Poisson's ratio ν, were estimated based on data com-
this study the plasticity model used was the well-
piled by Mellor (1975) and expanded on by Shapiro
documented DruckerPrager cap.
et al. (1997) (Fig. 21). Since the elastic modulus is a
The second material model, a crushable foam
strong function of density, varying over four orders
model, was chosen based on work by Johnson et al.
of magnitude, the elastic modulus should ideally be
(1992), where a crushable foam model was success-
modeled as a function of the density as a state vari-
fully applied to natural snow. Johnson used
able. For this study, however, Young's modulus was
PRONTO, an explicit finite element code from San-
held constant at a value equivalent to snow with a
dia National Lab, to simulate shock wave propaga-
density of 300400 kg/m3. At snow densities less
tion in one dimension. For tiresnow interaction,
however, the model must simulate snow deformation
than this, the elastic contribution will be minimal.
in a three-dimensional stress field. Initial simulations
using the ABAQUS crushable foam model showed
that no lateral deformation resulted from an axial
load. This issue prompted questions regarding the
deformation behavior of snow under a purely vertical
load and whether the predicted response from a
crushable foam model was suitable for snow in a
three-dimensional stress state. Experiments measur-
ing the lateral deformation of snow under a purely
vertical load were conducted in cooperation with the
Keweenaw Research Center in Houghton, MI (Shoop
and Alger 1998). Results indicate that vertical load-
ing of non-sintered snow causes primarily vertical
deformation. Fukue (1979) performed experiments
that showed that lateral deformation was minimal for
high load rates. Both of these studies support our
choice of crushable foam models for vehicle loading.
Material parameter determination
Extreme changes in snow properties with time, tem-
perature gradients, applied load, and deformation pro-
hibit model parameter acquisition from the same snow.
Therefore, approximations were made by estimating
parameters using test data from snow of similar charac-
teristics (density, age, snow type, and location). The
type of snow modeled was a fresh snow with a density
of 200250 kg/m3 at moderate temperatures (between
10 and 1C). Test data were gathered from the field
and from the literature to match this snow type as
closely as possible. Because data were gathered from
= Pulse propagation or flexural vibration at high
A
several studies, the snow was not exactly the same in
frequencies, 10 to 25C
crystal structure, but this effect is minimized in fresh
= Uniaxial compression, strain rate approximately 3
B
snow where sintering (bonding between snow grains)
103 to 2 102 s1, temperature 25C
has not occurred. Also, most of the tests were performed
= Uniaxial compression and tension, strain rate ap-
C1
on "lake effect" snow from the Keewenaw Peninsula in
proximately 8 106 to 4 104 s1, 12 to 15C
the Upper Peninsula of Michigan, which is a fairly con-
= Static creep test, 6.5 to 19C
C2
sistent snow type under normal winter conditions. A
= Complex modulus, 103 Hz, 14C
D
discussion of selecting the initial values of the material
= Static Young's modulus and quasi-static Pois-
K
parameters follows.
son's ratio
Elastic properties. For snow deformation under a
= Quasi-static measurements of Poisson's ratio
S
tire, the plastic deformation is much greater than the
elastic deformation; however, the elastic deformation
Figure 21. Compilation of Young's modulus and
contributes to energy losses during rolling resistance.
Poisson's ratio measurements on snow. [After
The basic elastic parameters, Young's modulus E and
Mellor (1975) and Shapiro et al. (1997)]
13
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