A macroscale model for low density snow subjected to rapid loading

8/8/2019 A macroscale model for low density snow subjected to rapid loading

1/19

A macroscale model for low density snow subjected to

rapid loading

Robert B. Haehnel*, Sally A. Shoop

US Army Engineer Research and Development Center, Cold Regions Research and Development Laboratory (ERDC-CRREL),

Hanover, NH, 03755, Germany

Received 3 September 2003; accepted 4 August 2004

Abstract

A Capped DruckerPrager (CDP) model was used to simulate the deformation-load response of a low density (150250 kg/

m3) snow being loaded at high strain rates (i.e., strain rates associated with vehicle passage) in the temperature range of 1 to10 8C. The range in the appropriate model parameters was determined from experimental data. The model parameters wererefined by running finite-element models of a radially confined uniaxial compression test and a plate sinkage test and comparing

these results with laboratory and field experiments of the same. This effort resulted in the development of two sets of model

parameters for low density snow, one set that is applicable for weak orbsoftQ snow and a second set that is representative of

stronger orb

hardQ

(aged or sintered) snow. Together, these models provide a prediction of the upper and lower bound of themacroscale snow response in this density range. Furthermore, the modeled snow compaction density agrees well with measured

data. These models were used to simulate a tire rolling through new fallen snow and showed good agreement with the available

field data over the same depth and density range.

D 2004 Elsevier B.V. All rights reserved.

Keywords: Snow; Low density; Finite element modeling (FEM); Capped DruckerPrager; Plastic constitutive law; Snow mechanics

1. Introduction

Understanding the mechanical properties of snow

subjected to rapid loading has application to manyareas of interest (Shapiro et al., 1997) including:

(1) Designing snow removal equipment.

(2) Predicting vehicle performance in snow.

(3) Applying to military, such as the ability of snow

to absorb projectile impacts and problems related

to snow-covered minefields.

Field evaluation of these technologies in snow can

be problematic owing to lack of reproducibility of

results in what is seemingly the bsameQ snow (e.g., the

same density and temperature). A typical example is

the difficulty of quantitatively evaluating the relative

performance of snow tires, even when each is tested

on the same track on the same day. It has long been

recognized that this is attributable to the variation in

0165-232X/$ - see front matterD 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.coldregions.2004.08.001

* Corresponding author.

E-mail address: [email protected]

(R.B. Haehnel).

Cold Regions Science and Technology 40 (2004) 193211

www.elsevier.com/locate/coldregions


2/19

mechanical properties of the snow with the micro-

structure of the pack, which evolves in time because

of changes in air temperature, humidity, sintering of

the snow particles over time, the work done to thesnow (e.g., vehicle passage), etc.

Characterization of snow in general, and low

density snow in particular (100250 kg/m3), is

problematic because of the range of falling snow

types and the rapid metamorphic changes that take

place in the snow once it is on the ground. Falling

snow can range from a feathery denderitic structure to

pellet-like sub-angular forms. The exact form of the

snow as it falls depends on the temperature, humidity,

and wind history that the flakes experience from the

time of their genesis until they reach the ground. Onceon the ground, the snow particles continue to

metamorphose due to ambient weather conditions

(e.g., wind, humidity, temperature, etc.). For example,

a snow cover that is initially dendritic in structure

(typically density of 100 kg/m3) subjected to wind

induced drifting is rapidly transformed to nearly

spherical particles that have a mean density of 300

400 kg/m3. In the absence of wind this same fresh

snow cover can undergo temperature-driven meta-

morphism that causes sublimation of the small crystal

structures and deposition of water vapor on larger

crystal structures (migration from a high surface area/

volume grain structure to a low surface area/volume

grain structure). Concurrently, pressure and temper-ature gradients within the snow pack lead to sintering

of the snow. These mechanisms also lead to consol-

idation of the snow pack, yet can yield a profoundly

different structure and snow strength for the same

density.

For example, Fig. 1 shows the wide range in

response of snow that has a density of approximately

150 kg/m3 subjected to uniaxial compression. This

plot is for snow collected in its pristine condition from

the field and then compressed in a radially confined

uniaxial compression test (Abele and Gow, 1975).This kind of scatter is typical for snow. The material

responds differently because of variations in sample

temperature, age of sample, snow microstructure, etc.,

not all of which are easily controlled or readily

measured in the field. Of particular interest is the

response of tests 1 and 2 in comparison to test 50 (Fig.

1). Tests 1 and 2 were done at a temperature of 7 8C,while test 50 was conducted at 3 8Cyet tests 1 and2 are the bsoftestQ samples in this density range, and

test 50 is among the bhardest.Q In this plot, it is clear

Fig. 1. Stressstrain curve for radially confined uniaxial compression tests of snow with a density range of 140160 kg/m3 (data from Abele and

Gow, 1975).

R.B. Haehnel, S.A. Shoop / Cold Regions Science and Technology 40 (2004) 193211194


3/19

that the strength of the snow does not simply decrease

with temperature and illustrates that characterization

of snow requires far more information than simply

density or temperature; a knowledge of the snowsmicrostructure (the type of snow grains present and

the strength of the bond between these grains) and its

influence on the structural properties (compressive

strength, elastic modulus, etc.) of the snow pack is

required.

For example, Fukue (1979) showed that the age

of the snow can have a significant effect on its

strength. In unconfined uniaxial compression tests

using manufactured snow, Fukue showed that, for

samples allowed to sinter at constant density, the

strength of the snow increased 10-fold over thecourse of 6 days. The influence of the snows

microstructure on its mechanical response is reported

by Armstrong (1980). His study of consolidation of

an alpine snow pack showed that a fine-grained,

sintered snow deformed at a rate 10 times greater

than an adjacent layer of depth hoar, though the

density of both layers were the same. Finally,

Voitkovsky et al. (1975) showed that the cohesion

strength of snow varies widely when plotted as a

function of density, yet the same cohesion data

plotted as a function of specific snow grain contact

surface (net area of inter-grain contacts per unit

volume) revealed a nearly linear relationship.

One way to side step the problem of poor

reproducibility of field results is to use virtual

prototyping techniques, in which the snow is repre-

sented using a computer model and the interaction of

the vehicle with the snow surface is simulated. This

provides a test bed using reproducible snow con-

ditions for a proscribed snow state. However, this

requires development of a material model for snow

that is representative of the snow response for an

appropriate set of conditions (e.g., snow density, age).Such a model was produced for compacted high

density snow (400500 kg/m3) by Meschke et al.

(1996). The model presented here is for low density,

undisturbed snow.

Terrain substrate subjected to wheel loads has

been represented using a wide variety of material

models, including elastic, non-linear elastic, viscoe-

lastic (Pi, 1988), and elasticviscoplastic (Saliba,

1990). Recent studies concentrate on using either

Capped DruckerPrager plasticity (Aubel, 1993,

1994; Fervers, 1994) or critical state models such

as the Bailey and Johnson (1989) soil compaction

model, implemented by Foster et al. (1995), or a new

critical state model (similar to Lade and Kim, 1995),implemented by Liu and Wong (1996). Comparisons

between the DruckerPrager and Cam-clay models

for tireterrain interaction have been published by

Meschke et al. (1996) for snow and by Chi and

Tessier (1995) for soil. Although Cam-clay is perhaps

the most widely used soil model, both of the above

studies comment on convergence problems when

using Cam-Clay. The choice of material model is

based on balancing the type of behavior desired in

the model with the information available to determine

model parameters.Meschke et al. (1996) and Miyori et al. (2002),

respectively, have produced material model parame-

ters for Cam-Clay and MohrCoulomb models of

packed snow (snow density of 400500 kg/m3). In

this work we present a material model that is

representative of low density snow (approximately

150250 kg/m3) subjected to short-term loading (i.e.,

creep response of the snow is not considered).

Recognizing that the strength of snow depends on

the snow microstructure, and not just density alone;

we have developed material models representative of

both weak and strong snow structures using a

modified Capped DruckerPrager (CDP) model as

implemented in the ABAQUS (HKS, 1998) finite

element computer code. Though the microscale

processes (e.g., breaking of bonds, sliding of grains)

are not explicitly modeled, this model is suitable for

capturing the large-scale compressive and shear

behavior of natural, fresh snow (Shoop, 2001; Shoop

et al., 1999a,b, 2001). A description of the model,

model parameter determination, and validation of the

material models with laboratory and field data

follows.

2. General concepts for plasticity models

The purpose of a plasticity model is to describe the

permanent deformation of a plastic material and has

the following basic components (Wood, 1990):

(1) Elastic properties to define the recoverable

deformation.

R.B. Haehnel, S.A. Shoop / Cold Regions Science and Technology 40 (2004) 193211 195


4/19

(2) A mathematical surface to define the yield

boundary between elastic behavior and plastic

material behavior.

(3) A plastic flow potential to mathematically definethe plastic deformation (also called a plastic flow

law).

(4) A hardening/softening rule defining the move-

ment (expansion or contraction) of the yield

surface during plastic deformation.

A good overview of the development of plasticity

theory and constitutive modeling of soil is given in

Scott (1985). Schofield and Wroth (1968) extended

plasticity theory to the critical state concept, defining

either contractile or dilatant deformation of porousmaterial as a function of its specific volume or void

ratio. In critical state theory, this rule is developed

around the concept of a bcritical state,Q where the

plastic shearing deformation occurs at a constant

volume. Perhaps the most famous critical state

model, the Cam-Clay model, was developed based

on the behavior of clays. The concepts are equally

applicable to defining the shearing and volumetric

behavior of granular materials, such as granular soils

or snow (Wood, 1990). Although the concepts are

applicable for both cohesive and granular materials,

the behavior of the granular materials has not been

explored as thoroughly, particularly regarding the

influence of deviatoric stress on the yield surface,

which is less clearly defined in soils but may take on

a much different shape than the yield surface of

metals.

Liu (1994) used the DruckerPrager plasticity

model for modeling the sliding of rubber blocks on

snow. Mundl et al. (1997) extended Lius work

using a multi-surface plasticity model to optimize

the snow behavior under both shear and compres-

sion. Their intent, however, was to simulateshearing forces of snow compacted on roads

(density of 500 kg/m3), which is a significantly

different material than that encountered during

cross-country mobility on fresh snow (density of

200300 kg/m3). For this study, the plasticity

model used was the well-documented Capped

DruckerPrager.

Cap plasticity models account for both the com-

pression and shearing of an isotropic material. The

modified Capped DruckerPrager (CDP) model has

the features of a critical state model (i.e., regions of

constant volume shear deformation, and compactive

dilatant flow). CDP uses non-associative flow on the

shear surface (i.e., the flow potential is not associatedwith the yield surface) and associated flow on the cap

surface.

3. Modified cap DruckerPrager model

The yield surface for the CDP material model is

described in terms of stress invariant functions of the

stress tensor, S (a term in bold face denotes a matrix

or tensor). The particular invariants used for the CDP

model as implemented in ABAQUS (HKS, 1998) aredefined as:

(1) The invariant, p, is the equivalent pressure

stress, or the negative octahedral normal stress, roct(Jaeger and Cook, 1969), which determines uniform

compression or dilation:

p roct 1

3r : I: 1

where I is the identity matrix and the operator b : Q

denotes a scalar product.

(2) The invariant, q, is the deviatoric stress, alsocalled the Mises equivalent stress, which determines

distortion:

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2S:S

r2

where S is the stress deviator

S r pI:

This form of the second invariant is analogousbutnot identicalto the octahedral shear stress put

forward by Jaeger and Cook (1969).

(3) The invariant, r, is

r

9

2SdS:S

!13

: 3

The bd Q operator denotes matrix multiplication.

In the formulation of the CDP model, the second

and third invariants are combined to create an



5/19

Fig. 2. (a) Modified DruckerPrager yield surface in deviatoric space compared with (b) other common yield surfaces ( HKS, 1998).



6/19

alternate expression, t, the deviatoric stress measure

(HKS, 1998),

t q2

1 1K

1 1K

!r

q

!33524 4where K is the flow stress ratio (ratio of tensile

strength to compressive strength) and describes the

yield dependence on the third stress invariant, defin-

ing the shape of the yield surface in the deviatoric

plane. For K=1, the yield surface is circular (von

Mises yield), as shown in Fig. 2a, and the failure

stress is the same in tension and compression. For the

surface to remain convex, K is limited to values of

0.778 to 1. For K=0.778, the CDP model can be usedto approximate the MohrCoulomb surface shown in

Fig. 2b.

The subsequent model parameters are defined in

the pressure-deviatoric plane (also called the meridia-

nal, pq, or pt plane). For the modified cap Drucker

Prager model used in this study, the yield surface is a

modified von Mises yield (i.e., the material constant

K=1.0); hence, Eq. (4) reduces to t=q, eliminating the

effects of the invariant, r. In the pt plane, the yield

surface has two major segments (Fig. 3):

(1) The DruckerPrager portion of the curve (anal-

ogous to the MohrCoulomb line) defines shear

deformation.

(2) The cap portion of the surface defines the

intersection with the pressure axis.

The following equations define the yield criteria in

each section of the yield surface. For DruckerPrager

shear or distortional failure

Fs tptanb d 0 5

where d is the DruckerPrager material cohesion and

b is the DruckerPrager material angle of friction.

These are analogous to MohrCoulomb cohesion, c,

and the internal angle of friction, /, respectively.

For the cap region of compactivedilatant failure

Fc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip pa

2

"Rt

1 a a=cosb

#2vuutR dpatanb 0 6

where a is a transition parameter, ranging typically

from 0.0 to 0.05, that smoothes the transition between

the shear failure and the cap failure, R is a material

parameter controlling the shape of the cap (or the cap

eccentricity parameter), and pa is the intersection of

the shear line and the cap (in absence of a transition

surface) and relates to the cap hardening behavior

according to

pa pb Rd1 Rtanb 7

where the mean hydrostatic pressure, pb, is a function

of the volumetric plastic strain, evolpl . This functional

relationship, pb=f(qvolpl ), is the hardening law which

Fig. 3. Modified cap DruckerPrager yield surface in the pt plane (HKS, 1998).



7/19

defines the pressurevolume relationship during com-

pression of the material at the cap failure surface.

The transition surface between the shear and the

cap failure is defined as

Ft

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip pa

2

"t

1

a

cosb

dpatanb

#2vuut a dpatanb 0

3.1. Hardening law

8

The pressurevolume relationships define bothhardening and softening through volume changes

based on how the cap portion of the yield surfaces

expands and contracts. The cap is generally spherical

or ellipsoidal, and the material either hardens or

softens by expanding or contracting the cap, respec-

tively. This behavior is defined in a pressurevolume

relationship called a hardening law. The pressure

volume relationship is often exponential and, there-

fore, can be modeled using an exponential hardening

law. In cases where exponential hardening is not a

good fit, as is common in soils, the hardening law can

also be represented in a piecewise linear approach

using the experimental data (e.g., Fig. 4) as a table of

pb and evolpl pairs. The piecewise approach is recom-

mended by HKS (1998) for a better fit to the data and

better model performance; this approach was used in

this study.

3.2. Plastic flow

The plastic flow is defined by an elliptical shaped,

flow potential surface. Flow is associative (normal to

the surface) in the cap region; therefore, the equation

for the flow surface is identical to the equation for the

cap yield surface. In the transition and shear region,

the flow is non-associative (flow potential is inde-

pendent of the failure surface), and the flow surface,

Gs, is defined as (HKS, 1998)

Gs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"pa ptanb

#2

"t

1 a a=cosb

#2vuut :9

4. Material parameter determination

Extreme changes in snow properties with time,

temperature gradients, applied load, and deformation

prohibit model parameter acquisition from the same

snow. Consequently, approximations were made by

estimating parameters using test data from snow of

Fig. 4. Piecewise linear models of the hardening law for low density snow-based on Abele and Gows (1975) experimental datafor fresh

(S200) and age hardened snow (H200). The rapid rise in pressure stress for high volumetric strain (N1.5) occurs as the snow is compacted to the

density of ice.



8/19

similar characteristics (density, age and snow type).

The type of snow modeled was a natural snow with

a density of 150250 kg/m3 at moderate temper-

atures (between 10 and 1 8C). Test data weregathered from the field and from the literature to

match this snow type as closely as possible. Because

data were gathered from several studies, the snow

was not exactly the same in crystal structure;

however, the cumulative result can be considered

an baverageQ snow response taken over all of the

available field and laboratory data. A discussion of

selecting the initial values of the material parameters

follows.

4.1. Elastic properties

For snow deformation, the plastic deformation is

much greater than elastic deformation; however, the

elastic deformation is not negligible. The basic elastic

parameters, Youngs modulus E and Poissons ratio m

were estimated based on data compiled by Shapiro et

al. (1997). Because the elastic modulus is a strong

function of density, varying over 4 orders of

magnitude, the elastic modulus should ideally be

modeled as a function of the density as a state

variable. For this study, however, Youngs modulus

was held constant at a value equivalent to snow

having a density of 300400 kg/m3. At snow densities

below this, the elastic contribution is considered

minimal. Based on this, a Youngs modulus in the

range of 120 MPa and a Poissons ratio of 0.3 was

used for initial model parameter selection.

4.2. Yield surface

The parameters defining the shear portion of the

yield surface for the DruckerPrager model can be

calculated from the MohrCoulomb cohesion, c, andthe internal angle of friction, /. Assuming plane strain

response and non-dilatant flow, we can calculate the

DruckerPrager material cohesion, d, and the

DruckerPrager material angle of friction, b from

(HKS, 2001)

tanb 1:73sin/ 10

d c1:73cos/: 11

Acquiring apparent values for cohesion and fric-

tion angle, cV and /V from shear loading of snow

under a vehicle tire and from a ring shear device is

discussed by Blaisdell et al. (1990) and Alger andOsborne (1989). Both methods were applied to field

measurements on undisturbed snow with a density

ranging from 60 to 250 kg/m3 and tem peratures from

2 to 16 8C. From these data, Shoop (2001)determined that for a snow density of about 200 kg/

m3, c=2.1443 kPa and /=8.98148. Because of the

nature of the test, these values may be more

representative of the interface shear. However, similar

values of cohesion (approximately 2030 kPa) were

reported in Shapiro et al. (1997). These values yield a

DruckerPrager d ranging from 3.7 to 72 kPa and branging from 15.28 to 22.58.

Parameters describing the shape of the cap were

adjusted based on model response. The flow stress

ratio, K, was held at 1.0. This agrees with data

presented in Shapiro et al. (1997), indicating nearly

equal values of compressive and tensile strength for

low-density snow. The cap eccentricity parameter, R,

was chosen based on typical values for earth materials

having a very steep compaction cap (R=0.1 to

0.0001). The transition surface radius, a, was set to

0.0 (no transition). The initial cap yield surface

position evolpl |0 was arbitrarily set to 0.001 to allow

initial softening as needed. These material parameters

were refined based on model response and compar-

ison to experimental data.

4.3. Hardening law

The hardening law was based on tabular data

obtained from compression test data published in

Abele and Gow (1975) using three different tests of

snow having similar density, along with values

calculated from the final pressuredensity pairs ofan additional 21 tests. As the measurements are from

one-dimensional compression or consolidation (oed-

ometer) tests, by application of a vertical load to a

sample within a rigid cylinder (i.e., radially confined

uniaxial strain), the calculation of the mean hydro-

static pressure, pb, is not straightforward. Two simple

approaches are to assume thatr1=r2=r3, which gives

pb=r1; alternately we can assume r2=r3=0, whichgives pb=1/3r1. These two cases bound the sol-ution. An initial hardening table was derived based on



9/19

the average of Abele and Gow tests on warm, 200 kg/

m3 snow (tests 10 and 29) using pb=r1 (H200 curvein Fig. 4); this table was modified within the range of

1/3r1zpbzr1 during model calibration.

5. Model calibration using laboratory and field

data

Finite element models simulating a radially con-

fined uniaxial compression test and a plate-sinkage

test were used to refine the material properties given

in the previous sections.

5.1. Uniaxial compression

For the uniaxial compression model, a single

element model was determined to be adequate after

multi-element models gave the same results. Fig. 5

gives a plot of all of the data from Abele and Gow

(1975) with a density ranging from 140 to 230 kg/m3

and temperature between 7 8C and freezing.Unloading was not measured in the laboratory experi-

ments. As with Fig. 1, we see a broad range in the

response of the snow in experimental data; this scatter

in the data is not explained by Abele and Gow (1975).

They mention that b[s]ome samples were stored at

constant temperature before testing,Q indicating that

aging of the samples indeed occurred, though it wasnot documented how long each individual sample sat

at constant temperature prior to testing. Furthermore,

there was no attempt to categorize the snow type of

each individual sample (e.g., grauple, dendritic, etc.);

without such information on sample age and micro-

structure it is impossible to determine the root cause

of the scatter. Nevertheless, it is clear that for the same

density range snow can behave very differently and

this can be attributed to the microstructure of the

snow, be it the structure of the snow when it falls or

the time-dependent metamorphic changes in snowstructure; thus, we seek to have the capability to

model both the soft and hard snow conditions evident

in low density snow. This was done by selecting two

sets of model parameters and hardening curves to

capture the range in response of the snow.

Fig. 5 compares the uniaxial experimental data and

finite element model results. The nomenclature used

for the CDP material models are bS200Q for weak or

soft snow in the density range of 200 kg/m3, and

bH200Q for strong or hard snow in the same density

Fig. 5. Comparison of model and experimental data for uniaxial compression tests on low-density snow. The data are from Abele and Gow

(1975): snow temperature ranges from 0 to 7 8C, snow density ranges from 140 to 230 kg/m3.



10/19

range. The agreement between modeled and measured

material behavior is reasonably good for the two

material models, with one model capturing the softer

response of Abele and Gow snow data and the secondmodel representing harder response of the snow. Table

1 gives the material parameters used for each of these

snow models. The hardening curves for each of these

models are also plotted in Fig. 4.

5.2. Plate sinkage

Comparisons with three plate sinkage tests are

presented: two field tests and one laboratory test. The

first field test used, designated bKRC Runway,Q

consisted of pushing a 20-cm (8-in.)-diameter rigidplate, originally at the surface, into 39.2 cm (15.5 in.)

of fresh snow with average density of 300 kg/m3

(Alger and Osborne, 1989). This was modeled using

a 2D axis-symmetric representation of the test

geometry, with vertical shear surfaces along the

snow/plate interface. The vertical frictional surfaces

allowed these elements to slide, closely mimicking

the clearly defined shear failure surfaces observed inthe field. The use of slip planes was a reasonable

compromise that allowed us to model this test

geometry without having excessive element distortion

(in the vicinity of the sharp edge of the plate) that

prevented the model running to completion. The slide

planes were modeled using a contact surface with

Coulomb friction of l=0.3. As the cohesion of low

density snow is very small (5 kPa), we found that

adding a failure criteria to this slip plane (i.e., the

bond between adjacent elements on the slip plane

would break if the shear stress at that point exceededthe cohesion strength, d, of the material) did not

significantly change the simulated loadsinkage

curves. This suggests that frictional sliding along this

failure plane dominates and including only the

frictional resistance along these slide planes is

adequate for modeling this case.

The plate was lowered by constraining the surface

nodes radially while displacing them into the snow,

effectively creating a no-slip contact between the plate

and snow. Both the S200 and H200 material

parameters were used to simulate this condition. The

deformed model of the field test is shown in Fig. 6.

Fig. 7 gives a comparison of the field data and the

model results. Both of the snow models do a fair job

of reproducing the field data: the S200 model follows

the low strain region of the curve but under predicts

the peak load; the H200 model does a very good job

of capturing the peak forces but it over predicts the

force at the low end (plate sinkage 00.2 m) of the

loading curve.

The second test, designated bKRC Texas field,Q

used the same size plate as above, but the overall

snow depth was 51.4 cm (20.2 in). The upper layer ofsnow was fresh snow with a density of 110 kg/m3, and

the lower layer was older snow with a density of 230

280 kg/m3 (Alger and Osborne, 1989). This was

modeled using the same basic geometry as the KRC

runway described above, except the snow depth used

in the model matched that of the field, 51.4 cm, and

the snow was modeled in three ways:

(1) The entire snow depth was modeled using H200

parameters.

Table 1

Parameters for the modified Capped DruckerPrager model of low

density snow (150250 kg/m3)

Parameter S200 H200

Youngs modulus (MPa), E 1.379 13.79

Poissons ratio, m 0.3

DruckerPrager cohesion (kPa), d 5

DruckerPrager angle of friction (deg), b 22.538

Cap eccentricity, R 0.02

Initial cap yield surface position, evolpl |0 0.001

Transition surface radius, a 0.0

Flow stress ratio, K 1.0

Average snow density (kg/m3) 200

Hardening law

Mean hydrostatic pressure (Pa) eplvol

S200 H200

113.76 113.76 0

0.017106

0.05106

0.5930.033106 0.1106 0.6690.067106 0.2106 0.8060.167106 0.5106 0.9440.333106 1.0106 1.0830.667106 2.0106 1.2990.933106 2.8106 1.4551.083106 3.25106 1.4752.0106 6.0106 1.502.0107 6.0107 1.514

The softer snow response is modeled using the bS200Q parameters

while the harder snow response is modeled with the bH200Q

material parameters.



11/19

(2) The entire snow depth was modeled using S200

parameters.

(3) The snow was modeled in layers, the upper 13.1

cm of the snow was modeled using the S200material parameters, and the remaining snow depth

was modeled using the H200 material parameters.

The results of these simulations are compared

with the field data in Fig. 8. Though all three models

provide reasonable agreement, the S200 model tends

under-predict the peak load, while the H200 model

tends to over-predict the forces at low sinkage

depths. The two-layer model does a better job of

capturing the peak forces while still following the

overall force trace. This simulation supports thenotion that there needs to be some consideration in

modeling that takes into account age hardening (or

microstructure) of snow where two samples of snow

that have similar or identical density differ in

strength owing to one being fresh snow and the

other having strengthened over time because of

sintering and metamorphism. In this simulation the

upper layer of fresh snow is weaker than theFig. 6. Deformed mesh for simulation of plate sinkage tests

performed on the KRC Runway: H200 model parameters used.

Fig. 7. Comparison of plate sinkage KRC Runway field test data (Alger and Osborne, 1989) ( S) to the model results (lines).



12/19

underlying old snow, and the differences in material

parameters attempt to reproduce this condition.

The laboratory plate sinkage test pushed a 20-cm

(8-in.)-diameter rigid plate into a cube of snow

606050 cm deep with an average initial densityof 180 kg/m3 (Shoop and Alger, 1998). This same

geometry was captured in the three-dimensional

model simulation of this test. The snow was modeled

Fig. 8. Comparison of plate sinkage KRC Texas field test data (Alger and Osborne, 1989) (S) to the model results (lines).

Fig. 9. Comparison of modeled (lines) and measured ( Shoop and Alger, 1998) ( S) plate sinkage results for laboratory experiments.



13/19

using both the S200 and H200 material parameters.

The model results are compared with the lab experi-

ment in Fig. 9. The H200 snow simulation does a very

good job of capturing the forcedisplacement curve

for a penetration of the plate into the snow greater

than 0.1 m. For plate penetration less than 0.1 m, the

Fig. 10. Modeled snow density (kg/m3) of plate sinkage test for Capped DruckerPrager material (laboratory test simulation at left and field test

simulation at right). Both models shown used the H200 material parameters (Shoop et al., 1999a,b).

Fig. 11. Measured displacement and snow density in laboratory plate sinkage test (after Shoop and Alger, 1998).



14/19

H200 model over predicts the force; the S200 model

in general is too soft and does not accurately

reproduce the data for these snow conditions.

In the model, snow density can be calculated fromthe volumetric plastic strain. The snow density values

from the models are shown in Fig. 10. These densities

can be compared to snow densities measured under

the plate, as shown in Fig. 11. The maximum

volumetric inelastic strain in the plate sinkage model

occurred immediately below the plate at values of

approximately 0.5, which is equivalent to a snow

density of 360 kg/m3, whereas in the experiment a

very dense (440520 kg/m3) bulb of snow was

observed extending approximately 5 cm below the

plate (Fig. 11). Below this, the modeled snow densityvaries from about 240 kg/m3 under the plate to about

215 kg/m3 near the ground. Yet, the measured density

shown in Fig. 11 is nearly constant at 290310 kg/m3

below the high density bulbother experimental

results from Shoop and Alger (1998) show that the

density under the bulb can vary more widely (e.g.,

270350 kg/m3) than that shown in Fig. 11. Thus, the

model does not clearly capture the high density bulb

observed in the laboratory experiments, but rather

appears to smear out the variation in density over the

entire snow depth. Yet, the predicted density of the

snow adjacent to the plate compares quite favorably

with the measured lab data with both model and

experiment demonstrating a slight rise in density due

to lateral compaction.

6. Discussion

The comparison of the uniaxial data to the model

results shows that the S200 and H200 models may not

be capable of capturing the specific response of a

particular snow microstructure, yet using these twomodels can be effective at bounding the snow

response in this density range, with S200 capturing

the lower bound associated with soft snow and H200

approaching the upper bound (hard snow). This

concept is further reinforced with comparisons to

plate sinkage tests. For both of the field cases (KRC

Runway and Texas Field) it appears that the two

models do a very good job of bounding the scatter in

the snow response, and the two-layer model of the

KRC Texas Field gives an average through the data.

The material parameters presented in Table 1 in

some respect dictate that these models would

provide bounding conditions. The elastic modulus

for the two models (S200 and H200) roughlyfollows the bounding range given by Shapiro et

al. (1997): 120 MPa. By increasing the elastic

modulus within this range. the snow can be made

harder and btuningQ of this parameter within this

range can help match specific field data. Likewise,

the hardening curves given in Table 1 are for the

bounds 1/3r1zpbzr1. These hardening tablescould also be modified within this range to match a

particular set of experimental data. However, the

ability of these models, used together, to capture

the upper and lower bound of the snow responsecan be of great use by providing the expected range

of snow response in this density range, not just the

average. Capturing the range of snow response,

when so little is known a priori about the age or

microstructure of the snow that will be encountered

in the field, is far more useful than having a single

mo de l t ha t m at ch es a s pe ci fi c s et o f s no w

conditions that may or may not be representative

of the field conditions at another time or location.

7. Wheel on snow application

Turning now to a prototyping application, we

developed a model of a wheel rolling through snow

and compared these results to measurements of the

performance of the CRREL instrumented vehicle

(CIV) rolling through fresh snow. The CIV, originally

a 1977 AMC Jeep Cherokee, is instrumented to

measure vertical, longitudinal, and lateral forces at

the tire interface, wheel speed at each wheel, true

vehicle speed, and steering angles. In this study we

used available data for the CIV documenting the tiresinkage depth (rut depth), the longitudinal resistance

force of the wheel as the vehicle moved through the

fresh snow, and the snow density surrounding the tire

rut (Blaisdell et al., 1990; Green and Blaisdell, 1991;

Richmond, 1995).

To reduce run time, the tire was modeled as a rigid

wheel (a rigid analytic surface). Fig. 12 shows the

model geometry and the resulting deformed snow

surface for 20-cm-deep snow. Taking advantage of the

symmetry of the problem, we modeled the simulation



15/19

in half space. No slip plane was needed to model this

case, as the rounded transition between the wheel

btreadQ and bsidewallQ appeared to prevent excessive

distortion of the elements in this region, allowing the

model to run to completion. The wheel motion was

simulated as follows:

(1) The wheel was allowed to settle unto the snow

surface under the influence of gravity and its

own weight.

(2) The hub of the wheel was accelerated to 2.24 m/

s (5 miles/h); concurrently, an angular velocity

was applied to the hub to assure zero slipbetween the snow surface and the wheel.

(3) Once the hub velocity reached 2.24 m/s, the

velocity was maintained constant for the remain-

der of the simulation.

The model was run using both the S200 and H200

material parameters. The performance of the CIV was

simulated for snow depths ranging from 5 to 50 cm.

The coefficient of friction between the wheel and the

snow was set at 0.3.

From the model we determined the steady state

reaction force at the hub (i.e., the resistance force to

pull the wheel forward) and the steady state wheel

sinkage depth into the snow. These steady state

values were obtained from the portion of the

simulation at which the hub was moving at a constant

velocity. The resistance force was normalized by the

vertical load on the hub (the sprung and unsprung

mass carried by that tire). These results, along with

the field data, are plotted in Fig. 13. The S200 model

does a very good job of predicting the depth that the

tire sinks into the snow (Fig. 13a) over the full range

of the available field data, yet the H200 model tendsto under predict the sinkage depth. This is not

surprising, as these data were taken in freshly fallen

snow and the S200 model should be more represen-

tative of this condition.

The normalized resistance force (Fig. 13 b) pre-

dicted by the S200 model is a bit high in comparison

to field data, yet it does a reasonable job of capturing

the upper envelope of the data. As such, it tends to be

conservative with respect to predicting the rolling

resistance forces on the tire. Again, because the data

Fig. 12. Model of a rigid wheel rolling through snow. Material parameters for S200 were used.



16/19

were taken in fresh snow, the S200 model provides a

better estimate of the rolling resistance than the H200

model.

Fig. 14 compares the deformation of the snow in

the model and field as well as the modeled and

measured snow density under the tire. The model does

a good job of reproducing the overall snow deforma-

tion shape and matches the compaction density of the

snow under the tire. Because the snow density varies

with depth in the field, yet was held constant in the

Fig. 13. Comparison of the CDP snow model predictions to field data for (a) sinkage and (b) normalized motion resistance.



17/19

Fig. 14. Comparison of (a) the field snow deformation shape (Richmond, 1995) to (b) the model and comparison of (b) the predicted snow

density under a tire to (c) density measurements from Richmond (1995). The snow depth is 19 cm. The model uses the S200 material

parameters. The black lines in the snow shown in (a) are chalk lines used to document the deformation of the snow as the wheel passes

through it.



18/19

model, it is more difficult to directly compare the

variation in density adjacent to the tire side wall. Yet,

it is clear that, in the field, there is some lateral

compaction of the snow adjacent the tire side wall,evidenced by the lateral deformation of the vertical

chalk lines in the snow (Fig. 14a); this lateral

compaction is predicted by the model as well. We

note that the snow density at the corners of the tire is

about 20% higher in the model than that measured in

the field.

8. Conclusions

The modified Capped DruckerPrager constitutivelaw was used for modeling the macroscale material

behavior of relatively warm (1 to 10 8C) lowdensity snow (150250 kg/m3) using the finite

element method. Available laboratory and field data

were used to obtain suitable material parameters and

calibrate the model. Two sets of material parameters

were developed, one for describing the behavior of

bsoftQ snow, and the second for describing the

response of bhardQ snow. The derivation of these

two cases for describing snow in this density range

help to capture the structure-dependent differences in

snow strength attributed to snow type and to age

hardening or sintering of the snow over time. These

models are not intended to represent a specific snow

microstructure, but rather provide description of the

baverageQ snow behavior in this density and temper-

ature range.

The snow models were compared to laboratory and

field measurements of three load cases: radially

confined uniaxial compression, plate sinkage tests,

and a wheel rolling through new fallen snow. This

illustrates the application and validation of the

constitutive model under a wide variety of geometriesand loading conditions. The agreement for all of these

geometries is very good. It is clear from a two-layer

simulation of plate sinkage tests conducted in the field

that the use of the soft snow (S200) parameters for the

top layer and the hard snow (H200) parameters for the

bottom layer improves the ability of the simulation to

reproduce the measured load trace. The plate sinkage

and wheel-on-snow models illustrate the ability to

also model the compaction density of the snow as it is

loaded.

Acknowledgements

This effort was funded by the U.S. Army High

Fidelity Ground Platform and Terrain MechanicsScience and Technology Objective, Terrain Mechanics

Models for All-Season Terrain, Work Unit. This work

was supported in part by a grant of computer time from

the DOD High Performance Computing Modernization

Program at the Engineer Research and Development

Center, Vickburg, MS and the Army Tank-automotive

and Armaments Command, Warren, Michigan.

References

Abele, G., Gow, A., 1975. Compressibility Characteristics of

Undisturbed Snow. Research Report 336. U.S. Army Cold

Regions Research and Engineering Laboratory, Hanover, NH.

Ahlvin, R., Shoop, S.A., 1995. Methodology for predicting for

winter conditions in the NATO reference mobility model. 5th

North American Conf. of the ISTVS. Saskatoon, SK, Canada,

May 1995, pp. 320 334.

Alger, R., 1988. Effect of snow characteristics on shear strength.

Contract Report No. DACA89-88-K-004. Keweenaw Research

Center, Michigan Technological University, Houghton, MI.

Alger, R., Osborne, M., 1989. Snow characterization field data

collection results. Final Report No. ACA89-85-K-002. Kewee-

naw Research Center, Michigan Technological University,

Houghton, MI.Armstrong, R.L., 1980. An analysis of compressive strain in

adjacent temperature-gradient and equi-temperature layers in a

natural snow cover. J. Glaciol. 26 (94), 283289.

Aubel, T., 1993. FEM simulation of the interaction between elastic

tyre and soft soil. 11th International Conference of the ISTVS,

Lake Tahoe, NV, vol. 2, pp. 791802.

Aubel, T., 1994. The interaction between the rolling tyres and soft

soil FEM Simulation by VENUS and validation. 6th European

Conf. Of the ISTVS, Vienna, vol. 1, pp. 169188.

Bailey, A.D., Johnson, C.E., 1989. A soil compaction model for

cylindrical stress states. Trans. ASAE 32 (3), 822825.

Blaisdell, G.L., Richmond, P.W., Shoop, S.A., Green, C.E., Alger,

R.G., 1990. Wheels and tracks in snow: validation study of the

CRREL shallow snow mobility model. CRREL Report No. 90-9. U.S. Army Cold Regions Research and Engineering

Laboratory, Hanover, NH.

Chi, L., Tessier, S., 1995. Finite element analysis of soil compaction

reduction with high flotation tires. 5th North American Conf. of

the ISTVS, Saskatoon, SK, Canada, pp. 167176.

Fervers, C.W., 1994. FE simulations of tyre-profile effects on

traction on soft soil. 6th European Conf. of the ISTVS, Vienna,

Austria, pp. 618633.

Foster, W.A., Johnson, C.E., Raper, R.L., Shoop, S.A., 1995. Soil

deformation and stress analysis under a rolling wheel. North

American Conf. of the ISTVS. Saskatoon, SK, Canada, May

1995, pp. 194 203.



19/19

Fukue, M., 1979. Mechanical Performance of Snow under Loading.

Tokai Univ. Press. (based on authors thesis, McGill University,

1977).

Green, C.E., Blaisdell, G.L., 1991. US Army Wheeled Versus

Tracked Vehicle Mobility Performance Test Program. Report no.

2: Mobility in shallow snow. U.S. Waterways Experiment

Station, Technical Report GL-91-7.

HKS (Hibbitt, Karlsson and Sorensen), 1998. ABAQUS Theory and

Users Manuals. Pawtucket, RI.

HKS (Hibbitt, Karlsson and Sorensen), 2001. Analysis of Geo-

technical problems with ABAQUS. ABAQUS Course Notes.

Pawtucket, RI.

Jaeger, J.C., Cook, N.G.W., 1969. Fundamentals of Rock Mechan-

ics. John Wiley & Sons, New York.

Lade, P.V., Kim, M.K., 1995. Single hardening constitutive model

for soil, rock and concrete. Int. J. Solids Struct. 32 (14),

19631978.

Liu, C., 1994. Traction Mechanisms of Automobile Tires on Snow.PhD dissertation, Vienna University of Technology. Vienna,

Austria.

Liu, C.H., Wong, J.Y., 1996. Numerical simulations of tiresoil

interaction based on critical state soil mechanics. J. Terramechs.

33 (5), 209222.

Meschke, G., Liu, C., Mang, H.A., 1996. Large strain finite-element

analysis of snow. J. Eng. Mech., July, 581591.

Miyori, A., Shiraishi, M., Yoshinaga, H., Naoaki, I., 2002.

Simulation of tire performance on snow. Proceedings Interna-

tional Tire Exhibition and Conference. Akron, OH.

Mundl, R., Meschke, G., Leiderer, W., 1997. Friction mechanism

of tread blocks on snow surfaces. Tire Sci. Technol. 25 (4),

245264.

Pi, W.S., 1988. Dynamic tire/soil contact surface interaction modelfor aircraft ground operations. J. Aircr. 25 (11), 1038 1044.

Richmond, P.W., 1995. Motion resistance of wheeled vehicles in

snow. CRREL Rpt. 95-7. U.S. Army Cold Regions Research

and Engineering Laboratory, Hanover, NH.

Richmond, P.W., Shoop, S.A., Blaisdell, G.L., 1995. Cold regions

mobility models. CRREL Report 95-1. U.S. Army Cold

Regions Research and Engineering Laboratory, Hanover, NH.

Saliba, J.E., 1990. Elasticviscoplastic finite-element program for

modeling tiresoil interaction. J. Aircr. 27 (4), 350357.

Schofield, A.N., Wroth, C.P., 1968. Critical State Soil Mechanics.

McGraw-Hill, London, England.

Scott, R.F., 1985. Plasticity and constitutive relations in soil

mechanics. J. Geotech. Eng. 11 (5), 563605.

Shapiro, L., Johnson, J., Sturm, M., Blaisdell, G., 1997. Snow

Mechanics: Review of the State of Knowledge and Applica-

tions. CRREL Report 97-3. U.S. Army Cold Regions Research

and Engineering Laboratory, Hanover, NH.

Shoop, S.A., 2001. Finite element modeling of tireterrain

interaction. PhD dissertation, University of Michigan.

Shoop, S., Alger, R., 1998. Snow deformation beneath a vertically

loaded plate. Proceedings, ASCE Cold Regions Specialty

Conference. Duluth, Minnesota.Shoop, S., Haehnel, R., Kestler, K., Stebbins, K., Alger, R., 1999.

Finite element analysis of a wheel rolling in snow. Proceed-

ings ASCE Cold Region Engineering Conference, Lincoln, NH,

pp. 519530.

Shoop, S., Haehnel, R., Kestler, K., Stebbings, K., Alger, R.,

1999. Snowtire FEA. Tire Technology International, June,

pp. 2025.

Shoop, S., Lacombe, J., Haehnel, R., 2001. Modeling tire perform-

ance for winter conditions, tire technology international. Annual

Review of Tire Material and the Tire Manufacturing Technol-

ogy, Dec., pp. 10 14.

Voitkovsky, K.F., Bozhinsky, A.N., Golubev, V.N., Laptev, M.N.,

Zhigulsky, A.A., Slesarenko, Yu.Ye., 1975. Creep induced

changes in structure and density of snow, internationalsymposium on snow mechanics. IAHSAISH Publication, vol.

114. Grindelwald, Switzerland, pp. 171179.

Wood, D.M., 1990. Soil Behavior and Critical State Soil Mechanics.

Cambridge Univ. Press, New York.


Date post:	09-Apr-2018
Category:	Documents
Upload:	anonymous-ecd5zr
View:	222 times
Download:	0 times

A macroscale model for low density snow subjected to rapid loading

Documents