times (blue curves). The simulation results with a rolling friction of 0.2 (dashed black line) and 0.3 (solid black line) are shown. ig. 10:
Shearing of hard snow. Comparison simulation with experiments. The experiment was repeated 11 times (blue curves). The simulation results with a rolling friction of 0.2 (dashed black line) and 0.3 (solid black line) are shown.
Discussion
Angle of repose
The angle of repose experiment enables to analyze the granular behavior of snow which results fromthe interaction of loose ice grains. The interaction of the loose ice particles is controlled mainly by ice-ice friction, the particle shape and sintering. The goal of this experiment is to calibrate the modelparameters rolling friction and particle-particle friction for the temperature of -5 °C. The rollingfriction parameter is a structural parameter and considers the non-sphericity of the snow grains. Theparticle-particle friction is a material parameter and corresponds to ice-ice friction. It is very difficultto determine the ice-ice friction from literature. Values between 0.01 and 0.7 (Schulson and Fortt,2012) can be found. The fitted value of 0.3 seems reasonable. The rolling friction is a simplification to consider the shape of the snow grains in the model. Aphysically more correct implementation would approximate the shape of real snow grains. In DEmodelling “clumps of spheres“ are often used to approximate the shape of the particles. However,this approach is more difficult to implement and computationally more expensive. Therefore we usedthe simple rolling friction to consider the non-sphericity of the snow grains. Sintering is not included in our model. However, it is known that sintering can have a big impact onthe mechanical behavior of snow at low strain rates (Reiweger et al., 2009). Also at high strain rates,intering might have an impact on the mechanical behavior. Fast sintering of ice on the sub-secondtimescale was described by Szabo and Schneebeli (Szabo and Schneebeli, 2007). Fast sintering istemperature dependent and occurs at temperatures between -15 °C and 0 °C. Fast sintering and itstemperature dependence is the reason why you can form snowballs only at warm temperatures closeto 0 °C. As sintering is not included explicitly in our model, it will be included in the fitted frictionparameter. However, without any time or velocity-dependence. With this calibration we can estimate the model parameters rolling friction and particle-particlefriction. This is useful since the estimation of the two parameters from particle shape and fromliterature is not possible.
Unconfined compression
The unconfined compression experiment was performed to calibrate the bond size in the model. Thisparameter is the most crucial parameter in the model. The bond size has a small influence on thesintered snow behavior: the thicker the bonds the stiffer the snow. But more importantly the bondsize together with the bond strength determines under which stresses the bonds break and how thesintered snow transforms to granular snow, both in the model and in real snow. Since we fixed thebond strength in the model to the strength of ice, we only have to calibrate the bond size. Bond size isa microstructure parameter. This means that different snow types have different bond sizes. It wouldbe elegant to measure the bond size in real snow and feed the model with these values. However,this approach seems not applicable for our model. First it is very difficult to measure the bond size insnow, and second it is unlikely that these bond sizes are valid in our model due to othersimplifications of the microstructure. Therefore we just used this parameter as a fitting parameterand used the unconfined compression experiment for calibration. In this experiment we vary thebond size without changing any other characteristics of the snow, like density and grain shape. Withincreasing sintering time the bonds grow thicker and the strength increases. Not only the strengthincreases, also the failure behavior changes. For a sintering time of two minutes the snow deformsideal plastically (blue curve in Fig. 5). With longer sintering time the snow becomes more brittle(green and red curve in Fig. 5). Not only the increasing strength of the snow can be reproduced byincreasing the bond size in the model, but also the different failure behavior. This shows how well thismodel is able to reproduce snow behavior. However, the sintered behavior is not reproduced well forthis experiment. For the red and green curves in Fig. 5 the modeled curves show a much stifferbehavior than the experimental curves. Most likely the reason for this difference is that in theexperiments the surface of the snow samples is not perfectly parallel to the plate which is pressed onthe snow. If the angular deviation is only 1° this will result in an additional strain of 7% until the twosurfaces are in full contact. Also the steepening shape of the red curve confirms this explanation. obbs and Mason (Hobbs and Mason, 1964) have shown experimentally and theoretically that bondsize increases with sintering time according to a power law with exponent 0.2. The dominantmechanism is vapor transport. The relation between bond size and sintering time shown in Fig. 6 alsofollows a power law with an exponent of about 0.2 for the faceted snow and 0.15 for the roundedsnow. This indicates that the fitted bond sizes behave realistically. Several studies about the sinteringof snow have determined a connection between some measured mechanical property and sinteringtime. Jellinek (Jellinek, 1959) found an exponent of 0.21 for the compressive strength. Van Herwijnenand Miller (Van Herwijnen and Miller, 2013) found an exponent of 0.18 for the penetration resistanceand argued that this is in good agreement with sinter theory of ice. However, linking the mechanicalproperty linearly to bond size is doubtful. Using our model we get a direct connection between bondsize and sintering time, which allows a comparison to sinter theory and thus a further confirmation ofthe model.
Direct shear experiment
The aim of the direct shear experiment is to further validate and challenge the model with a differentload case and with different snow densities. The challenge is to model this complex process withcompression and shear deformation with large strains, including the transformation from sintered togranular snow. The two snow types investigated, low- and high-density snow, show fundamentallydifferent behavior. The low-density snow is fluidized already when the compressive stress is applied,the following shear deformation of the granular snow resembles ideal plastic behavior. In contrast thehigh-density snow hardly deforms when the compressive stress is applied. The strength is higher thanthe applied stresses. Therefore the fluidization takes place during the shear deformation, showing thetypical peak stress at failure followed by granular behavior. The granular shear behavior, which isbasically snow-snow friction, is similar for both snow types with a shear stress of around 90 kPa at anormal stress of 170 kPa. This corresponds to a snow-snow friction of 0.53 or a friction angle of 28°.This agrees well with the avalanche rule of thumb saying that a minimum slope of 30° is required foravalanche release. Interesting is also the compression of the low-density snow. Up to about 50% of strain the snow isdensified with an almost constant stress. After this point the stresses suddenly increase significantly.The reason for this behavior is that the low density snow reaches a critical density of 600 kg/m after50% of strain. This density corresponds to the density of a random dense packing. Up to this densitysnow can be densified by the rearrangement of snow grains. After this point the grains themselveshave to be deformed, resulting in a much higher compression resistance. The model also shows thisbehavior, but the critical density is lower with 500 kg/m . A possible explanation for this difference ishat all particles have the same size in the model, while in real snow particle sizes are distributed andcan therefore reach a higher critical density. The model performs well in reproducing the experimental results with all its characteristics withoutfurther fitting of the model parameters. Even though the model parameters were obtained underdifferent load cases. This shows that the most important mechanical processes are included in themodel and that the model seems promising to simulate arbitrary high strain rate processes for a widerange of snow types with different densities. However, not all characteristics of the measured resultsare reproduced by the model. The stick-slip behavior of the low-density snow during shearing is notreproduced by the model. The reason for this stick-slip behavior might be periodic sintering andbreaking of snow grains or periodic jamming of irregular shaped snow grains. Both effects are notincluded in the model. As you would expect, the rolling friction parameter has no impact on thesintered snow behavior but on the granular behavior (dashed and solid black lines in Fig. 8-10). Thesimulations with a rolling friction of 0.2 correspond slightly better to the experimental results thanthe simulations with a rolling friction of 0.3. Simulation volumes and computing times
For all experiments the volume of the snow samples was in the order of magnitude of 10 cm . Thesevolumes correspond to about 10,000 particles with 1 mm diameter in the simulations. The computingtimes ranged from 20 minutes to 240 minutes on one processor depending on the number of timesteps. For the angle of repose simulation, which is without bonds, 14 million time steps are calculatedper second for one particle. For the simulations with bonds about 1 million time steps are calculatedper second for one particle. The time step was set to 10 -7 seconds for all simulations. To check the potential of the model for larger volumes a simulation with one million particles and 0.4million time steps was calculated on a cluster on 80 cores in 1320 minutes. The same simulation was3.7 times faster on four times more cores, showing the potential of the parallelization for largeproblems. Table 4 Computing times
Model
Conclusions
We have developed and verified a DE model for high strain rate deformations of snow. Thetransformation from sintered to granular snow is the key process for large and fast deformations ofsnow. The model includes this mechanism. Furthermore the model considers different characteristicsof the snow microstructure, like density or grain shape. Including relevant micromechanical processesand characteristics of different snow types is the key to create a snow model as general as possible. Inthe presented model the complex microstructure of snow is approximated by a “simple” spheresystem. Nevertheless, the model performs well in reproducing different experiments with differentsnow types and different load cases with one set of parameters. This versatility is the strongest pointof the model. Another strong point of the model is the free and easy availability of the model. Allpresented simulations can be found in the supplementary materials. Weaknesses of the model arethe missing study of temperature dependence, the missing implementation of sintering and thesimplified material model without creep deformation. However, these simplifications are a tradeoffbetween accuracy and calculation time, they keep the model simple and reduce the number ofparameters. Due to this simplicity and the fast and parallized liggghts-code, simulations with up totwo million particles can be solved in reasonable times. This enables the application of this model todifferent snow engineering problems, like mobility on snow or snow removal equipment, in futurework.